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Yarmouk University Publications
Deanship of Research and
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ABHATH AL-YARMOUK
Basic Sciences and Engineering
Refereed Research Journal
Vol. 16
No. 1
1428/2007
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TABLE OF CONTENTS
Articles in English
*
Use of X-ray Diffraction and Mud Master
Program in the Evaluation of Crystallite Size and
Size Distributions of Illite and Kaolinite at Various
Depths of Rock Formations in Jordan
Ibrahim M. Odeh, Waleed A.
Saqqa and Denis Eberl
1
*
Assessment of Background Radiation Dose due to
Radon in Kharja Village, Irbid Governorate,
Jordan
Khitam M Khasawinah and
Khalid M. Abumurad
19
*
Determination of the Specific Activities of
Radioactive Nuclides of Surface Soils of Some
HadhramoutвЂ™s Valleys in Yemen Using GammaRay Spectrometry
Awad Ali Bin- Jaza and
Abdulazize Omer Bazohair
27
*
Origin and Migration of Primordial Germ Cells in
Mosquitofish, Gambusia affinis
Janti S. Qar and Mohammad
A. Al-Adhami
39
*
The Role of Selenium and Lycopene in the
Protection of Mice Against Microcystin Toxicity
Saad Al-Jassabi and Ahmad
Khalil
49
*
Geoeletrical and Hydrochemical Investigations in
the Hallabat Area, North Jordan
Ghunaim S. Nasher, Hakam
A. Mustafa and Nabil
Abderahman
59
*
Environmental Impact Assessment of Hiswa Clay
Deposits, South Jordan
Talal Al-Momani
93
*
Performance Evaluation of One UMTS Cell by the
New Modelling Specification and Evaluation
Language MOSEL-2
Aymen Issa Zreikat
105
*
A Generic Parallel Map: Extending the List Map
Higher Order Function to Arbitrary Data
Structures
Mohammad M. Hamdan
123
*
Effect of Access Point Moving on the Wireless
Network Performance Using Opnet Modeler
Sameh H. Ghwanmeh
135
*
New Techniques for Improving H.263
Videoconferencing
Ammar M. Kamel and Moh'd
Belal. Al-Zoubi
147
Articles in Arabic
*
Application of Unbounded Linear Operators in
Hilbert Spaces
Ahmad Ajour and Rainer
Picard
1
ABHATH AL-YARMOUK: "Basic Sci. & Eng."
Vol. 16, No.1, 2007, pp. 1-17
Use of X-ray Diffraction and Mud Master Program in the Evaluation
of Crystallite Size and Size Distributions of Illite and Kaolinite at
Various Depths of Rock Formations in Jordan
Ibrahim M. Odeh*, Waleed A. Saqqa** and Denis Eberl ***a
Received on March 29, 2006
Accepted for publication on Nov. 5, 2006
Abstract
The mean crystallite size (CSmean), mean thickness (Tmean) and crystallite thickness
distributions (CTD) for illite (I) and kaolinite (K) have been determined for
representative Jordanian mudrock samples of different ages. MudMaster analyses of the
X-ray diffraction (XRD), based on the Bertaut-Warren-Averbach (BWA) method, have
been employed in the calculations. The CSmean and Tmean for illite show a slight increase
with a mean difference of +0.5nm and +1.4nm respectively with the increase of the
present-day stratigraphic thicknesses (depths) of rock formations. An increase in the
CSmean and Tmean of kaolinite proceeds only to certain depths (1.22km, 1.05km
respectively). Beyond these depths, the two means flip with a mean difference of 0.7nm, -2.1nm, respectively, when the depth reaches 1.6km. The shapes of crystallite
thickness distributions (CTD) for illite in all investigated mudrock samples are
asymptotic. The asymptotic shape more likely mirrors the overlapping reflections of
different generations of illite and mixed layer I-S phases. Shapes of CTD for kaolinite
vary between asymptotic, multimodal and lognormal. This is more likely due to the
presence of two generations of kaolinite growth coinciding with increasing depths of the
rock formations.
Keywords: X-ray diffraction, mudrock, kaolinite, illite, MudMaster, BWA-method,
crystallite size, asymptotic, multimodal, lognormal.
Introduction
The use of X-ray diffraction (XRD) to study the crystalline structure of clay
minerals, crystallite size - and shape of crystallite thickness distributions (CTDs) has
attracted considerable attention over the past years. Crystallite size is a measurement of
adjacent, repeating crystalline units. It is only equivalent to the grain size if the
individual grains are perfect single crystals free of defects, grain boundaries or stacking
faults. Crystallite size is determined by measuring the broadening of a particular XRD
В© 2007 by Yarmouk University, Irbid, Jordan.
* Department of Physics, , Yarmouk University, Irbid, Jordan.
** Department of Earth & Environmental, Faculty of Sciences, Yarmouk University, Irbid, Jordan.
*** U.S. Geological Survey, 3215 Marine St., Boulder, CO 80303-1066 USA.
Odeh, Saqqa & Eberl
peak associated with a particular planar reflection from within the crystal unit cell. It is
inversely related to the FWHM (full width at half maximum) of the peak. The narrower
is the peak, the larger the crystallite size. This is due to the periodicity of the individual
crystallite domains, in phase, reinforcing the diffraction of the X-Ray beam, resulting in
a tall narrow peak. If the crystal are defect free and periodically arranged, the X-Ray
beam is diffracted to the same angle even through multiple layers of the specimen. If the
crystallites are randomly arranged or have low degree of periodicity, the result is a
broader peak. Crystallite thickness refers to the mode of stacking two or more crystals of
a clay mineral. The Bertaut-Warren Averbach (BWA) method is adapted to study clay
minerals having a periodic structure along the c-axis. The XRD-peak shapes for the 00l
reflections are then analyzed to calculate the crystallite thickness distributions (CTDs).
The shape of the CTDs is used to infer crystal growth mechanism undergone by clay
minerals during diagenesis or alteration by applying a simulation technique. Dudek et al.
[1] pointed to several mathematical approaches in order to obtain information about
crystallite size from the XRD peak shapes. The Scherrer equation [2], variance method
[3, 4], Voigt method [5] and Bertaut-Warren-Averbach (BWA) method [6, 7] are applied
to determine crystallite size using the XRD spectra. The BWA-method is the most
universal because it can be used to measure the crystallite size, strain and crystallite size
distributions. The MudMaster, a computer program utilizing a version of the BWAmethod is applied to carry out such measurements [6, 7]. The evolution of illite as a
function of burial diagenesis can be better explained by applying the BWA-method [6, 7,
and 8]. The shapes of crystallite thickness distributions (CTD), asymptotic, lognormal
and multimodal, may provide information regarding nucleation and crystal growth
mechanisms of clay minerals [2, 6, 7, and 8].
The main task of the present study is to determine the mean crystallite size and
thickness of illite and kaolinite from the XRD data and to interpret the shapes of size
distributions from the 001 basal reflections in the XRD spectra. The MudMaster with the
aid of the BWA-method has been extensively utilized for this purpose.
Materials and Methods
Representative mudrock samples were selected from different sites in Jordan as
shown in Fig. (1). The samples represent the late Ordovician-early Silurian Batra
Member kaolinitic claystone (Batn el-Ghol area), black shale of late Permian Umm Irna
Formation (Dead Sea area), the gray-green kaolinitic claystone from the Lower
Cretaceous Kurnub Group (Mahis area), the green claystone of the Turonian Shueib
Formation (Tafila area), the red-brownish claystone of the Pleistocene and AlYamaniyya clay deposits (Al-Yamaniyya, Aqaba area). The abbreviations: SIL, UI,
MAH, TAF, and YAM in the text are derived from the original names of the studied
rock formations or their ages. They indicate the Silurian Batra Member claystone, Umm
Irna black shale, Mahis, Tafila and Al-Yamaniyya clay deposits respectively.
The samples were gently crushed using an agate mortar and pestle. They were then
treated with 12% H2O2 to remove organic matter. After oxidation, samples were wet
sieved using a test sieve of 45Вµm to separate coarser fractions. The clay fraction (<2 Вµm)
obtained from the pass 45Вµm fraction by sedimentation, centrifugation and dialysis was
2
Use of X-ray Diffraction and Mud Master Program in the Evaluation of Crystallite Size and
Size Distributions of Illite and Kaolinite at Various Depths of Rock Formations in Jordan
sedimented on a polished silicon substrate and left to dry at room temperature. After
mounting on the substrates, air-dried samples were heat treated separately at 350В°C and
550В°C. Vaporization of ethylene glycol was used to saturate clay samples at 60В°C for 16
hours. PVP-10 intercalated air-dried clay samples were analyzed by XRD for the
purpose of measuring the fundamental illite particle size and size distribution [9]. The
ratios of PVP to clay intercalations were as follows:
Sample
Ratio PVP : CLAY
YAM
4:1
TAF
3:1
MAH
2:1
UI
2:1
SIL
2:1
Clay fraction was suspended in distilled water (concentration 2.5mg clay/1ml H2O),
mixed with an aqueous solution containing 5mg PVP/1ml H2O and fully dispersed by
ultrasonic treatment. The proportion of clay to PVP used depended on the expandability
of the clay sample. The larger the proportion of smectite layers the more PVP was added
to obtain good intercalation [1]. The advantage of the PVP-10 intercalation was to
remove the effect of peak broadening related to swelling interface of the exposed basal
surfaces of illite. In this way, crystallite size can be accurately calculated by the BWAmethod. The use of silicon substrates improved the background of the XRD spectra [7, 9,
and10]. The 001 reflection of PVP-illite and kaolinite were used to measure mean
crystallite size (Csmean), mean crystallite thickness (Tmean) and crystallite thickness
distributions [6, 8, and 11]. XRD diffraction spectra were carried out using a Phillips
PW1710 Diffractometer equipped with copper tube, CuKО± radiation, and a theta
compensatory slit assembly. The diffractometer was operated under the following
experimental conditions: a voltage of 35 KV, beam current of 40mA , step size of 0.02
2Оё, counting time of 0.01sec/step, 1mm receiving slit and a scanning range of 2-65В° 2Оё.
The BWA-method adapted in the MudMaster computer program is based on Fourier
analysis of the interference function [2, 6].
Lithology of mudrocks
Fig (2) illustrates the graphic logs for the different rock formations. The Batra
Member of Late Ordovician-early Silurian (Batn el-Ghol, SE Jordan) consists of light
gray-green kaolinitic clays with Graptolite remains along bedding planes. Reddish
siltstone-fine sandstone beds intercalate the green clays with a total thickness of about
80m. The occurrence of Graptolite indicates marine origin [12]. The Umm Irna
Formation of late Permian (Dead Sea area) consists principally of two quartz arenite
sandstone facies (60 m. thick) with fining upward cycles. Each starts with pebbly
sandstone at the base, changes upward into fine sandstone, siltstone and silty claystone.
The uppermost two cycles are distinguished by the presence of reddish-brown and
occasionally dark gray paleosol horizons separated from each other by dark gray-black
3
Odeh, Saqqa & Eberl
shale beds with plant remains [13]. The Lower Cretaceous Kurnub Group (Mahis area)
consists of a thick varicolored sandstone sequence (в‰€200m.) interbedded with kaolinitic
clay beds. The Turonian Shueib Formation (Karak-Tafila area) consisting of green and
red gypsiferous claystone beds interbedded with gypsum changes near the top to
fossiliferous dolomitic limestone. The Al-Yamaniyya Pleistocene clay deposits consist
of varicolored (gray, green, yellow, red-brown) claystone interbedded with pebblycoarse-fine sandstone beds. Al-Yamaniya clay deposits were more likely deposited in a
nearshore environment [Note 1, 14, 15].
Experimental results
a. XRD analysis
The XRD diffraction patterns (Fig.3; Table 1) show that smectite (S) dominates
other clay minerals in YAM claystone samples. The other clay minerals are kaolinite
(K), illite (I), mixed layer illite-smectite (I-S), chlorite (Ch) and mixed layer illitechlorite (I-Ch). The 001 peak of S at 6.2В° 2Оё (14.25О‘В°) in air-dried samples expands to
5В° 2Оё (17.67О‘В°) after ethylene glycol saturation. The same peak collapsed upon heating
at 550В°C to 9.04В° 2Оё (9.8О‘В°). Peaks of Ch and mixed layer I-Ch are apparent in heated
samples as shown in Figure (3).
In TAF clays, a broad peak appears in air-dried mounts at 8.02В° 2Оё (11.02О‘В°). This
is more likely to be a mixture of вЂ�IвЂ™ and one-water layer S. After glycolation, the peak
shifts to 5.4В° 2Оё (16.37О‘В°) and a new 001 peak of illite appears at 8.8В° 2Оё (10О‘В°). When
heating at 550В°C, the broad peak at 11.02О‘В° collapsed to 10О‘В°. Kaolinite is also present
in the TAF green claystone.
In the UI black shale and MAH clay samples, asymmetric broad and low intensity
mixed-layer I-S peaks appear at 7.7В° 2Оё (11.5О‘В°) and 7.2В° 2Оё (12.3О‘В°) together with I
and K. The mixed layer I-S peaks collapsed upon heating (550В°C) to 8.8В° 2Оё (10О‘В°).
Illite and kaolinite are dominant in the SIL clays.
b. Crystallite size of illite and kaolinite versus depth
Table (2) summarizes the results of mean crystallite size (CSmean), mean thickness
(Tmean) and CTD shapes of illite and kaolinite. The CSmean (2.8nm-3.3nm) and Tmean
(3.8nm-5.2nm) for illite show a slight increase (+0.5nm, +1.4nm respectively) with
increasing depth of the rock formations (Fig.4.a). The CSmean and Tmean for kaolinite are
somewhat different. The means of the two values range between 3nm-10.7nm and
7.5nm-16.8nm, respectively. A slight increase in the CSmean of K (+ 0.7 nm) occurs when
the depth changes from shallow to 0.95km. This is followed by an abrupt increase from
3.7nm-9.0nm (+5.3nm) when depth is 1.05km.
The CSmean increases by 1.7nm when the depth changes from 1.05km to1.22km.
Beyond the 1.22km depth, the mean size flips (- 0.7 nm.) when the depth is 1.6km
(Fig.4.b). The Tmean for kaolinite increases from 7.5nm to 11.3nm (+3.8nm) going from a
shallow depth to 0.95km. The Tmean abruptly increases to 16.8nm (+5.5nm) at a depth of
1.05km and then eventually decreases to 14.7nm (-2.1nm) beyond this depth (Fig. 4.b).
4
Use of X-ray Diffraction and Mud Master Program in the Evaluation of Crystallite Size and
Size Distributions of Illite and Kaolinite at Various Depths of Rock Formations in Jordan
c. Crystallite thickness distributions (CTD) for illite and kaolinite
The crystallite thickness distributions (CTD) for air-dried PVP-illite and heated
(350В°C) kaolinite samples were determined using MudMaster analysis based on the
BWA- method ,area-weighted option [9]. Figures (5 and 6) illustrate the shapes of CTD
for PVP-I and K respectively. The CTD shapes in PVP-I was asymptotic in all samples.
This is more likely a result of overlapping reflections of different generations of illite
and I-S phases in clay deposits [16]. Another view point relates the asymptotic shape to
the great influence of simultaneous nucleation and crystal growth of illite [17, 18]. This
is only very likely, if illite in the samples represents one generation devoted to one
mechanism of formation [18]. However, the later explanation needs more investigation.
The CTD shapes of kaolinite vary from asymptotic (YAM and TAF clays) to multimodal
(MAH clays) and lognormal (UI and SIL clays). This variation reflects the occurrence of
two generations for kaolinite growth in response to changes in depth from shallow to
deep. Otherwise, the variation in the CTD can be explained on how the crystal growth
mechanism of kaolinite alters from a constant rate nucleation accompanied with crystal
growth (at shallow-moderate depths) to crystal growth without continuous nucleation (at
greater depths) [17, 18].
Discussion
Results of X-ray analyses give an initial impression on how the type and abundance
of clay minerals in the investigated mudrock samples vary with the present-day
stratigraphic thicknesses of rock formations (depths). Mineralogically, smectite
predominates other clay minerals in Al-Yamaniyya Pleistocene superficial clay deposits,
but deficient in the Tafila green claystone of the Turonian Shueib formation and the
Lower Cretaceous Mahis clay deposits occurring at depths around one km. Smectite
more likely changed to mixed layer I-S in the Tafila claystone. In the Lower Cretaceous
Mahis clays, kaolinite predominates other clay minerals and illite is less dominant.
Progressive illitization is possibly enhanced at depths over 1 km. Therefore, strong peaks
of illite and kaolinte were typical in the late Permian Umm Irna black shale (depth of
rock formation approximates 1.2km) and in the Upper Ordovician-Lower Silurian green
claystone of Batra Member (depth around 1.6km). The distinct appearance of illite peaks
in the Paleozoic mudrocks indicates diagenetic alterations of smectite and mixed layer SI precursors to illite. The degradation of smectite and its transformation to illite involve
incorporation of potassium (K+) ions into smectite structure and loss of interlayer water
[19, 20]. The compaction of mud through overburden pressure soon removes much of
the water contained in clay mineral lattices and water adsorbed onto clay surface or the
less common pore-filled water. At depths greater than one kilometer (temperature
в‰Ґ45В°C), 70-90% of the original volume of water is reduced to about 30%. Further
compaction through water loss requires temperature of about 100В°C (depths 2-4km)
where dehydration of clays takes place causing some changes in clay mineral types [21,
22}. Final compaction, leading to mudrock formation with only small percent water
needs much longer period of overburden pressure with elevated temperatures [22].
Despite the significant role of temperature and burial history in clay diagenesis, the role
of tectonic uplift and sediment unroofing cannot be ignored. Another criterion which
5
Odeh, Saqqa & Eberl
may complicate the origin of clay minerals is the possible occurrence of the same types
of clay minerals in more than one particular environment (terrestrial-marine). This is
despite differences in the physical and chemical conditions.
In this work, we consider the issue of primary (detrital) origin of clay minerals of
paramount importance. For instance, the coexistence of kaolinite and smectite in AlYamaniyya clay deposits can be explained on the basis of their primary origin after
chemical weathering of source materials. The latter minerals are very likely weathering
products of the very near Precambrian granites intruded by mafic dikes. Products of
weathering were then transported by streams draining granite terrains and finally
deposited in a lacustrine environment separated from sea by beach terraces [14, 15, and
23]. Leaching of iron and magnesium in source area leaves behind concentrations of
silicon and aluminum. This is favorable for kaolinite formation.
In the Tafila mudrocks, the presence of dolomite and gypsum beside the
argillaceous sediments signifies a deposition in a coastal sabkha environment after an
uplift at the beginning of the Turonian age. The tectonic event caused sea-level fall
resulting in the development of the sabkha environment [24]. The Tafila mudrocks were
more likely deposited during periodic influxes of fresh water which may facilitate
illitization process [e.g. 25]. The green type of Tafila clays were possibly developed by
leaching of nearby red clays and reduction of hematite under surface conditions. This
idea can be supported by the absence of chlorite (not detected by X-ray analyses) from
the green clays which can also impart green color for clay deposits in nature. We do
believe that Tafila clays were subjected to illitization as a result of surface wet-dry
cycles, evaporation, and fixation of K+ ions. The lack of sodium chloride (NaCl) in the
depositional medium (absence of halite in the rock sequence) enhances K+ ions fixation
in clays [26]. A burial depth of about one kilometer may help the transformation of
smectite to mixed layer I-S. Many researchers [e.g. 27, 28, 29, 30] believe that
illitization of smectite through I-S can take place under surface conditions when lake
water becomes alkaline and saturated with K+ ions whereby wet-dry cycles and
evaporation eventually lead to irreversible fixation of K+ ions and illite formation.
Mahis kaolinitic clay deposits that are a part of fluvial system sediments [31], have
two origins, detrital (primary) and authigenic. Khouri [23] mentioned neoformation of
kaolinite flakes adhered to quartz grains surfaces in Mahis clay sediments. Intensive
leaching of the detritus in acidic medium helps kaolinization aided by rainfall water.
Field observations confirmed a presence of carbonized plant remains in Mahis clays. The
breakdown of plant remains due to compaction releases CO2 and creates acidic pore
water. This initiates neoformation of kaolinite at depths greater than 1km (T>50В°C). The
minor content of I-S and the initial presence of illite in Mahis clays suggest advanced
stages of illitization at such depths. Coalified plant remains are also found in the Late
Permian Umm Irna black shale. This again prompts the creation of acidic medium in the
pore-filling water as a result of organic matter decay and CO2 release possibly by
thermal maturation at greater depths (>1200m). Accordingly, crystal growth of a primary
kaolinite can take place. The strong appearance of illite and kaolinite peaks present in
the XRD patterns supports the idea of precursor smectite illitization and the increase of
kaolinite crystallinity with depth. The great compaction of green claystone and
6
Use of X-ray Diffraction and Mud Master Program in the Evaluation of Crystallite Size and
Size Distributions of Illite and Kaolinite at Various Depths of Rock Formations in Jordan
sandstones of the Upper Ordovician-Lower Silurian Batra Member by overlying rocks
(present-day depth>1.6km) undoubtedly produced a set of physical and chemical
changes on these sediments. Hence, illitization process is continued and the crystallinity
of K is well developed.
Summary and conclusions
The present work revealed that diagenetic changes of clay minerals in the
investigated mudrock samples took place at various depths. The occurrence of smectite
gradually diminished with depth and was replaced by illite through an intermediate stage
of mixed-layer I-S. Transformations between clay minerals could be produced under
different depositional settings regardless of the evolved physical and chemical
conditions. For instance, the illitization of Tafila clays in a saline environment was
enhanced by wet-dry cycles, evaporation, and K+-ions fixation at surface conditions.
Burial depths and the increase of subaerial temperature were essential for illite
crystallization and the increase of kaolinite crystallinity. The asymptotic CTD shape in
illite indicates, at most, an overlap between different generations of illite and mixed
layer I-S in clays. The variations in CTD shape of kaolinite (asymptotic, multimodal, and
lognormal) explain the occurrence of two generations of kaolinite growth coincident
with the continual increase of the present-day depths of the rock formations.
Acknowledgments
The authors would like to thank Dr. F. Ababneh from Physics Department,
Yarmouk University for helping, partly, in operating the X-ray Diffractometer. We
extend our appreciation to the anonymous referees for their comments and critique in
advance.
7
вЂ«вЂЄOdeh, Saqqa & EberlвЂ¬вЂ¬
вЂ«Ш§пєіпє�пєЁпєЄШ§Щ… п»Јп»„п»ґпєЋп»“п»ґпє” пєЈп»ґп»®ШЇ Ш§п»·пє·п»Њпє” Ш§п»џпєґп»ґп»Ёп»ґпє” Щ€Ш§п»џп±Єп»§пєЋп»Јпєћ Ш§піЊпє¤п»®пєіпєђ )вЂЄ (MudMasterвЂ¬п»“п»ІвЂ¬
вЂ«ШҐп»іпє пєЋШЇ Ш§п»џпє¤пє п»ў Ш§п»џпє’п» п»®Ш±ЩЉ Щ€пє—п»®ШІп»іп»Љ Ш§п»·пєЈпє пєЋЩ… Ш§п»џпє’п» п»®Ш±п»іпє” п»џп»јп»іп»јп»іпє– Щ€Ш§п»џп»њпєЋЩ€п»џп»ґп»ЁпєЋп»іпє– п»‹п» п»° Ш§п»‹п»¤пєЋЩ‚вЂ¬
вЂ«п»ЈпєЁпє�п» п»”пє” п»џп» пє�п»њпєЋЩ€п»іп»¦ Ш§п»џпєјпєЁпє®п»іпє” п»“п»І Ш§п»·Ш±ШЇЩ†вЂЄ.вЂ¬вЂ¬
вЂ«Ш§пє‘пє®Ш§п»«п»ґп»ў п»Јпє¤п»¤пєЄ п»‹п»®ШЇШ©вЂЄ ШЊвЂ¬Щ€п»џп»ґпєЄ п»‹пє’пєЄ Ш§п»џп»�пєЋШЇШ± Ш§п»џпєґп»�пєЋ Щ€ШЇп»іп»ЁпєІ ШЇвЂЄ .вЂ¬Ш§пє‘пє®п»іп»ћвЂ¬
вЂ«п»Јп» пєЁпєєвЂ¬
вЂ«п»“п»�пєЄ пє—п»ў Ш§пєіпє�пєЁпєЄШ§Щ… п»Јп»„п»ґпєЋп»“п»ґпє” пєЈп»ґп»®ШЇ Ш§п»·пє·п»Њпє” Ш§п»џпєґп»ґп»Ёп»ґпє” Щ€Ш§п»џпє’пє®п»§пєЋп»Јпєћ Ш§п»џп»¤пє¤п»®пєіпєђ "п»ЈпєЄ п»ЈпєЋпєіпє�пє®" п»џпє�пє¤пєЄп»іпєЄ п»Јпє�п»®пєіп»‚вЂ¬
вЂ«Ш§п»џпє¤пє п»ў Ш§п»џпє’п» п»®Ш±ЩЉ Щ€п»Јпє�п»®пєіп»‚ Ш§п»џпєґп»¤пєЋп»›пє” Ш§п»џпє’п» п»®Ш±п»іпє”вЂЄ ШЊвЂ¬Щ€ШЈп»§п»¤пєЋШ· пє—п»®ШІп»іп»Љ Ш§п»џпєґп»¤пєЋп»›пєЋШЄ Ш§п»џпє’п» п»®Ш±п»іпє” п»џпє¤пє’п»ґпє’пєЋШЄ Ш§п»№п»іп» п» п»ґпє–вЂ¬
вЂ«Щ€Ш§п»џп»њпєЋШ¤п»џп»ґп»Ёп»ґпє– п»“п»І п»‹п»ґп»ЁпєЋШЄ пє—п»¤пєњп»ћ пє»пєЁп»®Ш± п»ѓп»ґп»Ёп»ґпє” п»ЈпєЁпє�п» п»”пє” Ш§п»·п»‹п»¤пєЋШ± п»Јп»¦ Ш§п»·Ш±ШЇЩ†вЂЄ .вЂ¬ШЈп»‡п»¬пє®ШЄ Ш§п»џпєЄШ±Ш§пєіпє” ШІп»іпєЋШЇШ© п»ѓп»”п»ґп»”пє” п»“п»ІвЂ¬
вЂ«п»Јпє�п»®пєіп»‚ Ш§п»џпє¤пє п»ў Ш§п»џпє’п» п»®Ш±ЩЉ Щ€п»Јпє�п»®пєіп»‚ Ш§п»џпєґп»¤пєЋп»›пєЋШЄ Ш§п»џпє’п» п»®Ш±п»іпє” п»џпє¤пє’п»ґпє’пєЋШЄ Ш§п»№п»іп» п» п»ґпє– пє—пє�п»ЁпєЋпєіпєђ п»Јп»Љ пє—пє°Ш§п»іпєЄ ШЈп»‹п»¤пєЋЩ‚ Ш§п»џпє�п»њпєЋЩ€п»іп»¦вЂ¬
вЂ«Ш§п»џпєјпєЁпє®п»іпє” Ш§п»џпє¤пєЋп»џп»ґпє”вЂЄ .вЂ¬ШЈп»ЈпєЋ пє‘пєЋп»џп»Ёпєґпє’пє” п»џпє¤пє’п»ґпє’пєЋШЄ Ш§п»џп»њпєЋШ¤п»џп»ґп»Ёп»ґпє–вЂЄ ШЊвЂ¬п»“пє�пєґпє�п»¤пє® Ш§п»џпє°п»іпєЋШЇШ© п»“п»І п»Јпє�п»®пєіп»‚ Ш§п»џпє¤пє п»ў Ш§п»џпє’п» п»®Ш±ЩЉ Щ€п»Јпє�п»®пєіп»‚вЂ¬
вЂ«Ш§п»џпєґп»¤пєЋп»›пєЋШЄ Ш§п»џпє’п» п»®Ш±п»іпє” ШҐп»џп»° ШЈп»‹п»¤пєЋЩ‚ пє—пєјп»ћ ШҐп»џп»° вЂЄ1.22вЂ¬п»›п»ўвЂЄ1.05 ШЊвЂ¬п»›п»ў п»‹п» п»° Ш§п»џпє�п»®Ш§п»џп»ІвЂЄ .вЂ¬п»іпє�п»ЁпєЋп»—пєє п»›п»ћ п»Јп»¦ п»Јпє�п»®пєіп»‚ Ш§п»џпє¤пє п»ўвЂ¬
вЂ«Щ€Ш§п»џпєґп»¤пєЋп»›пєЋШЄ Ш§п»џпє’п» п»®Ш±п»іпє” п»‹п»ЁпєЄп»ЈпєЋ п»іпєјп»ћ Ш§п»џп»Њп»¤п»– вЂЄ1.6вЂ¬п»›п»ўвЂЄ .вЂ¬п»іпє„пє§пє¬ п»Јп»Ёпє¤п»Ёп»° Ш§п»џпє�п»®ШІп»іп»Љ Ш§п»№пєЈпєјпєЋпє‹п»І п»џпєґп»¤пєЋп»›пє” пє‘п» п»®Ш±Ш§ШЄ Ш§п»№п»іп» п» п»ґпє– п»—п»ІвЂ¬
вЂ«пєџп»¤п»ґп»Љ Ш§п»џп»Њп»ґп»ЁпєЋШЄ Ш§п»џп»Ёп»¤п»‚ Ш§п»џп»¤пє¤пєЋШ°ЩЉ п»Јп»¤пєЋ п»іпєёп»ґпє® ШҐп»џп»° пє—пєЄШ§пє§п»јШЄ п»ЈпєЁпє�п» п»”пє” п»Јп»¦ ШЈпєџп»ґпєЋЩ„ Ш§п»№п»іп» п» п»ґпє– Щ€п»ЈпєЁпє�п» п»‚ Ш§п»џп»„пє’п»”пєЋШЄ ШҐп»іп» п»ґпє–вЂЄ-вЂ¬вЂ¬
вЂ«пєіп»¤п»ґп»њпє�п»ґпє–вЂЄ .вЂ¬ШЈп»ЈпєЋ пє‘пєЋп»џп»Ёпєґпє’пє” п»џпєёп»њп»ћ п»Јп»Ёпє¤п»Ёп»° пє—п»®ШІп»іп»Љ Ш§п»џпєґп»¤пєЋп»›пєЋШЄ Ш§п»џпє’п» п»®Ш±п»іпє” п»џпє¤пє’п»ґпє’пєЋШЄ Ш§п»џп»њпєЋШ¤п»џп»ґп»Ёп»ґпє–вЂЄ ШЊвЂ¬п»іпє�п»”пєЋЩ€ШЄ пє‘п»ґп»¦ Ш§п»џп»Ёп»¤п»‚вЂ¬
вЂ«Ш§п»џп»¤пє¤пєЋШ°ЩЉвЂЄ ШЊвЂ¬Щ€п»Јпє�п»ЊпєЄШЇ Ш§п»џп»Ёп»¤пєЋШ°Ш¬вЂЄ ШЊвЂ¬Щ€Ш§п»џп» п»®п»ЏпєЋШ±пє—п»¤п»І Ш§п»џп»„пє’п»ґп»Њп»ІвЂЄ ШЊвЂ¬Ш§п»·п»Јпє® Ш§п»џпє¬ЩЉ п»іп»Њпє°Щ‰ ШҐп»џп»° пє—п»Ёп»®Ш№ ШЈп»ѓп»®Ш§Ш± п»§п»¤п»® пє‘п» п»®Ш±Ш§ШЄвЂ¬
вЂ«Ш§п»џп»њпєЋЩ€п»џп»ґп»Ёп»ґпє– п»Јп»¤пєЋ п»іпє�п»®Ш§п»“п»– п»Јп»Љ ШІп»іпєЋШЇШ© ШЈп»‹п»¤пєЋЩ‚ Ш§п»џпє�п»њп»®п»іп»ЁпєЋШЄ Ш§п»џпєјпєЁпє®п»іпє”вЂЄ.вЂ¬вЂ¬
вЂ«вЂЄReferencesвЂ¬вЂ¬
вЂ«вЂЄDudek T., ЕљrodoЕ„ J., Eberl D.D., Elsass F. and Uhlik P., Thickness distribution ofвЂ¬вЂ¬
вЂ«вЂЄillite crystals in shales. Part 1: X-ray diffraction vs. high-resolution transmissionвЂ¬вЂ¬
вЂ«вЂЄelectron microscopy measurements. Clays and Clay Minerals, 50 (2002) 562-577.вЂ¬вЂ¬
вЂ«]вЂЄ[1вЂ¬вЂ¬
вЂ«вЂЄDrits V.A., ЕљrodoЕ„ J., and Eberl D.D., XRD measurement of mean crystalliteвЂ¬вЂ¬
вЂ«вЂЄthickness of illite and illite/smectite; reappraisal of the Kubler index and theвЂ¬вЂ¬
вЂ«вЂЄScherrer equation. Clays and Clay Minerals, 45 (1997) 461-475.вЂ¬вЂ¬
вЂ«]вЂЄ[2вЂ¬вЂ¬
вЂ«вЂЄWilson A. J.C., Proc. Phys. Soc. 80 (1962) 286; ibid. 81 (1963) 41.вЂ¬вЂ¬
вЂ«]вЂЄ[3вЂ¬вЂ¬
вЂ«вЂЄГЃrkai P., Merriman R.J., Roberts B., Peacor D.R. and TГіth M., Crystallinity,вЂ¬вЂ¬
вЂ«вЂЄcrystallite size and lattice strain of illite-muscovite and chlorite: comparison ofвЂ¬вЂ¬
вЂ«вЂЄXRD and TEM data for diagenetic to epizonal pelites. European Journal ofвЂ¬вЂ¬
вЂ«вЂЄMineralogy, 8 (1996)1119-1138.вЂ¬вЂ¬
вЂ«]вЂЄ[4вЂ¬вЂ¬
вЂ«вЂЄLangford J. I., J. Appl. Cryst. 11 (1978). 10-14.вЂ¬вЂ¬
вЂ«]вЂЄ[5вЂ¬вЂ¬
вЂ«вЂЄ8вЂ¬вЂ¬
Use of X-ray Diffraction and Mud Master Program in the Evaluation of Crystallite Size and
Size Distributions of Illite and Kaolinite at Various Depths of Rock Formations in Jordan
[6]
Eberl D.D., Drits V.A., ЕљrodoЕ„ J. and NГјsch R., MudMaster: a program for
calculating crystallite size distribuition and strain from the shapes of X-ray
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Drits V.A., Eberl D.D. and ЕљrodoЕ„ J., XRD measurement of mean thickness,
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Bertaut-Warren-Averbach technique. Clays and Clay Minerals, 46 (1998) 38-50.
[8]
Brime C., Eberl D.D. and ЕљrodoЕ„ J., Growth mechanisms of low-grade illites
based on the shapes of crystal thickness distributions. Schwarz. Mineral. Petrogr.
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[9]
Eberl D.D., NГјsch R. Е ucha, V. and Tsipursky S., Measurement of illite particle
thicknesses by X-ray diffraction using PVP-10 intercalation. Clays and Clay
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[10]
Bove D.J., Eberl D.D., McCarty D.K. and Meeker G.P., Characterization and
modeling of illite crystal particles and growth mechanism in a zoned
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1546-1556.
[11]
Е ucha V., Kraus I., Е amajovГЎ E. and Puskelova L. Crystallite size distribution of
kaolin minerals. Periodico di Mineralogia, 68 (1999) 81-92.
[12]
Abed A., Geology, Environment and water of Jordan (in Arabic). Jordanian
Geological Association, Scientific Books Series 1, Amman, (2000) 571.
[13]
Makhlouf I., Turner B. and Abed A.M., Depositional facies and environments in
the Permian Umm Irna Formation, Dead Sea area, Jordan. Sedimentary Geology,
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Abu Halima K., Mineralogy, chemistry and industrial studies of Al-Yamaniyya
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Amman (1993).
[15]
Saqqa W.A., Dwairi I.M. and Akhal H. M. Sedimentology, mineralogical
evaluation and industrial applications of the Pleistocene Al-Yamaniyyah clay
deposits, near вЂ�Aqaba, southern Jordan. Applied Clay Science 9, (1995) 443-458.
[16]
ЕљrodoЕ„ J., Precise identification of illite/smectite interstratifications by X-ray
powder diffraction. Clays and Clay Minerals, 28 (1980) 401-411.
[17]
Eberl D.D., Drits V.A. and ЕљrodoЕ„ J. Deducing growth mechanisms for minerals
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ЕљrodoЕ„ J., Eberl D.D. and Drits V.A. Evolution of fundamental-particle size
during illitization of smectite and implications for reaction mechanism. Clays and
Clay Minerals, 48 (2000) 446-458.
9
Odeh, Saqqa & Eberl
[19]
Dunoyer de Segonzac, G. The transformation of clay minerals during diagenesis
and low grade metamorphism: a review. Sedimentology, 15, (1970)281-346.
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metamorphism of argillaceous sediment: 1. Mineralogical and chemical evidence.
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[21] Tucker M., Sedimentary Petrology: An introduction. Blackwell Scientific
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Tucker M., Sedimentary Petrology: an introduction to the origin of sedimentary
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Jordan, Amman, (2002)116.
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Cretaceous gypsum horizon in west-central Jordan. Sedimentary Geology,
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Honty M., Uhlik P., Е ucha V., ДЊaploviДЌovГЎ M., FrancЕЇ J., Clauer N. and BiroЕ€
A., Smectite-to-illite alteration in salt-bearing bentonites (The east Slovak basin).
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[27]
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вЂ“ their diagenetic changes as an indicator of the paleoclimate. Geologische
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[28]
Eberl D.D., ЕљrodoЕ„ J. and Northrop H.R., Potassium fixation in smectite by
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and K.F. Hayes, editors). ACS Symposium Series 323, American Chemical
Society, (1986) 296-326.
[29]
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(1986).
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Report, Natural Resources Authority, Amman, 1990.
10
Use of X-ray Diffraction and Mud Master Program in the Evaluation of Crystallite Size and
Size Distributions of Illite and Kaolinite at Various Depths of Rock Formations in Jordan
Table (1) :Summary for XRD data between 3В°-15 В° 2Оё.
Name of
Sample
shifted to
Ethyleneglycolated
(2Оё) (dО‘В°)
5.0 17.67
collapsed toв†’
9.04
9.8
n.d.
в†’
n.d. n.d.
в†’
6.54
13.5
8.88
10
slightly
increased
to
8.68 10.10
slightly shifted
to
9.04
9.8
n.d.
n.d.
в†’
10.3 8.6
shifted to
9.04
9.8
12.42 7.13
в†’
12.2 7.25
в†’
12.62
7.02
12.48 7.09
в†’
12.34 7.17
в†’
Air-dried
(2Оё) (dО‘В°)
6.2
n.d.
YAM
TAF
14.25
8.02 11.02
12.32 7.18
6.92 12.7
MAH
UI
8.8
0.0
12.38
7.7
8.94
12.4
7.15
11.5
9.89
7.13
8.92
9.91
SIL
12.44 7.12
(dО‘В°)
(dО‘В°)
Heated (550В°C)
(2Оё) (dО‘В°)
Clay
mineral
S
I-Ch
.
I
I-S
Ch
K
destroyed
Mixture
of I and
one layer
S.
S
I
K
separated
to two
peaks:
1) 5.2 17.0
2) 8.8 10.0
1)collapsed to
2) unchanged
unchanged
12.32 7.18
в†’
1) 5.4 16.3
2) 7.2 12.3
1)collapsed to
2) в†’
9.04 9.78
12.38 7.15
7.7 11.5
8.94 9.89
12.4 7.13
unchanged
в†’
collapsed to
unchanged
в†’
8.8
10
destroyed
8.8
10
8.94 9.89
destroyed
I
K
I-S
I
K
9.02 9.8
unchanged
8.92
12.48 7.2
в†’
I
K
separated
to two
peaks:
slightly
shifted to
unchanged
unchanged
unchanged
unchanged
v. little
change
unchanged
8.8
8.8
10.0
10.0
destroyed
8.8
I-S
S
I-S
10.0
в†’
9.91
destroyed
(n.d. = non detected, S = smectite, I = illite, I-S = mixed layer illite-smectite,
Ch = chlorite, I-Ch= mixed layer illite-chlorite, K= kaolinite).
Table (2): D-spacing of the 001 reflection, means of crystallite size (CSmean) and
thickness (Tmean) and CTDs shapes of PVP-illite and kaolinite vs. depths.
(A) PVP-illite (air-dried samples)
Sample Depth(m) Mean size(nm) Thickness(nm) d-spacing(A пЂ©
YAM
TAF
MAH
UI
SIL
shallow
950
1050
1220
1600
(nm)
3.8
4
4
4.1
5.2
2.8
3
2.9
3
3.3
11
(A пЂ©
10.07
9.83
10.12
10.03
10.09
CTDs
shape
Asymptotic.
Asymptotic
Asymptotic
Asymptotic.
Asymptotic
Odeh, Saqqa & Eberl
(B) Kaolinite (heated samples at 350В°C )
Sample Depth (m) Mean size (nm) Thickness(nm) d-spacing(A пЂ© CTDs shape
YAM
TAF
MAH
UI
SIL
shallow
950
1050
1220
1600
3
3.7
8.9
10.7
10
7.5
11.3
16.8
14.7
14.7
12
7.13
7.17
7.17
7.13
7.16
Asymptotic
Asymptotic
Multimodal
Lognormal
Lognormal
Use of X-ray Diffraction and Mud Master Program in the Evaluation of Crystallite Size and
Size Distributions of Illite and Kaolinite at Various Depths of Rock Formations in Jordan
13
Odeh, Saqqa & Eberl
Fig.3. XRD-patterns (air-dried, ethylene glycolated, heated at 550В°C) of study samples.
S=smectite, I-S= mixed layer illite-smectite, I=illite, K=kaolinite, Ch =chlorite,
I-Ch= mixed layer illite-chlorite, Si = silicon wafer
14
Use of X-ray Diffraction and Mud Master Program in the Evaluation of Crystallite Size and
Size Distributions of Illite and Kaolinite at Various Depths of Rock Formations in Jordan
Mean (nm)
(a) Illite
6
Crystallite size
4
Crystallite
thickness
2
0
0
500
1000
1500
2000
Depth (m)
Mean (nm)
(b) Kaolinite
20
15
Crystallite size
10
Crystallite
thickness
5
0
0
500
1000
1500
2000
Depths (m)
Fig.4. Means of crystallite size and thickness for illite (a) and kaolinite (b) vs. depths.
15
Odeh, Saqqa & Eberl
(YAM)
0.6
F re q u en cy .
F r eq u e n c y .
0.8
0.4
0.2
0
0
5
10
15
0.6
0.5
0.4
0.3
0.2
0.1
0
20
(TAF)
0
5
10
15
20
Thickness (nm)
Thickness (nm)
Frequency.
0.7
0.6
0.5
0.4
(U I )
0.3
0.2
0.1
0
0
5
10
15
20
F requency.
Thickness (nm)
0.6
0.5
0.4
0.3
0.2
0.1
0
(SIL)
0
5
10
15
20
Thickness (nm)
Fig. 5. Shapes of PVP-Illite crystallite thickness distributions (CTDs) in the study
samples using the BWA-method (MudMaster).
16
Use of X-ray Diffraction and Mud Master Program in the Evaluation of Crystallite Size and
Size Distributions of Illite and Kaolinite at Various Depths of Rock Formations in Jordan
Frequency.
0.8
(YAM)
0.6
0.4
0.2
0
0
5
10
15
20
25
30
35
40
Thickness (nm)
Frequency
0.6
0.5
0.4
0.3
0.2
0.1
0
(TAF)
0
5
10
15
20
25
30
35
40
Thickness (nm)
Frequency
0.08
(MAH)
0.06
0.04
0.02
0
0
5
10
15
20
25
30
35
40
Frequency.
Thickness (nm)
0.15
(UI)
0.1
0.05
0
0
5
10
15
20
25
30
35
40
Frequency.
Thickness (nm)
0.08
(SIL)
0.06
0.04
0.02
0
0
5
10
15
20
25
30
35
40
Thickness (nm)
Fig. 6. Shapes of kaolinite crystallite thickness distributions (CTDs) in the study samples
using BWA-method (MudMaster).
17
ABHATH AL-YARMOUK: "Basic Sci. & Eng."
Vol. 16, No.1, 2007, pp. 19-26
Assessment of Background Radiation Dose due to Radon
in Kharja Village, Irbid Governorate, Jordan
Khitam M. Khasawinah and Khalid M. Abumurad* 1
Received on April 24, 2006
Accepted for publication on Sept. 28, 2006
Abstract
In this work, dwellings in Kharja village, north of Jordan were studied for background radiation doses due to radon using closed can techniques. The results obtained
reveal that the indoor radon levels almost equal or less than the national average (45
Bqm-3). In addition, the consumed water (collected rain water or precipitation)
characterized by moderate average radon concentration ~12.6 Bqв„“-1 which is less than
the action level that was set by the Europeans Countries and the Alternative Maximum
Contamination Level (AMCL) suggested by U.S.EPA for drinking water. Some
locations of the studied area were relatively characterized by high radon levels in soil air
(1.42 вЂ“ 1.85 kBqm-3). The estimated dose equivalent for Kharja village inhabitants due
to radon ranges from about 0.43 to 1.70 mSv, which is with in the acceptable range for
background radiations. The increase in lung cancer risk due to these doses is estimated
to be about 0.23 %. More studies should be carried out to look for other factors or
reasons to explain the many cancer incidents among the dwellers of Kharja village.
Keyword: Radon, Kharja Village, Jordan, Dose Equivalent.
Introduction
There is a strong epidemiologic evidence of the link between radon exposure and
lung cancer in the studies of underground miners at exposure only one or two order of
magnitude greater than typical lifetime exposure from indoor radon. Data from recent
studies [1, 2] provides direct evidence of a statistically significant association of indoor
radon exposure and lung cancer.
Radon is an inert radioactive gas which is produced naturally in three isotopes,
namely: actinon (219Rn), thoron (220Rn) and radon (222Rn).The mother radioactive series
of these elements are Actinium, Thorium and Uranium, respectively.
Environmentally speaking, 219Rn and 220Rn are immaterial.219Rn has half-life of
about 4 seconds and 220Rn has half-life of about 55 seconds. These half-lives limit their
spread and movement in the environment. 222Rn with half-life of about 3.82 days can
move from its cradle to far away places by diffusion and/or convection mechanisms.
В© 2007 by Yarmouk University, Irbid, Jordan.
* Department of Physics, Faculty of Science, Yarmouk University, Irbid 21163, Jordan.
Khasawinah and Abumurad
Soil and bedrocks beneath houses are considered to be the main source of radon in
the environment. Thus, radon can seep into dwellings through foundation cracks, wall
joints and openings for water and other services pipes.
Radon progeny's of epidemiological concern are the ones with short half-lives i.e.
polonium 218 and 214, these daughters can attached themselves to aerosol particles,
which if they inhaled can stick to the inner walls of airways and the inner lining layers of
the lung. Exposure for long times even for low radon levels may increase lung cancer
risk. In a study of EPA in the United States, radon in homes causes more deaths than
fires, drowning and airplanes accidents combined [3].
In this study, the area of Kharja, Governorate of Irbid, north of Jordan, was selected
for radon concentration measurement in order to correlate between radon level and the
many incidents of lung and other types of cancers; there are many reported cases of lung
cancer in the area of study compared to other parts of the Governorate. So, this study
aims to measure radon level in dwellings, soil air and water of Kharja region and hence
to calculate dose equivalent and lung cancer risk increase.
Experimental procedure
Solid state nuclear track detectors CR-39 with zero background were used in this
study. Those detectors were cut into 1.5cm x1.5cm pieces. For indoor study the
techniques of long term integrated passive closed can dosimeters were used. This type of
dosimeters was previously calibrated in the school of physics and space research at
Birmingham University, UK. The detectors were distributed in the selected houses
(group B) of Kharja where one or more cases of cancer were reported. Another group
of houses (group A) in the same village was selected as a control group. No cancer
incidents were reported among the inhabitants of this group. Two detectors for each
house were used, one was located in the living areas and the other was placed in the
bedrooms about 1m bellow the ceiling.
Water samples were collected from underground reservoirs (50 вЂ“ 80 m3 each) from
10 houses mentioned above. Small amounts (300 cm3) of these samples were placed in
drinking water bottles of 1в„“ volume. The CR-39 pieces (detectors) were fixed in the
inner bottoms of the bottle caps. The bottles were sealed tightly to prevent air exchange
and evaporation from water samples. Similar empty bottles were calibrated separately.
The detector height above the water sample was about 15cm.
Soil samples were collected from the gardens of the above mentioned houses
around 20cm beneath the ground surface. The samples were mixed thoroughly and
extraneous materials were removed. About 200 gm of each sample was transferred to a
bottle like those used for water and sealed with detectors inside. The detector was about
15cm above the surface of the soil sample. A total of 30 locations were selected for this
stud After exposure time of about 90 days the dosimeters and detectors were separated
from the sample cups and the sample bottles. There was no water droplets formed on the
surface of the detectors in the soil or water samples. The detectors were removed and
chemically etched using a 30% solution of KOH at a temperature of 70 oC for eight
hours. An optical microscope with a proper magnification was used to count the number
20
Assessment of Background Radiation Dose due to Radon in Kharja Village, Irbid Governorate, Jordan
of alpha tracks per unit area. The standard deviation of the data was taken as the
uncertainty in the reported results.
Calibration of dosimeters for water and soil studies
Twelve dosimeters were prepared for calibration process and three for the
background. The prepared dosimeters were placed in a chamber with standard radon
concentration (170 kBqm-3). Two detectors were removed from the chamber after 0.5, 1,
2, 4, 8, 16 hours. These irradiated detectors were chemically etched using 30% KOH at
70 oC for 8 hours. The track density, ПЃo, (tracks.cm-2) was measured and plotted against
the exposure time to as shown in Fig 1.
12000
Track Density (#/cm2)
10000
8000
6000
4000
2000
0
0
2
4
6
8
10
12
14
16
18
Exposure Time (hrs)
Fig. 1: Calibration curve for dosimeters used in water and soil samples.
Theoretical Background
Indoor study
To calculate radon concentration, the radon activity density in the can air was
determined by measuring the tracks density on the detector according to the following
relation:
C=
co t o ПЃ
ПЃ o t .................................................................................................. (1)
Where co, to ,ПЃo are radon level in the calibration chamber (90kBqm-3), exposure
time(48 h) and the surface density of tracks on the calibrated detector (3.032x104),
respectively. ПЃ and t are the measured surface density of tracks on exposed detectors and
the exposure time (2160 h), respectively.
21
Khasawinah and Abumurad
Water study
To calculate radon concentration, the radon activity density ca in the bottle air above
water sample was determined by measuring the tracks density on the detector according
to the following relation:
ca =
ПЃ 1
t Вµ
.................................................................................................... (2)
Where ПЃ is the average number of tracks on the detector, t is the exposure time
(2160h) of our sample and Вµ is the sensitivity of the detectors which equals to (3.688x103
(tracks.cm-2.h-1)/(Bqm-3)). However, Вµ was calculated by dividing the slope of Fig. 2 on
the radon concentration in the calibration chamber.
The radon activity density in water samples, cw, was calculated by using a model
proposed by Somogyi et al. [4]. The concentration of radon atoms exhaled can be
calculated according to the following relation
cw =
О»c a ht
l
.......................................................................................... (3)
Where О» is the decay constant of 222Rn =7.56 *10-3h-1, h is the distance from the
surface of water in the bottle to the detector, t is the exposure time and в„“ is the thickness
of the sample.
Soil study
To measure radon concentration and the exhalation rates, sealed bottle technique
was used.
The radon concentration A (l) Bq/m3 in the bottle air above the sample is related to
the aerial exhalation rate E(Bq/m2.h ) as
k в€’1
A(l ) = Ao k ........................................................................................... (4)
Where Ao is the radon concentration for very large relative thickness of the sample,
k is the geometrical correction factor for the sealed bottle
k = 1+
pd
l
tanh
hв€’l
d ................................................................................... (5)
1
2
Where p is the porosity of the sample, d is the diffusion length (= ( D / О» ) в‰€ 1)
[4] and the aerial exhalation rate is given by
22
Assessment of Background Radiation Dose due to Radon in Kharja Village, Irbid Governorate, Jordan
E (l ) = О».(h в€’ l ). A(l ) ................................................................................ (6)
Where О» is the decay constant and Ds (= 2 в€— 10
coefficient of radon.
в€’6
m 2 s в€’1 ) is the diffusion
One can calculate also the mass exhalation rate M (l ) at the surface of the sample
from the following equation:
M (l ) = О»
Where
Оґ
hв€’l
A(l )
hОґ
.................................................................................... (7)
is the mass density of the sample in crushed form?
The concentration A is given in terms of track density on the CR-39 detector by
equ. (2).
1Пѓ
Вµ t ........................................................................................................ (8)
A
Where Вµ was defined above, t is the exposure time (2160h) and Пѓ is the density of
=
tracks on the exposed detectors.
Results and Discussion
Indoor radon concentrations in the bed rooms in Kharja village were found in the
range of 16 - 90.0 Bqm-3 with an average of 32.9 Bqm-3 for group A and in the range of
14-186 Bqm-3 with an average of 66.1В±11 Bqm-3 for group B as shown in Table (1). The
highest concentration was found in a newly built house and the lowest level was found in
a house which was built on a rock foundation and has good ventilation. The average
radon level in group B more than twice of its average in group A. However, the average
radon level in Kharja village was about 43.3 В± 8.6Bqm-3. Assuming the indoor average
occupancy and the equilibrium factor are 0.8 and 0.45 respectively, one can calculate the
effective doses by using the dose conversion factor of 1.0 mSv per 40 Bqm-3 [5]. The
results of the average dose estimation for residents in group A and B from inhalation of
radon progeny show that the annual effective doses are 0.88 and 1.76 mSv, respectively.
From the data shown in Table (1), no positive associations of lung cancer cell type with
radon were observed. For example, in a house with lung cancer patient, radon levels in
the bedroom and in the living room were 78.3 and 24.9 Bqm-3, respectively; the bedroom
is almost always closed. But no lung cancer cases were reported among the dwellers of
other houses characterized by high radon level (186 Bqm-3) in spite of the inhabitants of
these two houses have almost the same living conditions, ages and family numbers.
However, there is no significant excess risk of lung cancer to more heavily exposed
residents. Further studies in other population groups are needed to clarify the
carcinogenic potential of indoor radon. Following Darby and his colleagues [2], the lung
cancer risk increase due to these doses was estimated to be about 0.28 %.
23
Khasawinah and Abumurad
Group
A
B
Group
A
B
Table 1: Indoor radon levels in different rooms of the studied houses.
Rooms
Max (Bqm-3)
Min (Bqm-3)
Avg (Bqm-3)
Dose (mSv)
Bed
90
16
32.9В±6.4
0.84
Living
20
13
17.0В±3.1
0.43
All Rooms
90
13
27.0В±5.2
0.69
Bed
186
14
66.1В±11.0
1.70
Living
91
9
32.2В±5.8
0.83
All Rooms
186
9
50.5В±9.8
1.29
Table 2: Radon levels in drinking water (rain collected water).
Max (Bql-1)
Min (Bql-1)
Avg (Bql-1)
Contribution to
indoor radon
7
21.3
4.9
6
6В±0.8
12.6В±2.6
0.6 Bqm-3
1.3 Bqm-3
Table (2) shows the results of radon concentrations in water consumed in the study
area. Most of the inhabitants use collected rain water from the roofs of their houses
during winter season. The radon levels range from 6 to 21 Bqв„“-1 with an average of
about 12.6 Bqв„“-1 for group B; while for group A the levels are almost the same. The over
all average of radon in water is about 10 Bqв„“-1, which is about the international accepted
level. EPA has set a maximum contamination level (MCL) of 11 Bqв„“-1 and an
Alternative Maximum Contamination Level (AMCL) of 150 Bqв„“-1 in the drinking water
at the United States [6], and European Countries has set reference levels range from
about 20 to 1000 Bqв„“-1. The upper limit considered in many European Countries as an
action level for radon in drinking water [7]. In spite of the fact that water contributes
only about 0.01% to the indoor air radon, these levels should be seriously considered
when cancers of gastrointestinal tract reasons are studied. Since stomach cancer
represents only about 7% while liver cancer represents more than 25% of all reported
cancer cases in the studied groups. Further studies are needed to illustrate the factors (if
any) affecting liver cancer.
Table 3 reflects the levels of radon in soil air; the aerial and mass exhalation rates
for soil samples from different houses in group B. Locations 1, 2, 3, and 4 almost have
the same moderate levels. But locations 5, 7, and 9 characterized relatively by high
levels, while location 6 has the lowest radon level. At least one or more inhabitants of
these locations have one kind of cancer incident.
Table3: Radon concentration, A, aerial, E, and mass, M, exhalation rates for soil samples from
gardens of different houses in group B.
M(Bqkg-1h-1)
Porosity
Location
A(kBqm-3)
E(Bqm-2h-1)
1
1.44
293
1.18
0.511
2
1.25
273
1.14
0.444
3
1.20
265
1.13
0.533
4
1.27
269
1.09
0.488
5
1.46
319
1.35
0.488
6
0.66
0.511
.560
138
7
1.85
377
1.46
0.444
8
1.42
331
1.56
0.533
24
Assessment of Background Radiation Dose due to Radon in Kharja Village, Irbid Governorate, Jordan
Conclusion
The geometric and arithmetic mean of radon activity in indoor of Kharja dwellings
were 36 and 50.5 Bqm-3, respectively. Opposite to what was expected, these values are
equal to or less than the mean values assessed in other Jordanian cities and villages [811]. In spite of the calculated cancer risk increase, this study did not find positive
associations between lung cancer and radon concentration in the studied samples. This
leads to examine other factors such as smoking and chemical air pollutants especially air
refreshers and insects' killers. Most consumed water (public and collected rain water) in
Kharja demonstrate low radon levels, since these waters were subject to aeration and
other processes which allow removal of dissolved radon and other gases. The average
radon level in drinking water in the study region is equal to the assumed world average
which is about 10 Bqв„“-1 [12-13] 1996). The soil air demonstrates, relatively, low radon
level compared to other locations in Jordan.
Acknowledgement
The authors acknowledge the financial support by grant number 2003/3 from the
Deanship of Scientific Research and Higher Education at Yarmouk University.
вЂ«ШЇШ±Ш§пєіпє” Ш§п»џпє пє®п»‹пє” Ш§п»№пє·п»ЊпєЋп»‹п»ґпє” Ш§п»џп»ЁпєЋпє—пє пє” п»‹п»¦ п»ЏпєЋШІ Ш§п»џпє®Ш§ШЇЩ€Щ†вЂ¬
вЂ«п»“п»І п»—пє®п»іпє” пє§пє®пєџпєЋ вЂ“ п»Јпє¤пєЋп»“п»€пє” Ш§Ш±пє‘пєЄ вЂ“ Ш§п»·Ш±ШЇЩ†вЂ¬
вЂ«пє§пє�пєЋЩ… пє§пєјпєЋЩ€п»§пє” Щ€пє§пєЋп»џпєЄ ШЈпє‘п»® п»Јпє®Ш§ШЇвЂ¬
вЂ«п»Јп» пєЁпєєвЂ¬
вЂ«п»џп»�пєЄ пє—п»ў п»“п»І п»«пє¬Щ‡ Ш§п»џпєЄШ±Ш§пєіпє” пє—п»Њп»ґп»ґп»¦ Ш§п»џпє пє®п»‹пє” Ш§п»»пє·п»ЊпєЋп»‹п»ґпє” Ш§п»џп»¤п»Ёпє’п»Њпєњпє” п»Јп»¦ п»ЏпєЋШІ Ш§п»џпє®Ш§ШЇЩ€Щ† Ш§п»џпє�п»І п»іпє�п»Њпє®Ш¶ п»џп»¬пєЋ пєіп»њпєЋЩ†вЂ¬
вЂ« ШЇп»џпє– Ш§п»џп»Ёпє�пєЋпє‹пєћ ШЈЩ† п»Јпєґпє�п»®п»іпєЋШЄвЂ¬.вЂ«п»—пє®п»іпє” пє§пє®пєџпєЋ п»“п»І пє·п»¤пєЋЩ„ Ш§п»џп»¤п»¤п» п»њпє” Ш§п»·Ш±ШЇп»§п»ґпє” Ш§п»џп»¬пєЋпє·п»¤п»ґпє” пє‘пєЋпєіпє�пєЁпєЄШ§Щ… п»ѓпє®п»іп»�пє” Ш§п»џп»®п»‹пєЋШЎ Ш§п»џп»¤п»ђп» п»–вЂ¬
вЂ« ШЈп»ЈпєЋ п»Јп»ґпєЋЩ‡ Ш§п»џпєёпє®ШЁ )Ш§п»џп»¤пє п»¤п»®п»‹пє” п»Јп»¦вЂ¬.(45 Bqm-3) вЂ«Ш§п»џпє®Ш§ШЇЩ€Щ† Ш§п»џп»¤п»Ёпє°п»џп»І п»іпєґпєЋЩ€ЩЉ пє—п»�пє®п»іпє’пєЋЩ‹ ШЈЩ€ ШЈп»—п»ћ п»Јп»¦ Ш§п»џп»¤пєґпє�п»®Щ‰ Ш§п»џп»®п»ѓп»Ёп»ІвЂ¬
вЂ« Щ€п»«п»® ШЈп»—п»ћ пє‘п»њпєњп»ґпє®вЂ¬.вЂ« пє—п»�пє®п»іпє’пєЋвЂ¬12.6 Bqв„“-1 вЂ«п»ЈпєЋШЎ Ш§п»џп»¤п»„пє® п»“п»І Шўпє‘пєЋШ±( п»“п»�пєЄ пє—п»¤п»ґпє°ШЄ пє‘пє�пє®п»›п»ґпє° п»Јп»Њпє�пєЄЩ„ п»џп» пє®Ш§ШЇЩ€Щ† пє‘п»¤пєЋ п»Јпє�п»®пєіп»„пє”вЂ¬
вЂ« ШЈп»ЈпєЋ п»“п»І п»«п»®Ш§ШЎ Ш§п»џпє�пє®пє‘пє” п»“п»�пєЄ Ш§пє—пєґп»¤пє– пє‘п»Њпєѕ Ш§п»џп»¤п»®Ш§п»—п»ЉвЂ¬.вЂ«п»Јп»¦ Ш§п»џпє¤пєЄ Ш§п»·п»‹п» п»° Ш§п»џп»¤п»�пє’п»®Щ„ п»‹пєЋп»џп»¤п»ґпєЋЩ‹ п»“п»ґп»¤пєЋ п»іпєЁпєє п»Јп»ґпєЋЩ‡ Ш§п»џпєёпє®ШЁвЂ¬
вЂ« пє‘п»ґп»Ёпє– Ш§п»џпєЄШ±Ш§пєіпє” ШЈЩ† Ш§п»џпє пє®п»‹пє” Ш§п»џп»¤п»њпєЋп»“пєЊпє” п»џп» пє�пє®Ш§п»›п»ґпє° Ш§п»џп»¤п»�пєЋпєіпє”вЂ¬.(1.42 вЂ“ 1.85kBqm-3) Щ‹вЂ«пє‘п»¤пєґпє�п»®п»іпєЋШЄ Ш±Ш§ШЇЩ€Щ† п»‹пєЋп»џп»ґпє” п»§пєґпє’п»ґпєЋвЂ¬
вЂ«( Щ€п»«пє¬Ш§ п»іп»Њпє�пє’пє® пєїп»¤п»¦ Ш§п»џп»¤пєЄЩ‰вЂ¬0.43 to 1.70mSv) вЂ«п»Јп»¦ Ш§п»џпє®Ш§ШЇЩ€Щ† п»“п»І Ш§п»џп»¤п»Ёп»„п»�пє” п»—п»ґпєЄ Ш§п»џпє’пє¤пєљ пє—пє�пє®Ш§Щ€Ш пє‘п»ґп»¦ Ш§п»џпє¤пєЄп»іп»¦вЂ¬
ШЊ0.23% вЂ« п»іпє°ШЇШ§ШЇ пє§п»„пє® Ш§п»»пє»пєЋпє‘пє” пє‘пєґпє®п»ѓпєЋЩ† Ш§п»џпє®пє‹пє” п»џп»¤пєњп»ћ п»«пє¬Щ‡ Ш§п»џп»¤пєґпє�п»®п»іпєЋШЄ п»Јп»¦ Ш§п»»пє·п»ЊпєЋШ№ пє‘пє¤п»®Ш§п»џп»ІвЂ¬.вЂ«Ш§п»џп»¤пєґп»¤п»®Ш п»џп»јпє·п»ЊпєЋШ№вЂ¬
вЂ«п»Јп»¤пєЋ п»іпє п»Њп»ћ п»«п»ЁпєЋЩѓ пєЈпєЋпєџпє” п»џп»¤пє°п»іпєЄ п»Јп»¦ Ш§п»џпєЄШ±Ш§пєіпєЋШЄ п»џп» пє’пє¤пєљ п»‹п»¦ ШЈпєіпє’пєЋШЁ ШЈпє§пє®Щ‰ п»џпє�п»”пєґп»ґпє® пєЈпєЋп»»ШЄ Ш§п»џпєґпє®п»ѓпєЋЩ† Ш§п»џп»ЊпєЄп»іпєЄШ© пє‘п»ґп»¦вЂ¬
.вЂ«пєіп»њпєЋЩ† Ш§п»џп»�пє®п»іпє”вЂ¬
25
Khasawinah and Abumurad
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Somogyi G., Hafez A., Hunyadi I. and Toth-Szilagyi M., Measurement of exhalation and
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[5]
UNSCEAR, Sources and effects of ionizing radiation. Report to the General Assembly,
United Nations, New York. (1997).
[6]
EPA, A citizen guide to radon (EPA Doc. No. 402-K-02-006), Washington, D.C. 20460.
(1994).
[7]
EC, Commission recommendation of 20 December 2001 on the protection of members of the
public to exposure of radon in drinking water supplies. 2001/928/euratom. Official Journal of
the European Commission, L344, (2001) 85-88.
[8]
Abumurad K. M., Kullab M. K., Al-Bataina B. A., Ismail A. M. and Lehlooh A., Estimation
of Radon Concenrations Inside Houses in some Jordanian Regions. Mu`tah J. for Research &
Studies, 9(5) (1994).
[9]
Abumurad K. M., Al-Bataina B., Ismail A., Kullab M. and Al-Eloosy A., A Survey of Radon
Levels in Jordanian Dwellings during an Autumn Season. Radiation Protection Dosimetry,
69(3)(1997) 221-226.
[10] Abumurad K. M., Chances of Lung Cancer due to radon exposure in Al-Mazar Al-Shamali,
Jordan. Radiat. Meas. 34 (2001) 537-540.
[11] Kullab M., Al-Bataina B., Ismail A., Abumurad K. M. and Ghaith A., 1997. A Study of
Radon-222 Concentration. Levels inside Kindergartens in Amman. Radiation Measurements,
28(1-6)(1997) 699-702.
[12] WHO, Indoor quality: a risk-based approach to health criteria for radon indoors. Report on
a WHO Working Group, Eilat, Israel, 28 march - 4 April 1993; EUR/ICP/CEH 108 (A).
WHO Regional Office for Europe. Denmark. (1996).
26
ABHATH AL-YARMOUK: "Basic Sci. & Eng."
Vol. 16, No.1, 2007, pp. 27-37
Determination of the Specific Activities of Radioactive Nuclides of
Surface Soils of Some HadhramoutвЂ™s Valleys in Yemen Using
Gamma-Ray Spectrometry
Awad Ali Bin- Jaza* and Abdulazize Omer Bazohair* a
Received on Sept. 14, 2005
Accepted for publication on Aug. 2, 2006
Abstract
In this research five surface soil samples were taken from some of HadhramoutвЂ™s
valleys in Yemen (Khrid , Arf , Huwayrah , Buwaish , Khirbah ) in the years from
(1999) to (2002) and analyzed using gamma-ray spectrometry. The spectra of samples
were measured using a multichannel analyzer (MCA) connected with measurement
system for this purpose. A high purity germanium(HpGe) detector with resolution of
(2.11keV) at gamma-line(1332keV) of radioactive source(Co-60) was used for detecting
(U238, Th232 , K40 , Cs137 ) in the samples. The results showed that the average specific
activities of radioactive nuclides in the samples for uranium ranged from(37.56 Bq/Kg)
to(46.58 Bq/Kg), and for thorium ranged from(37.93 Bq/Kg) to(47.28 Bq/Kg), and for
potassium ranged from(347.57 Bq/Kg) to(850.10 Bq/Kg), and for cesium ranged
from(13.01 Bq/Kg) to(15.86 Bq/Kg), and the measured precision of samples ranged
from(4.71%) to (9.23%).
Keywords: specific activity, surface soil, gamma-ray spectrometry
Introduction:
The soil surface is contaminated by the nuclear fallout caused by nuclear weapons
tests, which are still being carried out by some countries, nuclear accidents such as that
of Chernobyl nuclear reactor in 26 April 1986 that released a high radioactivity to the
atmosphere[6] and finally by cosmic rays .
The background natural radioactivity essentially consists of primary radiation
emanating from the uranium series ( U238, half-life = 4.5x109yr.), thorium series ( Th232,
half-life = 1.4x1010yr.), potassium ( K40, half-life = 1.3x109yr.), and the artificial fission
product is (Cs137, half-life = 30yr.). All these radiations are assimilated into plants and
body tissues of organisms and human beings.
В© 2007 by Yarmouk University, Irbid, Jordan.
* Physics Department, Faculty of Education- Mukalla.University of Hadhramout of science & Technology ,
Yemen.
Bin-Jaza and Bazohair
The environmental contamination is mainly caused by the spread of radioactive
materials on the soil surface samples .This contamination is of two kinds ; natural
contamination as in the case of uranium, thorium and potassium nuclides and artificial or
external contamination as in the case of cesium nuclide .
Most of people live in the regions of study use soil for agriculture and building, so
they afraid from the accumulations of its radiations. Consequently this study will provide
the concerned authorities in the Yemeni government with good information about the
natural and artificial radioactivity distribution in the Yemeni surface soils or its
contamination level.
Many studies were made to estimate the concentration of radio nuclides of U238,
Th , K40 and Cs137 in surface soils in different countries of the world[2,3,7 -17 ] .
232
Experimental Procedures:
Five soil surface samples were taken from some of HadhramoutвЂ™s valleys in Yemen
( Khrid , Arf , Huwayrah , Buwaish and Khirbah ) and illustrated in the map in fig(1) , in
the years from (1999) to (2002) , from an area of about 100cm2 and 5cm depth . These
samples were cleaned from stones and any other impurities and dried at 80ВєC for one
day, and grounded in the same mach. The samples were put in Marrinelli beaker, sealed
and then stored for a period of time without measurement so as to reach the natural radio
nuclides in an equilibrium state between the parent nuclei ( U238, Th232 ) and their
daughters of radon R222 and thoron R220 respectively .
The gamma spectrometry consists of horizontal high purity germanium (HpGe)
detector with sealed preamplifier, a bias high voltage supply, a linear amplifier, a
detector's shield made of (5cm) thick lead lined with aluminum, iron and copper plates
for x-ray degradation, and a multichannel analyzer (MCA) of Ortec model 6240B with
(1024) channel .
The (HpGe) detector is a Canberra model (7229N) with a crystal diameter of (7cm)
and a length of (7.5cm). It has a resolution of (2.11keV) at gamma-line (1332keV) for
(Co60 ) radioactive source .
28
Determination of the Specific Activities of Radioactive Nuclides of Surface Soils of Some HadhramoutвЂ™s
Valleys in Yemen Using Gamma-Ray Spectrometry
AL-HuwarahвЂ™s valley
KhridвЂ™s valley
ArfвЂ™s valley
BuwashвЂ™s valley
AL-Dees
AL-Shihir
KhrbahвЂ™s
Shohair
Arabia Sea
Mukalla
Fowah
0
20km
Fig (1) The location of valleys on the map
Results and Discussion:
Energy was calibrated by the least square fit method through the use of standard
radioactive sources with well- known energies and relative intensities to identify the
photo peaks present in the spectrum on the screen of (MCA)and recording the number of
channels corresponding to every gamma line (photo peak).
Efficiency calibration is necessary for activity measurements, so we used a sealed (
Eu152 ) radioactive source contained in Marrinelli beaker(*) [4],and has a long half-life of
(13.50yr.) and produces multilined spectra that include the major photo peaks usually
encountered in natural radioactivity measurements.
The efficiency of any photo peak included in the sample was calculated by
the following formula :
Efficiency (Оµ ) =
Cps
Г— 100(%) ................................................................(1)
A Г— IОі
Where : Cps вЂ“ is the count rate or count per second .
A- is the activity of reference radio nuclide in (Bq) at measurement .
IОі- is the intensity per decay (%) of photo emission at energy (E) .
(*)
Marrinelli beaker is a plastic beaker whose geometrical shape was designated for making the contents of
the sample near the effective region of the detector so as to allow high efficiency counting the gammarays which are emitted from the radio nuclides present in the sample.
29
0
0
350
700
1050
Bi214 (2204KeV)
(210Tl Pair Peak) (1919KeV)
0.01
(Tl210 +Ac228) (1650KeV)
K40 (1460KeV)
(Tl210+Bi212) (1080KeV)
Cs137(662KeV)
(Tl210+Ac228) (911KeV)
1.00
Tl208214
(583KeV)
Bi (609KeV)
Count x 1000
100.00
(U235+Th228+Ac227) (167KeV)
Bin-Jaza and Bazohair
1400
Channel No.
Fig (2) The typical Оі-ray spectrum of one of the studied soil sample
From the analysis of photo peaks of the studied soil samples shown in fig (2), we
see that there are many spectral lines. For this reason we choose the energy line
(609keV) for radio nuclide (Bi214) to indicate the presence of ( U238 ) in the soil samples ,
also we choose another energy line (583keV) for radio nuclide ( Tl208 ) to indicate the
presence of ( Th232 ) in the soil samples and we calculated its specific activities in the soil
samples according to the following causes :
(1) There is no interference of the chosen energy lines with the other energies of
radionuclides.
(2) They have high relative intensities.
(3) The statistical errors of their radioactivities are very low.
To measure the specific activities of radio nuclides in the soil surface samples , five
soil samples in Marrinelli beaker were measured by the detector (HpGe) for (40000)sec.
Because of the low radioactivity level.
The specific activities of radio nuclides U238, Th232 , K40 and Cs137 in (Bq/kg) of the
soil samples were calculated as following[13] :
30
Determination of the Specific Activities of Radioactive Nuclides of Surface Soils of Some HadhramoutвЂ™s
Valleys in Yemen Using Gamma-Ray Spectrometry
S . A.( Bq / kg ) = 1000 Г—
(C / t c )
............................................................................(2)
ОµI Оі m
Where : C- is the net count of the photo peak in (Bq) .
tc- is the time count of measurement (sec.) .
Оµ- is the counting efficiency at the specific energy (%) .
IОі- is the intensity per decay (%) of photon emission at the energy (E) .
m- is the mass of the sample (gm) .
Table(1) shows the specific activities of radio nuclides in different Yemeni soil
samples .
The detection limit (D.L.) of our method is calculated by Currie equation [5] as
following:
D.L. = 2.77 В± 3.29 B.G.
Concentration(C )( Bq / kg )
.................................(3)
A
Where : D.L.- is the detection limit .
B.G.- is the background (cps.) .
C.-is the concentration of radio nuclide of the sample (Bq/kg) .
A- is the net peak area (cps.) .
Table(2) shows the detection limit of our method in the studied surface soil
samples.
Table(3) shows the specific activities of ( U238, Th232 , K40 ,Cs137 ) in some countries
of the world and comparing with Yemeni surface samples .
The precision of the different radio nuclides in the samples was calculated by the
following equation :
R.S .D.(%) =
Пѓ
x
Г— 100 .......................................................................................(4)
Where : R.S.D.- is the relative standard deviation (precision) .
x - is the average concentration.
Пѓ
вЂ“ is the standard deviation .
Table(4) shows the precision analysis of radio nuclides (U238, Th232 , K40 ,Cs137 ) in
different studied samples .
31
Bin-Jaza and Bazohair
Table(1) The specific activities of radio nuclides in Yemeni surface soil samples
No. of
sample
Name of
valley
12345No. of
sample
Khrid
Arf
Huwayrah
Buwaish
Khirbah
Name of
valley
2000
2001
Uranium-238 concentration
( Bq/kg )
43.63В±3.42
40.1.3В±298
48.91В±4.22
38.79В±4.11
35.47В±3.14
42.64В±3.86
41.35В±3.18
38.91В±2.75
45.82В±3.99
36.34В±2.84
35.62В±2.40
40.69В±3.07
45.09В±4.33
43.84В±3.58
51.07В±4.81
Thorium-232 concentration
( Bq/kg )
12345No. of
sample
Khrid
Arf
Huwayrah
Buwaish
Khirbah
Name of
valley
38.62В±2.84
34.71В±3.09
43.41В±3.01
47.49В±1.41
42.97В±4.01
53.23В±4.62
45.80В±0.72
40.66В±3.72
48.62В±4.77
41.61В±2.03
39.92В±3.08
45.29В±3.64
37.95В±3.01
34.64В±2.71
40.68В±3.55
Potassium-40 concentration
( Bq/kg )
40.23В±3.34
45.41В±2.95
43.79В±4.19
42.63В±3.71
38.46В±3.09
12345No. of
sample
Khrid
Arf
Huwayrah
Buwaish
Khirbah
Name of
valley
428.81В±17.99 409.84В±16.31 470.61В±15.83
375.26В±11.62 356.17В±10.92 382.34В±12.01
346.38В±15.23 320.94В±7.62
365.43В±14.52
618.74В±24.51 584.52В±18.30 662.34В±21.56
874.38В±29.35 761.65В±22.50 892.73В±25.62
Cesium-137 concentration
( Bq/kg )
440.23В±16.20
360.64В±13.40
357.53В±14.01
596.71В±20.70
871.62В±22.03
12345-
Khrid
Arf
Huwayrah
Buwaish
Khirbah
14.71В±0.32
13.68В±0.53
13.31В±0.52
16.23В±074
12.46В±0.27
15.23В±0.21
15.74В±0.13
14.49В±0.09
16.35В±0.14
13.38В±0.08
1999
14.23В±01.25
13.45В±0.21
13.18В±0.19
16.03В±0.62
12.29В±0.26
32
13.64В±0.38
12.93В±0.29
12.79В±0.12
14.84В±0.54
13.91В±0.37
2002
45.64В±3.04
40.82В±2.78
41.74В±3.62
37.57В±2.84
46.31В±4.05
Average
concentration
В± S.D.
44.58В±3.68
39.43В±3.0 7
41.96В±2.8 6
37.56В±2.2 4
46.58В±3.1 6
Average
concentration
В± S.D.
39.24В±3.62
47.28В±4.38
44.72В±3.35
42.36В±2.25
37.93В±2.49
Average
concentration
В± S.D.
437.37В±25.46
368.60В±12.26
347.57В±19.40
615.58В±34.24
850.10В±59.70
Average
concentration
В± S.D.
14.45В±0.68
13.95В±1.23
13.44В±0.73
15.86В±0.69
13.01В±0.77
Determination of the Specific Activities of Radioactive Nuclides of Surface Soils of Some HadhramoutвЂ™s
Valleys in Yemen Using Gamma-Ray Spectrometry
Table(2) The detection limit of some surface soil sample
No. of
sample
Radio nuclide
Energy of radio
nuclide ( keV )
Concentration of
radio nuclide
(Bq/kg)
Detection Limit
(D.L.) (Bq/kg)
1-
Bi-214
609
44.58В±3.68
4.52
Tl-208
583
39.24В±3.62
3.68
K-40
1460
437.37В±25.46
24.18
Cs-137
662
14.45В±0.68
1.62
Bi-214
609
41.96В±2.86
4.01
Tl-208
583
44.72В±3.35
3.45
K-40
1460
347.57В±19.40
16.74
Cs-137
662
13.44В±0.73
0.71
Bi-214
609
46.58В±3.16
3.95
Tl-208
583
37.93В±2.49
2.67
K-40
1460
850.10В±59.70
33.49
Cs-137
662
13.01В±0.77
0.84
3-
5-
Table(3) The specific activities of radio nuclides in some countries of the world and
compared with Yemeni surface soil samples
Serial
No.
Country
1-
The specific activities of radio nuclides (Bq/kg)
No. of
reference
U-238
Th-232
K-40
Cs-137
West Malaysia
51.80В±10.36
70.30В±36.63
-
87.88В±15.83
[7],[8]
2-
U.S.A.
38.85В±17.76
36.26В±17.02
472
82.51В±38.11
[12],[15]
3-
Bangladesh
88.10В±4.80
68.20В±5.28
256.40В±16.30
7.50В±1.62
[10],[11]
4-
Finland
37
43
1043
147.35В±74.65
[14]
5-
Iraq
34.89В±17.66
8.78В±1.10
579.50В±156
3.23В±1.51
[9],[16]
6-
Tunis
17.10В±2.54
19.54В±3.47
284.20В±55.45
8.72В±7.57
[13]
7-
Taiwan
30
44
462
<D.L.
[17]
8-
Hong Kong
73
73
482
<D.L.
[3]
9-
Australia
24.79В±24.61
32.56В±19.98
172.79В±156
<D.L.
[2]
10-
Yemen
42.02В±3.67
42.31В±3.84
523.84В±210.66
14.14В±1.10
This work
33
Bin-Jaza and Bazohair
Table(4) The precision analysis of radio nuclides in some studied samples
x В±Пѓ
No. of sample
Radio nuclide
Energy ( keV )
1-
Bi-214
609
44.58В±3.68
8.25
Tl-208
583
39.24В±3.62
9.23
K-40
1460
437.37В±25.46
5.82
Cs-137
662
14.45В±0.68
4.71
Bi-214
609
46.58В±3.16
6.78
Tl-208
583
37.93В±2.49
6.56
K-40
1460
850.10В±59.70
7.02
Cs-137
662
13.01В±0.77
5.92
5-
(Bq/kg)
R.S.D. (%)
The results showed [ table (1) ] that the average specific activities of radio nuclides
in the various valleys Yemeni surface soil samples for uranium ranged from ( 37.56
Bq/kg) to (46.58 Bq/kg), for thorium ranged from ( 37.93 Bq/kg) to ( 47.28 Bq/kg), for
potassium ranged from ( 347.57 Bq/kg) to ( 850.10 Bq/kg) and for cesium ranged from (
13.01 Bq/kg) to ( 15.86 Bq/kg). But the final specific activities averages of these radio
nuclides in the Yemeni samples were ( 42.02В±3.67 ) Bq/kg for uranium , ( 42.31В±3.84)
Bq/kg for thorium, ( 523.84В±210.66 ) Bq/kg for potassium and ( 14.14В±1.10) Bq/kg for
cesium as shown in table (3) .
Table (3) shows that the specific activities of these radio nuclides in the Yemeni
samples is more than their specific activities in some countries and in some cases it is
less than that in some other countries, however the specific activities of these radio
nuclides in the Yemeni samples is small and within the possible range[6]. Also from
table(3) we see that the specific activities of potassium in the Yemeni soil samples is
high except when it is compared with that in Iraq and Finland . This indicates that the
floods or torrents in HadhramoutвЂ™s valleys come from the near by mountains which are
very rich in potassium[1]. The presence of cesium-137 in Yemeni environment results
from the seasonal winds which come to Yemen from the southern oceans ( the Indian
and the southern Atlantic oceans ) as a result of the different atmospheric pressures in
the area . From table(1) we see that the specific activities of 238U, 232Th and 40K are more
than in 137Cs because the presence of U,Th,K in soil surfaces are terrestrial, whereas the
presence of Cs is aerial. In year 2001 fall heavily rains on Hadhramout more than the
years 1999,2000,2002 , so the precipitation of radio nuclides U,Th,K and Cs in its
valleys are larger. From table(3) the average specific activity of cesium-137 in Yemeni
soil samples is small in comparison with some countries, but it is very small in
comparison with some countries of the world near the nuclear accident regions which
reached more than ( 300Bq/kg ) according to IAEA report[6]. All these radio nuclides
are transferred to the soil surface by the factor of erosion and rains.
34
Determination of the Specific Activities of Radioactive Nuclides of Surface Soils of Some HadhramoutвЂ™s
Valleys in Yemen Using Gamma-Ray Spectrometry
The detection limits of radio nuclides in the studied samples were good as shown in
table(2). The results for measuring radionuclides in soil samples were obtained
according to the best or optimum practical procedures by using:
(1) Large geometry for the detector crystal in detecting more number of falling
radionuclides on it by the samples.
(2) Long measuring time ( 40000sec.) .
(3) Large weight of the studied samples .
Conclusions
From this research, we deduce the following:
(1) The presence of radionuclides in soil samples of the valleys of Hadhramout which
resulted from the volcanic mountains chain surrounding the valleys.
(2) The average specific activities of radio nuclides in the studied soil samples is less
than the international concentration level.
(3) The specific activities of radio nuclides in Yemeni soil samples is within the
permissible range, so there is no dangerous level as pollution in soils taking into
consideration that most of the houses in Hadhramout were built and they are still
being built with these sands. There is also no danger in the use of these sands in
agriculture.
(4) The analytical precision for radio nuclides analysis in Yemeni soil samples is
statistically good because its value is less than ( 10%) .
(5) The detection limits of radio nuclides in the studied samples are good .
(6) The radio nuclides in the studied samples were measured according to optimum
practical procedures such as: (a) the large geometry of detecting samples (b) long
measuring time (c) large weight of samples .
(7) The high concentration of potassium in the Yemeni soil samples can be
economically exploited.
(8) There should be coordination with the concerned authorities in the Yemeni
government to provide them with information about the natural and artificial radio
activities distribution in the Yemeni soil as well as its contamination level .
35
Bin-Jaza and Bazohair
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вЂ«п»“п»І п»«пє¬Ш§ Ш§п»џпє’пє¤пєљ пє—п»ў пє—пє¤п» п»ґп»ћ пє§п»¤пєІ п»‹п»ґп»ЁпєЋШЄ п»Јп»¦ Ш§п»џпє�пє®пє‘пє” Ш§п»џпєґп»„пє¤п»ґпє” ШЈпє§пє¬ШЄ п»Јп»¦ пє‘п»Њпєѕ ШЈЩ€ШЇп»іпє” пєЈп»Ђпє®п»Јп»®ШЄ п»“п»ІвЂ¬
вЂ« Щ… пє‘пєЋпєіпє�пєЁпєЄШ§Щ… пє—п»�п»Ёп»ґпє”вЂ¬2002 вЂ« Щ€вЂ¬1999 вЂ« пє§пє®пє‘пє” ( п»“п»І Ш§п»џп»”пє�пє®Ш© пє‘п»ґп»¦вЂ¬ШЊ вЂ« пє‘п»®п»іпє¶вЂ¬ШЊ вЂ« пєЈп»®п»іпє®Ш©вЂ¬ШЊ вЂ« п»‹пє®ЩЃвЂ¬ШЊ вЂ«Ш§п»џп»ґп»¤п»¦ Щ€п»«п»І ) пє§пє®ШЇвЂ¬
(MCA) вЂ« п»—п»ґпєґпє– Ш§п»·п»ѓп»ґпєЋЩЃ Ш§п»№пє·п»ЊпєЋп»‹п»ґпє” п»џп» п»Њп»ґп»ЁпєЋШЄ пє‘п»®пєіпєЋп»ѓпє” п»Јпє¤п» п»ћ п»Јпє�п»ЊпєЄШЇ Ш§п»џп»�п»Ёп»®Ш§ШЄвЂ¬. вЂ«Ш§п»џпє�пє¤п» п»ґп»ћ Ш§п»џп»„п»ґп»”п»І п»·пє·п»Њпє” п»›пєЋп»ЈпєЋвЂ¬
вЂ«( Ш°Щ€ п»—пєЄШ±Ш©вЂ¬HpGe) вЂ« Ш§пєіпє�пєЁпєЄЩ… п»›пєЋпє·п»’ Ш§п»џпє пє®п»ЈпєЋп»§п»ґп»®Щ… п»‹пєЋп»џп»І Ш§п»џп»Ёп»�пєЋЩ€Ш©вЂ¬. вЂ«Ш§п»џп»¤пє®пє—пє’п»‚ пє‘п»¤п»Ёп»€п»®п»Јпє” п»—п»ґпєЋШі п»Јпє�п»њпєЋп»Јп» пє” п»џп»¬пє¬Ш§ Ш§п»џп»ђпє®Ш¶вЂ¬
вЂ«( п»џп» п»њпєёп»’ п»‹п»¦вЂ¬Co60 )вЂ«( Ш§п»џп»ЁпєЋпє—пєћ п»Јп»¦ Ш§п»§пє¤п»јЩ„ Ш§п»џп»¤пєјпєЄШ± Ш§п»џп»¤пєёп»ЉвЂ¬1332keV) вЂ«( п»‹п»ЁпєЄ Ш§п»џпєЁп»‚ Ш§п»џп»њпєЋп»Јп»ІвЂ¬2.11keV) вЂ«пє—пє¤п» п»ґп» п»ґпє”вЂ¬
вЂ« ШЈп»‡п»¬пє®ШЄ Ш§п»џп»Ёпє�пєЋпє‹пєћ ШЈЩ† п»Јп»ЊпєЄЩ„ Ш§п»џп»ЁпєёпєЋШ· Ш§п»№пє·п»ЊпєЋп»‹п»ІвЂ¬. вЂ« ( п»“п»І Ш§п»џп»Њп»ґп»ЁпєЋШЄвЂ¬Cs137, K40 , Th232 ,U238 ) вЂ«Ш§п»џп»Ёп»®ЩЉ Ш§п»џп»¤пєёп»Њпє”вЂ¬
(37.56Bq/Kg) вЂ« пє‘п»ґп»¦вЂ¬U238 вЂ«Ш§п»џп»Ёп»®п»‹п»І п»џп»¬пє¬Щ‡ Ш§п»џп»Ёп»®Щ‰ Ш§п»џп»¤пєёп»Њпє” п»“п»І Ш§п»џп»Њп»ґп»ЁпєЋШЄ пє—пє®Ш§Щ€пєЈпє– п»‹п» п»° Ш§п»џпє�пє®пє—п»ґпєђ п»џп»Ёп»€п»ґпє® Ш§п»џп»ґп»®Ш±Ш§п»§п»ґп»®Щ…вЂ¬
вЂ«( Щ€п»џп»Ёп»€п»ґпє® Ш§п»џпє’п»®пє—пєЋпєіп»ґп»®Щ…вЂ¬47.28Bq/Kg)вЂ«( Щ€вЂ¬37.93Bq/Kg) вЂ« пє‘п»ґп»¦вЂ¬Th232 вЂ«(Щ€п»џп»Ёп»€п»ґпє® Ш§п»џпєњп»®Ш±п»іп»®Щ…вЂ¬46.58Bq/Kg)вЂ«Щ€вЂ¬
(13.01Bq/Kg) вЂ« пє‘п»ґп»¦вЂ¬Cs137 вЂ«( Щ€п»џп»Ёп»€п»ґпє® Ш§п»џпєґп»ґпє°п»іп»®Щ…вЂ¬850.10Bq/Kg)вЂ«( Щ€вЂ¬347.57Bq/Kg) вЂ« пє‘п»ґп»¦вЂ¬K40
. (9.23%) вЂ«( Щ€вЂ¬4.71%) вЂ« Щ€п»—пєЄ пє—пє®Ш§Щ€пєЈпє– ШЇп»—пє” Ш§п»џп»�п»ґпєЋШі п»џп» п»Њп»ґп»ЁпєЋШЄ пє‘п»ґп»¦вЂ¬ШЊ (15.86Bq/Kg)вЂ«Щ€вЂ¬
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37
ABHATH AL-YARMOUK: "Basic Sci. & Eng."
Vol. 16, No.1, 2007, pp. 39-48
Origin and Migration of Primordial Germ Cells in Mosquitofish,
Gambusia affinis
Janti S. Qar and Mohammad A. Al-Adhami *a
Received on May 4, 2006
Accepted for publication on Sept. 12, 2006
Abstract
The origin and migration of primordial germ cells (PGCs) were studied in six
embryonic stages of mosquitofish, Gambusia affinis (Cyprinidontiformes, Teleostei). In
the earliest stage studied here (having 1pair of somites), the PGCs are observed in
splanchnic layer of the lateral plate mesoderm. Depending on their morphology alone,
this may be one of the earliest stages at which PGCs could be observed. No worth
mentioning change was observed at six pairs of somites. PGCs aggregate in the proximal
part of the lateral plate in embryos having 9 and 12 pairs of somites and are found in the
splanchnic mesoderm coating the gut in embryos with 14-16 pairs of somites. In
embryos having white-grey eyes, PGCs are found on either side of the dorsal mesentery
and start gathering underneath the pronephric duct. Finally, the PGCs reach the germinal
ridge and reside there in embryos with black eyes and reduced yolk sac. The present
study clearly indicates that PGCs in G. affinis are of mesodermal origin. It also indicates
that their migration from the lateral plate to the germinal ridges is composed of at least
three steps; two of which are active and one is passive. The results are in general
agreement with those observed for other teleosts, particularly zebrafish.
Keywords: primordial germ cells, teleosts, mosquitofish.
Introduction
It is well established that the primordial germ cells (PGCs) are the only source of
sex cells in vertebrates [1]. Reports on the origin of PGCs are, however, conflicting even
in related species [2, 3]. In nearly all species studied so far, these cells are first recorded
in places other than the germinal ridge. Later on, they migrate to reside in the gonadal
tissue [4, 5].
Among vertebrates, the origin, migration and characterization of PGCs are best
studied in amphibians [6, 7]. PGCs are of mesodermal origin in anuran amphibians [8]
and of endodermal origin in urodelian amphibians [3].
В© 2007 by Yarmouk University, Irbid, Jordan.
* Department of Biologiccal Sciences, Faculty of Sciences, Yarmouk University, Irbid, Jordan.
Qar and Al-Adhami
The origin of PGCs in teleosts is almost neglected, apart from a few early
publications [9, 10]. Later studies are confined to the description and differentiation of
PGCs at late embryonic stages, i.e. those having already migrated to the gonads [11, 12,
13]. Recently, the origin, characterization and migration of PGCs in zebrafish has been
the focus of intensive work [4, 14, 15, 16, 17].
This study is an attempt to throw light on the morphological description, site of
origin and route and mode of migration of PGCs in the mosquitofish, Gambusia affinis,
and comparing the results with similar observations on other teleost species.
Materials and Methods
Embryonic stages of the ovoviviparous, G. affinis (Cyprinidontidae, Teleostei) were
used in this study. To obtain different embryonic stages, pregnant females kept in
aquaria at 20-22oC were sacrificed by an overdose of MS222 and dissected at various
intervals. The embryos were liberated from the ovarian follicles and placed in
physiological saline using watchmaker forceps. They were then characterized according
to Kunz [18] before fixation in alcoholic Bouin's fluid.
To avoid yolk brittleness and obtain thin, serial sections (4 Вµm thick), Steedman's
ester wax technique [19] was used for embedding. Dehydration and fixation were carried
in changes of cellosolve (2-ethoxyethanol) as follows:
70%: 8h; 90%: overnight; three changes of pure cellosolve: 3h each; cellosolveester wax 50:50 overnight. The sections were stained with hematoxylin and eosin.
Infiltration was made in three changes (one hour each) of pure wax. Stains used were
Weigert's hematoxylin counterstained with van Gieson. This stain gives better
differentiation of PGCs than the routine stains.
Results
A few isolated, PGCs are observed in embryos having a single pair of somites.
They are sporadically found in the proximal part of the presumptive lateral plate
mesoderm at the level of, and posterior to, the differentiated somites (Fig. 1). A PGC can
be easily distinguished from the neighboring mesodermal cells by its remarkably bigger
size (Table 1) and weaker staining affinities. Its nucleus is large, round , vesicular, and
often deeply notched. Many chromatin granules underlie the nuclear envelope. One or
two nucleoli are also observed.
In embryos with six pairs of somites, the PGCs are still confined to the proximal
part of the lateral plate or its forerunner at the level of the differentiated somites and the
undifferentiated mesoderm posterior to them. They are totally lacking in regions anterior
to the first pair of somites. PGCs are still isolated and a cross section rarely shows more
than one cell on either side (Fig. 2).
40
Origin and Migration of Primordial Germ Cells in Mosquitofish, Gambusia affinis
Table 1. Numbers and dimensions of PGCs at different developmental stages of G.
affinis (Average of Seven Samples Each).
Stage
Number of PGCs
Diameter of PGCs (in Вµm)
(Number of
somites)
Left side
Right side
Total
1
18
35
53
14.06
6
24
27
51
14.10
9
34
40
74
14.89
12
38
32
70
14.68
14-16
47
40
87
14.73
white-grey eyes
190
180
370
14.54
Embryos with 9 pairs of somites and eye lens placodes show a clear increase in
numbers of PGCs (Table 1). Cross sections through the levels of posterior somites show
up to seven PGCs on at least one side of the embryo. These are either linearly arranged
or clustered into bunches (Figs. 3, 4). The increase of cells per section is not only due to
the overall increment of their total number, but also due to their migration along the
longitudinal axis towards the mid trunk region.
The PGCs are found in the proximal splanchnic mesoderm of the lateral plate. This
location, however, is almost confined to the posterior somites. PGCs at this stage seem
to be absent at the level of the anterior few somites. In rare observations, few PGCs may
be seen in the core of a somite. No significant change in embryos with 12 pairs of
somites is observed (Fig. 5).
In embryos with 14-16 pairs of somites and a single pair of aortic arches, some
PGCs are observed in the splanchnic mesoderm enclosing the gut at the levels where
lateral body folds have formed (Fig. 6). Posterior to the body folds, the PGCs are still
seen in the proximal ends of the splanchnic mesoderm (Figs. 7, 8).
At the stage when three aortic arches and white-grey eyes are developed, PGCs are
found on either side of the dorsal mesentery of the gut or at the future site of the
germinal ridge underneath the pronephric duct (Figs. 7, 8). This stage witnesses a sharp
increase in the number of PGCs. At the stage with black eyes and reduced yolk sac, well
defined early gonads are observed. PGCs are seen within these gonads. They are
sheathed by flattened cells in some individuals (Figs. 9, 10). It is concluded that these
gonads are ovaries.
Discussion
The findings of the present work on the origin and migration of the PGCs in
mosquitofish are in general accordance with results obtained in other teleosts,
particularly in zebrafish [5, 6]. Yet, these findings raise a few, interesting points. These
points are dealt with separately.
41
Qar and Al-Adhami
Origin: The results of this study are in favor of the widely accepted proposal that
PGCs in chordates originate somewhere away from the gonads [1, 3, 20]. In G. affinis,
PGCs are first seen in embryos with a single pair of somites. This is one of the earliest
stages involved in similar studies [9, 10, 14, 15]. The earliest stage of zebrafish at which
Braat et al. [14] were able to identify PGCs depending on their morphology alone (light
and electron microscopy) was the five somite stage.
The site of origin of PGCs in G. affinis recorded in this study is the proximal part of
the lateral plate mesoderm. This observation is in accordance with early observations on
the teleosts Fundulus heteroclitus [21],Cottus bairdii [22], Platypoecilus maculatus [9]
and Micropterus sal [10]. Nieuwkoop and Sutasurya [3], however, suggest that PGCs in
teleosts originate in the extraembryonic endoderm of the yolk sac. Accordingly, they
suggest a close phylogenetic relationship of teleosts with urodele amphibians in which
the same site of PGCs origin as teleosts has been recorded. Recently, Braat et al. [14]
proposed that PGCs formation might require a close contact with the mesoderm and/or
endoderm. This proposal might explain the contradictory observations on the origin of
PGCs in teleosts.
Migration: The present work indicates a stepwise migration of PGCs from their site
of origin to the germinal ridges. The process seems to involve a passive and an active
modes of migration. PGCs migrate from their site of origin in the lateral plate along
almost the entire embryo to aggregate in the mid-posterior trunk region at the level of the
pronephroi. This step is most likely an active one [23]. From their new location, they
passively migrate to the splanchnic mesoderm that coats the endodermal gut and to the
dorsal mesentery of the gut. This seems to take place via the enfolding of the splanchnic
mesoderm (formation of lateral body folds).
Such a mechanism and route of PGCs migration has early been suggested by Wolf
[9]. Most workers [6, 13, 24] overlook early developmental stages and begin their
investigation with stages having PGCs in the dorsal mesenteries or in the primordial
gonads. It is suggested here that the PGCs actively migrate from the dorsal mesenteries
to the gonads. Stepwise migration of PGCs in zebrafish has been well studied and
documented [4, 17]. During the active migration, PGCs possess filipodia and
pseudopodia, but these were not observed during the present study because these
structures are of very brief duration and persist for a minute or even shorter than that.
Morphology and count: The morphological description (shape, staining affinities
and measurements) given here for PGCs is in general accordance with that supplied by
other workers [11, 13, 14]. At early stages of G. affinis studied here, the number of
PGCs range from 30 to74 per individual embryo. The distribution of PGCs on either side
of the embryo is likewise variable. This variation is also observed in other teleost species
[9, 13] and in amphibians [25]. PGCs counts in the different embryonic stages studied
here show a slow, but a continuous proliferation during migration.
Finally, it is suggested here that the study of the origin and migration of PGCs in
vertebrates can be very valuable in a number of aspects including the mechanism of
differentiation of these cells and other types of cells, cellular migration and phylogenetic
relationships between the different vertebrate taxa.
42
вЂ«вЂЄOrigin and Migration of Primordial Germ Cells in Mosquitofish, Gambusia affinisвЂ¬вЂ¬
вЂ«п»Јп»Ёпєёпє„ Щ€п»«пє пє®Ш© Ш§п»џпєЁп»јп»іпєЋ Ш§п»џпє пє®пє›п»®п»Јп»ґпє” Ш§п»·Щ€п»џп»ґпє” п»“п»І пєіп»¤п»њпє” Ш§п»џпє’п»Њп»®Ш¶ вЂЄGambusia affinisвЂ¬вЂ¬
вЂ«пєџпєЋп»§пє�п»І пє»п»јШ Ш§п»џпєЄп»іп»¦ п»—пєЋШ± Щ€п»Јпє¤п»¤пєЄ ШЈп»Јп»ґп»¦ Ш§п»·п»‹п»€п»¤п»ІвЂ¬
вЂ«п»Јп» пєЁпєєвЂ¬
вЂ«ШЇШ±Ші п»§пєёп»®ШЎ Щ€п»«пє пє®Ш© Ш§п»џпєЁп»јп»іпєЋ Ш§п»џпє пє®пє›п»®п»Јп»ґпє” Ш§п»»Щ€п»џп»ґпє” п»“п»І пєіпє– п»Јпє®Ш§пєЈп»ћ пєџп»Ёп»ґп»Ёп»ґпє” п»џпєґп»¤п»њпє” Ш§п»џпє’п»Њп»®Ш¶ вЂЄGambusiaвЂ¬вЂ¬
вЂ«)вЂЄ .affinis (Gyprinidontiforme, TeleosteiвЂ¬п»џп»®пєЈп»€пє– Ш§п»џпєЁп»јп»іпєЋ Ш§п»џпє пє®пє›п»®п»Јп»ґпє” Ш§п»·Щ€п»џп»ґпє” п»“п»І Ш§п»џп»„пє’п»�пє” Ш§п»џпє¤пєёп»®п»іпє”вЂ¬
вЂ«п»џпєјп»”п»ґпє¤пє” Ш§п»џп»¤п»ґпє°Щ€ШЇШ±Щ… Ш§п»џпє пєЋп»§пє’п»ґпє” п»“п»І пєџп»Ёп»ґп»¦ Ш°ЩЉ ШІЩ€Ш¬ Щ€Ш§пєЈпєЄ п»Јп»¦ Ш§п»џпє’пєЄп»іп»ЁпєЋШЄ )вЂЄ (somitesвЂ¬Щ€п»«п»® ШЈпє»п»ђпє® п»Јпє®пєЈп» пє” ШЇШ±пєіпє–вЂ¬
вЂ«пє§п»јЩ„ п»«пє¬Ш§ Ш§п»џпє’пє¤пєљвЂЄ .вЂ¬Щ€Ш§п»‹пє�п»¤пєЋШЇШ§ п»‹п» п»° Ш§п»џпєјп»”пєЋШЄ Ш§п»џп»¤п»€п»¬пє®п»іпє” п»“п»�п»‚вЂЄ ШЊвЂ¬п»“п»�пєЄ пє—п»њп»®Щ† п»«пє¬Щ‡ ШЈпє»п»ђпє® п»Јпє®пєЈп» пє” пєіпє п» пє– п»“п»ґп»¬пєЋ Ш§п»џпєЁп»јп»іпєЋвЂ¬
вЂ«Ш§п»џпє пє®пє›п»®п»Јп»ґпє” Ш§п»·Щ€п»џп»ґпє” пє‘п»ґп»¦ Ш§п»·пєіп»¤пєЋЩѓ Ш§п»џп»Њп»€п»¤п»ґпє”вЂЄ .вЂ¬Щ€п»џп»ў п»іпєёп»¬пєЄ Ш§п»џпє п»Ёп»ґп»¦ Ш°Щ€ пєіпє– ШЈШІЩ€Ш§Ш¬ п»Јп»¦ Ш§п»џпє’пєЄп»іп»ЁпєЋШЄ ШЈЩЉ пє—п»ђп»ґпє® п»іпє¬п»›пє®вЂЄ.вЂ¬вЂ¬
вЂ«Щ€п»“п»І Ш§п»·пєџп»Ёпє” Ш°Щ€Ш§ШЄ вЂЄ 9вЂ¬ШЈЩ€ вЂЄ 12вЂ¬ШІЩ€пєџпєЋ п»Јп»¦ Ш§п»џпє’пєЄп»іп»ЁпєЋШЄвЂЄ ШЊвЂ¬пє—пє п»¤п»Њпє– Ш§п»џпєЁп»јп»іпєЋ п»Јп»®пєїп»®Ш№ Ш§п»џпєЄШ±Ш§пєіпє” п»“п»І Ш§п»џп»Ёп»¬пєЋп»іпє” Ш§п»џпєЄШ§п»§п»ґпє”вЂ¬
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вЂ«)вЂЄ.(Danio rerioвЂ¬вЂ¬
вЂ«вЂЄ43вЂ¬вЂ¬
Qar and Al-Adhami
References
[1]
Timmermans L. P. M., Origin and differentiation of primordial germ cells in
vertebrates especially fishes. Netherlands J. Zool. 46 (1996) 147-162.
[2] Humphrey R. R., The primordial germ cells of Hemidactylium and other Amphibia.
J. Morph. Physiol. 41 (1925) 1-43.
[3] Nieuwkoop P. D. and Sutasurya L. A., Embryological evidence for a possible
polyphyletic origin of the recent amphibians. J. Embryol. exp. Morphol. 35(1)
(1976) 159-167.
[4] Weidinger G., Wolke U., KoprГјnner M., Klinger M. and Raz E., Identification of
tissue and patterning events required for distinct steps in early migration of
zebrafish primordial germ cell. Devel. 126 (1999) 5295-5307.
[5] Dumstrei K., Mennecke R. and Raz E., Signaling pathways controlling primordial
germ cell migration in zebrafish. J. Cell Sci. 177 (2004) 4787-4795.
[6] Al-Mukhtar K. A. K. and Webb A. C., An ultrastructural study of primordial germ
cells, oogonia and early oocytes in Xenopus laevis. J. Embryol. exp. Morph. 26(2)
(1971) 195-216.
[7] Heasman J., Mohun T. and Wylie C. C., Studies on the locomotion of primordial
germ cells from Xenopus laevis in vitro. J. Embryol. exp. Morph. 42 (1977) 149161.
[8] Ljri K. and Egami N., Mitotic activity of germ cells during normal development of
Xenopus laevis tadpoles. J. Embryol. exp Morph. 34(1975) 687-694.
[9] Wolf L. E., The history of the germ cells in the viviparous teleost Platypoecilus
maculates. J. Morph. Physiol. 52(1)(1931) 115-153.
[10] Johnston P. M., The embryonic history of the germ cells of the large mouth black
bass, Micropterus salmoides. J. Morph. 88(1951) 471-542.
[11] Satoh N., An ultrastructural study of sex differentiation in the teleost Oryzias
latipes. J. Embryol. exp. Morphol. 32(1)(1974) 195-215.
[12]
Hamaguchi S. A., Light microscopic and electron microscopic study on the
migration of primordial germ cells in the teleost Oryzias latipes. Cell Tissue
Res. 227(1982) 139-151.
[13] Parmentier H. K. and Timmermans L. P. M., The differentiation of germ cells and
gonads during development of carp (Cyprinus carpio L.). A study with
anti-carp sperm monoclonal antibodies. J. Embryol. exp Morph. 90(1985) 13-32.
[14] Braat A. K., Speksnijder J. E. and Zivkovic D., Germ line development in fishes.
Int. J. Dev. Biol. 43(1999) 745-760.
44
Origin and Migration of Primordial Germ Cells in Mosquitofish, Gambusia affinis
[15]
Weidinger G., Wolke U., KoprГјnner M., Thisse C., Thisse B. and Raz E.,
Regulation of zebrafish primordial germ cell migration by attraction towards an
intermediate target. Devel. 129(2002) 25-36..
[16] Raz E., Primordial germ-cell development: the zebrafish prospective. Nat. Rev.
Genet. 4(2003) 690-700.
[17] Blaser H., Eisenbeiss S., Neumann M., Reichman-Freid M. Thisse B., Thisse C.
and Raz E., Transition from non-motile behavior to direct migration during early
PGC development in zebrafish. J. cell Sci. 118(2005) 4027-4038.
[18] Kunz Y. W., Histological study of greatly enlarged pericardial sac in the embryo of
the viviparous teleost Lebistes reticulates. Rev. Suisse. Zool. 78(1)(1971) 187207.
[19] Steedman H. F. Section Cutting in Microscopy. Oxford Co. London.(1960).
[20] Otani S., Maegawa S. Inoue K. Arai K. and Yamaha E., The germ cell lineage
identified by vas-nRNA during the embryogenesis in goldfish. Zool. Sci. 19:
(2005) 519-526.
[21] Richards A. and Thompson J., The migration of the primary sex cells of Fundulus
heteroclitus. Biol. Bull. 40(1921) 325-348.
[22] Hann H. W., The history of germ cells of Cottus bairdii, Girard. J. Morph. Physiol.
43(1935) 427-449.
[23] Doitsidu M., Reichman-Fred M., Stebler J., Koprunner M., Dorries J., Meyer D.,
Esguerra C. V., Leung T. and Raz, E., Guidance of primordial germ cell migration
by the chemokine SDF-1. Cell 111(2002) 647-659.
[24] Shibata N. and Hamaguchi S., Electron microscopic study of the blood-testis
barrier in the teleost Oryzias latipes. Zool. Sci. 3(1986) 331-338.
[25] Kita Y. and Wakahara M., Cytological analyses of factors which determine the
number of primordial germ cells (PGCs) in Xenopus laevis. J. Embryol. exp.
Morph. 90(1985) 251-265.
45
Qar and Al-Adhami
Legends of Figures
Figures 1 and 2. Cross section through the first somite level of embryos with one pair
and six pairs of somites, respectively. A single PGC (arrow) is seen in the proximal
part of the lateral plate (thick arrow). The endoderm (open arrow) is still stretched
as a flattened layer. Abbreviations: N: neural tube; Nc: notochord; S: somite;
E:ectoderm; Y:yolk. (X: 200)
Figure 3. Embryo with 9 pairs of somites. A cross section through the level of the third
somite (S). A cluster of primordial germ cells (arrows) are seen in the proximal part
of the lateral mesoderm. Thick arrow: roling up endoderm. (X: 250)
Figures 4 and 5. Embryo with 9 pairs of somites. Cross sections through the level of the
posterior somites showing clusters of PGCs (arrows) in the proximal parts of the
lateral plate. Thick arrows: endoderm; S: somite; other abbreviations as in figure 3.
(X: 150)
Figure 6. Embryo with 14-16 pairs of somites. A cross section through the level of the
pronephroi. A PGC (arrow) is seen in the splanchnic mesoderm of the intestine (I).
The notochord (Nc) and dorsal aorta (DA) are indicated. (X: 250).
Figures 7 and 8. Embryo with white to grey eyes. A cross section through the level of the
pronephroi. PGCs (arrows) can be seen on either side of the dorsal mesentery or
extending underneath the pronephric duct (asterisks). (X: 250).
Figures 9 and 10. Embryo with black eyes reduced yolk sac. A cross section through the
level of the early gonads (G). Abbreviations: BV: blood vessel; I: intestine; OC:
ossifying cartilage; P: pancreas. (X: 250).
46
Origin and Migration of Primordial Germ Cells in Mosquitofish, Gambusia affinis
47
Qar and Al-Adhami
48
ABHATH AL-YARMOUK: "Basic Sci. & Eng."
Vol. 16, No.1, 2007, pp. 49-57
The Role of Selenium and Lycopene in the Protection of
Mice Against Microcystin Toxicity
Saad Al-Jassabi and Ahmad Khalil* 1
Received on March 29, 2006
Accepted for publication on Aug. 10, 2006
Abstract
The aim of this study was to investigate the potential benefits of dietery
supplementation of selenium and or lycopene as antioxidants on microcystin toxicity in
mouse liver. Microcyitin-LRS (MC-LR) was isolated from Microcystis aeruginosa
blooms in King Talal Reservoir (Jordan). The LD50 for MC-LR, measured
intraperitoneally in Balb/c mouse was found to be 56 Вµg/kg body weight. Mice were
pretreated for two weeks either with sodium selenite (1.5 Вµg / mouse/ day), or with
lycopene (10 mg / mouse / day) before an intraperitoneal injection (i.p.) of 75 Вµg/kg of
MCвЂ“LR. Selenium supplementation was found to provide some protection to the action
of the toxin, while lycopene supplementation showed an excellent protection (about four
times as much as that observed for selenium). In both cases , and compared to the toxincontrol group, the liver damage caused by toxin was reduced as reflected in the reduction
of serum alanine transaminase(ALT) and lipid peroxidation level as well as prevention
of glycogen loss compared to non-supplemented toxin- treated mice. The increased
level of liver glutathione peroxidase (GPX) activity following selenium or lycopene
supplementation may indicate the possible route of selenium or lycopene protection in
the mice.
Keywords: microcystin; lycopene; selenium; King Talal Reservoir
Introductions
The contamination of aquatic ecosystems as a consequence of human activities is a
well established fact. In direct pollution processes like eutrification favors cyanobacterial
blooms, some of which are toxin producers [1-3]. These toxins are believed to be
products of the secondary metabolism of cyanobacteria, and it has been hypothesized
that they constitute a chemical defence against herbivores [4].
MicrocystinвЂ“LR (MCвЂ“LR), one of the toxins produced by these cyanobacteria, is a
potent hepatotoxin [5, 6]. This toxin is produced principally by Microcystis aeruginosa
and it is a potent hepatotoxin [7]. It is transported specifically into the liver by multi
specific bile acid transporters [8,9], thereby inducing severe intrahepatic hemorrhaging
necrosis and apoptosis [10,11]. MC- LR specifically inhibits serine/threonine protein
В© 2007 by Yarmouk University, Irbid, Jordan.
* Department of Biological Sciences, Yarmouk University, Irbid вЂ“ Jordan.
Al-Jassabi and Khalil
phosphates (PP1 and PP2A), which results in the disruption of many important cellular
processes [12-15]. Recently, M.aeruginosa was collected from King Talal Reservoir in
Jordan for the first time [16]. In this study, the bioactivity of the isolated MC-LR was
provided by phosphates inhibition and the HPLC analysis confirmed the identity of the
toxin.
MC-LR causes oxidative stress and increased reactive oxidative species ( ROS ), as
well as lipid peroxidation concomitant with an increase in lactate dehydrogenase(LDH )
release from primary rat hepatocytes [9,17]. Later work indicated a biphasic response in
cellular glutathione levels in primary rat hepatocytes exposed to MCвЂ“LR [17]. Sky
Herman et al [19] showed that hydrophobic antioxidants provide some protection against
lethal dose of MCвЂ“LR. Lycopene was defined as a nonvitamine active carotenoid that
has high oxygen radical scavenging and quenching capacity [20]. It is very beneficial in
the tissue to reduce the risk of adverse oxidative reactions that produce hydroxy radicals
and peroxides [14, 21].
Selenium is another antioxidant that appears to have a protective effect against
oxidative stress when provided prior to exposure to pro-oxidant [9, 22]. Selenium
supplementation has been shown to increase the activity of the selenoenzymes,
glutathione peroxidase (GPX) and thioredoxin reductase [23], both are important in the
reduction of lipid peroxides hydro.
The aim of this study is to investigate the role of each of selenium and lycopene
pretreatment on MCвЂ“LR induction of liver injury in mice.
Materials and Methods
Chemicals
All chemicals used in this study were purchased from Sigma Co., otherwise
indicated. All the chemicals were of analytical grade.
Animals
Five to seven weeks old male albino Balb/c mice (average weight 30 g.) were used
in this study. All animals were maintained under controlled conditions and on diet ad
labium and water in the animal house unit at Yarmouk University.
Microcystis cells
Microcystis aeruginosa cells were collected from selected sites of King Talal
Reservoir during June, July, August and September of 2003 (twice a month). Isolated
cells were cultivated in a culture medium as recommended by Lehtimaki [24].
Toxin extraction
The toxin was extracted from the freez-dried cells using the method of Gehringer et
al [9].
50
The Role of Selenium and Lycopene in the Protection of Mice Against Microcystin Toxicity
LD50 determination
The LD50 of the toxin extract was determined according to the up вЂ“ down method of
Farwell et al [25], which reduces the number of mice used and the quantity of toxin
required.
Toxicological studies
Thirty mice were divided into six groups (five mice each). Group 1 was the control
group with mice receiving neither toxin nor selenium or lycopene supplementation;
group 2 received selenium supplementation (1.5 Вµg of sodium selenite / mouse /day) for
two weeks prior to sacrification; group 3 received lycopene supplementation (10 mg/
mouse/day) for two weeks prior to sacrification; group 4 received a single i.p. dose of
toxin (75 Вµg/kg); group 5 received toxin (same amount) after two weeks of selenium
supplementation (same amount given to group 2) and group 6 received the same amount
of toxin after two weeks of lycopene supplementation (same amount given to group 3).
All toxin вЂ“treated mice were sacrificed 24 h. after injection of the toxin. The blood was
collected and the serum stored at вЂ“ 70Co for alanine transaminase (ALT) determination,
while the liver was perfused with Hanks buffered saline to remove excess blood. Liver
tissues were homogenized using (Ultra Turrax homogenizer ) in buffer ( 0.01 M Tris вЂ“
HCl pH7.8, 0.2 mM Dithiothreitol (DTT) and protease inhibitors : (7.5 mM PMSF , 2.5
mM EDTA, 3.25 ВµM bestatin, 2.5 ВµM eupeptic and 0.75 ВµM apoprotinin). The
homogenate was centrifuged and the supernatant stored at -70oC for the glutathione-Stransferase (GST) and glutathione peroxidase (GPX) assays.
ALT and lipid peroxide determination
ALT was determined in the serum sample and lipid peroxide in the liver sample
were performed according to Gehringer method [9].
Glycogen assay
A 0.5 g section of liver was homogenized in 1 ml of distilled water [26]. The
homogenate was diluted and added to 0.2% anthrone reagent, boiled for 10 min and read
at 620 nm as previously reported [27].
GST and GPX determination
GST assay and GPX determination were carried out according to the method
recommended by Gehringer et al [9].
Protein phosphates (PP1) activity
Protein phosphates (PP1) activity was determined by measuring the rate of colour
production from the dephosphorylation of para-nitro-phenolphosphate (ПЃ NPP) substrate
as a function of time using the micro titer plate reader according to Ane and Carmichael
method [28].
51
Al-Jassabi and Khalil
Results
The results showed that the liver weight (as % of body weight) increased
significantly (6.68) in toxin-treated mice ( without any supplementation) ( group 4 )
when compared with non-toxin treated control groups ( group 1, 2 and 3 ).The weight
decreased significantly in groups that were toxin-treated with selenium or lycopene
supplemented ( groups 5 and 6 ) as shown in Table (1).
In this study we found that the approximate intraperitoneal LD50 of the microcystin
(from M. aeruginosa of King Talal Reservoir) extract for the Balb/c mice was 56 Вµg
toxin/ kg mouse. Therefore, as recommended by earlier workers [10], the toxin dose
75Вµg/kg body weight, which is expected to kill all experimental mice, was chosen for
further experiments.
The average serum ALT levels per ml were calculated as shown in Table (1). The
average ALT value for the experimental group 4 which received toxin without any
supplementation showed a significant increase (pв‰¤ 0.05). The ALT values decreased in
those groups received supplementation especially in case of group 6 which received
lycopene.
All livers of control groups (1, 2 and 3) contained normal levels of glycogen, while
toxin-treated group (group 4) showed a decrease in liver glycogen level. Group 5 and 6
(selenium and lycopene supplemented mice receiving toxin) showed a smaller decrease
in the amount of glycogen in the liver (18.80 mg/g liver and 20.22 mg/g liver,
respectively) than mice in group 4 not receiving supplementation (7.66 mg/g liver) as
shown in Table (1).
The average of thiobarbituric acid (TBA) values were calculated and plotted as
shown in Table (1). The average TBA value for experimental group (group 4) treated
with toxin showed a significant increase (p в‰¤ 0.05) in lipid peroxidation levels when
compared with control groups (group 1, 2 and 3). In contrast, toxin-treated groups (
group 5 and 6 ) receiving supplementation at 1.5 Вµg selenium / mouse/ day , or 10 mg /
mouse/day lycopene showed significantly lower values( pв‰¤ 0.05) than the toxin-treated
group receiving no supplementation (Table 1).
The average GST levels per ml of liver homogenate are as shown in table (1). The
levels were significantly lower in toxin-treated mice (group 4) than the control ones, and
significantly increased in toxin-treated mice with the supplementation of either selsnium
(group 5) or lycopene (group 6). In comparison with the other groups, the levels of the
GPX activity were increased after supplementation of either selenium (group 2) or
lycopene (group 3).
Discussion
In the present study we have confirmed the identity and bioactivity of the isolated
toxin that have been reported before [16]. To avoid differences that may result from sex,
only male mice were used. The i.p. LD50 for this toxin when given to Balb/c mice was
found to be within the range reported in the literature [10]. The signs of toxin вЂ“ induced
52
The Role of Selenium and Lycopene in the Protection of Mice Against Microcystin Toxicity
injury such as hemorrhage, cell death, and apoptosis were observed [3, 9, 11, and 25].
The expected biochemical changes in serum ALT, liver glycogen content, TBA values,
GPX and GST activities were also observed in our study.
In our results, less severe liver damage was observed in groups 5 and 6
(supplemented with selenium and lycopene, respectively) than that occured in group 4
(toxin control). In the latter group, the wt. of the liver inceased as shown in table (1)
which confirms the occurrence of liver damage, and thus led to the accumulation of
fluids [11]. Further support for the ability of selenium or lycopene to protect the liver
from toxin injury was observed in the favorable changes in the biochemical markers
serum ALT, liver glycogen content, TBA values, GPX and GST activities with selenium
or lycopene treatment prior to toxin administration.
It was shown before [14] that MC-LR was a strong Reactive Oxygen Species
(ROS) inducer in hepatocytes of rat because of cytoskeleton disruption and
mitochondrial alterations. It has been well documented that ROS can affect the
microtubules, intermediate filaments and microfilaments organization due to their ability
to oxidize the cytoskeleton proteins or disturb the intracellular thiol balance [14, 29].
Ding et al [18] showed that the disruption of the cytoskeleton led to the ALT leakage,
suggesting that the disruption of the cytoskeleton could have played a crucial role in
microcystin- induced hepatotoxicity.
Our results showed a significant decrease in glycogen in the liver of toxin control
mice (group 4) when compared with other groups (Table 1). MC-LR is known to cause
the depletion of glycogen in the liver, possibly via the direct inhibition of glycogen
synthase an indirect activation of glycogen phosphorylase [9, 30].
In this study, it was shown that the levels of lipid peroxidation were increased in
toxin-treated mice (group 4) as reflected in increased TBA values because of the severe
liver damage, necrotic change, hemorrhaging, and apoptosis [17, 31, 32]. These levels
were decreased in group 5 and 6 (supplemented with selenium and lycopene
respectively), presumably via the ability of selenium and lycopene to quench free
radicals.
A significant decrease in liver GPX activity was observed in toxin-treated mice
(group 4), while the activity of GPX in selenium control (group 2) and lycopene control
(group 3 ) were significantly high comparing with other groups as shown in Table 1.
Our results showed increased level of GST activity in toxin-treated mice (group 4)
which confirms the work of Pietsch et al [33]. GST is a phase II detoxification enzyme
known to be involved in the detoxification of MC-LR [34]. It was found by [9, 35] that
the inhibition of GST occurs by increased level of lipid peroxidation (Table 1). Stahl and
Sies [36] indicated that lycopene is an a cyclic carotenoid containing a conjugated
double bonds arranged in the all-trans form, thus, it acts as antioxidant through physical
and chemical quenching of singlet oxygen and traps peroxyl radicals. Therefore, it can
be concluded that selenium and lycopene taken as a dietary supplement could confer
some protection against MC-LR in contaminated water sources. Finally, the relation
53
вЂ«вЂЄAl-Jassabi and KhalilвЂ¬вЂ¬
вЂ«вЂЄbetween the enzyme glutathione peroxidase, glutathione reductase and catalase needsвЂ¬вЂ¬
вЂ«вЂЄfurther experimentation.вЂ¬вЂ¬
вЂ«ШЇЩ€Ш± Ш§п»џпєґп»ґп» п»ґп»Ёп»ґп»®Щ… Щ€Ш§п»џп» п»ґп»њп»®пє‘пІ” п»“п»І пєЈп»¤пєЋп»іпє” Ш§п»џп»”пєЊпє®Ш§Щ† п»Јп»¦ Ш§п»џпє�пєґп»¤п»ў пє‘пєЋпіЊпєЋп»іп»њпє®Щ€пєіпєґпє�пІ”вЂ¬
вЂ«пєіп»ЊпєЄ Ш§п»џпє пєґпєЋпє‘п»І Щ€ШЈпєЈп»¤пєЄ пє§п» п»ґп»ћвЂ¬
вЂ«п»Јп» пєЁпєєвЂ¬
вЂ«п»«пєЄп»“пє– п»«пє¬Щ‡ Ш§п»џпєЄШ±Ш§пєіпє” ШҐп»џп»° Ш§пєіпє�п»�пєјпєЋШЎ Ш§п»µпє›пєЋШ± Ш§п»џп»¤п»”п»ґпєЄШ© п»№пєїпєЋп»“пє” Ш§п»џпєґп»ґп» п»ґп»Ёп»ґп»®Щ… Щ€Ш§п»џп» п»ґп»њп»®пє‘п»ґп»¦ п»›п»¤п»®Ш§ШЇ п»ЈпєЋп»§п»Њпє” п»џп» пє�пє„п»›ЩЂпєґпєЄвЂ¬
вЂ«ШҐп»џЩЂЩЂЩЂп»° п»ЏЩЂЩЂЩЂпє¬Ш§ШЎ Ш§п»џп»”пєЊЩЂЩЂЩЂЩЂпє®Ш§Щ† п»“ЩЂЩЂЩЂп»І Ш§п»џпє�п»�п» п»ґЩЂЩЂЩЂЩЂп»ћ п»ЈЩЂЩЂЩЂп»¦ пєіЩЂЩЂЩЂЩЂп»¤п»ґпє” Ш§п»џп»¤пєЋп»іп»њпє®Щ€пєіЩЂЩЂЩЂпєґпє�п»ґп»¦ п»“ЩЂЩЂЩЂп»І Ш§п»џп»њпє’ЩЂЩЂЩЂЩЂпєЄвЂЄ .вЂ¬Щ€п»џп»�ЩЂЩЂЩЂпєЄ пє—ЩЂЩЂЩЂЩЂп»ў Ш§пєіЩЂЩЂЩЂпє�пєЁп»јШµ пєіЩЂЩЂЩЂЩЂп»¤п»®Щ…вЂ¬
вЂ«Ш§п»џп»¤пєЋп»іп»њпє®Щ€Ш±пєіпє�п»ґп»¦ п»Јп»¦ Ш§п»џпє’п»њпє�пє®п»іпєЋ Ш§п»џпєЁп»Ђпє®Ш§ШЎ вЂ“ Ш§п»џп»¤пє°Ш±п»—пє” п»Јп»¦ п»§п»®Ш№ Ш§п»џп»¤пєЋп»іп»њпє®Щ€пєіпєґпє�пєІ Ш§Ш±п»іпє п»Ёп»®ШІШ§ Ш§п»џп»¤п»®пєџп»®ШЇШ© п»“ЩЂп»І п»Јп»ґЩЂпєЋЩ‡ пєіЩЂпєЄвЂ¬
вЂ«Ш§п»џп»¤п» ЩЂЩЂп»љ п»ѓЩЂЩЂп»јЩ„вЂЄ .вЂ¬п»“п»�пє’ЩЂЩЂп»ћ пєЈп»�ЩЂЩЂп»¦ п»“пєЊЩЂЩЂпє®Ш§Щ† Ш§п»џпє�пє ЩЂЩЂпєЋШ±ШЁ пє‘пє пє®п»‹ЩЂЩЂпєЋШЄ п»ЈЩЂЩЂп»¦ Ш§п»џп»¤пєЋп»іп»њпє®Щ€пєіЩЂЩЂпєґпє�п»ґп»¦вЂЄ ШЊвЂ¬пє—ЩЂЩЂп»ў ШҐп»‹п»„ЩЂЩЂпєЋШЎ Ш§п»џп»¤пє п»¤п»®п»‹ЩЂЩЂпє” Ш§п»·Щ€п»џЩЂЩЂп»° п»ЈЩЂЩЂп»¦вЂ¬
вЂ«Ш§п»џп»”пєЊпє®Ш§Щ† пєџпє®п»‹пє” п»Јп»¦ пєіпєЋп»»п»§пєЋп»іпє– Ш§п»џпєјп»®ШЇп»іп»®Щ… )вЂЄ 1.5вЂ¬п»Јп»ґп»њп»®п»Џпє®Ш§Щ… п»џп»њп»ћ п»“пєЋШ± п»іп»®п»Јп»ґпєЋ(вЂЄ .вЂ¬ШЈп»ЈЩЂпєЋ Ш§п»џп»¤пє п»¤п»®п»‹ЩЂпє” Ш§п»џпєњпєЋп»§п»ґЩЂпє”вЂЄ ШЊвЂ¬п»“п»�ЩЂпєЄ ШЈп»‹п»„п»ґЩЂпє–вЂ¬
вЂ«п»»п»іп»њп»®пє‘п»ґп»¦ Щ€п»џп» п»”пє�пє®Ш© п»§п»”пєґп»¬пєЋ Щ€пє‘пє�пє®п»›п»ґпє° вЂЄ 10вЂ¬п»Јп»ґп» п» п»ґп»ђпє®Ш§Щ… п»џп»њп»ћ п»“ЩЂпєЋШ± п»іп»®п»Јп»ґЩЂпєЋвЂЄ .вЂ¬Щ€п»џп»�ЩЂпєЄ пє‘п»ґп»ЁЩЂпє– Ш§п»џп»Ёпє�ЩЂпєЋпє‹пєћ Ш§п»џпє�ЩЂп»І пє—ЩЂп»ў Ш§п»џпє¤ЩЂпєјп»®Щ„ п»‹п» п»ґп»¬ЩЂпєЋвЂ¬
вЂ«ШЈЩ† пє—пє°Щ€п»іЩЂЩЂпєЄ п»ЏЩЂЩЂпє¬Ш§ШЎ Ш§п»џп»”пєЊЩЂЩЂпє®Ш§Щ† пє‘п»¬ЩЂЩЂпє¬Ш§ Ш§п»џп»¤п»�ЩЂЩЂпєЄШ§Ш± п»ЈЩЂЩЂп»¦ Ш§п»џЩЂЩЂпєґп» п»ґп»Ёп»ґп»®Щ… п»—ЩЂЩЂпєЄ п»Јп»Ёпє¤п»¬ЩЂЩЂпєЋ пє‘п»ЊЩЂЩЂпєѕ Ш§п»џп»®п»—пєЋп»іЩЂЩЂпє” п»ЈЩЂЩЂп»¦ пє—ЩЂЩЂпє„пє›п»ґпє® п»«ЩЂЩЂпє¬Ш§ Ш§п»џп»ЁЩЂЩЂп»®Ш№ п»ЈЩЂЩЂп»¦вЂ¬
вЂ«Ш§п»џпєґп»¤п»®Щ… вЂЄ.вЂ¬вЂ¬
вЂ«Щ€пє‘пєЋп»џп»¤п»�пєЋпє‘ЩЂЩЂп»ћ вЂЄ ШЊвЂ¬ШЈп»‡п»¬ЩЂЩЂпє®ШЄ п»ЈЩЂЩЂпєЋШЇШ© Ш§п»џп»јп»іп»њЩЂЩЂп»®пє‘п»ґп»¦ пє—ЩЂЩЂпє„пє›п»ґпє®Ш§ШЄ п»ЈЩЂЩЂп»ЂпєЋШЇШ© п»џЩЂЩЂпєґп»¤п»ґпє” п»ЈЩЂЩЂпєЋШЇШ© Ш§п»џп»¤пєЋп»іп»њпє®Щ€пєіЩЂЩЂпєґпє�п»ґп»¦ пє—пє’п» ЩЂЩЂп»Ћ ШЈШ±пє‘ЩЂЩЂп»Љ п»ЈЩЂЩЂпє®Ш§ШЄвЂ¬
вЂ«пє‘п»�ЩЂЩЂпєЄШ± пє—ЩЂЩЂпє„пє›п»ґпє® Ш§п»џЩЂЩЂпєґп»ґп» п»ґп»Ёп»ґп»®Щ…вЂЄ .вЂ¬Щ€пє‘п»¤п»�пєЋШ±п»§ЩЂЩЂпє” Ш§п»џп»Ёпє�ЩЂЩЂпєЋпє‹пєћ Ш§п»џпє�ЩЂЩЂп»І пєЈЩЂЩЂпєјп» п»ЁпєЋ п»‹п» п»ґп»¬ЩЂЩЂпєЋ п»ЈЩЂЩЂп»¦ Ш§п»џп»”пєЊЩЂЩЂпє®Ш§Щ† Ш§п»џпє�ЩЂЩЂп»І пєЈп»�п»ЁЩЂЩЂпє– пє‘пєЋп»џЩЂЩЂпєґп»ў пє‘ЩЂЩЂпєЄЩ€Щ† ШЈЩ† п»іЩЂЩЂпє�п»ўвЂ¬
вЂ«ШҐп»‹п»„пєЋпє‹п»¬пєЋ Ш§Щ‰ п»Јп»ЂпєЋШЇШ§ШЄ ШЈп»›пєґпєЄШ©вЂЄ ШЊвЂ¬п»џп»®пєЈп»† Щ€пє‘пєёп»њп»ћ Щ€Ш§пєїЩЂпєў пє—ЩЂпє„пє›п»ґпє® п»›ЩЂп»ћ п»ЈЩЂп»¦ Ш§п»џЩЂпєґп»ґп» п»ґп»Ёп»ґп»®Щ… Щ€Ш§п»џп»јп»іп»њЩЂп»®пє‘п»ґп»¦ п»‹п» ЩЂп»° Ш§п»џпє�п»�п» п»ґЩЂп»ћ п»ЈЩЂп»¦вЂ¬
вЂ«пє—ЩЂЩЂпє„пє›п»ґпє® Ш§п»џп»¤пєЋп»іп»њпє®Щ€пєіЩЂпєґпє�п»ґп»¦ Ш§п»џп»¤пє�п» ЩЂЩЂп»’ п»·п»§ЩЂЩЂпєґпє пє” Ш§п»џп»њпє’ЩЂЩЂпєЄ Щ€Ш°п»џЩЂЩЂп»љ пє‘пєЋп»§пєЁп»”ЩЂЩЂпєЋШ¶ п»ЈЩЂЩЂпєґпє�п»®п»іпєЋШЄ ШЈп»§ЩЂЩЂпє°п»іп»ў Ш§п»»п»»п»§ЩЂЩЂп»ґп»¦ пє—ЩЂЩЂпє®Ш§п»§пєІ Ш§п»Јп»ґп»Ёп»ґЩЂЩЂпє° п»“ЩЂЩЂп»ІвЂ¬
вЂ«п»Јпєјп»ћ Ш§п»џпєЄЩ…вЂЄ ШЊвЂ¬Щ€п»›пє¬п»џп»љ п»Јпєґпє�п»®Щ‰ ШЈп»›пєґпєЄШ© Ш§п»џп» п»ґпє’п»ґЩЂпєЄШ§ШЄ ШҐпєїЩЂпєЋп»“пє” ШҐп»џЩЂп»° п»Јп»ЁЩЂп»Љ пє—пє¤ЩЂпє®Ш± Ш§п»џп»њп»јп»іп»њЩЂп»®пєџп»ґп»¦ п»ЈЩЂп»¦ Ш§п»џп»њпє’ЩЂпєЄ Щ€Ш§п»џЩЂпє¬ЩЉ п»іпє¤ЩЂпєјп»ћвЂ¬
вЂ«п»‹пєЋШЇШ© пє‘пєґпє’пєђ пє—п» п»’ Ш§п»џп»њпє’пєЄвЂЄ .вЂ¬п»›п»¤пєЋ ШҐЩ† ШІп»іпєЋШЇШ© п»Јпєґпє�п»®Щ‰ п»§пєёпєЋШ· ШЈп»§пє°п»іп»ў Ш§п»џп»њп» п»®пє—пєЋпє›пєЋп»іп»®Щ† пє‘п»ґпє®Щ€п»›пєґп»ґпєЄп»іпє° п»“ЩЂп»І Ш§п»џп»њпє’ЩЂпєЄ п»‹п»�ЩЂпєђ пєЈп»�ЩЂп»¦вЂ¬
вЂ«Ш§п»џпєґп»ў п»“п»І пєЈпєЋп»џпє” Ш§п»џп»”пєЊпє®Ш§Щ† Ш§п»џпє�п»І пє—п»ў ШҐпєїЩЂпєЋп»“пє” п»›ЩЂп»ћ п»ЈЩЂп»¦ Ш§п»џЩЂпєґп»ґп» п»ґп»Ёп»ґп»®Щ… ШЈЩ€ Ш§п»џп»јп»іп»њЩЂп»®пє‘п»ґп»¦ ШҐп»џЩЂп»° п»ЏЩЂпє¬Ш§ШЎ п»›ЩЂп»ћ п»Јп»Ёп»¬ЩЂпєЋ п»—ЩЂпєЄ п»іЩЂпєёп»ґпє® ШҐп»џЩЂп»°вЂ¬
вЂ«Ш§п»µп»џп»ґпє” Ш§п»џп»¤пє¤пє�п»¤п» пє” п»џпє�пє„пє›п»ґпє® п»›п»ћ п»Јп»Ёп»¬п»¤пєЋ п»“п»І пєЈп»¤пєЋп»іпє” Ш§п»џп»”пєЊпє®Ш§Щ† п»Јп»¦ п»«пє¬Щ‡ Ш§п»џпєґп»¤п»®Щ… вЂЄ.вЂ¬вЂ¬
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56
The Role of Selenium and Lycopene in the Protection of Mice Against Microcystin Toxicity
Table 1: Summary of the biochemical effects of selenium and lycopene supplementation in
Balb/c mice receiving single lethal dose of microcystin.
Group 1
Group 2
Group 3
Group 4
Group 5
Group 6
( saline )
(selenium
(lycopene
(toxin
(selenium
(lycopene
(1)
control)
control)
control)
+ toxin)
+ toxin)
Liver( %
4.22В±0.117
4.39В±0.105
4.52В±0.077
6.68В±0.022
4.40В±0.064
4.32В±0.066
Body wt.)
(2)
ALT
0.68В±0.004
0.44В±0.002
0.39В±0.006
3.16В±0.015
0.53В±0.006
0.42В±0.006
20.40В±0.198
22.50В±0.188
23.30В±0.108
7.66В±0.099
18.81В±0.117
20.22В±0.109
0.020В±0.001
0.023В±0.006
0.022В±0.002
0.052В±0.004
0.027В±0.001
0.024В±0.001
1.84В±0.008
6.22В±0.079
7.88В±0.062
1.27В±0.008
2.28В±0.035
2.08В±0.022
20.05В±0.166
37.8В±0.011
38.08В±0.198
46.66В±0.208
32.22В±0.221
36.89В±0.217
(U/ml)
Glycogen
( mg/g)
TBA
Values (3)
GPX
Activity
( U/ml )
GST
Activity
(U/ml)
(1) Each group consists of 5 mice.
(2) Values represent the meansВ± standard errors of the mean of two experiments.
(3) Values read spectrophotometrically according to Gehringer et al.[9].
57
ABHATH AL-YARMOUK: "Basic Sci. & Eng."
Vol. 16, No.1, 2007, pp. 59-91
Geoeletrical and Hydrochemical Investigations in the Hallabat Area,
North Jordan.
Ghunaim S. Nasher, Hakam A. Mustafa and Nabil Abderahman*a
Received on Nov. 22, 2005
Accepted for publication on Sept. 12, 2006
Abstract
Geoelectrical and hydrochemical investigations were carried out in the Hallabat
area, north of Jordan, to study the groundwater salinization, and to assess the quality of
groundwater for its suitability for domestic and irrigation purposes. The area falls under
a semi-arid climate and water extraction is from basalt and limestone. Interpretations of
electrical survey data show that a low resistivity layer is present at depths ranging from
55 вЂ“ 130 m, with lowest resistivity values (less than 20в„¦.m) and highest depth found at
the central part of the study area. The data also show the continuity of this layer to the
surface which reflects the role of irrigation return flow for salinity build up. Chemical
analyses of the groundwater samples show that, the mean concentration of the cations is
in the order Na+ > CaВІ+ > MgВІ+ > K+ while for anions is in the order Cl ВЇ > HCO3ВЇ >
SO4ВІВЇ >NO3ВЇ. Three major water types occur in the study area namely alkaline earth
waters with increased portion of alkalies with prevailing chloride which is characterized
by high concentration of chloride and represents 61.9%, alkaline waters with prevailing
chloride which represent 33.3%, and alkaline earth waters with prevailing chloride
which represent 4.8%. High TDS levels accompanied with high nitrate levels were
observed in water from private wells in the cultivated land. The pollution with respect to
NO3ВЇ, Cl ВЇ, which are the main sources for salinity build up, is mainly attributed to the
extensive use of fertilizers, and excessive pumping from private wells. Most of the
sampled water is not suitable for drinking purposes due to the high NO3ВЇ content. It can
be used for irrigation purposes as the sodium hazard is low but it needs treatment to
reduce salinity that ranges from medium to very high hazard.
Keywords: geoelectrical, hydrochemical, pollution, Hallabat, Jordan, Amman
Formation, Wadi Es вЂ“ Sir Formation, Basalt.
В© 2007 by Yarmouk University, Irbid, Jordan.
* Department of Earth and Environmental Sciences, Faculty of Science, Yarmouk University, Irbid, Jordan.
59
Nasher, Mustafa and Abderahman
Introduction
The study area forms a part of the Hallabat catchment area which is,
hydrogeologically, considered as a part of the Zarqa Basin.
The investigation of this area is important due to: Its large population, the wide
distribution of its cultivated land which is considered as one of the major irrigated areas
in Jordan, its high rates of groundwater recharge, the industrial activities, and the
importance of shallow aquifers consisting of Upper Cretaceous cherty and calcareous
rocks, and Upper Tertiary and Quaternary basaltic rocks.
Groundwater abstraction from both private and government wells in the study area
began early 1960s. Agricultural activity is the major groundwater user in this area.
Continual withdrawl has resulted in problems such as regional decline of groundwater
levels and degradation of water quality. With passage of time salt concentration in the
water increased and became a real problem facing the domestic uses and irrigation in this
area.
The study area (Fig. 1), is located in the northern part of Jordan, some 45 km to the
northeast of the capital city Amman, it forms a part of the Hallabat catchment area. It is
defined by the coordinates E 277 вЂ“ E 285 and N 1165 вЂ“ N 1172 (Palestine Grid, PLG).
Many geological, hydrogeological, hydrochemical, environmental, and geophysical
studies were carried out on the study area for different purposes since the sixties of the
last century:
Fig.1: Location map of the study area.
60
Geoeletrical and Hydrochemical Investigations in the Hallabat Area, North Jordan
Mac Donald et al, [1-3] and Abu Obeid [4,5] studied the water resources in the
northeastern part of Jordan including the study area. They found that the water is of good
quality and can be used for irrigation and the lowering of water table in this area to
overpumping.
Al вЂ“ Zabet [6] conducted a study of two вЂ“ dimensional finite element model for
groundwater flow of Dhuleil вЂ“ Hallabat region. This model was run using the water level
of 1992 as a base level and with
a pumping rate of 30 MCM/a to predict the
fluctuation of the water levels in 2012. He found that the drawdown will be 8 meters in
1992 and will reach up 13.25 meters by 2012.
Salameh [7] investigated the water quality degradation in different areas in Jordan
including the study area and Bajjali [8] studied the building up of salinity in groundwater
over time in Dhuleil, Hallabat and Samra areas to the west of the study area. they found
that changes in the groundwater quality in Dhuleil area are attributed mainly to irrigation
return flows and to salinization of aquifers as a result of overpumping and also the
infiltration of the semi-treated waste water of Khirbet Es-Samra into the aquifer.
Al-Bis [9] applied aeromagnetic, gravity, electrical resistivity (VESs),
electromagnetic (VLF) methods on the upper aquifer system to the west of the study area
to study the subsurface structures to determine the relationship between structures and
groundwater movements.
The present study aims at determining the depth and chemical characteristics of the
groundwater, the areal water вЂ“ quality distribution and the factors that affect the water
quality in the study area.
Methodology
The present study involves geoelectrical measurements and hydrochemical
investigations.
The geoelecrical resistivity survey in Hallabat area was carried out in Septemper
2003. All measurements were taken using an ABEM- Terrameter SAS- 300B of the
Department of Geology, University of Jordan.
The survey is of Vertical Electrical Sounding (VES) type. This configuration is
widely used to measure vertical changes in the resistivity of the subsurface rocks below
given points on the earthвЂ™s surface. A sum of 12 VES profiles was conducted with half
spacing between current electrodes ranging between 400m to 800m (Fig.2).
The maximum separation of current electrodes was affected by the accessibility
situation.
61
Nasher, Mustafa and Abderahman
Fig. 2: Distribution of all vertical electrical sounding ( VES) in the study area.
For hydrochemical investigation twenty one water samples from twenty one
production wells were collected in October, 2003. These samples were taken after
operating the wells of about 2 вЂ“ 4 hours. The samples were stored in clean polyethylene
bottles, that have been washed by diluted HCl and by distilled water in the laboratory. In
the field, the bottles are well washed by sampled water, adequately labeled and
preserved in the refrigerator until they were taken to the laboratory for chemical analysis.
pH and electrical conductivity (EC) values were measured in the field using pHmeter and EC-meter respectively.
The coordinates of the wells (Fig. 3) were measured in the field by using
Maggellan Global Positioning System (GPS).
The laboratory analyses of the collected groundwater samples were performed in
the laboratories of the Yarmouk University.
The determination of chloride (ClВЇ) and Alkalinity which is related to bicarbonate
(HCO3ВЇ) was determined by titration. Sulfate (SO4ВІВЇ) and Nitrate (NO3ВЇ) determination
were done by spectrophotometery .
62
Geoeletrical and Hydrochemical Investigations in the Hallabat Area, North Jordan
Determination of the major cations CaВІ+, MgВІ+, K+, and Na+ were done by using
atomic absorption spectrophotometer (ASS).
Fig. 3: Distribution of studied wells in the study area.
stratigraphy
The area east of Mafraq-Azraq-Wadi Sirhan referred to as the northeastern desert of
Jordan comprises two morphological units, the first is the NE Basalt Plateau and the NE
Limestone plateau (Fig.4). The Northeastern Limestone Plateau extends from the eastern
margin of the basalt plateau with elevations of about 625 m (a.s.l) in the WNW to more
than 800 m (a.s.l) in the ENE.
The rocks exposed in the area consist mainly of limestone, marl, chalk, and
alluvium, and volcanics ranging in age from Late Cretaceous to Recent, and belong to
Ajlun, Belqa and Plateau Groups [2,10].
Ajlun Group
Only the Shueib and Wadi Es-Sir Formations are exposed in the study area. The
Shueib formation is exposed in the southern part of the study area. It consists of yellow,
brown, blue вЂ“ grey and black marl, containing
63
Nasher, Mustafa and Abderahman
Fig.4: Geology of the study area (as coated from Bajjali, [8]).
Sometimes pyrite, with marly limestones and occasional thin crystalline and shelly
limestone. its thickness reaches more than 50 meters [11]. The Wadi Es-Sir is Turonian
age. It consists of grey, white and pinkish crystalline limestone overlying
porcellanceous, marly and chalky limestone. The massive limestone forming the top 20
meters of the formation shows open jointed and strongly developed karstic weathering in
outcrop [11]. There is transition into the underlying Shueib Formation. Its thickness
ranges between 70 and 110 meters.
Belqa Group
The lower lying Umm Ghudran Formation (B1) consists mainly of white and pink
chalk, marl and occasional marly limestone with a total thickness between 15 and 35
meters [2,4].
The younger Amman Formation (B2) consists of chalk, marl, massive chert, with
relatively thin, white crystalline and shelly limestone. Only the lower part of this
formation is exposed. However, its whole thickness is estimated to be 47 meters [12].
Plateau Group
Plateau Group of Early Eocene to Late Pleistocene age, comprises the Older
Alluvial Formation, the basalt, and the Younger-Recent Alluvium[2].
The Older Alluvial formation underlies the oldest basalt flow and is intercalated
with the different younger basalt flows. It consists mainly of silts and clays in addition to
gravels.
64
Geoeletrical and Hydrochemical Investigations in the Hallabat Area, North Jordan
The basalt appears in massive, vesicular, and amygdaloidal texture, and with
occasional scoriaceous zones. A petrological study of the basalt indicates that the
basaltic flows in Jordan are in general similar, differences in mineralogical composition
have been found. The rocks consist essentially of ( 42 to 68%) plagioclase , 25% olivine,
9 to 23% pyroxene, and 2 to 9% magnetite, with calcite as a very common secondary
mineral [13].
The basalt flows separated by paleosoil and alluvial deposits consist of grey, brown,
silty clays, with limestone and chert fragments and also beds of volcanic ash. There are
six lava flows in north Jordan [13]. The characteristics of each flow differ texturally
from one to other. The thickness of the basalt ranges between less than 50 meters in the
western part of the study area to more than 100 meters in the east (Fig. 5).
Fig. 5: Isopach map of the basalt in the study area ( after BGR-WAJ, [14])
The Younger вЂ“ Recent Alluvium Formation occurs widely over the basalt. It
consists mainly of coarse well-cemented outwash gravels, sands, river gravels, scree, hill
wash, and soil with limited thickness of 0 вЂ“ 14 m. These deposits are hydraulically
connected to the upper surface of water bodies and with underlying permeable beds of
Upper Cretaceous or younger basaltic rocks.
Structure
Most of the structures within the sedimentary rocks are trending NE вЂ“ SW. To the
north of the study area, the structures are hidden completely beneath the basalt flows.
Generally, the NW вЂ“ SE trending tensional fractures saved as feeding dykes for the flood
basalt which had filled the major synclinal structures plunging ESE. Drilling of
exploration wells in the vicinity of the study area suggests the presence of a major fault,
which is probably an easterly extension of this flexure. Drilling activities have stopped in
most of the wells before penetrating the whole sequence of basalt so the whole structures
are not known. However, low dips and gentle folding the most marked structural
features are the abrupt flexures and associated faults.
65
Nasher, Mustafa and Abderahman
From the linear characteristics and extensions, the major trend of the lineaments in
the study area is NE вЂ“ SW (Fig.6).
Two fold systems occurred in and around the study area trending NE вЂ“ SW, and
NW вЂ“ SE [2]. The first fold system belongs to the Amman вЂ“ Hallabat structure system,
pronounced gentle anticline, striking NE вЂ“ SW, plunge at Qaser Hallabat toward east to
northeast under the plateau basalt [15,16]
The second fold system occupies the southern part of the study area and strike NW
вЂ“ SE, with dip of the axial planes towards the NW.
Fig. 6: Rose diagram showing the trend of the linear features delineated from landsat
TM images for Dhuleil вЂ“ Hallabat area ( After Al-Bis,[9]).
climate
The study area experiences a semiarid climate which is defined by hot вЂ“ dry
weather in summer and cold вЂ“ humid in the winter with an average annual rainfall
varying from 120 mm to 170 mm. The monthly average temperatures vary from 8 ВєC in
January to over 26 ВєC in July, and the maximum recorded temperature was 45ВєC [1,17].
Rainfall occurs between October and May, the highest rainfall occurs between December
and March, while the average annual evaporation exceeds 89 % [17]. The residual water
66
Geoeletrical and Hydrochemical Investigations in the Hallabat Area, North Jordan
forms a runoff in the valley from December to March ( e.g Wadi Dhuleil, Wadi ErRukban).
Hydrogeology
The hydrogeology of the study area is controlled by the geological set up, which
also controls the piezometry, occurrence and movement of the groundwater, and the
distribution of productive areas in the aquifers.
Three aquifer systems are recognized in the study area:
Wadi As-Sir Aquifer
It is an excellent aquifer especially when it is structurally deformed parts because
the resulting fissures and joints have been enlarged by solution. The potentiality of this
formation depends whether the formation lies below or above the zone of saturation. The
yield is poor to moderate ranging from 5 mВі/hr to 54 mВі/hr. Transmissibilities
determined at several locations indicate generally that this formation shows poor ability
for transmitting water. The base of this formation is about 400 meters above mean sea
level (asl) or less (Fig. 7). There is no individual base map for A7 and B2, because there
is a hydraulic connection between them [18].
Fig. 7: Structural contour map of the base of A7/B2 aquifer [14].
Amman Aquifer
This formation underlies the basalt, and can be considered as a good aquifer, it can
yield water readily to the wells. The Amman formation (B2) is encountered only in few
wells. Because of the deep erosion in pre вЂ“ basalt times, therefore the aquifer has
probably limited lateral extension [11].
67
Nasher, Mustafa and Abderahman
The Older Alluvial and Basalt Aquifer
It represents the main aquifer in the study area. The old alluvial formation lies
below or within the basaltic sequences as interflows marker, it ranges in thickness from
0 to 32 m. This formation can be considered as good conductor for groundwater or a
good reservoir depending on lithology.
The basalt is scoriaceous and jointed. A zone of scoriaceous basalt reaching a
thickness of 45 meters is considered to be the primary aquifer. The transmissivity and
the specific yield values are considered to be very high indicating that the scoriaceous
zone is highly developed and uniformly jointed. High transmissivity and yield wells
sometimes characterize the scoriaceous zone by partial or complete loss of circulation
during drilling. The basalt aquifer is made up of a series of semi вЂ“ independent channels
or pipes, lying side by side, and it is possible to pump from one such pipe without
affecting the other.
Groundwater Movement
Generally, groundwater movement depends on the hydraulic conductivity and
hydraulic gradient. It moves from high pressure (head) to low pressure (head) to reach
the stable condition. In figure (8) the detailed water levels map in the study area is
illustrated, before heavy pumping started. The depth to water table varied from 20
meters in the far west to 106 meters in the northeast [19]. The flow net configuration of
the groundwater in the Dhuleil and the surrounding areas is highly influenced by the
pumping history. The groundwater flows show a general trend to the south and west
direction .
Fig. 8:Groundwater flow pattern of the study area (Raikes and Partners 1972)
Geoelectrical Survey
The direct current (DC) resistivity method relies on the application of OhmвЂ™s Law
V=I*R (where V is potential difference, I gives current, and R denotes resistance) to
constrain the composition, structure, and/or hydrology of the subsurface. R is related to
apparent resistivity ПЃa through both the path length L and the cross-sectional area A
68
Geoeletrical and Hydrochemical Investigations in the Hallabat Area, North Jordan
through which the current flows. DC resistivity surveys are usually conducted by
electrifying two electrodes and measuring the potential across another pair of electrodes.
Various electrode configurations which can be used in resistivity surveying. The
apparent resistivity ПЃa which is measured by the array depends on the geometry of the
electrodes.
Fig. 9: Electrode Configurations. A) Wenner array , B) Schlumberger array.C1, C2 are
current electrodes, and P1, P2 are potential electrodes.
Data Processing
The first step in the resistvity data analysis is to calculate the apparent resistivity ПЃa
corresponding to each measured resistance ( в€†V/ I ) for each profile. The apparent
resistivities ПЃa are calculated using the following formula which depends on the
electrodes arrangements:
ПЃa = K * R
Where
ПЃa : apparent resistivity in ( в„¦.m ).
R
: Resistance in ( в„¦ ).
K
: The geometrical factor which depends on the arbitarily chosen distances
between electrode locations.
The calculated apparent resistivities are plotted against half electrode distance
(AB/2) on log вЂ“ log papers. The utilized field layout includes changing potential
electrodes separation (MN), so that, the field data are obtained in the form of
overlapping segments, each segment is being for a fixed MN dipole due to
inhomogeneities near electrodes M and N. These overlapping segments are processed to
obtain a single continuous field curve.
69
Nasher, Mustafa and Abderahman
Interpretation
Both qualitative and quantitative methods were used to interpret the resistivity
measurements.
Qualitative Interpretation
Iso- Resistivity contour maps
The surface isoresistivity contour map is constructed by contouring the values of
apparent resistivities at AB/2 = 20m (Fig.10). Through the investigation of the surface
distribution of the surface rocks and variation in the apparent resistivity of the surface
layer, it can be noticed that, the northern part of the study area is characterized by high
resistivity values due to presence of Quaternary basalt. To the south of the study area
there is a sharp decrease in the resistivity values. The occurrence of inland saline surface
waters and salt accumulations which are common phenomena in arid and semiarid
regions as occur in the study area, is the main cause of the sharp decrease in resistivity
values. Also three iso вЂ“ resistivity contour maps were constructed for AB/2 values of
100, 200, and 400 m (Figs. 11, 12 and 13), respectively. They show lateral variations
values that suggest the presence of water bearing sediments. The spatial distribution of
low resistivity zone shown on the maps indicates the observed contamination area
clearly at the spacing AB/2, increases from 200 m to 400 m. It also shows a general
decrease in resistivity values toward the central part of the study area. The comparison
between these maps and hydrochemical data indicates that the variation in resistivity
values is most probably due to the salinization of groundwater.
Long.
279
280
281
282
283
284
171
170
Lat.
169
168
167
166
165
Fig.10: Iso-resistivity contour map (in в„¦.m)for AB/2=20m
70
Geoeletrical and Hydrochemical Investigations in the Hallabat Area, North Jordan
Long.
279
280
281
282
283
284
171
170
Lat.
169
168
167
166
165
Fig.11: Iso-resistivity contour map (in в„¦.m) for AB/2= 100m
Long.
279
280
281
282
283
284
171
170
Lat.
169
168
167
166
165
Fig.12: Iso-resistivity contour map (in в„¦.m) for AB/2=200m
71
Nasher, Mustafa and Abderahman
Long.
279
280
281
282
283
284
171
170
Lat.
169
168
167
166
165
Fig.13:Iso-resistivity contour map (in в„¦.m) for AB/2=400m.
Quantitative Interpretation
The quantitative interpretation is concerned with calculations of the true resistivity
and the thickness of the layers. This process depends on applying ПЃa and the half
electrode spacing (AB/2). The final resistivity structures of sounding data were
determined applying the automatic interpretation method [ 20].
Figure. (14) shows an example of sounding curve and its interpretation in the
northern part of the study area utilizing previously method.
a)
b)
Fig. 14: An example of sounding curve (a), and its interpretation (b) in the northern part
of the study area.
72
Geoeletrical and Hydrochemical Investigations in the Hallabat Area, North Jordan
Representations of the Quantitative Interpretation
This is achieved by drawing a contour map between the sounding stations which are
distributed along two profiles at different directions. There are some geologic
informations that were obtained from drilled boreholes but most of the boreholes are
quite distant from the sounding point and the circumference of the study area. Therefore,
these informations were considered as background geological information, incorporated
in interpretation of sounding data. Following is a description of the geoelectric sections:
Geoelectric section (A вЂ“ AВЇ )
Seven ( VESвЂ™s) were measured along this section ( Fig. 2), nearly in N вЂ“ S
direction. The length of this profile is about 6.5 km starting at sounding VES-1 ( about
200m north to the well AL 2597) and ending almost east of Hallabat ( 100m west of well
AL 1093).
Interpretation of geoelectrical section (Fig.15) indicates that the first layer has
resistivity values of about 5 вЂ“ 519 в„¦.m and extends to about 1 m вЂ“ 19 m depth from the
surface; it is interpreted as top soil. The resistivity values tend to decrease with depth
beneath all VESs of this section except VES-1. This may be attributed partly to the dry
nature of the surface and partly because of the presence of fresh water that is just added
to the cultivated land. The high resistivity values were found in the northern part of the
study area beneath VES-1 which can be attributed to the presence of Quaternary basalt.
The second layer stretching between VES-1 and VES-3 has resistivity values between
34 вЂ“ 248 в„¦.m with thickness of 14 m вЂ“ 78 m; it can be related to basalt intercalated with
clay beds.
The third layer reveals a low resistivity value that decreases with depth to 16 в„¦.m
and a thickness of about 40 вЂ“ 70 m. This zone correlates with fractured limestone
saturated with saline water. The conductive layer overlies the moderate resistive layer
which has resistivity values that increase with depth in the range 50 - 180 в„¦.m. This
layer seems to correlate probably with hard crystalline limestone. The low resistivity
zone which corresponds to the salinization is especially thick in the central portion of the
survey lines, as it is shown clearly beneath VES-4 and VES-5.
73
Nasher, Mustafa and Abderahman
Fig.15: Geoelectrical cross вЂ“ section A вЂ“ AВЇ .
Geoelectric section (B вЂ“ BВЇ )
Five geoelectric soundings were measured along this profile ( Fig.2) in the N вЂ“ S
direction. The length of this section is about 4.5 km. The profile starts at VES-8 about
200m east of well AL 1111 and ends at VES-12.
74
Geoeletrical and Hydrochemical Investigations in the Hallabat Area, North Jordan
The resistivity and thickness of the layers are illustrated in ( Fig.16). The
interpreted layers differ from the northern part towards the southern part of the study
area. The analysis of this section reveals that the first layer has resistivity values in the
range 3 вЂ“ 95 в„¦.m, it extends to about 4.1вЂ“14m depth from the surface, and can be
related to top soil. The resistivity values of this layer decrease with depth; this can be
attributed to the dry nature at the surface. Near the surface the soil contains fresh water
without dissolved salts which increase during the movement of return flow downward.
Because wells in this area are found in cultivated land, top soil contains a large quantity
of water. Therefore, a correlation between top soil and sounding data and boreholes data
is difficult. In the northern part of this section, the second layer between VES-9 and
VES-10 has resistivity values of about 25 вЂ“ 135 в„¦.m with thickness of 20 вЂ“ 81 m, it is
considered as basalt interbedded with clay. The resistivity of the third layer ranges from
13 вЂ“ 40 в„¦.m. This value seems to be low compared to the ordinary rock types in the
study area. This seems to result from the presence of saline water. The evidence of
salinization of water is clearly shown in the interpretation of VES-9. In this
interpretation, the layer of low resistivity extends to depth exceeding 130 m. In the
fourth layer there is a continous increase in resistivity values, which exceed 160 в„¦.m, it
can be correlated with water unsaturated basalt water in the northern part and massive
limestone toward the southern part.
Comparing the two geoelectrical sections, the overall resistivity structures are
somewhat similar and reveal clearly the continuity to the surface which leads to the
increase of water salinity due to the irrigation return flow especially in the central
portion of the study area.
The extent and magnitude of groundwater salinization occurs at variable depths and
also has variable resistivity values. The depth of this layer varies from 40 m to more than
130 m with resistivity values in the range 13 в„¦.m to 72 в„¦.m ( Figs. 17 and 18 ),
respectively.
75
Nasher, Mustafa and Abderahman
.m
.m
.m
Fig. 16: Geoelectrical cross вЂ“ section B - BВЇ
The low resistivity layer (Fig.18) extends in two main zones with different depths.
The first zone covers extensive parts of the area specially the central part and extends to
the southeast with depth ranging between 90 вЂ“ 130 m; it is characterized by the lowest
76
Geoeletrical and Hydrochemical Investigations in the Hallabat Area, North Jordan
resistivity values 19 вЂ“ 30 в„¦.m (Fig. 17). The second zone is located in the southwestern
and northeastern parts with depth ranging between 30 вЂ“ 70 m.
Long.
279
280
281
282
283
284
171
170
Lat.
169
168
167
166
165
Fig.17:Depth to low resistivity layer (in m).
Long.
279
280
281
282
283
284
171
170
Lat.
169
168
167
166
165
Fig.18:Isoresistivity contour map (in в„¦.m )for the low resistivity layer.
77
Nasher, Mustafa and Abderahman
Hydrochemistry
The conditions of population growth, industrial, irrigation and other human
activities (addition of fertilizers, biocides вЂ¦etc) lead to increase of water demand as well
as deterioration of water quality. An example is the chemical fertilizers that dissolve in
the irrigation water and may end up in the aquifer. accordingly, the quality of
groundwater would be affected. This results in a major increase in cations and anions
contents in water.
The determination of the quality of groundwater is essential for assessing the
suitability of water for the various drinking, domestic, industrial and agricultural
purposes.
Some of these constituents are present in large quantities, others are secondary or
minor. These constituents include the cations ( Na+, K+, CaВІ+, MgВІ+ ) and anions ( ClВЇ,
HCO3ВЇ, SO4ВІВЇ, NO3ВЇ ). These ions are considered the most important parameters for
determining the water quality.
Results and Interpretations of Hydrochemical data
The results of the analyses of groundwater samples are summerized in (Table 1).
The ionic concentrations are presented in milligram per liter (mg/l). The mean
concentration of the cations is in the order Na+ > CaВІ+ > MgВІ+ > K+ while for anions it is
in the order ClВЇ > HCO3ВЇ > SO4ВІВЇ > NO3ВЇ.
The statistical analyses were performed on the different analyzed and calculated
parameters using MINITAB(1998).
The Total Dissolved Solids (TDS) of water samples were found to correlate directly
in different degrees with ions (Fig. 23). The relationships between TDS and Cl and Mg
are nearly perfect, with a correlation coefficient being very high at 0.984 and 0.965
respectively. There are also strong correlation between TDS and NO3 ,Ca and Na with
the correlation coefficients ( r ) 0.837, 0.749, and 0.883 respectively. In addition to the
overall salinity dependence on Cl ВЇ concentration, the dominance of the Cl ВЇ was also
illustrated by performing linear regression analysis of the concentration of Na+, CaВІ+, and
MgВІ+ against the ClВЇ concentration, which indicates Cl ВЇ, Na+, CaВІ+, and MgВІ+ are the
main chemical constituents and also NO3ВЇ that increase TDS of groundwater and the
previous cations were present in a chloridic form (Fig.19).
78
Parameters
Well
Name
AL 1093
AL 1095
AL 1097
AL 1109
AL 1111
AL 1113
AL 1762
AL 1763
AL 2353
AL 2567
AL 2571
AL 2574
AL 2587
AL 2594
AL 3285
AL 3286
AL 3305
AL 3306
AL 3445
AL****
AL 2597
EC
Вµs/cm
930
659
1027
2050
1280
2450
1119
699
2010
727
1710
752
2260
2970
1235
1295
711
730
859
3000
2110
pH
value
8.24
8.24
7.27
7.83
7.7
7.98
7.97
8.21
7.89
8.13
8.04
8.2
7.7
7.51
7.93
7.96
8.1
8.05
8.18
7.8
7.34
TDS
mg/l
595.2
421.76
657.28
1312
819.2
1568
716.16
447.36
1286.4
465.28
1094.4
481.28
1446.4
1900.8
790.4
828.8
455.04
467.2
549.76
1920
1350.4
385.045
141.027
271.348
610.541
644.591
819.005
293.723
189.378
662.04
160.361
747.281
260.056
1171.09
993.126
259.347
276.378
155.122
167.367
183.94
1504.9
1036.87
TH
mg/l
100.8
25.3
51.5
105.21
142.2
140.15
46.1
36.67
119.84
32.23
181.8
63.99
284.1
80.96
55.08
56.71
28.15
30.26
38.48
361.7
220.9
CaВІ+
32.45
18.97
34.78
84.76
70.51
114.3
43.53
23.83
88.4
19.46
71.41
24.41
112.4
192.86
29.67
32.83
20.67
22.37
21.4
146.5
118.2
MgВІ+
Table 1: Chemical analyses of groundwater samples (ions in mg/l).
79
96.18
68.77
78.43
158.7
91.17
161
102.35
56.58
132.25
80.73
161.1
68.84
145.2
211.6
142.14
146.74
74.29
77.51
95.45
151.8
146.3
NaВ№+
0.81
6.65
7.82
13.29
4.927
14.08
8.99
7.04
12.9
5.47
4.671
0.669
10.24
17.6
6.26
7.43
5.87
6.65
7.82
10.9
9.706
KВ№+
144
90.89
105.72
86.62
120
86.26
100.04
100.56
89.06
134.2
140
150
130
76.25
142.13
148.84
106.75
98.82
134.2
120
130
HCO3В№+
188
131.35
298.21
474.28
334
629.66
233.24
112.89
501.61
119.28
440
191
680
784.55
255.96
268.74
129.58
144.84
159.75
910
590
ClВ№ВЇ
3
19.86
29.76
172.32
21.9
162.72
83
60.48
130.56
43.68
4.7
3
13.5
248.64
79.68
87.84
33.6
36.96
55.68
6.1
5.5
SO4ВІ+
Geoeletrical and Hydrochemical Investigations in the Hallabat Area, North Jordan
40.35
18.7
24.25
73.49
85.1
106.69
28.79
21.39
53.6
12.21
61.65
31.8
122.9
75.92
12.62
10.83
14.97
17
7.94
127.1
126.33
NO3В№ВЇ
1.87
2.67
2.19
2.92
1.5
2.57
2.73
1.92
2.36
2.88
2.24
1.87
1.77
3.06
3.94
3.95
2.71
2.74
3.21
1.82
1.85761
SAR%
34.32
52.82
39.93
37.20
23.50
30.99
44.32
41.06
31.45
53.22
30.20
36.66
20.52
32.66
54.99
54.30
52.13
51.38
54.17
19.47
23.11
Na%
Nasher, Mustafa and Abderahman
Fig. 19: The relation between. a) Cl and TDS, b) NO3ВЇ and TDS, c) MgВІ+ and ClВЇ, d) Na
and ClВЇ, e) CaВІ+ and ClВЇ (all concentrations in mg/l).
Water Types
The major ion compositions of twenty-one representative water analyses were
plotted on trilinear diagram of Piper [21]. According to these plots (Fig. 20), and based
on Langguth's [22] classification of water types, the groundwater in the study area can be
classified into three types:
Вѓ
Alkaline earth waters with increasing portion of alkalis and prevailing chloride.
This type is characterized by a high concentration of chloride and represents 61.9%
from the collected water samples.
Вѓ
Alkaline waters with prevailing chloride. This type of waters represents 33.3% from
the collected water samples.
Вѓ
Alkaline earth with prevailing chloride, which represents only 4.8% of the waters.
These high concentrations of Cl ВЇ in groundwater in the area suggest a long history
of evaporation.
80
Na
CO
3
Geoeletrical and Hydrochemical Investigations in the Hallabat Area, North Jordan
M
g
K
HC
O3
+
+
Fig. 20: Trilinear diagram showing the chemical composition of water, shows the
different water types with prevailing chloride ion.
Physical Parameters
The measured pH вЂ“ values of the water in the present study range from 7.27 to 8.24,
thus classified as alkaline. Electrical conductivity (EC) is the ability of substances to
conduct electrical current. The lowest measured EC value was found to be about 650 Вµs/
cm in well AL 1095, while the highest was about 3000 Вµs/ cm in well Al ****.
The (TDS) in the study area vary in the range of 421.8 mg/l (in the well Al 1095)
to 1920 mg/l (in the well Al ****). The groundwater in the study area falls within fresh (
TDS<1,000 mg/l) to brakish ( TDS> 1,000 mg/l) type of water [23].
The salinity of all wells exhibits with Cl ВЇ and NO3ВЇ concentrations significant and
positive relationships, where there is a highly positive correlation between TDS вЂ“ ClВЇ ( r
= 0.984) and TDS вЂ“ NO3ВЇ ( r = 0.925) (Fig. 19) and can be perfectly quantified.
81
Ghunaim S. Nasher; Hakam A. Mustafa; Nabil Abderahman
The TDS in the study area (Fig.21) increases toward the central eastern of this area.
This is compatible with the presence of farmland in this part, so that this increase may
be, as also postulated by Schiffler [24], due to the excessive pumping, and it is believed
that continued re-irrigation of the soil from the groundwater aquifer is causing a build up
in salinity. Various techniques are used for irrigation including sprinkler and open
channel. The latter makes the water subject to intensive evaporation. Irrigation water
evaporating from the soil and the roots of plants precipitates salts above and below the
soil surface. Constant irrigation from the groundwater of the same area continues to
dissolve the accumulated salts from the soil. The salt is flushed and washed out
repeatedly from the soil causing infiltration of the solute into the subsurface thereby
increasing the salinity of groundwater. This practice has continued for more than three
decades near the productive wells in the area.
Fig. 21: The concentration map of TDS in the study area showing the highest
concentration in an area extending NW вЂ“ SE of the upper half of the study area.
Chemical Parameters
Calcium is the most abundant dissolved cationic constituent of groundwater, and is
responsible for water temporal hardness. Ca concentration ranges from 25.3 mg/l (in the
82
Geoeletrical and Hydrochemical Investigations in the Hallabat Area, North Jordan
well Al 1095) to 361 mg/l (in the well Al ****). The correlation coefficient of CaВІ+
versus HCO3ВЇ, Na+, and MgВІ+ is 0.148, 0.51, and 0.7, respectively. Therefore, the
concentration of CaВІ+ is mainly attributed to association with carbonates and to
plagioclase and mafic minerals found in basalt [25]. CaВІ+ has also high positive
correlation coefficient with NO3ВЇ ( 0.892); it can be attributed to association of CaВІ+ in
fertilizers. Ca comes also from plagioclase.
The minimum value of Magnesium is about 18.97 mg/l in well Al 1095 and the
maximum about 192.86 mg/l in well Al 2594. The major source of increasing
magnesium in the groundwater is the ion exchange of minerals in rocks and soils by
water. In volcanic rocks groundwater derives MgВІ+ from the minerals like olivine and
pyroxene.
The concentration of sodium ranges from 68.77 mg/l (in well Al 1095) to 211.6
mg/l (in well Al 2594). The increase of Na+ probably comes from the decomposition of
plagioclase. An increase of sodium is also assumed to result from cation exchange of
calcium and magnesium ions by water-rock interaction. For this reason, there is a
positive correlation between these cations.
K+ is considered as the minor trace cation in the groundwater; it is less than Na+,
CaВІ , MgВІ+. Potassium in the groundwater is derived salty rocks used in industry. It is
also used as an agricultural fertilizer, this can lead to significantly higher potassium
concentrations in groundwater below cultivated areas [26].
+
Generally, potassium content decreases in the groundwater because it can be easily
adsorbed by clays. The potassium analysis shows concentration of potassium ranges
from 0.669 mg/l (in well Al 2574) to 17.6 mg/l (in well Al 2594).
The concentration of chloride in the study area is high; it is increased many times in
comparison with other major cations and anions. The high concentration values of
chloride in groundwater is the result of dissolution of chloride minerals accumulated
either in the soil due to high evaporation rate in this area, or deposited in the subsurface
formations. It ranges from 131.35 mg/l (in well Al 1095) to 910 mg/l (in the well Al
****). Chloride causes in concentrations above 250 mg/l corrosion, while above 400
mg/l it causes a salty taste in water [27].
The high chloride concentrations suggest that the chemical character of the water is
influenced by recharge from meteoric water, weathering and subsequent release of ions
from soil
Alkalinity measurements are routinely conducted as part of standard water вЂ“ quality
determintations. Typically, these values are taken to be an indication of the concentration
of bicarbonate ( HCO3ВЇ) or carbonate (CO3ВІВЇ) in the sample [28-31].
The concentration of carbonate ranges from 86.26 mg/l (in well Al 1095) to 148.84
mg/l (in the well Al 3286). The concentration values are high near limestone and low
around volcanic rocks. This increase is attributed to the lithology changes.
Sulfate is abundant in most groundwater. It enters to the groundwater through the
weathering of sedimentary rocks which contain sulfate minerals or by dissolution of
83
Ghunaim S. Nasher; Hakam A. Mustafa; Nabil Abderahman
evaporite deposits. Further additions of sulfate to groundwater arise from the leacheable
sulfate from fertilizers. It may cause detectable tastes at concentrations of 300 вЂ“ 400 ml/l
[27].
The highest concentration of sulfate was found in the well Al 2595 with about
248.64 mg/l, while the lowest concentration was 3 mg/ l in the wells Al 2574 and Al
1093.
High nitrate concentrations in water supplies are potenial hazards to infant health,
as recommended by the WHO [32]. The susceptibility of infants to nitrate has been
attributed to their high intake relative to body weight, to the presence of nitrate-reducing
bacteria in the upper gastrointestinal tract, and to the greater ease of oxidation of fetal
hemoglobin [32].
The nitrate concentrations in the study area range from 10.83 mg/l (in the well Al
3286) to 127.1 mg/l (in the well Al ****). The average groundwater nitrate
concentration for the study area was 51.125 mg/l, and in ten samples out of the twentyone water samples ( 47.6%) exceeded the maximum permissible level making
groundwater unsuitable for drinking purposes. The increased usage of fertilizers can be
considered as the main source of nitrate in the study area which greatly increases the
concentration of nitrate especially below the cultivated lands.
Water Quality
The purpose of this section is to characterize the waters for both domestic as well as
for irrigation purposes.
Domestic purposes:
Two parameters Total Hardness ( TH ) and WHO [33] and Jordanian standards [36]
were used to determine whether water is suitable for domestic purposes;
The Total Hardness (TH) is the sum of calcium and magnesium concentration.
These ions react with soap to form precipitates. Hard water are unsatisfactory for
household cleaning purposes, so that, water softening processes are needed for removal
of hardness [34].
All wells except well Al 1095 are characterized by hard to very hard water
according to Sawyer and McCarty [35]. The high increase in TH in the study area can be
attributed to the dissolution of carbonate rocks and washing of soils during the
movement of groundwater. Water with TH greater than 80 mg/l can not be used for
domestic purposes, because it coagulates soap water. All wells have TH values greater
than the permissible value for domestic purposes; therefore, this water need softening to
eliminate the effect of TH.
The results of chemical analyses are compared with the international guide lines
limits of drinking water of the World Health Organization [33] and the Jordanian
drinking water standard. Most well waters are unsuitable for domestic purposes due to
the high NO3ВЇ and ClВЇ content, in addition to the high TDS as well.
84
Geoeletrical and Hydrochemical Investigations in the Hallabat Area, North Jordan
Irrigation Water Criteria
Numerous parameters are used to define irrigation water quality. Two criteria were
used in this study for evaluating irrigation water quality; total soluble salt content
(salinity hazard) and the relative proportion of sodium cations (Na+) to other cations
(sodium hazard).
The main problem with high sodium concentration is its effect on soil permeability
and water infiltration. Sodium also contributes directly to the total salinity of the water
and may be toxic to sensitive crops such as fruit trees. The sodium hazard of irrigation
water is estimated by the sodium absorption ratio (SAR). This is the proportion of
sodium to calcium plus magnesium in the water. The Sodium Adsorption Ratio (SAR) is
calculated as:
SAR =
Na
* 100
_________
в€љ(Ca + Mg )/2
( Todd, 1980).
All ionic concentrations are expressed in milliequivalent per liter.
High values for SAR imply a hazard of sodium replacing adsorbed calcium and
magnesium, and this replacement is damaging to the soil structure.
The SAR was plotted on the USA salinity laboratory diagram in which the SAR
appears as an index for sodium hazard (S) and EC as an index of salinity hazard (C)
(Fig.22). The water classes according to SAR method are shown in (Table 1). The waters
were found mostly confined in three classes of water types: C2-S1, C3-S1 and C4-S1
which means medium to very high salinity hazards and low sodium alkalinity hazards.
Medium вЂ“ salinity вЂ“ low вЂ“ sodium waters are useful for irrigation purposes, whereas
most of wells water in the study area has high to very high вЂ“ salinity вЂ“ low вЂ“ sodium
hazard and needs adequate drainage to overcome salinity problems for irrigation
purposes.
The sodium concentration is important in classifying the water quality, where the
sodium content is usually expresed in terms of percent sodium which is defined
according to Todd (1980) by the equation:
Na% =
Na + K
* 100
Ca + Mg + Na + K
All ionic concentrations are expressed in milliequivalent per liter.
85
Ghunaim S. Nasher; Hakam A. Mustafa; Nabil Abderahman
Fig. 22: Diagram for classification of irrigation waters (Richards, 1954).
The irrigation water containing high proportion of sodium will increase the
exchange of sodium content of the soil, affecting the soil permeability, and texture; it
makes the soil hard to plough and unsuitable for seedling emergence. If the percentage
of sodium is above 50% in irrigation water, calcium and magnesium exchange with
sodium, thus causing deflocculation and impairment of tilth and permeability of soils
[37]. A sodium percentage of more than 60% is considered unsafe for irrigation.
The sodium content in the groundwater, is regarded as low and the water can be
used for most crops and for irrigation on almost all soils according to the classification
of Wilcox [38].
Discussion
According to the geoelectrical sections (Figures 15 and 16), the high resistivity
values in the northern part of the study area can be attributed to the presence of
Quaternary basalt. Comparing the two geoelectrical sections, the overall resistivity
structures are somewhat similar, the low resistivity zone which corresponds to the zone
of interest to us ( salinization zone) is especially thick in the central part of the study area
( VESs 4, 5 and 9) (Figures 19 and 20), and reveals clearly the continuity to the surface
which leads to the increase of water salinity due to the irrigation return flow especially in
the central portion of the study area.
86
Geoeletrical and Hydrochemical Investigations in the Hallabat Area, North Jordan
The low resistivity layer in the study area which reflects the extent and magnitude
of groundwater salinization occurs at variable depths and also has variable resistivity
values ( Figures. 17 and 18), respectively. The greater depth in the central part of the
study area is most probably due to fractures ( faults) . The depth of this layer varies from
40 m to more than 130 m with resistivity values in the range 13 в„¦.m to 72 в„¦.m.
The hydrochemical investigation indicates that the water type falls within fresh to
brakish type of water. The areal distribution of many of the dissolved chemical species,
TDS and EC in the groundwater shows systematic variations in the centeral portion
towards the northwest due to the fracturing possibility in this direction. The very high
Cl ВЇ contents in groundwaters in the study area suggest a long history of evaporation, and
the pollution with respect to NO3ВЇ is mainly attributed to the extensive use of fertilizers
due to irrigation return flow. These are the two main sources for the salinity build up of
the groundwaters in the study area.
The hydrochemical results relatively conform with the geoelectrical results and
reveal clearly the pattern of salinity distribution of groundwater in the study area.
Conclusions
The combination of interpretted electrical resistivity data and hydrochemical data
successfully established the nature and configuration of salinization of groundwater in
the Hallabat area. The study shows the following results:
вЂў
The resistivity values are high in the northern and southrn parts of the study area
and decrease gradually toward the central part. The high resistivity values in the
northern part reflect the presence of basaltic rocks.
вЂў
The low resistivity values (16 вЂ“ 70 в„¦.m) of the third geoelectric layer represent
approximately the water bearing sediments and show clearly the conducting with
the land surface. This reflects the role of the irrigation return flow which caused the
salinity build up.
вЂў
As the depth increases the true resistivity values decrease, and the geoelectrical
layer of the low resistivity is detected at depth ranging from 55 вЂ“ 130 m. The depth
to low resistivity layer increases toward the central part of the study area.
вЂў
The groundwater in the study area falls under fresh ( TDS<1,000 mg/l) to brakish (
TDS> 1,000 mg/l) type of water.
вЂў
Most of the waters are unfit for drinking because of the high NO3ВЇ, and ClВЇ
contents, TH and TDS contents are as well high.
вЂў
Since SAR is low and salinity hazard is variable (from medium to very high) most
of well waters need suitable drainage to overcome salinity problems for irrigation
purposes.
вЂў
The increase in salinity is attributed to the increasing of NO3ВЇ, and ClВЇ salts of Na+,
CaВІ+, and MgВІ+, which is indicated by linear correlation between the cations Na+,
CaВІ+, and MgВІ+ and the anion Cl ВЇ.
87
Ghunaim S. Nasher; Hakam A. Mustafa; Nabil Abderahman
вЂў
The increasing of salinity of groundwater came close to private wells.
вЂў
Hydrochemically, these groundwaters are classified into three types;
вЂў
-
Alkaline earth waters with increasing portion of alkalis and prevailing chloride.
This type is characterized by a high concentration of chloride and represents
61.9% from the collected water samples.
-
Alkaline waters with prevailing chloride . this type of waters represents 33.3%
from the collected water samples.
-
Alkaline earth with prevailing chloride, which represents only 4.8% of the
waters.
The present groundwater problems of overdraft and salinity increase are largely
caused by inaccurate early estimates of the resource potentiality and inadequate
management practices.
Recommendations
вЂў
As the quality of groundwater is a general problem, the government authorities
strongly advised to protect water supply over the long term.
вЂў
The groundwater quality management should be attempted in the study area.
вЂў
Decrease the use of fertilizers and use the modern method in irrigation.
вЂў
Closing of wells yielding brakish water.
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88
Geoeletrical and Hydrochemical Investigations in the Hallabat Area, North Jordan
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91
ABHATH AL-YARMOUK: "Basic Sci. & Eng."
Vol. 16, No.1, 2007, pp. 93-103
Environmental Impact Assessment of Hiswa Clay
Deposits, South Jordan.
Talal Al-Momani* 1
Received on April 24, 2006
Accepted for publication on Dec. 10, 2006
Abstract
The results indicated that minimum negative environmental impact of exploration,
prospecting, testing, processing and treatment stages for the Hiswa clay deposit expected in the
present time if mining and processing take place in the Hiswa area, South Jordan.
This study indicated that the mining area is suitable to extract kaolinite as a major product,
taking into consideration the availability of the raw material, energy supply, water supply,
transportation facilities, and the absence of any rare or protected flora and fauna.
Keywords: Environmental Impact Assessment (EIA); Clay Deposits; Hiswa; Jordan.
Introduction:
Clays have always played a major role in human life. Kaolin is one of the most
common examples of clay minerals and important natural industrial substances. This is
largely because of their wide-ranging properties, high resistance to atmospheric
conditions, and easy access to their deposits near the earthвЂ™s surface and low price
(Konta, 1995)[1]. Kaolin is a commercial term to describe white clay composed
essentially of the clay mineral kaolinite. The term is typically used to refer to both the
raw clay and the refined commercial product. Kaolin is a naturally occurring white
mineral. It is relatively inert when mixed with other materials and has unique flow
characteristics when incorporated or mixed with liquids or placed in fluid systems such
as paints and paper industry. In addition, raw kaolin has few practical uses. Industrial
beneficiation is necessary in order to separate non-clay minerals from the kaolin.
Clay industry in common with open cast mining has some basic environmental
effects on the surrounding area. There are visual and other impacts of both clay
extraction and processing. The whole process may interrupt the existing land uses and
may pose difficult restoration problems if permanent change in land use is to be avoided.
In general, the exploration phase involves little or no permanent environmental damage.
The impact of mineral exploitation on the environment depends upon factors such
as mining procedure, local hydrologic conditions, climate, rock types encountered, size
В© 2007 by Yarmouk University, Irbid, Jordan.
* Department of Earth and Environmental Sciences, Faculty of Science, Yarmouk University, Irbid, Jordan.
Al-Momani
of operation, topography, and many more interrelated factors. Furthermore, the impact
varies with the stage of development of the resources.
According to Ward (1992)[2]; John et. al. (1994)[3]; Peter and Riki (1995)[4]; and
Larry (1996)[5], environmental impact assessment (EIA) constitutes one of the prime
instruments for development planning and resource management.
This paper sets the groundwork for the environmental impact assessment (EIA) of
mining activities of the Hiswa area, South Jordan. The main benefits of introducing
environmental impact assessment (EIA) are as follows:
-
Achieving a balance between environmental protection and economic development;
-
Better environmental resources management and minimization of wasted time,
money and raw materials;
-
Prevention of pollution and improvement in health and safety standards; and
-
Improved environment for future generations.
Location and Geological Setting of the Study Area
The study area lies in Aqaba Region, and it is situated in southern Jordan. The area
is located 20 km to the southeast of Ad-Disa Village at a distance of about 100 km east
of Aqaba (Figure 1).
The principal geological sequences cropping out in the study area are of Paleozoic
age. Three formations are cropping out in the study area, with a general dip towards the
southeast. These are from bottom to top (Khalil, 1994) [6]:
1- Umm Sahm Sandstone (Bedded brownish weathered sandstone) Formation.
2- Hiswa Sandstone (Graptolite sandstone) Formation.
3- Dubaydib Sandstone (Sabellarifex sandstone) Formation.
The Hiswa Sandstone (Graptolite Sandstone Formation) of Khreim Group is
subdivided into the Shale Member and the Sandstone-Siltstone Member. The Shale
Member consists of kaolinite of various colors, which range from light gray, dark gray,
violet, brown-red, and red.
Environmental Impact Assessment (EIA) Guidelines
The environmental impact assessments (EIA) of mining activities were investigated
by many authors in different areas around the world. This study is based on the available
data related the environmental impact assessment (EIA) guidelines issued by Patnaik
(1990)[7]; Trivedy and Sinha (1990)[8]; Johnson and Sides (1991)[9]; Environmental
Protection Agency (EPA) (1995)[10]; Sadar, (1996)[11]; and the General Corporation
for the Environmental Protection (1999)[12].
94
Environmental Impact Assessment of Hiswa Clay Deposits, South Jordan
Results and Discussion
Table (1) summarized the environmental impact assessment (EIA) of mining
activities of the Hiswa area.
Exploration and Prospecting
The environmental impact of exploration, prospecting and testing stages for the
Hiswa clay deposit are usually non significant, but in some cases the mentioned stages
may involve, for instance, drilling operations with significant noise and vibration.
Selection of the Mine Site and Site Preparation
The determination of the most suitable location of the mine site is critical. The clay
deposits in Hiswa Formation - South Jordan were first investigated by Jordanian
Company for Mining and Processing of Kaolin and Feldspar in 1995. The preliminary
investigations have recommended an exploration program for kaolin deposits in the area.
According to the economic viability and environmental impacts of clay deposits
from Hiswa area, this area is suitable to extract kaolinite as a major product; this is based
on the availability of raw materials; energy supply; water supply; appropriate
transportation facilities. In addition, the absence of any rare or protected flora and fauna
and the absence of negative impact on human settlements.
On the other hand, the company has started mining the area since 1997 on a small
mining scale for the production of commercial kaolin. In addition, the company carried
out the following activities: Land acquisition and transformation; construction
excavation of access secondary roads; construction of transmission lines and pipelines;
construction of offices, housing and other services for the labor forces; construction of
material storage; construction of electrical energy supply; and construction of water
supply.
Mining Operations and Transportation Impact
The Jordanian Company for Mining and Processing of Kaolin and Feldspar has
constructed a suitable plant for mining and simple processing operations (breaking,
crushing, screening, and loading).
Most of the study area is readily accessible and is crossed by the Aqaba-MaвЂ™anAmman Highway in the northwest, by the Aqaba - Batn El Ghul - Hisa railway from the
northwest to northeast, and by the El Rashdiah - QaвЂ™Disi - Ram asphaltic road.
Currently, transportation services mainly associated with the Port of Aqaba, the road
transport link to Amman, airport of Aqaba and the railway services (Figure 1). The
present road situation is quite satisfactory with the exception of necessary improvements
to the Hiswa-MaвЂ™an road and some secondary roads. See Table (1).
95
Al-Momani
Figure (1): Location map of the study area.
96
Environmental Impact Assessment of Hiswa Clay Deposits, South Jordan
Table (1): Environmental impact assessment of Hiswa clay deposits (Based on Sadar
(1996)[11]).
Impacting Actions
Environmental Parameters
Air quality
Water resources
Water quality
Noise and vibration
Land use
Vegetation
Human settlement
Health
Infrastructure and support services
Employment
Places of tourist or
archaeological importance
KEY:
+ sign shows beneficial impact
- sign shows adverse impact
вЂў sign shows impact must be
monitored
and mitigated
Ої sign shows no impact
* Pre-mining Phase
A вЂ“ Land acquisition and
transformation
B вЂ“ Civil works construction
C вЂ“ Erection of mechanical and
mining equipment
Pre-mining
Phase*
A B
C
Ої Ої
Ої
Ої Ої
Ої
Ої Ої
Ої
Ої
+ +
+
Ої Ої
Ої
+ +
+
Ої Ої
Ої
+ +
+
+ +
+
D
вЂў
Ої
вЂў
+
Ої
+
вЂў
+
+
E
Ої
Ої
вЂў
Ої
Ої
Ої
Ої
вЂў
Ої
Ої
F
вЂў
Ої
вЂў
Ої
Ої
Ої
Ої
вЂў
Ої
Ої
G
Ої
Ої
Ої
+
+
Ої
+
+
+
+
H
Ої
+
+
Ої
+
+
+
+
+
+
I
вЂў
Ої
Ої
+
Ої
вЂў
вЂў
+
+
J
Ої
Ої
Ої
Ої
Ої
Ої
+
+
+
+
K
+
+
+
Ої
+
+
+
+
+
+
Ої
Ої
Ої
Ої
Ої
Ої
Ої
Ої
Ої
Ої
Ої
Operational Phase**
** Operational Phase
D - Mining operations
E - Disposal of liquid effluent
F- Disposal of solid wastes on land for
reclamation
G - Housing provision
H - Provision of water, sewage, electricity
and other civic amenities
I - Transportation
J - Medical facilities
K - Land reclamation
Water Supply Impact
Ground water resources are available in the region. According to Salameh
(1996)[13], Jafr groundwater basin is subdivided into Jafr and Disi-Mudawwara aquifer
systems. The study area lies within the Disi aquifer system. The Disi aquifer consists of
medium- to - fine grained sandstones with a total thickness of about 1000 meters. The
average precipitation over the area is around 80 mm/year, with an average potential
evaporation of 4000 mm/year. The groundwater in the Disi area is unconfined and lies at
a depth of around 80 meters below the surface. At present, it is generally accepted that
the groundwater in Disi area does not receive major replenishment. This means that the
groundwater in the aquifer is fossil and that its extraction is at the expense of storage.
97
Al-Momani
In order to meet the needs of commercial kaolinites, low-grade kaolinite is fed to
beneficiation plants to produce high-grade kaolinite. The intensive use of water for wet
processing and production of clay deposits cause a significant water shortage.
Accordingly, water must be recycled in the beneficiation plants, but when it becomes
rich in impurities, the water must be treated for reuse.
Pollution control and recycling of water, which emphasizes the need for
delineation, water protection areas in order to safeguard groundwater resources.
Air Quality and Dust Impact
The main air pollution problem in the surface mining is dust. The dust pollution
may have significant impacts on vegetation, soil, equipment, and most importantly, on
human health.
There are various methods available to prevent or minimize dust emission. These
include the use of cyclones, filters, and other pollution control technologies, water
sprays, and paving roads. These methods are considerable choices, which will be helpful
to eliminate the dust impact and to conserve the environmental medium in Hiswa area
almost clean.
Noise and Vibration Impact
Drilling and heavy machinery are used regularly at mining sites, resulting in
potentially harmful amounts of noise pollution. Miners subject to high noise levels for
extended periods of times may become permanently deaf. The European Community has
set limits for daily exposure at 85 decibels with no peak sounds above 140 decibels
(Environmental Protection Agency, 1995)[10].
Noise levels can be monitored and steps should be taken to reduce volumes where
appropriate. Ear plugs or other noise-dampening devices may be appropriate for
employees in some cases.
Ecology (Wildlife), Agriculture and Land Use
The Royal Society for Conservation of Nature have indicated that there are no
species of wildlife or plant life in the study area that have any protected national or
international status. Accordingly, no unique habitat or species will be lost. On the other
hand, agricultural activities are negligible at present in the study area.
Tourism, Archaeology and Heritage
Tourism plays an important role in the national economy. The most important
tourist sites in South Jordan with national and international significance are AqabaвЂ™s
beaches and marine life, the ancient Nabatean city of Petra and the unique desert
landscape of Wadi Rum. These sites will not be affected by mining activities of Hiswa
area.
There is no archaeological site in the study area. The major archaeological site in
the region is Wadi Rum (approximately 30 km) from the study area. The Hiswa mining
and processing activities will not affect the site.
98
Environmental Impact Assessment of Hiswa Clay Deposits, South Jordan
Drainage and Topography Impacts
Most of the study area has an altitude of 860 to 940m. The eastern part has a lower
relief compared to the western part. The topography of the area shows decrease in the
eastern direction. Accordingly, land reclamation is highly important after the life-time of
mining stage.
Socio-economics Impacts (e.g. population, employment):
The Hiswa clay deposit area is deserted. Mining activity in the area will attract
workers and would contribute to reducing unemployment. In addition, the increase of
income is expected for some people from the surrounding areas. See Table (1).
Safety
A serious problem associated with mineral resources development is the possible
release of trace elements in hazardous concentrations into the environment. Trace
elements such as cadmium, cobalt, copper, lead, molybdenum, and others when leached
from mining wastes and become concentrated in water, soil, or plants, may be toxic or
may cause diseases to people and other animals who drink the water, eat the plants, or
use the soil (Keller, 1985[14] Patnaik, 1990[7]; Montogomery, 1995[15]; Murck et al.,
1996[16]).
Table (2): Minimum, maximum, and average values for the different trace elements of
the whole rock samples of Hiswa clay deposits compared with the standard
shales and the deep-see clays.
Trace
elements
(ppm)
Ba
Hiswa clay deposits
Maximum
Average
(ppm)
(ppm)
421
264
Standard
Deep-sea
shales*
clays*
(ppm)
(ppm)
580
2300
0.3
0.43
Cd
5
18
13
3.98
0-11**
0.1-1**
Co
8
52
26
8.96
19
74
17.80
90
90
Cr
50
115
75
30-590**
90**
45
250
Cu
55
154
104
28.67
18-120**
250**
Ni
63
225
123
35.61
68
225
20
80
Pb
45
220
139
41.67
16-50**
80**
Rb
43
117
95
15.03
140
-Sr
47
161
86
22.50
300
180
V
n.d
n.d
n.d
--130
120
95
165
Zn
30
195
109
44.64
18-180**
165**
* (Forster and Wittmann, 1983[17] ; Bowie and Thornton, 1985[18] ; and Montogemry, 1995[15]).
** (Keller, 1985[14] ).
Minimum
(ppm)
150
Standard
deviation
58.06
Table (2) shows the trace elements (Ba, Co, Cr, Cu, Ni, Rb, Sr, V, and Zn) in the
studied whole rock samples are low when compared with the standard shales and the
deep-see clays. In addition, lead and cadmium (Pb and Cd) occur relatively in high
99
Al-Momani
amounts. It is important to relate trace elements to human use (Tables 3 and 4).
Accordingly, geochemical studies are vital to determine the prevalence of such elements,
so that proper human adjustments can be made. In addition, monitoring the presence of
high concentration of trace elements in water results from mine wastes and smelters is
highly important.
Table (3): Comparison between the different trace elements (ppm) of the Hiswa clay
deposit and different commercial clays.
Trace
elements
(ppm)
Ba
Clays*
Natural
Average Clay*
Average
(Hiswa Clay)
Standard
shales**
Commercial
Pharmaceutical
248
147
75
580
264
Cd
0.18
0.02
1.3
0.3
13
Co
13.3
5
28
19
26
Cr
81.6
5.4
96.8
90
75
Cu
16.7
4.71
154
45
104
Ni
40
5
324
68
123
Pb
11.9
8.01
27
20
139
Rb
83
11
32
140
95
Sr
695
100
42
300
86
300
V
109
24
749
130
127
Zn
61
9
58
95
109
130
95
18-180***
580
0.3
0-11***
19
90
30-590***
45
18-120***
68
20
16-50***
140
* (Mascola et al., 1999) [19].
** (Forster and Wittmann, 1983; Bowie and Thornton, 1985; and Montogemry, 1995).
*** (Keller, 1985).
Table (4): Trace elements guidelines for soil quality; water quality and average body
concentration.
Trace
elements
Ba
Cd
Co
Cr
Cu
Ni
Pb
Rb
Sr
V
Zn
*Guidelines for soil quality
Levels for
agricultural use
(Вµg/gram)
750
3
40
750
150
150
375
--200
600
Levels for
industrial use
(Вµg/gram)
2,000
20
300
800
500
500
1,000
---1500
* (Dublanc, 1998) [20] ** (Montogomery, 1995). [15]
100
*Guidelines for water
quality
Levels for irrigation use
(Вµg/litre)
**Average
body
concentration
(ppm)
10
50
100
200
200
200
--100
2,000
-0.7
0.02
0.03
1.0
0.1
----33
Environmental Impact Assessment of Hiswa Clay Deposits, South Jordan
Conclusion
Mining and processing of mineral resources may have a considerable impact on the
land, water and, air. What must be done is to develop resources with a minimum of
adverse impact. Minimum environmental impact of exploration, prospecting, testing,
processing and treatment stages for the Hiswa clay deposit expected in the present time
if mining and processing take place in the Hiswa area.
The mining area is suitable to extract kaolinite as a major product, taking into
consideration the availability of the raw material, energy supply, water supply,
transportation facilities, and the absence of any rare or protected flora and fauna.
Many problems and adverse effects can be mitigated by integrating environmental
consideration into the overall planning and by the development of balanced, well-worded
and enforceable planning conditions.
The pollution control essentially involves good environmental management in
mining activities combined with adequate training and education of the workers as well
as other people in society. In addition, monitoring the presence of high concentration of
trace elements in water results from mine wastes and smelters is highly important.
Jordan is rich in its industrial rocks and minerals. The problem is that most of the
natural commodities are not evaluated for their industrial use and environmental impacts.
The exploration and evaluation of Jordanian raw materials for industrial use is an
important step for encouraging new industries. At the same time, the environmental
impact assessment, environmental planning and environmental legislation and
enforcement of future mining, beneficiation, and industrial use must be carried out for
these commodities. On the other hand, the demand for clay mineral resources in Jordan
is going to increase. Accordingly, planning, management, training, and education must
be given a top priority. In addition, one of the most important strategies of kaolin
production concerning the future generations is the environmental impact of mining and
processing. Environmental protection and land reclamation must take a special attention.
101
Al-Momani
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вЂ«п»›п»¤пєЋ ШЈпє·пєЋШ±ШЄ Ш§п»џпєЄШ±Ш§пєіпє” ШҐп»џп»° ШЈЩ† п»Јп»Ёп»„п»�пє” Ш§п»џпє�п»ЊпєЄп»іп»¦ п»Јп»ЁпєЋпєіпє’пє” п»»пєіпє�пєЁпє®Ш§Ш¬ Ш§п»џп»њпєЋЩ€п»џп»ґп»ЁпєЋп»іпє– Щ€Ш°п»џп»љ п»џпє�п»®п»“пє® Ш§п»џп»¤п»®Ш§ШЇ Ш§п»џпєЁпєЋЩ…вЂ¬
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.вЂ«Ш§п»џп»¤пє¤п»¤п»ґпє” ШЈЩ€ Ш§п»џп»ЁпєЋШЇШ±Ш©вЂ¬
References
[1]
Konta J., Clay and Man: Clay Raw Materials in the Service of Man. Applied Clay
Science, 10(1995)275-335.
[2]
Ward M. H., Mining and the Environment. Mineral Industry International,
(1992) 33-41.
[3]
John G., Riki T. and Andrew C., Introduction to Environmental Impact
Assessment : Principle and Procedure, Process, Practice and Prospects. UCL,
London, (1994).
[4]
Peter M. and Riki T., Methods of Environmental Impact Assessment. UCL,
London, (1995).
[5]
Larry W. C., Environmental Impact Assessment. McGraw-Hill, New York,
(1996).
[6]
Khalil B., The Geology of Ad-Disa Area (Qannassiya), Map Sheet No 3149 III.
Bulletin 26. Geology Directorate, Geological Mapping Division, Natural
Resources Authority (NRA), Amman, Jordan, (1994).
[7]
Patnaik L. N., Environmental Impact of Industrial and Mining Activities. Ashish
Publishing House, New Delhi, (1990).
[8]
Trivedy R. K. and Sinha M. P., Impact of Mining on Environment. Ashish
Publishing House, New Delhi, (1990).
102
Environmental Impact Assessment of Hiswa Clay Deposits, South Jordan
[9]
Johnson M. S. and Sides A., Environmental Assessment of New Mining Projects
вЂ“ Code of Practice for Appointment and Manegement of Environmental
Consultants. Mining Industry International, (1991) 13-17.
[10]
Environmental Protection Agency., Profile of the Non-fuel, Non-metal, Mining
Industry. EPA 310-R95-011, USA, (1995)
[11]
Sadar M. H., Environmental Impact Assessment. Carleton University Press,
Canada, (1996).
[12]
General Corporation for the Environment Protection.,
Environmental Impact Assessment. GCEP, Amman, (1999).
[13]
Salameh, E., Water Quality Degradation in Jordan (Impact on Environment,
Economy and Future Generations Resources Base). The Higher Council of
Science and Technology, Amman, (1996).
[14]
Keller E. A., Environmental Geology. Charles, E. Merrill Publishing Company,
USA, (1985).
[15]
Montogomery C. W., Environmental Geology. Wm. C. Brown Publisher, USA,
(1995).
[16]
Murck B. W., Skinner B. J. and Porter S. C., Environmental Geology. John Wiley
and Sons, UK, (1996).
[17]
Forstner U. and Wittmann G. T. W., Metal Pollution in the Aquatic Environment,
2nd. Edition. Springer-Verlag. Berlin, (1983).
[18]
Bowie S. H. U. and Thornton I. (editors), Environmental Geochemistry and
Health. 1st. edition. D. Reidel Publishing Company, Holland, (1985).
[19]
Mascolo N., Summa V. and Tateo F., Characterization of Toxic Elements in Clay
for Human Healing Use. Applied Clay Science, 15 (1999) 491-500.
[20]
Dublanc E., ArgentinaвЂ™s Mining and Environmental Legislation. Industrial
Minerals, (1998) 98-104.
103
Directives
for
ABHATH AL-YARMOUK: "Basic Sci. & Eng."
Vol. 16, No.1, 2007, pp. 105-122
Performance Evaluation of One UMTS Cell by the New Modelling
Specification and Evaluation Language MOSEL-2
Aymen Issa Zreikat* 1
Received on April 22, 2006
Accepted for publication on Oct. 1, 2006
Abstract
In this paper, performance evaluation of one UMTS cell is presented. The UMTS
stands for Universal Mobile Telecommunication Systems, the third generation of mobile
networks. The queuing model of one UMTS cell is suggested and solved for the first
time using MOSEL-2 language, which is the new version of MOSEL, developed by the
group of MOSEL in the University of Erlangen, Germany. The pre-processor rule of
MOSEL-2 is used in this paper, which is useful to solve the model using both
exponential and non-exponential distributions. The effect of important parameters on
UMTS performance is studied in this analysis, such as the variability of service time, the
availability of the codes and the effect of the arrival rate, О». It is shown that the
mentioned parameters have a major effect on UMTS performance and should be taken
into consideration by the network designers. Based on the above, critical performance
measures such as blocking probability and utilization are found and discussed.
Keywords: MOSEL-2 language, Performance Modelling, Performance Measures,
Queuing Model, UMTS systems.
Introduction
European Telecommunications Standard Institute (ETSI), found in 1998, is
working in Europe to develop technical standards for UMTS. The Third Generation
Partnership Project (3GPP) is a co-operation between international standard bodies and
third generation mobile telephony technical specifications [1-3]. The UMTS is one of
the Third Generation (3G) cellular systems being developed within the framework
defined by the ITU (International Telecommunications Union) and known as IMT-2000
(International Mobile Telephony, 3rd generation networks are referred to as IMT-2000
within ITU). It is a realization of a new generation of broadband multi-media mobile
telecommunications technology.
The UMTS technology demands a very high amount of system simulation and
modelling, not only because of the involved novelty inherent to the interaction between
the traditional IP systems and the new W-CDMA technology used in the air interface,
but also because of the complexity of the UMTS radio interface. The new technology
В© 2007 by Yarmouk University, Irbid, Jordan.
* Department of Information Technology, MuвЂ™tah University, Al-Karak, Jordan.
Zreikat
offers improvements from a radio resource allocation perspective. Extensive efforts are
necessary to optimize the main algorithms and to efficiently plan the network to be
deployed. The interaction between typical IP and wireless worlds requires simulation of
the complete end-to-end system modelling, to determine the network performance and
the quality of service available to UMTS customers at the application level.
In previous works [4-10], the performance of the UMTS cell is studied either by
simulation [4][5], or by finding the analytical solution [6-10], without presenting the
queuing model. Significantly, in this paper, the queuing model of a single UMTS cell is
presented and the numerical solution of different zones is found. The shared buffer
queuing system with different virtual zones is solved numerically using the Modelling
Specification and Evaluation Language (i.e. MOSEL-2).
The rest of the paper is organized as follows. In Section 2, the queuing model is
presented. The set of modelling assumptions are listed in Section 3. MOSEL-2
environment is given in Section 4. In Section 5, the rule pre-processor of MOSEL-2 is
presented. The description of the model by MOSEL-2 language is presented in Section
6. The numerical results and discussion are in Section 7, followed by the conclusions and
the suggestions for the future work in Section 8.
Queuing Model
The multiple queuing system of the UMTS cell is given in Figure 1. This model is a
SRXR (GE/GE/1)/N system which consists of R virtual zones. The SRXR (GE/GE/1)/N
system is such a system that the overall inter-arrival times and service times at a RГ—R
switch queue, are heterogeneous and GE (General Exponentially) distributed where each
output port has a single server and the total capacity of the system is given. Each virtual
equal zone is separated by a single channel and each zone includes a queue and a
service. These queues are dependent on each other since the total number of connections
in all zones cannot exceed nmax. Namely, in a permissible system state:
R
в€‘
i =1
n i в‰¤ n m a x , where ni denotes the number of active connections in zone i.
106
Performance Evaluation of One UMTS Cell by the New Modelling Specification and Evaluation
Language MOSEL-2
R
The condition:
в€‘
i =1
n i в‰¤ n m ax
Figure 1: The shared buffer queuing model SRXR(GE/GE/c)/nmax of a UMTS system
with R virtual single-channel zones.
Modelling Assumptions
The following assumptions are considered in the analysis:
вЂў
Only one cell is considered.
вЂў
Ideal free space propagation of the radio signal.
вЂў
The interference factor to be equal for all code combinations.
вЂў
Users within the considered cell are all using the same radio service.
вЂў
No movement is considered.
вЂў
The power gains of node B antenna and mobile equipment antenna to be 1.
вЂў
The call arrival process is a Poisson process.
вЂў
An exponentially distributed and general exponentially distributed burst size within
each active connection is assumed for both the arrival and the service times.
вЂў
The overall traffic intensity over the cell area is О».
107
Zreikat
The cell area is virtually divided into R concentric zones of equal areas. Only four
zones are considered in this analysis. When the number of zones increases, the state
space increases dramatically, which leads to the state space explosion problem.
Notations
О» is the mean arrival rate of calls in the system .
О»i is the mean arrival rate of calls going to Zone i, О»i = О» Г— Pi (i = 1, 2,..., R ).
Pi is the probability that the call going to Zone i, Pi = 1/ R (i = 1, 2, ..., R ).
C2a is the SCV(Squared Coefficient of Variation) of the inter-arrival time distribution of
those calls entering to the system .
C2ai is SCV of the inter-arrival time distribution of those calls entering to Zone i (i =
1,2,вЂ¦,R).
C2si is the SCV of the transmission time distribution of Zone i,
C 2 si = C 2 s (i = 1, 2,..., R).
Вµi is the mean departure rate of Zone i, Вµi = Вµ
(i = 1,2,вЂ¦,R).
Mosel-2 Environment
The Modelling Specification and Evaluation Language version 2 (i.e. MOSEL-2) is
similar to the predecessor MOSEL, which is developed at the university of Erlangen,
Germany [11].
The main objectives behind MOSEL-2 language are to:
вЂў
Provide system oriented language.
вЂў
Have easy to understand and learn the language.
вЂў
Be universally applicable, but with independent evaluation method.
вЂў
Have a language which comprises translators for several (already existing), welltested and good performance evaluation tools.
-The following 6 steps are to describe the work-flow of the performance evaluation
procedure using MOSEL-2:
1.
Inspecting the real system where the modeler builds a high-level, textual system
description using the MOSEL2 specification language and specifies the desired
performance and reliability measures.
2.
The MOSEL2 environment checks the input file for syntactical and semantical
errors. If no error occurs, the MOSEL2 model gets translated to the system
description of the evaluation tool selected by the user.
108
Performance Evaluation of One UMTS Cell by the New Modelling Specification and Evaluation
Language MOSEL-2
3.
The chosen tool is started by MOSEL2 environment.
4.
The tool evaluates the model by using numerical analysis methods or discrete-event
simulation.
5.
The evaluation tool saves its results in one or more tool-specific output files.
6.
The MOSEL2 environment provides the results in a single textual and graphical
file.
Figure 2 shows the above 6 steps explained graphically.
Introducing General Distributions: the rule pre-processor
In the old version of MOSEL, only Markovian models are allowed. Usually nonMarkovian models have to be evaluated using discrete-event simulation, which is a very
time-consuming task. It is now possible to enable an approximate numerical analysis for
a subclass of general distributions where their two first moments are: the mean and the
variance. These are called subclass empirical distributions (EMP), because mean and
variance can be obtained quite easily from measurements [12].
Figure 2: Modeling and analysis process in MOSEL2 environment [12].
It is possible to use an empirical distribution directly in MOSEL-2 by applying:
EMP (mean, var), where mean is the mean and var is the variance of the firing timeвЂ™s
distribution. The rule can then be substituted by one or several sub-rules depending on
the square coefficient of variation (SCV or c2) shown in Eq. (1):
109
Zreikat
2
c =
var
( mean)
2
............................................................................................... (1)
If c2 equals one, the empirical distribution is equivalent to an exponential
distribution. If it is greater than one, the distribution can be approximated by a general
exponential distribution. If c2 is smaller than one, the distribution can be approximated
by a hypo-exponential distribution.
The following MOSEL-2 rule can be represented by the Petri Net shown in Figure3.
FROM p1 TO p2 EMP (mean, var);
The pre-processor rule has been applied in MOSEL-2 program (Figure 4, lines 21,
24-25).
EMP (mean, var)
Figure 3: Example for transition with empirical distribution firing times.
MOSEL-2 Description
The queuing model of Figure 1 can be
follows. (See Figure 4).
solved numerically by MOSEL-2 as
The Mosel-2 description can be divided into 7 parts:
1.
Constant and Parameter part (lines 2-15).
2.
Node part (line 16).
3.
Function part (unused in this example).
4.
Condition (Assertion) part (line 17).
5.
Rule part (lines 18-25).
6.
Result part (lines 26-29).
7.
Picture part (lines 30-40).
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Performance Evaluation of One UMTS Cell by the New Modelling Specification and Evaluation
Language MOSEL-2
/*------------- UMTS.mos -------------------------*/
1 /*------------- Contants and parameter part---------*/
2 CONST
R:= 4;
3 PARAMETER
N:= 16,32,64;
4 @<1..R>{CONST p#:=1.0/R;}
5 @<1..R>{CONST
N#:=p#*N;}
6 PARAMETER u:= 1,2,3,4,5,6,7,8,9,10;
7 PARAMETER lambda:=1,5,10;
8 @<1..R>{CONST mue#:= lambda/u;}
9 CONST
Ca:=1; // SCV of inter-arrival time
10 PARAMETER Cs:=5,10,50; // SCV of service time
11 @<1..R>{CONST lambda#:=p#*lambda;}
12 @<1..R>{CONST mean_lambda#:=1.0/lambda#;}
13 @<1..R>{CONST mean_mue#:=1.0/mue#;}
14 @<1..R>{CONST var1# := Ca * (mean_lambda#*mean_lambda#);}
15 @<1..R>{CONST var2# := Cs * (mean_mue#*mean_mue#);}
/*---------------NODES------------------------------- */
16 @<1..R>{NODE S#[N#];}
/*--------------------Assertion-----------*/
17 ASSERT NOT (@<1..R>"+"{S#}) > N;
/*-----Transition Rates (Rules Part) -------------------*/
18
IF (@<1..R>"+"{S#}) < N
19
THEN
20
{
21
@<1..R>{FROM EXTERN TO S#
EMP(mean_lambda#, var1#);}
22
}
23
@<1..R>{IF S#>0
24
FROM S#
TO EXTERN
25
EMP(mean_mue#, var2#);}
/*--------- Results ------------------------------*/
26
@<1..R>{PRINT MEAN_S# := MEAN(S#);}
27
PRINT MEAN_T := @<1..R>"+"{MEAN_S#};
28
PRINT blk_T := PROB((@<1..R>"+"{S#}) >= N);
29
PRINT util := (MEAN_T/N);
30
//----PICTURES------------------31
PICTURE "BLOCKING"
32
PARAMETER u
33
XLABEL "Traffic load, u [Erlang]"
34
YLABEL "Blocking"
35
CURVE blk_T
36
PICTURE "UTILIZATION"
37
PARAMETER u
38
XLABEL "Traffic load, u [Erlang]"
39
YLABEL "Utilization"
40
CURVE util
Figure 4: MOSEL-2 Description
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Zreikat
The numerical solution in Figure 4 is explained by the following steps:
вЂў
The assumed number of zones, R = 4 (line 2).
вЂў
The assumed number of codes, N = SF x 2 = 16, 32 or 64 (line 3). In MOSEL-2, if
we have more than one constant, the word вЂњPARAMETERвЂќ is used instead of the
word вЂњCONSTвЂќ.
вЂў
Line 4 defines the probabilities (p1вЂ¦p4) given in Figure 1. p1=p2=p3=p4 = Вј.
вЂў
The maximum size of each node is defined in line 5 as follows:
N1 = p1x N , N2 = p2 x N, N3 = p3 x N, N4 = p4 x N.
вЂў
The traffic load is defined in line 6 as follows: u = 1,2,3,4,5,6,7,8,9,10.
вЂў
вЂњLambdaвЂќ in line 7 can take different values in turn: 1, 5 or 10.
вЂў
Accordingly, the service rate for each server is computed in line 8.
вЂў
Lines 9 and 10 assume different values to the square coefficient of variation for
both the arrival (i.e. Ca) and service times (i.e. Cs).
вЂў
Lines 11-13, the mean inter-arrival and mean service-time is computed. They are
used in lines 14 and 15 to compute the variance (i.e. var) in Eq. (1).
вЂў
Line 16 defines the nodes. As we have assumed that R = 4 zones, then 4 nodes
should be defined with their maximum capacities. But the total capacity of all nodes
should not exceed the total number of codes, N. This condition is tested in line 17
using the вЂњASSERTIONвЂќ concept of MOSEL-2.
вЂў
The transition rates of the queuing model of Figure 1 is written in MOSEL-2 in
lines 18-25. This part defines the call admission control of the system. Line 18
checks whether there are available codes in the system. The transition triggers only
if there are available codes in the system, otherwise the call is blocked. If the
condition in line 18 is true, then the transition starts at line 21. As the вЂњloopвЂќ
concept is used here, the transition happens to each zone in turn. For example: Line
21 can be translated as:
FROM EXTERN TO
S1 EMP (mean_lambda1, var11);
FROM EXTERN TO
S2 EMP (mean_lambda2, var12);
FROM EXTERN TO
S3 EMP (mean_lambda3, var13);
FROM EXTERN TO
S4 EMP (mean_lambda4, var14);
вЂњFROM EXTERNвЂќ means that the call moves from the external population
(outside) to the given zone (zone 1,вЂ¦,zone 4) with rates defined by the pre-processor
rule defined in Section 5 (i.e. EMP (mean, var)). In such a case, the rate can be
112
Performance Evaluation of One UMTS Cell by the New Modelling Specification and Evaluation
Language MOSEL-2
exponential or non-exponential, depends on the square coefficient of variation for both
the arrival and service time (Ca and Cs, respectively).
вЂў
In Lines 23-25, if there is a call in a given node and this call ends its service, then
the call moves from the given node to the outside world with the transition rate
which is given by the pre-processor rule defined above. Certainly, this applies to all
nodes in the system.
вЂў
The вЂњresult part" is given in lines 26-29. It uses the word вЂњPRINTвЂќ to compute the
mean of each server at line 26. In line 27, the total mean is computed. In line 28, the
blocking probability is computed and in line 29, the total utilization of the system is
computed.
вЂў
The last part of the numerical solution is the вЂњpicture partвЂќ. In this part, the IGL
program defined above is used to generate the required figures. In lines 30-35, the
figure for the вЂњblocking probabilityвЂќ is generated, while, lines 36-40 generate the
вЂњutilizationвЂќ figure. In each figure, the same parameters are defined.
вЂњPARAMETERвЂќ gives the x-axis values of the given parameter (u is given here).
вЂњXLABELвЂќ names the x-label, followed by the label name between double quotes.
вЂњYLABLEвЂќ names the y-label and вЂњCURVEвЂќ defines the curve values of its
followed name.
A more detailed description of MOSEL-2 can be found in [15]. In order to avoid
repetition, the вЂњloopвЂќ concept is used in the above MOSEL-2 program. For example, line
4 in the above program:
4 @<1..R>{CONST p#:=1.0/R;}
following 4 statements:
CONST
p1 := 1.0/R;
CONST
p2 := 1.0/R;
CONST
p3 := 1.0/R;
CONST
p4 := 1.0/R;
could also be written in MOSEL-2 by the
But repeating these four statements without using the "loop" concept is intractable
and time-consuming.
MOSEL-2 program can be written in any text editor, but the program has to be
saved as вЂњfilename.mosвЂќ similar to the name which is written on top of the above
MOSEL-2 program (i.e. UMTS.mos).
MOSEL-2 program can then be compiled with the following DOS command:
> mosel2 вЂ“options filename.mos
The option вЂќ-optionвЂќ tells MOSEL-2 to translate the model to a specified evaluation
tool:
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Zreikat
(i.e. вЂњ-csвЂќ for SPNP [12][13], вЂњ-msвЂќ for MOSES [14] or вЂњ-TsвЂќ for TimeNET
[15][16]. If the program is compiled successfully without any error, then MOSEL-2 will
automatically generate two output files (i.e. result file вЂњfilename.resвЂќ and IGL file
вЂњfilename.iglвЂќ). This is accomplished by the second option вЂњ-sвЂќ which orders MOSEL-2
to generate the suitable outputs. One can then browse the result file to see the results of
different performance measures. Additionally, to visualize the results by IGL file, the
following command line is used:
> igl filename.igl
Additionally, the comments in MOSEL-2 can be written between вЂњ/* */вЂќ or after
вЂњ//вЂќ. For example, lines 1 and 9 of Figure 4 are examples of using different comments
in MOSEL-2 language.
Numerical Results and Discussion
The numerical results are divided into four groups with two figures in each group.
One figure is for the blocking probability and the other is for the utilization. The effect
of the three main parameters are discussed in this analysis. The assumed parameters are
given in Table 1.
The effect of the variability of service time
Four figures are considered in this group (Figures 5-8). In the first two figures
(Figures 5 and 6), two values for the square coefficient of variation of the service time
(i.e. Cs = 1, 5) are considered, while the other parameters are left as they are mentioned
in Table 1. It can be seen from Figures 5 and 6 that for higher Cs (Cs = 5), we gain better
blocking probability but worse utilization. It is worth mentioning here that when Cs = 1,
this means that the service time is exponentially distributed, while when Cs is greater
than one, then the service time is a general exponential distribution. This factor should
be taken into consideration when the UMTS performance is studied.
It can be seen from Figures 5 and 6 that at traffic load 4 the blocking probability is
0.10 and the utilization is about 0.94 when Cs = 5. While it is about 0.24 blocking and
about 0.88 utilization when Cs = 1, which means that the exponential distribution service
time gives us worse blocking probability but better utilization. This means that the
change in the service time has a dramatic effect on the performance especially on the
blocking probability, even for higher traffic load, as it has been explained above. On the
other hand, the effect on the utilization is positive at lower traffic load values (i.e. up to 3
Erlang), as shown in Figure 6. It starts to get worse for higher load values because as the
traffic load increases, the variability of the service time neglects the effect of the
utilization when the variability of the arrival time is equal to one (i.e. Ca = 1).
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Performance Evaluation of One UMTS Cell by the New Modelling Specification and Evaluation
Language MOSEL-2
Table 1: Assumed Default Parameters.
Parameters
Spreading factor, SF (N = SF * 2)
Number of Zones, R
Square Coefficient of Variation of inter-Arrival time, Ca
Square Coefficient of Variation of inter-Arrival time, Cs
Traffic load, u [Erlang] = О»/Вµ
Values
8, 16, 32
4
1
1, 5, 10, 50
1, 2,вЂ¦,10
5, 10
Arrival rate to the whole cell, О»
Figures 7 and 8 show the blocking probability and the utilization for different
values of Cs (Cs = 5, 10, 50). It can be seen from these figures that the variability of the
service time, Cs, has a clear effect on the performance of both the blocking probability
and the utilization. The higher Cs, the better the performance. According to this, the
network designers are facing a very important issue when planning the network, because
choosing the suitable value of Cs is an optimization issue. However, this is out of the
scope of this paper.
The effect of the availability of the codes
Figures 9 and 10 show interesting results. Figure 9 shows the blocking probability,
while Figure 10 shows the utilization for different number of available codes (i.e. N =
16, 32, 64). Very interesting results can be seen in Figure 9 where it explains clearly that
when the traffic is low (i.e. up to 4 Erlang), there is no effect of the number of codes on
the performance since the effect of the interference between users is much higher than
the effect of the number of codes. It can be seen that at traffic load 4 Erlang, the graph
behaviour starts to change as we become close to the base station and hence, the density
of the traffic increases, which means that the effect of the number of codes starts at this
point. Additionally, the increase in the number of codes has a positive effect on both the
blocking probability and the utilization of the UMTS. This behaviour can be seen clearly
in both Figures 9 and 10.
The effect of the arrival rate to the whole cell О»
In Figures 11 and 12, two different values of О» are considered (i.e. О» = 5, 10). It can
be noticed from these two figures that when О» is increasing, both the blocking
probability and the utilization are also increasing. The two figures show clearly that the
whole arrival rate has an effect on the performance of UMTS network and designers.
This should be taken into account as an important planning issue.
Conclusions and Future Work
This paper presents a numerical solution of one UMTS cell using the MOdeling
Specification and Evaluation Language (MOSEL-2). It is shown in this paper that it is
possible to find the numerical solution by MOSEL-2 for any system represented either
by queuing model or Petri Net with different square coefficient of variations. Thus, it is
possible with the new version of MOSEL (i.e. MOSEL-2) to solve any queuing model
with exponential and non-exponential distributions by applying the pre-processor rule
115
Zreikat
mentioned above. The effect of different parameters on the performance of the UMTS
such as: the variability of the service time, the variability of the arrival rate and the
availability of the codes, were shown and discussed. It is shown that all of these
parameters affect the performance of the UMTS network and should be considered by
the network operators and designers.
Only 4 zones are considered in the work of this paper, because when the number of
zones is increasing, the state space is also increasing. This leads to the state space
explosion problem and therefore, in order to consider higher number of zones, we need
to run MOSEL-2 program on machines with very high specifications. It would be
possible to do this in the future. Therefore, in the future work of this paper, higher
number of zones will be considered. Additionally, in order to cope with the requirements
of the new technology of multimedia services (i.e. IP based services) in the UMTS, it is
possible with MOSEL-2 to have firing rates of different distributions (i.e. Pareto
distribution). It would be interesting to have Pareto arrival and service time distributions
to the above type of queuing system.
вЂ« Щ€Ш§пєЈпєЄШ© пє‘п»®Ш§пєіп»„пє” п»џп»ђпє” Ш§п»џп»Ёп»¤пє¬пєџпє” Щ€Ш§п»џп»®пє»п»’вЂ¬UMTS вЂ«пє—п»�п»ґп»ґп»ў Ш§п»·ШЇШ§ШЎ п»џпєЁп» п»ґпє”вЂ¬
(MOSEL-2) вЂ«Щ€Ш§п»џпє�п»�п»ґп»ґп»ўвЂ¬
вЂ«ШЈп»іп»¤п»¦ п»‹п»ґпєґп»° ШІШ±п»іп»�пєЋШЄвЂ¬
вЂ«п»Јп» пєЁпєєвЂ¬
вЂ« п»«п»® ШҐпє§пє�пєјпєЋШ± п»·п»§п»€п»¤пє” Ш§п»№пє—пєјпєЋп»»ШЄвЂ¬UMTS .UMTS вЂ«п»“п»І п»«пє¬Щ‡ Ш§п»џп»®Ш±п»—п»Є Ш§п»џпє’пє¤пєњп»ґпє” пє—п»ў пє—п»�п»ґп»ґп»ў Ш§п»·ШЇШ§ШЎ п»џпєЁп» п»ґпє”вЂ¬
(Queuing вЂ« п»џп»�пєЄ пє—п»ў ШҐп»—пє�пє®Ш§Ш Щ€пєЈп»ћ Ш§п»џп»Ёп»¤п»®Ш°Ш¬ Ш§п»џпєјп»”п»ІвЂ¬.вЂ« Ш§п»џпє п»ґп»ћ Ш§п»џпєњпєЋп»џпєљ п»Јп»¦ Ш§п»џпєёпє’п»њпєЋШЄ Ш§п»џпєЁп» п»®п»іпє”вЂ¬,вЂ«Ш§п»џпєЁп» п»®п»іпє” Ш§п»џп»ЊпєЋп»џп»¤п»ґпє”вЂ¬
вЂ«( Ш§п»џпє�п»І пє—п»Њпє�пє’пє® п»§пєґпєЁпє” пєџпєЄп»іпєЄШ© п»Јп»¦ п»џп»ђпє”вЂ¬MOSEL-2) вЂ« п»·Щ€Щ„ п»Јпє®Щ‡ пє‘пєЋпєіпє�пєЁпєЄШ§Щ… п»џп»ђпє”вЂ¬UMTS вЂ« п»џпєЁп» п»ґпє”вЂ¬Model)
(pre- вЂ« ШҐпєіпє�пєЁпєЄЩ… п»Јпє’пєЄШЈвЂ¬.вЂ« п»“п»І Ш§п»џп»¤пєЋп»§п»ґпєЋвЂ¬Erlangen вЂ«( Ш§п»џпє�п»І пє—п»ў пє—п»„п»®п»іпє®п»«пєЋ пє‘п»®Ш§пєіп»„пє” п»“пє®п»іп»– п»Јп»¦ пєџпєЋп»Јп»Њпє”вЂ¬MOSEL)
вЂ«( Щ€Ш§п»џпє¬ЩЉ п»іпєґпєЋп»‹пєЄ п»“п»І пєЈп»ћ Ш§п»џп»Ёп»¤п»®Ш°Ш¬ пє‘пєЋпєіпє�пєЁпєЄШ§Щ… п»›п»ћ п»Јп»¦ Ш§п»џпє�п»®ШІп»іп»ЉвЂ¬MOSEL-2) вЂ« Ш§п»џп»¤пє�п»®п»“пє® п»“п»І п»џп»ђпє”вЂ¬processor)
вЂ« п»џп»�пєЄ пє—п»ў ШЇШ±Ш§пєіпє” пє—пє„пє›п»ґпє® п»Јпє п»¤п»®п»‹пє”вЂ¬. (Exponential and non-exponential distribution) вЂ«Ш§п»·пєіп»І Щ€п»Џп»ґпє® Ш§п»·пєіп»ІвЂ¬
вЂ« п»ЈпєЄЩ‰ пє—п»®п»“пє®вЂ¬,(service time) вЂ« Ш§п»џпє�п»ђп»ґпє® п»“п»І ШІп»Јп»¦ Ш§п»џпєЁпєЄп»Јпє”вЂ¬:вЂ« п»Јпєњп»ћвЂ¬UMTS вЂ«п»Јп»¦ Ш§п»џп»¤пє�п»ђп»ґпє®Ш§ШЄ Ш§п»џп»¤п»¬п»¤пє” п»‹п» п»° ШЈШЇШ§ШЎ пє§п» п»ґпє”вЂ¬
вЂ« Щ€п»џп»�пєЄ пє—пє’п»ґп»¦ ШЈЩ† п»џп»¬пє¬Щ‡ Ш§п»џп»¤пє�п»ђп»ґпє®Ш§ШЄ пє—пє„пє›п»ґпє® пєџпє¬Ш±ЩЉ п»‹п» п»° ШЈШЇШ§ШЎвЂ¬.О» (arrival rate) вЂ« Щ€пє—пє„пє›п»ґпє® п»Јп»ЊпєЄЩ„ Ш§п»џп»®пє»п»®Щ„вЂ¬, codes
,вЂ« пє‘п»ЁпєЋШЎШЈ п»‹п» п»° п»ЈпєЋ Ш°п»›пє® пєіпєЋпє‘п»�пє‚вЂ¬.UMTS вЂ« Щ€п»іпє пєђ ШЈЩ† п»іпє†пє§пє¬ пє‘п»Њп»ґп»¦ Ш§п»№п»‹пє�пє’пєЋШ± п»Јп»¦ п»—пє’п»ћ п»Јпєјп»¤п»¤п»І пє·пє’п»њпє”вЂ¬UMTS вЂ«пє§п» п»ґпє”вЂ¬
(blocking вЂ« ШҐпєЈпє�п»¤пєЋп»џп»ґпє” ШҐпєЈпє’пєЋШ· Ш§п»џп»¤п»њпєЋп»џп»¤пє”вЂ¬:вЂ«пє—п»ў ШҐп»іпє пєЋШЇ Щ€п»Јп»ЁпєЋп»—пєёпє” пє‘п»Њпєѕ Ш§п»џп»¤пє�п»ђп»ґпє®Ш§ШЄ Ш§п»џпє¤пє®пєџп»Є п»џп» п»Ёп»€пєЋЩ… п»Јпєњп»ћвЂ¬
.вЂ«( п»џп» п»Ёп»€пєЋЩ…вЂ¬utilization) вЂ« Щ€ ШЇШ±пєџпє” Ш§п»№пєіпє�пєЁпєЄШ§Щ… Ш§п»·п»“п»Ђп»ћвЂ¬probability)
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Performance Evaluation of One UMTS Cell by the New Modelling Specification and Evaluation
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Performance Evaluation of One UMTS Cell by the New Modelling Specification and Evaluation
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Figure 5: Blocking against traffic load for (Cs = 1, 5, Ca = 1)
Figure 6:Utilization against traffic load for (Cs =1, 5 Ca = 1)
119
Zreikat
Figure 7: Blocking against traffic load for different Cs values
Figure 8: Utilization against traffic load for different Cs value
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Performance Evaluation of One UMTS Cell by the New Modelling Specification and Evaluation
Language MOSEL-2
Figure 9: Blocking against traffic load for different N values
Figure 10: Utilization against traffic load for different N values
121
Zreikat
Figure 11: Blocking against traffic load for different values of О»
Figure 12: Utilization against traffic load for different values of О»
122
ABHATH AL-YARMOUK: "Basic Sci. & Eng."
Vol. 16, No.1, 2007, pp. 123-134
A Generic Parallel Map:
Extending the List Map Higher Order Function
to Arbitrary Data Structures
Mohammad M. Hamdan* 1
Received on July 12, 2005
Accepted for publication on Feb. 28, 2006
Abstract
The standard map higher order function is commonly used by the parallel functional
programming community to apply a function to the elements of a list data structure in
parallel. In this work, a generic parallel map was developed. The design and
implementation of the generic parallel map was based on the integration of a sequential
generic functional programming language, a data distribution algorithm, a parallel
master slave template and generic sends. The new generic parallel map can take a
function and apply it on any data type. Results are shown for parallelizing a ray tracer
using the generic parallel map. Speedup results are good.
Keywords: functional programming, parallel processing, generic programming, data
distribution, templates, ray tracing, genetic algorithms.
Introduction
There is a big need within the scientific computing community for programming
methodologies that facilitate the development of computer application that can run
sequentially and possibly in parallel. These methodologies are required as programmers
need to focus on the problem being solved rather than looking at how to speed their
program. Therefore, the idea of skeletons or templates or even patterns [1,2] was
welcomed by the scientific computing community because of their implicit parallelism.
However, the work on templates has primarily focused on parallelizing lists. Most
of the problems had to be converted from their original data structure representation into
a list. For example, a vector can be represented using a list and a tree can be represented
using a list of lists. This conversion process is expensive as it requires the programmer to
think of the problem in a different format and the algorithm had to be modified to work
on the new data structure.
Previous work on generic functions provided either sequential functions such as
PolyP [3] and Generic Haskell [4] that can work on any data type or parallel functions
В© 2007 by Yarmouk University, Irbid, Jordan.
* Department of Computer Science, Faculty of IT, Yarmulke University, Jordan.
Hamdan
using an explicit approach such Janus [5] and extending C++ standard template library
[6]. These approaches are compared to the work in this paper in Section 4.
Using the power of generic programming [7,8] which is available in Fish [9] and
the basic ideas in parallel algorithmic skeletons enabled us to implement a generic
parallel map (called gpmap). The gpmap can be used with complex data structures and
its underlying parallel implementation will do the work in parallel.
The rest of the paper is organized as follows. Section 2 describes how the generic
parallel map was engineered. The results are shown in Section 3. In Section 4,
comparison to related work is outlined and finally Section 5 concludes and outlines
future work.
Engineering a Generic Parallel Map
The generic parallel map was a result of the integration of a host language (Fish), a
generic sequential map, a static data distribution algorithm, a static master slave template
and generic sends. The following subsections will explain each part in detail.
The Host Language
A generic function is a function which is able to handle any data structure
containing any kind of numerical data such as integers, floats, Boolean, lists вЂ¦ etc. It is
polymorphic in the choice of the structure used to hold the data. To explain the idea of
generic programming [3, 5], take a look at the following example:
equal 1 2
equal 3.4 6.7
equal true false
equal [1,2,3] [1,2,3]
As we can see, arguments for the function equal can be of different types. There is
one implementation for the function and it can take many different data types as
arguments. This is similar to the idea of overloading functions and templates in C++
[10]. However, we are interested in a functional language rather than an imperative or an
object-oriented language due to the advantages of using functional languages such as the
declarative style which matches our goal of implicit parallelism.
The host language for this work is a well known functional programming language
called Fish [9]. Fish stands for Functional = Imperative + Shape and it is designed to be
an array programming language. Its main features are the expressive power of functional
languages and the efficient execution of imperative languages. The power in Fish comes
from static shape analysis [11], which is done at compile-time in order to reduce
functional programs to simple imperative forms. Moreover, it supports generic
programming using the idea of constructor calculus [12] which is based on generic
pattern matching.
124
A Generic Parallel Map: Extending the List Map Higher Order Function to Arbitrary Data Structures
Fish was chosen for the gpmap due to the availability of the source code for the
compiler and its support for generic programming. The Fish code shown below,
illustrates how to define a binary tree with two data types: real for leaf nodes and integer
for root and intermediate nodes. Its graphical visualization is shown in Fig. 1.
datatype btree (a,b) = leaf a | node b : btree(a,b) : btree (a,b)
let Tree = node 4 (node 6 (leaf 3.2)
(node 10 (leaf 4.8) (leaf 1.3))
The tree can hold mixed data types such as integers and reals by using the
constructor calculus [12]. The constructor calculus allows generic pattern matching to
branch on any constructor of any type.
4
10
6
5
3.2
2.5
4.8
1.3
3.7
Fig.1: The tree data structure with mixed data types
A Generic Sequential Map
In FISh there is a built in generic sequential map that can apply a function or more
to a data structure that contains more than one data type. To illustrate how the generic
map works, an example is shown below. In the code we define two functions: the first
function fun1 takes an integer and returns an integer while the second function fun2
takes a real and returns a real. The generic map can take a tuple of these two functions
and apply them to the values stored in the tree. The resulting tree is shown in Fig. 2.
let fun1 x = x + 3
let fun2 y = y +. 4.5
map (fun1,fun2) Tree
Tree = node 7 (node 9 (leaf 7.7)
(node 8 (leaf 7.0) (leaf 8.2)))
(node 13 (leaf 9.3) (leaf 5.8))
125
Hamdan
7
9
13
7.77
8
9.3
7.0
5.8
8.2
Fig 2: Applying a tuple of two functions to the mixed data type tree using the sequential
generic map.
Data Distribution
The basic idea of data distribution [13] is to take a data structure and distribute it
across a given number of processors. In case of an array or a list this is an easy task
where we divide the number of elements over the number of processors. Each processor
will get its piece of the array accordingly. The advantages of data distribution are load
balancing and improve data locality. Nonetheless, few processors might get less or more
data than the others and this might cause load imbalance. However, the problem with
data distribution is how to distribute complex data structures such as trees. There are two
approaches: the first one is to flatten the data structure to bytes which is quite expensive
and looses locality of data and might have extra overhead due to load imbalance and the
second one is generic which is not expensive and preserves locality of data but difficult
to implement.
A compile-time algorithm was developed that distributes a tree across a given
number of processors. We illustrate in Fig. 3 a distribution for the tree data structure that
was defined in section 3.1 and a flow chart for the algorithm is shown in Fig. 4.
4
Processor 2
Processor 1
10
6
3.2
4.8
5
2.5
1.3
3.7
Fig. 3: Distributing the tree data structure across two processors.
126
A Generic Parallel Map: Extending the List Map Higher Order Function to Arbitrary Data Structures
Start
Find size of Tree
Take a sub-tree
If size of subtree is small
enough according to
work to be assigned to
every processor ?
No
Yes
Assign the sub-tree to
the processor with the
least amount of work
so far
Put the sub-tree on the root
processor
Yes
More sub-trees ?
No
End
Fig 4. Flow chart for the distribution algorithm.
The result of the algorithm is a list of lists of pieces as shown in Fig. 5. Each sublist will be assigned to a processor and hopefully amount of work is balanced across all
processors.
127
Hamdan
6
4
4.8
5
3.2
10
2.5
1.3
3.7
Fig. 5: Generating a list of lists of unprocessed pieces.
The result of applying the gpmap on the tree that was defined in Section 2.1 would
be as shown in Fig. 6. The final step is to reconstruct the tree from the processed list of
lists of processed pieces.
9
7
9.3
8
7.7
13
7.0
5.8
8.2
Fig. 6: The processed list of lists of piece
The Static Master Slave Template
Template-oriented parallel programming is a well methodology for parallel
programming. It has two components: a high-level interface for templates that the
programmer can use in his program in order to parallelize it and a low-level
implementation that will be used to parallelize the high-level interface. The user is not
responsible for any coordination, messages, load balancing or any other parallel
programming issues. All these are taken care of using the pre-implemented template.
The programmer is only responsible for choosing the suitable template and instantiate it
with the needed function and data. The choice can be from a variety of templates such
as the master slave, pipeline, divide and conquer and geometric parallelism. These
templates correspond to the common algorithms that are normally used in developing
parallel applications. To illustrate the methodology lets take the map Higher-Order
Function (HOF) as an example. The user may in his program apply a function to the
elements of a list. This can be easily done using the map HOF as follows:
fun square x = x * x;
map square [1, 2, 3, 4, 5] =>
[square 1, square 2, square 3, square 4, square 5] =>
[1, 4, 9, 16, 25]
128
A Generic Parallel Map: Extending the List Map Higher Order Function to Arbitrary Data Structures
We may note that the square function can be applied to each element in the list
simultaneously as each element is independent from the other one. This type of implicit
parallelism can be easily exploited by the compiler using a master slave template. The
parallelizing compiler will replace the map function with the master slave template in the
generated object code. Once the program is executed the square function will be applied
to the elements of the list in parallel.
We have implemented modified version of the static master salve template for the
generic parallel map. The master sends a list of pieces to each worker. Each slave will
receive its list of pieces, process it and send the result back to the farmer. Note that the
farmer after the send process will do work as well. Once the master finishes its
computation then it is ready to receive the result from the workers. The pseudo-code for
the master and slave template are shown below:
The Master Alg.
1) Send the nth list of pieces to the nth worker
2) Process the 1st list of pieces
3) Receive the kth list of pieces from the kth worker
4) Build a list of lists of processed pieces
The Worker Alg.
1) Receive the nth list of pieces from Master
2) Process the nth list of pieces
3) Send the list of processed pieces to Maste
Generic Sends
The gpmap requires generic sends. Therefore, it was required to add functions to be
used by the master slave template that can send and receive complex data structures.
Since any Fish data structure can be translated using the shape information into one or
up to four basic arrays (int, float, char, Boolean). Simply, to send arbitrary data
structures: either send the four arrays using separate messages or combine the four arrays
into one array and transmit the data structure using one send message. The receive
process is similar. It important to note that there is no need to flatten or unflatten a data
structure at run-time.
Using the message passing interface library (MPI) [14] it was possible to combine
the four arrays and send them as one big message. This was achieved by defining a new
data structure composed from the four arrays using MPIвЂ™s built-in datatypes. The new
data structure is then send using a standard send routine.
Results
To test the generic parallel map, we have implemented a ray tracer in Fish. The ray
tracer is a well known example that has been used by the parallel functional
129
Hamdan
programming community as an application to test their methodologies and techniques.
Therefore, the program was converted from ML [15] to Fish. The original program was
taken from Impala suite [16] for parallel benchmark programs.
The basic idea of the program is to generate a 2D image from a 3D scene which
consists of spheres. In the resulting 2D image each pixel in the grid will be colored
according to traced ray for that pixel. During the tracing of a ray, all intersections of this
ray with the given objects are computed. The ray is reflected once an intersection is
found and the color of the intersection point is computed based on the strength of the
ray. The program was tested at Heriot-Watt University's Beowulf cluster which is based
on Intel Celeron 533MHz, 128KB cache, 128 of DRAM and 5.7GB of IDE disk. The
workstations are connected through a 100 Mb/s fast Ethernet switch.
Fig 7 shows the execution time and speedup for the ray tracing of an image (512 x
512) consisting of 9 Spheres and 2 light sources. Speedup are results are good and better
than those presented in [17] up to 6 processors. However, there is drop in performance
when 8 processors where used. This might be caused by the increase in size of the
resulting object code and possibly less utilization of the cache. It is important to note that
this is caused by the FISh compiler as it generated bigger code size as we increase the
number of processors. It is possible to solve this issue by adding run-time shape analysis
and not relying completely on compile time analysis. The resulting ray traced spheres are
shown in Fig. 8.
Ray Tracer Execution Time
Ray Tracer Speedup
140
6
5
100
Speedup
Time (seconds)
120
80
60
40
4
3
2
1
20
0
0
1
2
3
4
6
1
8
2
3
4
6
Processors
Processors
Fig. 7: The execution time and speedup for the ray tracer program.
130
8
A Generic Parallel Map: Extending the List Map Higher Order Function to Arbitrary Data Structures
Fig. 8: The result of applying the ray tracer on a scene Consisting of 9 spheres and 2
light sources.
Comparison with other Models
Table 1 presents a comparison between gpmap and other models. It is important to
note that gpmap differs from most of these models as it is based on a functional language
and it has the advantage of the implicit parallelism that is available in HOFs. The other
implementations are semi-explicit and rely on the programmer to parallelize the
sequential application. Therefore, it is expected that gpmap will be preferred by
programmers as it encapsulates implicit parallelism and the user does not have to worry
about communication, coordination and any parallel processing issues.
Table 1: Comparison between gpmap and other models.
Models
gpmap
Janus
PolyP
Generic
Haskell
PMLS
[18]
GpH
[19]
Eden
[20]
Functional or
Imperative
Functional
Imperative
Imperative
List Data
Structure
Yes
Yes
Yes
Generic Data
Structures
Yes
No
No
Supports
Parallelism
Yes
No
No
Parallelism
Type
Implicit
N/A
N/A
Run-time Support
Skeleton-library
N/A
N/A
Functional
Yes
Yes
No
N/A
N/A
Functional
Yes
No
Yes
Implicit
Skeleton-library
Functional
Yes
No
Yes
Semi-Implicit
Graph-Reduction
Functional
Yes
No
Yes
Semi-Implicit
Graph-Reduction
131
Hamdan
5 Conclusions and Future Work
It was possible to design and implement a generic parallel map that can apply a
function to arbitrary data structures. It was engineered from different components: a host
language, a generic programming, master slave template, data distribution and generic
sends. It was used on a decent example (the ray tracing of spheres) which consisted of a
slightly complex data structure due to the lack of description of applications using
generic programming languages [21]. It is planned to use the generic parallel map for
further examples. Also it is possible to think of adding cost models with real hardware
parameters to the generic distribution.
It is important to note that a good application for the gpmap would be in the
parallelization of genetic algorithms Gas [22]. To explain, in a GA there are different
types of representations such as binary, float, arithmetic, trees вЂ¦ etc according to the
encoded problem. The GA operators such as the crossover operator is nearly the same
operation regardless of the representation method. Therefore, a generic crossover could
be implemented that works on different datatypes (representations) and used a function
argument for gpmap to applied in parallel on the population strings.
Acknowledgment
The author would like to thank Dr. C. B. Jay and Dr. Q. T. Nquyen from University
of Technology, Sydney, Australia for access to the Fish compiler and the useful
discussions during the development of this work. Also, the author would like to thank
Dr. Greg Michaelson from Heriot-Watt University for giving the author access to HeriotWatt's Beowulf cluster.
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132
A Generic Parallel Map: Extending the List Map Higher Order Function to Arbitrary Data Structures
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134
ABHATH AL-YARMOUK: "Basic Sci. & Eng."
Vol. 16, No.1, 2007, pp. 135-145
Effect of Access Point Moving on the Wireless Network
Performance Using Opnet Modeler
Sameh H. Ghwanmeh*1
Received on Oct. 23, 2005
Accepted for publication on July 30, 2006
Abstract
The effect of moving the access point on the performance of the wireless network
has been presented using an advanced network simulator. The Network Performance has
been shown via a series of simulation tests with different access points locations. The
simulation results are presented showing satisfactory network performance when the
access point is chosen to include maximum possible number of users over the RF
coverage area, geographical nature of the considered area and the number of access
points.
Keywords: access points, wireless networks, remote access, performance.
Introduction
Wireless local area networks (WLAN) enables people to communicate with anyone,
anywhere, at anytime, using a range of multimedia services. The exponential growth of
cellular telephone and mobile systems coupled with the spreading of laptop computers
indicate a bright future for such networks.[1] The next stage of its development will be
in complementing or replacing the network infrastructure that was traditionally wired as
well as enabling network infrastructures. However as the communications field become
a key factor in many other fields such as medical, teaching, business, information
technology and other, the need for the communication to be anywhere without the
problem of cables and the position of the user was increased and applying the wireless
network become more important and essential in this fields, thus this help to spread these
technology as fast as possible. WLAN technologies employ infrared, laser, spread
spectrum, microwave, and satellite for supporting voice, video, and data transmission in
local area and wider area environments [2,3].
The wireless networks gives the user to use any applications regardless to its source
or vendors and the type of the application which can be data, video, real-time
applications or others. However, the WLAN performance and quality of transmission are
key factors in spreading and usage of such technologies [4,5]. This paper deals with the
WLAN performance techniques based on advanced network simulator, OPNET modeler.
В© 2007 by Yarmouk University, Irbid, Jordan.
* Director, Computer and Information Centre. Computer Engineering Department, Yarmouk University
211-63, Irbid, Jordan. E-mail: sameh@yu.edu.jo
Ghwanmeh
The effect of moving the access point on the performance of the wireless network has
been presented. The Network Performance has been shown via a series of simulation
tests with different access points locations.
WLAN Configurations
Wireless LAN configurations range from extremely simple to very complex. The
simplest WLAN is an independent, peer-to-peer configuration where two or more
devices with wireless adapters connect to each other, as depicted in Fig. 1. Peer-to-peer
configurations are often called ad hoc networks since they do not require any
administration or pre-configuration [1]. They also do not require the use of an access
point, as each adapter communicates directly to another adapter without going through a
central location.
Figure 1 Peer-to-peer WLAN configuration.
Peer-to-peer networks are very useful when a group of users need to communicate
with one another in an unstructured way. A wireless network can also use an access
point (AP). The access point acts like a hub, providing connectivity for the wireless
computers (Fig. 2). It can connect the wireless LAN to a wired LAN, allowing wireless
computer access to LAN resources, such as file servers or existing Internet Connectivity.
Figure 2 WLAN configuration with access point.
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Effect of Access Point Moving on the Wireless Network Performance Using Opnet Modeler
In a corporate environment, many access points can work together to provide
wireless coverage for an entire building or campus. The coverage area from each access
point is called a cell.
2.2 Setting up a WLAN Access Point
Overlapping AP Configuration
When adjacent access points are located close enough to each other, parts of the
coverage area of access points may overlap with each other. Figure 3 shows that when
two adjacent access points, A and B, are located close enough to each other, a part of the
coverage area of access point A overlaps that of access point B.
Figure 3 Overlapping Access Points.
The overlapping area has two very important attributes: any workstation situated in
the overlapping area can associate and communicate with either AP A or AP B and any
workstation can move continuously through the overlapping coverage areas without
losing its network connection [2].
Non-Overlapping AP Configuration
Wireless LANs congested with many network users and heavy traffic load may
require non-overlapping, multi-AP configuration. In a non-overlapping, multi-AP
configuration, several APs are installed in the same location (Fig. 4).
Figure 4 Non-overlapping AP configuration.
In non-overlapping configuration, each AP has the same coverage area, creating a
common coverage area that increases aggregate throughput. Any workstation in the
overlapping area can associate and communicate with any AP covering that area.
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Description of the Simulation Tool
The OPNET (Optimized Network Engineering Tool) can be best described as a set
of decision support tools, providing a comprehensive development environment for the
specification, simulation and performance analysis of communication networks,
computer systems and applications, and distributed systems. Discrete event simulations
are used as the means of analyzing system performance and their behaviour. This
sophisticated package comes complete with a range of tools which allows us to specify
models in great detail, identify the elements of the model of interest, execute the
simulation and analyze the generated output data. The OPNET simulator has many
features such as object orientation and hierarchical modelling [6,7].
OPNET provides four editors that are used to develop a representation of a system
being modeled. These editors, the Network, Node, Process and Parameter Editors, are
organized in a hierarchical fashion as shown in Fig. 5. This hierarchical organization
supports the concept of model reuse. Models developed at one layer can be used by
another model at a higher layer. The Network Domain deals with the specification of the
physical topology of a communications network. The basic building block is a node. An
underlying model, developed using the node editor, defines the specific capabilities of
each node. Each node instance has a set of parameters or characteristics that can be set to
customize the node's behavior. The number of node models that can be used in a
network model is unrestricted. The Node Domain deals with the specifications of the
communication devices created and interconnected at the network level. These nodes
correspond to communicating devices such as personal computers, workstations, file
servers, printers, bridges, routers or switches. Process Domain created using the process
editor, which are used to describe the behavior of processor and queue modules within
the node domain. These models are used to simulate software subsystems, such as a
communication protocol, and also to model hardware subsystems, such as the memory of
a switching device. Each process that executes in a queue or processor module is an
instance of a particular process model. Process models are expressed in a language called
Proto-C. OPNET's Proto-C consists of state transition diagrams (STDs), a library of
kernel procedures, and the standard C programming language. The high level of
flexibility offered by the Proto-C language enables almost any type of task, collection of
specific data and definition of any protocol is performed [8].
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Effect of Access Point Moving on the Wireless Network Performance Using Opnet Modeler
Figure 5. Simulation tool: the four OPNET editors.
Move-Test Description
The move-test solution can be considered as a basic step in building wireless
network and optimizing it. Wireless service providers need to optimize their networks, as
new access points installed, new buildings constructed or other condition changed.
An overview of the optimization process is shown in Fig. 6. Move-testing is the
first step in the process, with the goal of collecting measurement data as it relates to a
user's location. Once the data have been collected over the desired RF coverage area, the
data is outputted to a post-processing software tool. Engineers can use the postprocessing and collection tools to identify the causes of potential RF coverage or
interference problems can be solved. Once the problems, causes, and solutions are
identified, steps are performed to solve the problem [9].
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Ghwanmeh
Figure 6 Optimization Process.
The key characteristic of building wireless network is the WLAN Coverage. The
WLAN will lose its reliability if it cannot respond to the user requests [10]. The wireless
service providers will not give chance to this disadvantage to occur using Move- test
solution. Before installing the access point we have to perform site evaluation to
determine an appropriate location for the new access point by transmitting a continues
signal from the undertest location and measure it using mobile receiver which moves
over the coverage area to determine the locations of poor RF coverage.
Move-Test Simulation Results
The WLAN testing framework consists of a network contains workstations with
wireless NICs, and a single access point to enable the workstations to access the
connected wired network which provides several services such as FTP, HTTP and
Email. We note that the workstations are distributed in different locations, the
workstations in the left side are group_1 and the others are group_2. First, the access
point is set in location 1 as shown in Fig. 7.
Figure 7 Access point location 1.
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Effect of Access Point Moving on the Wireless Network Performance Using Opnet Modeler
Figure 8 Simulation results of AP location 1.
Because all workstations are in group_2 are out of the transmitting and receiving
RF rang of the access point, which shown as green cycle with radius of 1160.5m (Fig. 7).
The data dropped at this location mainly caused by the workstations of group_2 because
they are out of the coverage. The buffer size Specifies the maximum size of the higher
layer data buffer in bits. Once the buffer becomes full, the data packets arrived from
higher layer will be discarded until free space is available, the buffer size of the OPNET
wireless nodes is 256000 bit. It can be noted that the wireless LAN delay is
approximately constant since the location of each workstation is fixed respect to the
Access point. We can also note that the values of the throughput and load are equal at
any given execution time since the number of bits that are stored in the higher layer
queue in transmitter nodes (which reflect the load) is equal to the number of bits that
received in the receiver node (Fig. 8).
Figure 9 Access point location 2.
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Ghwanmeh
Figure 10 Simulation results of AP location 2.
In the second test the access point is set at location 2 (Fig. 9) which is closer to
group_2. From Fig. 10 it can be noted that the number of data dropped in the WLAN
increases due to increasing the number of workstations that are out of the RF coverage
range. Other statistics are mainly collected from group_2 workstations.
In the third test the Access point is moved to a newly location (Fig. 11). At this
location, the access point is distant from both groups and few workstations are included
in the RF coverage range of the access point. Hence, the data dropped reaches the
maximum value. It can be seen that the media access delay increases by increasing the
load of the wireless network (Fig. 12).
In the fourth test, the AP is set at location 4 (Fig. 13). As it can be seen from Fig.
14, this test demonstrates no data dropped from both groups, which improves the
reliability of the wireless network. Further the throughput reaches its maximum value
compared with previous test locations.
Figure 11 Access point location 3.
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Effect of Access Point Moving on the Wireless Network Performance Using Opnet Modeler
Figure 12 Simulation results of AP location 3.
Figure 13 Access point location 4.
Figure 14 Simulation results of AP location 4.
143
Ghwanmeh
Conclusions
The OPNET advanced simulator has been used to study the effect of access-point
moving on the WLAN performance. Simulation results indicate that we can greatly
improve the WLAN performance. The location of the access point plays a very
important role in enhancing the performance of the WLAN and choosing unsuitable
location will prevent some users from accessing the network. The simulation results
presented showing satisfactory WLAN performance when the access point is chosen to
include maximum possible number of users over the RF coverage area, geographical
nature of the considered area and the number of access points.
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145
ABHATH AL-YARMOUK: "Basic Sci. & Eng."
Vol. 16, No.1, 2007, pp. 147-161
New Techniques for Improving H.263
Videoconferencing
Ammar M. Kamel and Moh'd Belal Al-Zoubi* 1
Received on Aug. 2, 2006
Accepted for publication on Feb. 18, 2007
Abstract
Two-way real time videoconferencing techniques have become the big challenge of
the Internet revolution. The ITU-T H.263 is designed for very low-bit rate coding, which
is concerned with real-time two-way communication. Motion estimation\compensation
is the core of H.263 video coding. Different motion estimation techniques were
developed to enhance the ability and efficiency of H.263 technique for video
compression.
In this paper, two new techniques have been developed; Thresholding Half-pixel
and OddEven motion search techniques. The first technique increases the image quality
of video frames. The second one minimizes the encoding delay time through eliminating
the searching points in each searching window of the video frame.
The extracted results proved the effectiveness of the proposed techniques in
comparison with the well-known motion search techniques in terms of delay time, image
accuracy and compression ration.
Keywords: Videoconferencing, Motion estimation, H.263, Image/video compression...
Introduction
Digital video communications technology has impressed the lives of people
everywhere in the world. Digital video has an inherently high bandwidth (i.e., a digitized
video signal requires a very high data rate for transmission). In order to store and
transmit this information effectively, it is necessary to develop techniques for video data
compression; otherwise, raw video may require large storage space and bandwidth [6,
11, 12].
Special techniques, for example (MPEG1, MPEG2, MPEG4, H.261 and H.263),
which take the characteristics of the video into account, can compress the video with a
high compression ratio [8].
H.263 is mostly used for dialogue mode applications in network environments. for
example, video telephony and video conferencing[5, 7, 8]. The resulting continuous bit
rate is eminently suitable for todayвЂ™s wide area networks operating with ISDN, leased
lines, and some other connections. The H.263 ITU coding standard for low bit rate
В© 2007 by Yarmouk University, Irbid, Jordan.
* Department of Computer Information Systems, University of Jordan, Amman, Jordan.
Kamel and Al-Zoubi
videoconferencing is based on a hybrid motion estimation/compensation video coding
method [6, 13, 14].
Hybrid video compression exploits temporal redundancy by motion compensation
and spatial redundancy by DCT transformation. Motion estimation is a major part in
such video compression systems. Many block matching search techniques (such as
Three-Step search, 2D-Logarithm search, Conjugate search, and Full search) have been
developed to reduce the complexity of the motion estimation that is associated with
exhaustive searches [1, 2, 3, 4, 9, 10, 12]. These techniques reduce the video frame total
searching points, which greatly affect the encoding delay time. The encoding delay time
and image accuracy are major problems that limit the H.263 efficiency. The encoding
delay time significantly contributes to the video conferencing interactivity achievement
through decreasing the elapsed time in the video compression. On the other hand,
increasing the image accuracy directly affects the video compression ratio.
In this paper, two new techniques are developed; Thresholding Half-pixel and
OddEven motion Search (OES) to enhance the efficiency of the H.263
videoconferencing technique. The first technique increases the image quality of video
frames. The second one decreases the total elapsed time of video compression (encoding
the delay time).
H.263-Motion Estimation-compensation: Proposed Improvements
In this section, two considerably improved techniques are developed to enhance the
efficiency of the H.263 technique. This is done through enhancing the data rate and
reducing the transmission delay limitations. The techniques also improve the video
sequence quality.
The first technique (called Thresholding Half-pixel technique) greatly intensifies
the performance of the traditional half-pixel technique by adding a threshold value which
controls the clarity and smoothness of each of the pixel groups included from which
video frames are composed.
The second technique (called OddEven search technique) employs a new search
strategy to match similar blocks from video successive frames and exclude the similar
blocks. This technique achieves an efficient compression process. These techniques are
explained below..
Thresholding Half-Pixel Accuracy Improvement
H.263 uses a sub-pixel accuracy for motion compensation instead of using a loop
filter to smooth the anchor (reference) frame as in H.261. The half-pixel technique
achieves the image smoothing by considering a pixel and its neighbors, eliminating any
extreme values (high frequency-noise) in this group. The values of half-pixel are found
using bilinear interpolation [6].
Half-pixel technique does not take into account the color distance relationship
among pixels directly, but rather produces values representing half the distance among
these pixels.
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New Techniques for Improving H.263 Videoconferencing
In the current study, we improved the technique by adding an equilibrium factor to
achieve a high level of smoothness among anchor frame pixels through computing
threshold value.
The threshold value stands for the average difference between two pairs of pixels,
which reduces the noisy pixelвЂ™s values (not correlated pixels) resulted from the half-pixel
technique. This makes the pixel values of each group closer and enhances the video
anchor frames.
The improvement of the half-pixel accuracy is explained in Fig. (1) and equation
(2.1) [8]:
where A, B, C, and D are grouped pixels and There is an additional value used to
reduce the extreme half-pixel values. According to equation (2.1), the threshold value
will decrease the color distance for group (a, b, c, and d) and produce a new group of
pixels with a suitable color approximation.
OddEven Search Technique (OES)
The main idea of the proposed OddEven Search (OES) algorithm is to compute the
cost function (typically, the MAD function is used [12]) for the odd blocks of y-axis and
for the even blocks of x-axis finding the best match.
Where
F (i,j) represents an (m x n) macroblock from the current frame,
G (i,j) represents the same macroblock from a reference frame (past or future),
(Dx, Dy) a vector representing the search location.
The matching candidate block contains the minimum value of the computed cost
function values in the search windows of the previous frame. This block represents the
most similar block for the selected block in the current frame of the video sequence.
The matching process starts by calculating the cost function of the odd locations of
the y-axis and even locations of the x-axis in the search window. The location that
produces the smallest cost function becomes the center location for the next step, and the
search range is reduced by half.
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Kamel and Al-Zoubi
The matching process of the OES algorithm will continue until the minimum
computed cost function value is equal to or less than the threshold value previously
determined to achieve the matching process.
The threshold value is determined depending on the requirements of the
applications that use the proposed algorithm. The threshold value controls the speed and
efficiency of the block matching process. This effectively affects the H.263 bit rate and
the encoding delay time.
Fig. (2a) shows the block matching process on the y-axis. Fig. (2b) shows the block
matching process on the x-axis.
OddEven Control Parameters
This technique uses two control values to produce the best and fastest block
matching process. These control values are:
Information Density Control Value: A value is determined representing the number of
bits from which the maximum color value of the matched block is composed. This value
represents the amount of color information distributed on the matched block. The
following formula is used to compute this value:
MADThreshold = пЈ®Log 2 ( Max( SBlk ))пЈ№
where SBlk represents the current frame candidate block for the matching process.
The computed value of the MADThreshold is compared to the minimum MAD
computed values. If the selected MAD minimum value is less than or equal to the
MADThreshold value, the matching process will be terminated. The block of the
minimum MAD value is the approximate match of the SBlk.
Loop Step Size Control Value: Three values are used (0.5, 0.7, or 0.9) to reduce the
matching process comparison on the searching window blocks to find the best match.
The 0.5 value divides the loop iteration by 2; this makes the OddEven technique
compare half of the searching window blocks. When the value is 0.7, the OddEven
technique compares one third of the searching window blocks. However, when the value
is 0.9, the OddEven technique compares all of the searching window blocks.
Experimental Results
In this section, four well-known techniques are evaluated and compared with the
proposed OddEven Search (OES) technique: Full Search (FS), Conjugate search, 2-D
logarithmic search, and Three-Step Search (3SS). Several results are constructed and
analyzed according to the total delay time, the average of PSNR, and the total of the
compression ratio. In addition, another proposed Thresholding Half-pixel technique is
compared with the classical Half-pixel technique.
The proposed and the four well-known techniques are reviewed in this study. The
obtained results have been developed using MATLAB 6.5 and a computer system with a
PIII processor and 128 MB memory.
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New Techniques for Improving H.263 Videoconferencing
Image Accuracy Evaluation
The proposed Thresholding Half-pixel technique was simulated and compared with
the traditional Half-pixel technique. PSNR and eRMS are used as two measures utilized to
evaluate the image quality. Results illustrated in Table (1) indicate that the proposed
technique has achieved a clear increase in the PSNR compared with the PSNR of the
traditional Half-pixel technique. Increasing the PSNR means that the enhanced
decompressed image maintains color properties close to those of the original image.
Fig. (3a) represents an image after compression and decompression using JPEG
technique, while Fig. (3b) represents the histogram of the decompressed image.
Fig. (4a) denotes the decompressed image after applying the traditional Half-pixel
technique. Fig. (4b) represents the histogram of the decompressed image.
Fig. (5a) represents the decompressed image after applying the proposed Thresholding
Half-pixel technique. Fig. (5b) shows that the color distribution of the image is
better than that of Fig. (4b).
The Evaluation of Block Matching Search Techniques
The proposed OddEven Search (OES) algorithm is simulated using the luminance
component of the first 10 QCIF frames of (Foreman, Miss America, Employer) video
sequence. These sequences consist of a large amount of information and different scenes
movement. The Mean Absolute Difference (MAD) is used as the cost function to
calculate the similarity between the matched blocks. The maximum displacement in the
search area is (В±7) pixels in both horizontal and vertical directions for a block size, of 16
Г—16.
The proposed OES technique uses one of three different threshold values that
control and affect the search process. These values are (0.5, 0.7, and 0.9).
The performance comparison between OES, FS, Conjugate search, 2-D logarithmic,
and 3SS in terms of the total delay time are shown in Tables (2), (5) and (8).
Table (8) shows that the proposed OES algorithm that uses three different threshold
values (0.5, 0.7, and 0.9) achieved good results. This was obtained through decreasing
the elapsed time for the matching process in comparison with the well-known
techniques, particularly, the 3SS algorithm, which had the second less delay time. Tables
(2) and (5) show that the proposed OES algorithm that uses two different threshold
values (0.5, 0.7) achieved the best results in comparison with the 3SS algorithm.
Tables (3), (6) and (9) give some PSNR statistical comparisons between the
proposed technique and each of the four well-known techniques. Tables (6) and (9)
illustrate that the proposed OES algorithm that uses the three different threshold values
(0.5, 0.7, and 0.9) resulted in the highest PSNR in comparison with the FS. Tables (4),
(7) and (10) show another comparison according to the Compression ratio (Cr). Tables
(4), (7) and (10) show that the proposed OES algorithm that uses the three threshold
values (0.5, 0.7, and 0.9), has come near to the value of 3SS.
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Kamel and Al-Zoubi
Discussion
In this study several improvements on the parameters that hinder the H.263
CODEC efficiency are performed in order to improve the image accuracy and
transmission delay time. These improvements are done by Thresholding and traditional
Half-pixel techniques in order to enhance the video frame accuracy. This considerably
improves the performance of H.263 motion estimation/compensation.
The improvement results indicate the following:
1.
When computing the PSNR of the decompressed image applying both the proposed
Thresholding Half-pixel technique and the traditional Half-pixel technique, it
appeared to be 39.025 for the proposed technique and 38.976 for the traditional one.
The higher PSNR of the proposed technique indicates remarkable image accuracy
and clarity.
2.
When testing a decompressed image and applying the proposed technique, eRMS
appeared to be 0.0223 compared with 0.0255 for the traditional Half-pixel
technique. The nearer the eRMS value is to zero, the closer the decompressed image
is to the original one.
The other improvement deals with the block matching search time. Using the
proposed OES of one of the threshold values (0.5, 0.7, or 0.9) gave higher results in
comparison with the well-known four search methods. The elapsed time for the block
matching search and determining the Motion Vectors (MV) have been decreased
considerably, which in turn would decrease the encoding delay time and increase the
efficiency of H.263 CODEC. The results of testing the four well-known techniques (FS,
Conjugate, 2-D logarithmic, and 3SS) and the proposed (OES) indicate a reverse
relationship between the PSNR and the Compression ratio (Cr). Maintaining equilibrium
between the PSNR and the Cr and increasing both of them would result in a great
improvement in H.263. It can be noticed that:
1.
Using both the proposed OES and the Thresholding Half-pixel resulted in a
noticeable increase in the PSNR compared with the PSNR of the traditional FS
technique, which is characterized with a high PSNR value.
2.
Increasing the PSNR using the two proposed techniques (Thresholding Half-pixel
and OES), a fair Compression ratio (Cr) was maintained in comparison with the 3SS
technique.
Conclusion
In this study, two algorithms have been developed: Thresholding half-pixel
technique and OddEven technique. These algorithms have enhanced video compression.
The idea of the first algorithm is to increase the image clarity and accuracy, while the
second one is concerned with decreasing the elapsed time of video compression and
increasing the compression ratio.
Testing these two techniques gave good results, improving the performance of the
H.263 technique significantly. Applying the proposed technique (Thresholding Half152
New Techniques for Improving H.263 Videoconferencing
pixel) resulted in high accuracy and clarity in the decompressed images compared with
the traditional Half-pixel technique. The proposed OES technique has remarkably
decreased the elapsed time of video compression in comparison with the four wellknown techniques (FS, Conjugate, 2-D logarithmic and 3SS). Using three different
threshold values by OES resulted in a fast block matching process; decreasing the
encoding delay time clearly. Choosing simple scenes with few moving objects (called
Head and Shoulder) increases Cr and decreases the video compression delay time. Each
motionless object is considered similar to an object in the same location in the next
frame. These blocks can be substituted without computing the Motion Vector (MV) for
each block.
вЂ« п»џп» п»¤пє†пє—п»¤пє®Ш§ШЄ Ш§п»џп»”п»ґпєЄп»іп»®п»іпє”вЂ¬H.236 вЂ«пє—п»�п»Ёп»ґпє” пєџпєЄп»іпєЄШ© п»џпє�пє¤пєґпІ” пє§п»®Ш§Ш±ШІп»Јп»ґпє”вЂ¬
вЂ«п»‹п»¤пєЋШ± п»›пєЋп»Јп»ћ Щ€ п»Јпє¤п»¤пєЄ пє‘п»јЩ„ Ш§п»џпє°п»‹пє’п»ІвЂ¬
вЂ«п»Јп» пєЁЩЂЩЂпєєвЂ¬
.вЂ«( пє—пє¤пєЄп»іпєЋЩ‹ п»›пє’п»ґпє®Ш§Щ‹ п»“п»І п»‹пєЋп»џп»ў Ш§п»№п»§пє�пє®п»§пє–вЂ¬H.236 Videoconferency) вЂ«пє—п»¤пєњп»ћ пє—п»�п»Ёп»ґпє” Ш§п»џп»¤пє†пє—п»¤пє®Ш§ШЄ Ш§п»џп»”п»ґпєЄп»іп»®п»іпє”вЂ¬
вЂ« Ш§п»џпє�п»�п»Ёп»ґпє”вЂ¬.вЂ«Щ€п»—пєЄ пє—п»ў пє—п»„п»®п»іпє® пє—п»�п»Ёп»ґпє�п»ґп»¦ пєџпєЄп»іпєЄпє—п»ґп»¦ п»џп»Ђп»ђп»‚ п»«пє¬Ш§ Ш§п»џп»Ёп»®Ш№ п»Јп»¦ Ш§п»џпє’п»ґпєЋп»§пєЋШЄ п»“п»І п»«пє¬Ш§ Ш§п»џпє’пє¤пєљвЂ¬
вЂ«( Щ€Ш§п»џпє�п»І п»Јп»¦ пє·пє„п»§п»¬пєЋ пє—пє¤пєґп»ґп»¦ п»§п»®п»‹п»ґпє” пє»п»®Ш± Ш§п»џп»”п»ґпєЄп»іп»® ШЈпє›п»ЁпєЋШЎ п»‹п»¤п» п»ґпєЋШЄ Ш§п»џп»Ђп»ђп»‚вЂ¬Thresholding Half-pixel) вЂ«Ш§п»·Щ€п»џп»°вЂ¬
вЂ«( п»џпє�п»�п» п»ґп»ћ Ш§п»џп»®п»—пє– Ш§п»џп»јШІЩ… п»џп» пє�пєёп»”п»ґпє® Щ€Ш§п»џп»Ђп»ђп»‚ п»‹п»¦ п»ѓпє®п»іп»–вЂ¬OddEven motion search techniques) вЂ«Щ€Ш§п»џпєњпєЋп»§п»ґпє”вЂ¬
ШЊвЂ« Щ€п»—пєЄ ШЈпє›пє’пє�пє– п»§пє�пєЋпє‹пєћ Ш§п»џпє’пє¤пєљ п»“пєЋп»‹п» п»ґпє” Ш§п»џп»„пє®п»іп»�пє�п»ґп»¦ Ш§п»џпє пєЄп»іпєЄпє—п»ґп»¦вЂ¬.вЂ«пєЈпє¬ЩЃ пє‘п»Њпєѕ Ш§п»џпє’п»ґпєЋп»§пєЋШЄ Ш§п»џп»¬пєЋп»Јпєёп»ґпє” ШЈпє›п»ЁпєЋШЎ п»‹п»¤п» п»ґпєЋШЄ Ш§п»џпє’пє¤пєљвЂ¬
.вЂ«п»‹п»¦ п»ѓпє®п»іп»– п»Јп»�пєЋШ±п»§пє” Ш§п»џп»Ёпє�пєЋпє‹пєћ п»Јп»Љ пє—п»�п»Ёп»ґпєЋШЄ п»‹пєЋп»џп»¤п»ґпє” п»Јп»Њпє®Щ€п»“пє” п»“п»І п»«пє¬Ш§ Ш§п»џп»¤пє пєЋЩ„вЂ¬
References
[1]
ALexis M., Guobin S., Ming L., Oscar C. and Ishfaq A., вЂњA New Predictive
Diamond Search Algorithm for Block Based Motion EstimationвЂќ. Proceeding of
Visual Communication and Image Processing (VCIP-2000). Perth. Australia,
(2000).
[2]
Bagni D., de Haan G. and Riva V., вЂњMotion Compensated Post-Processing for
Low-bit Rate Videoconferencing on ISDN Lines,вЂќ in Proc. of the ICCE'97,
(1997) 28-29.
[3]
Cheung C. and Po L., вЂњA Hierarchical Block Matching Algorithm Using Partial
Distortion Measure,вЂќ in Proc. of IEEE International Conference of Image
Processing. 3, Santa Barbara. CA. USA, (1997) 606-609.
[4]
Gharavi H. and Mills M., вЂњBlock Matching Motion Estimation Algorithm вЂ“ New
Results,вЂќ IEEE Transactions on Circuits System, 37 (1990) 649-651.
153
Kamel and Al-Zoubi
[5]
Halsall F., Computer networking and the internet, 5th ed., Addison-Wesley,
(2005).
[6]
ITU-T Draft Recommendation H.263, вЂњVideo Coding for Low Bit Rate
CommunicationвЂќ, (1996).
[7]
Konrad J., вЂњVisual Communications of Tomorrow: Natural, Efficient and
FlexibleвЂќ, IEEE Communications Magazine, 39(1) ( 2001) 126вЂ“133.
[8]
Li Z. and Drew M., вЂњFundamentals of MultimediaвЂќ, Prentice Hall, (2004).
[9]
Po L. and Ma W., вЂњA Novel Four-Step Search Algorithm for Fast Block Motion
Estimation,вЂќ IEEE Trans., on Circuits and Systems for Video Technology 3, 6
(1996) 313-317.
[10]
Poynton C., вЂњDigital Video and HDTV Algorithms and InterfaceвЂќ, Morgan
Kaufmann, (2003).
[11]
Srinivasan R. and Rao K., вЂњPredictive Coding Based on Efficient Motion
Estimation,вЂќ IEEE Transaction and Communication, 33 (1985) 888-896.
[12]
Tartagni M., Leone A. and Guerrieri R., A Low-Power Block-Matching Cell for
Video Compression, 27 (3) (2001) 259-271.
[13]
Wang Y., Ostermann J. and Zhang Y., вЂњVideo Processing and CommunicationsвЂќ,
Prentice Hall, (2002).
[14]
Yun Q. and Huifang S., Image and Video Compression for Multimedia
Engineering, 1st Ed. CRC Press: USA, (2000).
154
New Techniques for Improving H.263 Videoconferencing
List of Tables
Table (1): PSNR and eRMS comparison criteria between the proposed Thresholding
Half-pixel technique and the well-known Half-pixel technique
Image Technique
Proposed Thresholding Half-pixel
Half-pixel
PSNR
39.025
38.976
eRMS
0.0223
0.0255
Table (2): Elapsed delay time comparison between the four well-known techniques
and the proposed OddEven technique for Forman video sequence
Frame
1
I
2
P
3
P
4
P
5
P
6
P
7
P
8
P
9
P
10
P
Total
Delay
Time
Full
search
FS
11.257
ms
31.826
ms
30.404
ms
30.363
ms
30.493
ms
30.364
ms
30.304
ms
30.714
ms
30.494
ms
31.335
ms
287.554
ms
Conjugate
search
2-D
logarithmic
search
11.257 ms
11.257 ms
6.469 ms
7.251 ms
6.470 ms
7.250 ms
6.479 ms
7.261 ms
6.469 ms
7.210 ms
6.440 ms
7.190 ms
6.479 ms
7.201 ms
6.480 ms
7.200 ms
6.459 ms
7.210 ms
6.479 ms
7.231 ms
69.481 ms
76.261 ms
ThreeStep
Search
(3SS)
11.257
ms
6.489
ms
6.449
ms
6.429
ms
6.419
ms
6.419
ms
6.439
ms
6.439
ms
6.500
ms
6.429
ms
69.369
ms
Proposed
(OddEven)
search
threshold =
0.5
Proposed
(OddEven)
search
threshold
= 0.7
Proposed
(OddEven)
search
threshold =
0.9
11.257 ms
11.257 ms
11.257 ms
6.158 ms
6.329 ms
6.499 ms
6.149 ms
6.249 ms
6.479 ms
6.169 ms
6.229 ms
6.489 ms
6.159 ms
6.259 ms
6.499 ms
6.149 ms
6.239 ms
6.500 ms
6.139 ms
6.229 ms
6.469 ms
6.168 ms
6.239 ms
6.500 ms
6.149 ms
6.229 ms
6.490 ms
6.178 ms
6.239 ms
6.479 ms
66.675 ms
67.498 ms
69.661 ms
Table (3): PSNR comparison between the four well-known techniques and the
proposed OddEven technique for Forman video sequence
Frame
Average
of
PSNR
Full
search
FS
Conjugate
search
2-D
logarithmic
search
ThreeStep
Search
(3SS)
Proposed
(OddEven)
search
threshold
= 0.5
Proposed
(OddEven)
search
threshold
= 0.7
Proposed
(OddEven)
search
threshold
= 0.9
39.778
38.222
39.298
38.499
39.433
39.603
39.484
155
Kamel and Al-Zoubi
Table (4): Compression ratio (Cr) comparison between the four well-known
techniques and the proposed OddEven technique for Forman video sequence
Frame
Compression
ratio
(Cr)
Full
search
FS
Conjugate
search
2-D
logarithmic
search
ThreeStep
Search
(3SS)
Proposed
(OddEven)
search
threshold =
0.5
Proposed
(OddEven)
search
threshold =
0.7
Proposed
(OddEven)
search
threshold =
0.9
Total Cr
45.527
49.822
59.693
64.520
62.114
64.388
62.865
Table (5): Elapsed delay time comparison between the four well-known techniques
and the proposed OddEven technique for Miss America video sequence
Frame
1
Full search
FS
I
10.756 ms
2
P
3
P
4
P
5
P
6
P
7
P
8
P
9
P
10 P
Total
Delay
Time
28.691 ms
29.323 ms
29.022 ms
28.571 ms
29.242 ms
29.443 ms
29.242 ms
29.032 ms
29.101 ms
Conjugate
search
10.756
ms
6.459 ms
6.309 ms
6.309 ms
6.309 ms
6.319 ms
6.319 ms
6.329 ms
6.329 ms
6.299 ms
67.737
ms
272.423 ms
Three-Step Proposed
2-D
Proposed
logarithmic search
(OddEven) (OddEven)
search
3SS
search
search
threshold = threshold =
0.5
0.7
10.756
10.756
10.756
10.756
ms
ms
ms
ms
7.211 ms
6.549 ms
6.099 ms
6.309 ms
7.831 ms
6.890 ms
6.349 ms
6.249 ms
7.110 ms
6.309 ms
6.209 ms
6.269 ms
7.110 ms
6.339 ms
6.109 ms
6.259 ms
7.131 ms
6.339 ms
6.098 ms
6.249 ms
7.110 ms
6.349 ms
6.098 ms
6.279 ms
7.131 ms
7.040 ms
6.088 ms
6.259 ms
8.051 ms
6.339 ms
6.119 ms
6.239 ms
7.121 ms
6.339 ms
6.129 ms
6.229 ms
76.562
ms
69.249
ms
66.054
ms
Proposed
(OddEven)
search
threshold = 0.9
67.097
ms
10.756 ms
6.599 ms
6.610 ms
6.600 ms
6.610 ms
6.589 ms
6.600 ms
6.609 ms
6.609 ms
6.630 ms
70.212 ms
Table (6): PSNR comparison between the four well-known techniques and the
proposed OddEven technique for Miss America video sequence
Frame
Full
search
FS
Conjugate
search
2-D
logarithmic
search
ThreeStep
search
3SS
Proposed
(OddEven)
search
threshold =
0.5
Proposed
(OddEven)
search
threshold =
0.7
Proposed
(OddEven)
search
threshold =
0.9
Average
of PSNR
41.289
40.157
41.143
40.907
41.518
41.370
41.311
Table (7): Compression ratio (Cr) comparison between the four well-known
techniques and the proposed OddEven technique for Miss America video sequence
Frame
Compression
ratio
(Cr)
Full
search
FS
Conjugate
search
2-D
logarithmic
search
ThreeStep
Search
(3SS)
Total Cr
34.364
34.609
43.686
45.453
156
Proposed
(OddEven)
search
threshold
= 0.5
44.190
Proposed
(OddEven)
search
threshold
= 0.7
45.335
Proposed
(OddEven)
search
threshold
= 0.9
45.26
New Techniques for Improving H.263 Videoconferencing
Table (8): Elapsed delay time comparison between the four well-known techniques
and the proposed OddEven technique for Employer video sequence
Frame
Full
search
FS
1.
I
2.
P
3.
P
4.
P
5.
P
6.
P
7.
P
8.
P
9.
P
10.
P
Conjugate
search
11.076
ms
31.315
ms
31.225
ms
31.546
ms
31.736
ms
31.455
ms
31.625
ms
31.465
ms
31.034
ms
31.796
ms
Total
Delay
Time
294.273
ms
2-D
logarithmic
search
11.076
ms
11.076 ms
6.719 ms
7.621 ms
6.559 ms
7.451 ms
6.549 ms
7.470 ms
6.559 ms
7.451 ms
6.559 ms
7.441 ms
6.559 ms
7.430 ms
6.520 ms
7.441 ms
6.550 ms
8.332 ms
6.550 ms
7.430 ms
70.200
ms
79. 143 ms
ThreeStep
Search
(3SS)
11.076
ms
6.609
ms
6.530
ms
7.421
ms
6.509
ms
6.560
ms
7.380
ms
6.579
ms
6.559
ms
7.401
ms
72.624
ms
Proposed
(OddEven)
search
threshold
= 0.5
Proposed
(OddEven)
search
threshold
= 0.7
Proposed
(OddEven)
search
threshold
= 0.9
11.076 ms
11.076 ms
11.076 ms
6.179 ms
6.269 ms
6.559 ms
6.179 ms
6.279 ms
6.519 ms
6.189 ms
6.299 ms
6.509 ms
6.148 ms
6.309 ms
6.549 ms
6.168 ms
6.299 ms
6.540 ms
6.159 ms
6.289 ms
6.559 ms
6.159 ms
6.289 ms
6.569 ms
6.169 ms
6.289 ms
6.559 ms
6.139 ms
6.309 ms
6.539 ms
66.565 ms
67.707 ms
69.978 ms
Table (9): PSNR comparison between the four well-known techniques and the
proposed OddEven technique for Employer video sequence
Frame
Average
of
PSNR
Full
search
FS
Conjugate
search
2-D
logarithmic
search
ThreeStep
search
3SS
Proposed
(OddEven)
search
threshold =
0.5
Proposed
(OddEven)
search
threshold =
0.7
Proposed
(OddEven)
search
threshold =
0.9
39.951
38.715
39.899
39.613
40.001
40.042
40.062
Table (10): Compression ratio (Cr) comparison between the four well-known
techniques and the proposed OddEven technique for Employer video sequence
Frame
Compression
ratio
(Cr)
Full
search
FS
Conjugate
search
2-D
logarithmic
search
ThreeStep
Search
(3SS)
Total Cr
57.760
43.11
64.588
67.45
157
Proposed
(OddEven)
search
threshold
= 0.5
67.41
Proposed
(OddEven)
search
threshold
= 0.7
65.556
Proposed
(OddEven)
search
threshold
= 0.9
66.164
Kamel and Al-Zoubi
The Figures
Fig. (1): The Thresholding Half-pixel technique
Fig. (2a): The OddEven search procedure. Points at (0,-3), (-2,-3), and (-2,-2) are found
to give the minimum dissimilarity in steps 1,2, and 3 respectively.
158
New Techniques for Improving H.263 Videoconferencing
Fig. (2b): The OddEven search procedure. . Points at (-2, 0), (-2, 2), and (-3, 2) are
found to give the minimum dissimilarity in steps 1, 2, and 3 respectively.
159
Kamel and Al-Zoubi
Fig. (3): (a) Applying the JPEG technique on the decompressed image. (b) Image
histogram.
Fig. (4): (a) Applying the traditional Half-Pixel technique on a decompressed Image. (b)
Image histogram
Fig. (5): (a) Applying the proposed Thresholding Half-Pixel technique on a
decompressed Image. (b) Image histogram.
160
New Techniques for Improving H.263 Videoconferencing
Fig. (6): Samples of Foreman video frames. (a) Current frame. (b) Previous frame. (c)
The difference blocks between the current and previous frames using the proposed
OddEven Search (OES) technique.
Fig. (7): Samples of Miss America video frames. (a) Current frame. (b) Previous frame.
(c) The difference blocks between the current and previous frames using the Three-Step
Search (3SS) technique.
Fig. (8): Samples of Employer video frames. (a) Current frame. (b) Previous frame. (c)
The difference blocks between the current and previous frames using the 2-D
logarithmic technique.
161
вЂ«п»‹пє п»®Ш± Щ€ пє‘п»ґп»њпє®ШЇвЂ¬
2
( )
elements of a basis of H H . Finally, we find the relation between the residual spectrum of
that operator and its norm and we present some example too.
Reference
[1]
Conway J., Acourse in Functional Analysis, printed in the United States of
America Springer-Verlag New York ,inc, 1990.
[2]
Julian M., WsД™p do Analizy Funconalney, Panstwowe Wydawnictwo
Naukowe.Warszwa, 1976.
[3]
Pedersen M., Functional Analysis in Applied Mathematics and Engineering,
Chapman Hall crc by Crcpressllc, 2000.
[4]
Picard R., Evolution Equstions as Operator Equations in Lattices of Hilbert
Spaces, Glasnik Matematicki,Vol.35 (55) (2000),III-136
[5]
Walter R., Real And Complex Analysis, New Y0rk: McGraw-Hill: 1966.
[6]
Wawrzynczyk-Antonl "Group Rrpresesentions and Special Functions" D.Reidel
Publishing Company Dordrecht-Luncaster Company Copyright by Pwn-Polish
Scientific Publishers Warszawa,1984.
[7]
Yosida K., Functional Analysis, 2dprint Springer-Verlag Berlin. Holmberg. New
York, 1966.
24
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{Ei }iв€ћ=1 вЉ‚ H 2 (H ) .....................................................................вЂ«п»—пєЋп»‹пєЄШ©вЂ¬
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K ..............................................................вЂ«пєЈп»�п»ћ пє—пєЋЩ… п»‹пєЄШЇЩЉ п»Јпє®п»›пєђ ШЈЩ€ пєЈп»�п»ґп»�п»ІвЂ¬
H ..................................................................................вЂ«п»“п»ЂпєЋШЎ п»«п» пє’пє®ШЄвЂ¬
H 2 (H)
H S ................................................................ вЂ«п»“п»ЂпєЋШЎ п»«п» пє’пє®ШЄ пє·п»¤пє–вЂ¬
вЂ«п»Јпє п»¤п»®п»‹пє” п»Јпє†пє›пє®Ш§ШЄ пє§п»„п»ґпє” п»Јп»ђп» п»�пє” п»Јп»Њпє®ЩЃ п»›п»ћ п»Јп»¬пєЋ Щ€ п»Јпє®Ш§п»“п»�п»Є п»‹п» п»° п»Јпє п»¤п»®п»‹пє” п»›пєњп»ґп»”пє” п»“п»ІвЂ¬
H
(
)
H
H
Hr (H ) ШЊ p
ШЊ H! (H) ................................................. (11) вЂ«Ш§п»§п»€пє® Ш§п»џпєјп»”пє¤пє”вЂ¬
I .............................................................вЂ«п»Јпє п»¤п»®п»‹пє” ШЈШЇп»џпє” ЩЊп»—пєЋпє‘п» пє” п»џп» п»ЊпєЄ п»‹п» п»° Ш§п»·п»›пєњпє®вЂ¬
A, B
r
A, B
в€ћ
Ae i
i =1
ri
= C r2 в€‘
p
,
Be i
ri
, A, B
в€ћ
пЈ« 1 пЈ¶
= C в€‘ пЈ¬ 2 p пЈ· Aei , Bei
пЈё
i =1 пЈ i
2
p
#
H
= C 2 в€‘ fi
iв€€I
A, B ! = C
2
!
H
2
,
Aei , Bej
в€ћ
в€‘
i =1
Aei Bei
,
i! i!
H
H
Application of Unbounded Linear Operators
in Hilbert Spaces
Ahmad Ajour and Rainer Picard
Abstract
H (H ) to be the class of all closed unbounded linear operators A
*
such that the domains of A and its adjoint operator A contain a basis of the Hilbert space H
2
and they are also dense in H. We show that H (H ) is a Hilbert space with a norm related to a
basis of H and we show that, this norm is independent of the chosen basis. We study the
2
2
orthogonality of the subspace of H (H ) and we write any linear operator of H (H ) as the
In this paper, we define
2
sum of two orthogonal operators. Moreover, we rewrite that operator as an infinity sum of
23
вЂ«п»‹пє п»®Ш± Щ€ пє‘п»ґп»њпє®ШЇвЂ¬
вЂ«вЂЄ.вЂ¬вЂ¬
вЂ«вЂЄ=0вЂ¬вЂ¬
вЂ«вЂЄjxdxвЂ¬вЂ¬
вЂ«вЂЄв‹… 2ПЂвЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄПЂвЂ¬вЂ¬
вЂ«)!вЂЄв€‘ в€« ( jвЂ¬вЂ¬
вЂ«вЂЄjв€€Z в€’ПЂвЂ¬вЂ¬
вЂ«вЂЄ=CвЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«!вЂ¬
вЂ«вЂЄHвЂ¬вЂ¬
вЂ«вЂЄПЂ2вЂ¬вЂ¬
вЂ«вЂЄ3вЂ¬вЂ¬
вЂ«вЂЄMej в€‚e jвЂ¬вЂ¬
вЂ«вЂЄ,вЂ¬вЂ¬
вЂ«!вЂЄjвЂ¬вЂ¬
вЂ«!вЂЄjвЂ¬вЂ¬
вЂ«Щ€ п»іп»¤п»њп»¦ ШЈЩ† п»§п»њпє�пєђ ШЈп»іп»ЂпєЋ вЂЄ в€‚ В± О»I , M ! = 0 :вЂ¬п»џп»њп»ћ вЂЄ О»вЂ¬п»Јп»¦ Ш§п»џпє¤п»�п»ћ вЂЄKвЂ¬вЂ¬
вЂ«=вЂ¬
вЂ«вЂЄПЂ2вЂ¬вЂ¬
вЂ«)!вЂЄв€‘3в‹… ( jвЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«в€‘вЂ¬
вЂ«вЂЄjв€€ZвЂ¬вЂ¬
вЂ«вЂЄjв€€ZвЂ¬вЂ¬
вЂ«вЂЄ=CвЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«!вЂ¬
вЂ«вЂЄM, в€‚ ! = CвЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«!вЂ¬
вЂ«Ш§п»џп»¤п»�пє�пє®Щ† ШЁ )] вЂЄ([в€’ ПЂ ,ПЂвЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄ.LвЂ¬вЂ¬
вЂ«п»§пє�пєЋпє‹пєћ Ш§п»џпє’пє¤пєљ Щ€ ШЈп»«пєЄШ§п»“п»ЄвЂЄ .вЂ¬п»џп»�пєЄ пє‘п»ґп»ЁпєЋ п»“п»І п»«пє¬Ш§ Ш§п»џпє’пє¤пєљ ШҐп»Јп»њпєЋп»§п»ґпє” Щ€ пєџп»®ШЇ п»“п»ЂпєЋШЎ п»«п» пє’пє®ШЄ п»Јп»¦ Ш§п»џп»¤пє†пє›пє®Ш§ШЄ Ш§п»џпєЁп»„п»ґпє” п»Џп»ґпє®вЂ¬
вЂ«Ш§п»џп»¤пє¤пєЄЩ€ШЇШ© пє‘пєёп»њп»ћ п»‹пєЋЩ… Щ€ ШҐп»Јп»њпєЋп»§п»ґпє” пєЈпєґпєЋШЁ п»§п»€п»ґп»ў п»«пє¬Ш§ Ш§п»џп»¤пє†пє›пє® п»Јп»Љ п»›пє�пєЋпє‘пє” Ш§п»џп»¤пє†пє›пє® п»Јп»¦ пє»п»’ Ш§п»џп»¤пє†пє›пє®Ш§ШЄ Ш§п»џп»¤п»Њпє®ЩЃ п»‹п» п»°вЂ¬
вЂ«пє·п»њп»ћ п»Јпє п»¤п»®Ш№ пє§п»„п»І п»џп»¤пє†пє›пє®п»іп»¦ ШҐпєЈпєЄШ§п»«п»¤пєЋ п»Јпє¤пєЄЩ€ШЇ п»‹п» п»° п»Јпє п»¤п»®п»‹пє” пє—п»Њпє®п»іп»’ Ш§п»џп»¤пє†пє›пє® Ш§п»·пє»п» п»ІвЂЄ .вЂ¬пє‘п»ґп»ЁпєЋ Щ€пєџп»®ШЇ п»—пєЋп»‹пєЄШ© п»Јп»¦вЂ¬
вЂ«Ш§п»џп»¤пє†пє›пє®Ш§ШЄ Ш§п»џпєЁп»„п»ґпє” Ш§п»џп»¤пє�п»ЊпєЋп»ЈпєЄШ© п»“п»І п»«пє¬Ш§ Ш§п»џп»”п»ЂпєЋШЎвЂЄ ШЊвЂ¬п»›п»¤пєЋ пє‘п»ґп»ЁпєЋ Щ€пєџп»®ШЇ п»‹п»јп»—пє” пє‘п»ґп»¦ п»§п»€п»ґп»ў п»«пє¬Ш§ Ш§п»џп»¤пє†пє›пє® Щ€ пє‘п»ґп»¦ Ш§п»џп»„п»ґп»’вЂ¬
вЂ«Ш§п»џп»¤п»Ёп»”пєјп»ћ п»џп»¬пє¬Ш§ Ш§п»џп»¤пє†пє›пє®вЂЄ.вЂ¬пєЈпєЋЩ€п»џп»ЁпєЋ Ш§п»џпєЄпє§п»®Щ„ ШҐп»џп»° п»§п»€пє®п»іпє” Ш§п»џп»„п»ґп»’вЂЄ.вЂ¬вЂ¬
вЂ«Ш§п»‹пє�п»�пєЄ ШЈЩ† п»џп» пє’пє¤пєљ ШЈп»«п»¤п»ґпє” п»“п»І Ш§п»»п»џп»њпє�пє®Щ€п»§п»ґпєЋШЄ Щ€ п»“п»І п»§п»€пє®п»іпє” Ш§п»џп»њп»ўвЂЄ.вЂ¬вЂ¬
вЂ«Ш§п»џп»њп» п»¤пєЋШЄ Ш°Щ€Ш§ШЄ Ш§п»џпєЄп»»п»џпє” )вЂЄ(KeywordsвЂ¬вЂ¬
вЂ«п»Јпє†пє›пє® пє§п»„п»І вЂЄA,B..................................................................................вЂ¬вЂ¬
вЂ«п»Џп»јп»—пє” п»Јпє†пє›пє® пє§п»„п»ІвЂ¬
вЂ«вЂЄAвЂ¬вЂ¬
вЂ«вЂЄ:вЂ¬Щ€ п»«п»® ШЈпє»п»ђпє® пє§п»„п»І п»Јп»ђп» п»– п»іпє¤п»®ЩЉвЂ¬
вЂ«вЂЄAвЂ¬вЂ¬
вЂ«вЂЄA .....................................вЂ¬вЂ¬
вЂ«п»Јпє†пє›пє® пє§п»„п»І Щ€ п»Јпє¤пєЄЩ€ШЇ п»іп»Њпє�пє’пє® Ш§п»џпє пє°ШЎ Ш§п»·Щ€Щ„ п»Јп»¦ вЉҐвЂЄAD .................................. A = AB + AвЂ¬вЂ¬
вЂ«вЉҐвЂЄAвЂ¬вЂ¬
вЂ«п»Јпє†пє›пє® пє§п»„п»І п»Џп»ґпє® п»Јпє¤пєЄЩ€ШЇ ШҐЩ† п»џп»ў п»іп»њп»¦ п»Јп»ЊпєЄЩ€п»ЈпєЋЩ‹ п»«п»® Ш§п»џпє пє°ШЎ Ш§п»џпєњпєЋп»§п»І п»Јп»¦ вЉҐвЂЄA = AB + AвЂ¬вЂ¬
вЂ«Ш§п»џп»¤пє†пє›пє® Ш§п»џп»¤пє®Ш§п»“п»– п»џп» п»¤пє†пє›пє® вЂЄA* ....................................... ......... ................ AвЂ¬вЂ¬
вЂ«вЂЄв€’ 1вЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄ1вЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄв€’вЂ¬вЂ¬
вЂ«вЂЄпЈ¶вЂ¬вЂ¬
вЂ«вЂЄпЈ·вЂ¬вЂ¬
вЂ«вЂЄпЈёвЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄiвЂ¬вЂ¬
вЂ«вЂЄfвЂ¬вЂ¬
вЂ«в€‘вЂ¬
вЂ«вЂЄiв€€ IвЂ¬вЂ¬
вЂ«вЂЄпЈ« nвЂ¬вЂ¬
вЂ«вЂЄ2пЈ¶вЂ¬вЂ¬
вЂ«вЂЄcn = пЈ¬ в€‘ f i пЈ·вЂ¬вЂ¬
вЂ«вЂЄпЈ i =1вЂ¬вЂ¬
вЂ«вЂЄпЈёвЂ¬вЂ¬
вЂ«вЂЄ,вЂ¬вЂ¬
вЂ«вЂЄпЈ«вЂ¬вЂ¬
вЂ«вЂЄC := пЈ¬вЂ¬вЂ¬
вЂ«вЂЄпЈвЂ¬вЂ¬
вЂ«вЂЄ,вЂ¬вЂ¬
вЂ«вЂЄ1вЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄв€’вЂ¬вЂ¬
вЂ«вЂЄ1вЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄв€’вЂ¬вЂ¬
вЂ«вЂЄпЈ« в€ћ 1 пЈ¶вЂ¬вЂ¬
вЂ«вЂЄCr = пЈ¬ в€‘ 2 пЈ·вЂ¬вЂ¬
вЂ«вЂЄпЈ i =1 r пЈёвЂ¬вЂ¬
вЂ«вЂЄ,вЂ¬вЂ¬
вЂ«вЂЄ1вЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄпЈ« в€ћ 1 пЈ¶вЂ¬вЂ¬
вЂ«вЂЄпЈ« 1пЈ¶вЂ¬вЂ¬
вЂ«вЂЄпЈ·вЂ¬вЂ¬
вЂ«в€‘ вЂЄ, Cp = пЈ¬ в€‘ 2 p пЈ· C ! = пЈ¬пЈ¬вЂ¬вЂ¬
вЂ«вЂЄ2 пЈ·вЂ¬вЂ¬
вЂ«(вЂ¬
вЂ«)вЂ¬
вЂ«вЂЄiвЂ¬вЂ¬
вЂ«!вЂ¬
вЂ«вЂЄпЈ i =1вЂ¬вЂ¬
вЂ«вЂЄпЈёвЂ¬вЂ¬
вЂ«вЂЄпЈ i =1 i пЈёвЂ¬вЂ¬
вЂ«вЂЄв€’вЂ¬вЂ¬
вЂ«в€ћвЂ¬
вЂ«п»Јпє п»¤п»®п»‹пє” пє—п»Њпє®п»іп»’ Ш§п»џп»¤пє†пє›пє® вЂЄD( A) ................................................................. AвЂ¬вЂ¬
вЂ«п»Јпє†пє›пє® пє—п»”пєЋпєїп» п»І вЂЄв€‚ ................................................................................вЂ¬вЂ¬
вЂ«вЂЄ22вЂ¬вЂ¬
вЂ«пє—п»„пє’п»ґп»�пєЋШЄ п»“п»І п»“п»ЂпєЋШЎШ§ШЄ п»«п» пє’пє®ШЄ п»џп» п»¤пє†пє›пє®Ш§ШЄ Ш§п»џпєЁп»„п»ґпє” п»Џп»ґпє® Ш§п»џп»¤пє¤пєЄЩ€ШЇШ©вЂ¬
A в€€ H 2 (H )
:вЂ«п»§п»њпє�пєђвЂ¬
2
A в€’ О»I
2
= Оµ Ih
2
ОµI
=
H
( A в€’ О» )h
=
H
2
h
#
2
2
h
H
2
Оµ
=
H
2
= Оµ
H
2
h
#
h в€€ D( A) вЂ«( п»џп»њп»ћвЂ¬2
вЂ«Щ€вЂ¬
H
2
Оµ
>
2
h
h
4
2
H
ПЂ
пЈ±
пЈј
2
[
]
H = L пЈ®в€’ ПЂ ,ПЂ пЈ№ = пЈІ f : в€’ ПЂ ,ПЂ в†’ C, measurable, в€« f ( x) dx < в€ћпЈЅ вЂ« п»џп»ґп»њп»¦вЂ¬.4вЂ«п»ЈпєњпєЋЩ„вЂ¬
в€’ПЂ
пЈі
пЈѕ
2
в€‚ f (x ) =
:вЂ«пє‘пєЋп»‹пє�пє’пєЋШ±вЂ¬
df ( x ) :вЂ« в€‚ пє‘пєЋп»‹пє�пє’пєЋШ±вЂ¬: D(в€‚ ) вЉ‚ L2 ([в€’ ПЂ , ПЂ ]) в†’ L2 ([в€’ ПЂ , ПЂ ]) : вЂ«п»§п»Њпє®ЩЃ Ш§п»џп»¤пє†пє›пє®вЂ¬
dx
M : L2 ([в€’ ПЂ , ПЂ ]) в†’ L2 ([в€’ ПЂ , ПЂ ]) :
Щ‹вЂ«ШЈп»іп»ЂпєЋвЂ¬
вЂ« п»›пєњп»ґп»”пє”вЂ¬D(в€‚) = C1([в€’ ПЂ ,ПЂ ]) вЂ« Щ€вЂ¬L
2
2
A! =C
2
!
2
H
2
!
C
([в€’ ПЂ , ПЂ ])
в€‘
iв€€Z
Aei
i!
2
в€‘
jв€€Z
пЈ« в€ћ Ae
= C пЈ¬в€‘ i
пЈ¬ i =1 i!
пЈ
2
2
i!
H
пЈ« eijx
пЈ¬пЈ¬
d
ПЂ
2ПЂ
= в€« пЈ
dx
в€’ПЂ
в€‚ej
eijx
, j в€€ Z = {0,В±1,В±2,...}
2ПЂ
2
:вЂ« п»§пєґпє�п»Њп»¤п»ћ Ш§п»џп»Ёп»€п»ґп»ў Ш§п»џп»¤п»Њп»„п»° п»“п»ІвЂ¬. L ([в€’ ПЂ , ПЂ ]) вЂ«п»“п»ІвЂ¬
H ! (H ) вЂ«( п»“п»І Ш§п»џп»”п»ЂпєЋШЎвЂ¬22 )
H
в€ћ
+в€‘
i =1
Aei
i!
пЈ¶ 2
1
пЈ·, Ci! =
в€ћ
пЈ·
1
HпЈё
2в€‘ 2
i =1 (i!)
2
2
пЈ¶
пЈ·пЈ·
пЈё dx
2
2
:вЂ« = ! в€‚ п»џпє¬Ш§ п»§п»њпє�пєђвЂ¬C
ПЂ
2
!
в€‘
в€‚e j
j!
jв€€Z
M
!
=C
в€‘
jв€€Z
2
:вЂ«Щ€ п»Јп»Ёп»Є п»§пє пєЄвЂ¬
H
2
ij dx
j2
j2
2
= C в€‘ в€« 2 = C в€‘ 2 = 2C! в€‘ 2 < в€ћ.
j! H
jв€€Z в€’ПЂ ( j!) ПЂ
jв€€Z ( j!)
jв€€Z ( j!)
2
!
2
!
2
2
!
вЂ«Щ€вЂ¬
вЂ« п»—пєЋп»‹пєЄШ© п»“п»ІвЂ¬e j (x) =
2
. M , в€‚ ! = ? вЂ« пє›п»ў п»§пє¤пєґпєђвЂ¬M
2
вЂ«п»§п»Њпє®ЩЃвЂ¬
{ei}iв€€Z вЂ« ШҐп»§п»Є п»Јп»Њпє®Щ€ЩЃвЂ¬. x в€€ [в€’ ПЂ ,ПЂ ] вЂ«( п»Јп»¦ ШЈпєџп»ћ п»›п»ћвЂ¬Mf )(x ) = xf (x )
:вЂ«пє‘пєЋп»‹пє�пє’пєЋШ±вЂ¬
в€‚e j
вЂ«Ш§п»џп»¤пє†пє›пє®вЂ¬
Me j
j!
2
ПЂ
=C
2
!
H
xeijx dx
в€‘ в€« ( j!) ПЂ
2
j в€€Z в€’ ПЂ
21
C
= C2
!
!
ПЂ
= ? вЂ«п»§пє¤пєґпєђ Ш§п»µЩ†вЂ¬
x 2 dx
в€‘
2
в€«
jв€€Z в€’ ПЂ 2ПЂ ( j!)
вЂ«п»‹пє п»®Ш± Щ€ пє‘п»ґп»њпє®ШЇвЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«п»§пє’пє®п»«п»¦ Ш§п»µЩ†вЂЄ= 1,в€Ђi, j в€€ N :вЂ¬вЂ¬
вЂ«)в€— в€— в€—(вЂ¬
вЂ«вЂЄ EiejвЂ¬п»“п»ІвЂ¬
вЂ«вЂЄHвЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«пє‘пєЋп»»пєіпє�п»”пєЋШЇШ© п»Јп»¦вЂ¬
вЂ«вЂЄHвЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄ 1 = E i e1вЂ¬п»§пє пєЄвЂЄ:вЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄHвЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄHвЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄ+ f2вЂ¬вЂ¬
вЂ«вЂЄHвЂ¬вЂ¬
вЂ«п»џпє’пє®п»«пєЋЩ† вЂЄ= 1, в€Ђi, n в€€ NвЂ¬вЂ¬
вЂ«(вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«Щ€ п»›пє¬п»џп»љвЂ¬
вЂ«п»Јп»¦вЂ¬
вЂ«вЂЄ#вЂ¬вЂ¬
вЂ«вЂЄ#вЂ¬вЂ¬
вЂ«вЂЄHвЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄEi e1вЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄ= f1вЂ¬вЂ¬
вЂ«вЂЄfiвЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«в€‘вЂ¬
вЂ«вЂЄi =1вЂ¬вЂ¬
вЂ«Щ€ пє‘пєЋп»»пєіпє�п»”пєЋШЇШ© п»Јп»¦ Ш§п»џп»¤пєґпєЋЩ€Ш§Ш© )в€— в€— в€—( п»§пє пєЄвЂ¬
вЂ«вЂЄnвЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄ Ei enвЂ¬п»џпєЄп»іп»ЁпєЋ вЂЄ1 = Ei Hn = cn2 в€‘ f j Ejej :вЂ¬вЂ¬
вЂ«вЂЄHвЂ¬вЂ¬
вЂ«вЂЄ#вЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄHвЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄij=11вЂ¬вЂ¬
вЂ«вЂЄn в€’1вЂ¬вЂ¬
вЂ«вЂЄj =1вЂ¬вЂ¬
вЂ«Ш§п»џп»¤пєґпєЋЩ€п»іпє�п»ґп»¦вЂ¬
вЂ«(вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«Щ€ Ш°п»џп»љ п»Јп»¬п»¤пєЋ п»іп»њп»¦ вЂЄ. i, j в€€ NвЂ¬вЂ¬
вЂ«вЂЄ= cn2в€’1 в€‘ f j Ei e jвЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄEi e1, Eje1вЂ¬вЂ¬
вЂ«вЂЄ.вЂ¬вЂ¬
вЂ«вЂЄ 1 = Ei e2вЂ¬Щ€ п»›пє¬п»џп»љ п»§п»њпє�пєђ п»“п»І ) вЂЄH 2 (H2вЂ¬вЂ¬
вЂ«вЂЄ0 =c22 f1 Eie1,Eje1 + f2 Eie2,Eje2вЂ¬вЂ¬
вЂ«вЂЄ0 = Ei e2 , E j e2вЂ¬вЂ¬
вЂ«вЂЄHвЂ¬вЂ¬
вЂ«вЂЄHвЂ¬вЂ¬
вЂ«вЂЄ 1 = c2 f1 Eie1 H + f2 Eie2вЂ¬п»џпє¬Ш§ п»§п»њпє�пєђвЂЄ:вЂ¬вЂ¬
вЂ«вЂЄE i e2вЂ¬вЂ¬
вЂ«вЂЄHвЂ¬вЂ¬
вЂ«вЂЄHвЂ¬вЂ¬
вЂ«вЂЄ . HвЂ¬п»§пє®Щ‰ ШЈЩ†вЂ¬
вЂ«вЂЄ= 0, в€Ђi, j в€€ N &i в‰ j & k = 1вЂ¬вЂ¬
вЂ«Щ€ п»›пє¬п»џп»љ п»“п»І ) вЂЄ H (HвЂ¬п»џпєЄп»іп»ЁпєЋ вЂЄ) :вЂ¬вЂ¬
вЂ«)вЂ¬
вЂ«) вЂЄ(H 1вЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄE ie1вЂ¬вЂ¬
вЂ«= вЂЄ 1вЂ¬Щ€вЂ¬
вЂ«п»§пє¤пєјп»ћвЂ¬
вЂ«Ш§п»џпєґпєЋпє‘п»�пє�п»ґп»¦вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄ#вЂ¬вЂ¬
вЂ«вЂЄ1 = Ei H n в€’1вЂ¬вЂ¬
вЂ«вЂЄ= 1, в€Ђ i , n в€€ NвЂ¬вЂ¬
вЂ«п»‹п» п»°вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄHвЂ¬вЂ¬
вЂ«вЂЄ. E i enвЂ¬вЂ¬
вЂ«вЂЄ 0 = Eien , EjenвЂ¬п»Јп»¬п»¤пєЋ п»іп»њп»¦ вЂЄ n в€€ N :вЂ¬Щ€ вЂЄ . в€Ђ i, j в€€ N & i в‰ jвЂ¬п»§п»њпє®Ш± Ш§п»џп»њпє�пєЋпє‘пє”вЂЄ:вЂ¬вЂ¬
вЂ«вЂЄEiek , EjekвЂ¬вЂ¬
вЂ«вЂЄnвЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄkвЂ¬вЂ¬
вЂ«вЂЄв€‘fвЂ¬вЂ¬
вЂ«вЂЄk =1вЂ¬вЂ¬
вЂ«вЂЄ 0 = cnвЂ¬Щ€вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄ#вЂ¬вЂ¬
вЂ«вЂЄE i ek , E j ekвЂ¬вЂ¬
вЂ«вЂЄn в€’1вЂ¬вЂ¬
вЂ«вЂЄ0 = c n2 в€’1 в€‘ f kвЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄk =1вЂ¬вЂ¬
вЂ«п»Јп»¦ Ш§п»џп»¤пєґпєЋЩ€п»іпє�п»ґп»¦ п»§пє¤пєјп»ћ п»‹п» п»° Ш§п»џп»¤п»„п» п»®ШЁвЂЄ.вЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«п»§п»€пє®п»іпє” Ш§п»џп»„п»ґп»’вЂЄ .вЂ¬п»§п»”пє®Ш¶ вЂЄ {ei }i =1вЂ¬п»—пєЋп»‹пєЄШ© п»“п»І п»“п»ЂпєЋШЎ п»«п» пє’пє®ШЄ вЂЄ. Aв€€H (H) ШЊ HвЂ¬вЂ¬
вЂ«в€ћвЂ¬
вЂ«вЂЄ (1вЂ¬ШҐШ°Ш§ п»›пєЋЩ† вЂЄ Aei = О»ei , i = 1,2,... :вЂ¬п»“пє€Щ† вЂЄA # = О»вЂ¬вЂ¬
вЂ«вЂЄ (2вЂ¬ШҐШ°Ш§ п»›пєЋЩ† )вЂЄh в€€ D( AвЂ¬вЂ¬
вЂ«вЂЄHвЂ¬вЂ¬
вЂ«вЂЄhвЂ¬вЂ¬
вЂ«вЂЄ Ah = (О» + Оµ )h ,вЂ¬Щ€ вЂЄ Оµ > 0вЂ¬п»“пє€Щ†вЂЄ:вЂ¬вЂ¬
вЂ«вЂЄОµвЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«>вЂ¬
вЂ«вЂЄHвЂ¬вЂ¬
вЂ«вЂЄв€’ О» )hвЂ¬вЂ¬
вЂ«вЂЄ(AвЂ¬вЂ¬
вЂ«=вЂ¬
вЂ«вЂЄHвЂ¬вЂ¬
вЂ«вЂЄhвЂ¬вЂ¬
вЂ«вЂЄ#вЂ¬вЂ¬
вЂ«вЂЄA в€’ О»IвЂ¬вЂ¬
вЂ«Ш§п»џпє’пє®п»«пєЋЩ†вЂЄ (1 .вЂ¬ШҐШ°Ш§ п»›пєЋЩ† вЂЄ Aei = О»ei , i = 1,2,... :вЂ¬п»“пє€Щ† вЂЄ= О» , i = 1,2,...вЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄfiвЂ¬вЂ¬
вЂ«в€ћвЂ¬
вЂ«в€‘вЂ¬
вЂ«вЂЄi =1вЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄ= О»вЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄHвЂ¬вЂ¬
вЂ«вЂЄAe iвЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄfiвЂ¬вЂ¬
вЂ«в€ћвЂ¬
вЂ«в€‘вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«Щ€ п»Јп»Ёп»Є п»§пє пєЄвЂЄ:вЂ¬вЂ¬
вЂ«вЂЄi =1вЂ¬вЂ¬
вЂ«вЂЄ20вЂ¬вЂ¬
вЂ«вЂЄ= О»вЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄ#вЂ¬вЂ¬
вЂ«вЂЄ. AвЂ¬вЂ¬
вЂ«вЂЄHвЂ¬вЂ¬
вЂ«вЂЄAeiвЂ¬вЂ¬
вЂ«п»§пє’пє®п»«п»¦вЂ¬
вЂ«Ш§п»µЩ†вЂ¬
вЂ«пє—п»„пє’п»ґп»�пєЋШЄ п»“п»І п»“п»ЂпєЋШЎШ§ШЄ п»«п» пє’пє®ШЄ п»џп» п»¤пє†пє›пє®Ш§ШЄ Ш§п»џпєЁп»„п»ґпє” п»Џп»ґпє® Ш§п»џп»¤пє¤пєЄЩ€ШЇШ©вЂ¬
= B в€— o A, I B
A, B
=C
#
2
в€ћ
в€‘
2
fi
i =1
в€ћ
.
#
= C 2 в€‘ fi
2
i =1
Ae i , Be i
I Aei , Aв€— o Bei
= I A , Aв€— o B
в€ћ
Ei , E j
= C2 в€‘ fk
#
ШЊ H 2 (H ) вЂ«п»—пєЋп»‹пєЄШ© п»“п»ІвЂ¬
вЂ«Щ€вЂ¬nв€€ N
:
#
2
ik =1
= E в€—j o Ei , I j
#
H
.
E в€—j o Ei ek , I j ek
H
.
.вЂ«( пє—пє’пє®п»«п»¦ пє‘п»Ёп»”пєІ Ш§п»џп»„пє®п»іп»�пє”вЂ¬27 ) вЂ«( Щ€вЂ¬26 ) вЂ«ШҐЩ† Ш§п»џп»¤пєґпєЋЩ€Ш§Ш©вЂ¬
{Ei }iв€ћ=1 вЉ‚ H 2 (H )
вЂ«п»Јп»¬п»¤пєЋ п»іп»њп»¦вЂ¬
H
{ei }iв€ћ=1
вЂ« Щ€вЂ¬H вЂ«п»—пєЋп»‹пєЄШ© п»“п»І п»“п»ЂпєЋШЎ п»«п» пє’пє®ШЄвЂ¬
0 = Ei en , E j en
Eie j
вЂ«Щ€вЂ¬
#
H
.5вЂ«п»§п»€пє®п»іпє”вЂ¬
= 1, в€Ђi, j в€€ N :вЂ«п»‹п»ЁпєЄпє‹пє¬вЂ¬
. в€Ђ i, j в€€ N & i в‰ j
{
}
:Щ‹вЂ« Щ€ п»§п»Њпє®ЩЃ Ш§п»џп»”п»ЂпєЋШЎ ШЈпє‘п»ЂпєЋвЂ¬Hn := hn = ( x1, x2 ,...xn ,0,0...) : h = {xi }i =1 вЂ« п»§п»Њпє®ЩЃвЂ¬.вЂ«Ш§п»џпє’пє®п»«пєЋЩ†вЂ¬
в€ћ
пЈ±
H 2 (Hn ) = пЈІAH n : A в€€ H 2 (H ) : AH n
пЈі
пЈ« n
2пЈ¶
cn2 = пЈ¬ в€‘ f i пЈ·
пЈ i =1
пЈё
H
2
(H 1 ) вЉ‚
H
(H 2 ) вЉ‚
#
n
2
2 пЈј
= cn2 в€‘ fi Aei H пЈЅ
i =1
пЈѕ
1
2
H1 вЉ‚ H2 вЉ‚ ... вЉ‚ Hn вЉ‚ ...H
вЂ«Щ€вЂ¬
2
в€’
2
... H
2
(H n ) вЉ‚
... H
2
(H )
:вЂ«п»§пє®Щ‰ ШЈЩ†вЂ¬
:вЂ«Щ€ п»§п»јпєЈп»†вЂ¬
2
n
пЈ«в€ћ 2
2
2 пЈ¶
limпЈ¬в€‘ fi Aei в€’ в€‘ fi Aei H пЈ· = 0 : вЂ« п»·Щ†вЂ¬lim AHn
пЈ·
nв†’в€ћпЈ¬
nв†’в€ћ
i =1
H
пЈё
пЈ i=1
2
. lim cn f i
n в†’в€ћ
19
2
AHn
2
H
2
#
= A#
= A # :вЂ«Щ€ п»«пє¬Ш§ п»іпє†ШЇЩЉ ШҐп»џп»°вЂ¬
вЂ«п»‹пє п»®Ш± Щ€ пє‘п»ґп»њпє®ШЇвЂ¬
1
вЂ« в€€ в€‚ п»·п»§п»Є п»Јп»ђп» п»– Щ€ п»§п»€п»ґп»¤п»Є п»Јпє¤пєЄЩ€ШЇ п»Јп»¦ ШЈпєџп»ћвЂ¬H p (H ) вЂ«п»‹п»ЁпєЄпє‹пє¬вЂ¬
2
. H вЂ«п»“п»ІвЂ¬
1
:вЂ« п»‹п»ЁпєЄпє‹пє¬ п»§пє пєЄ ШЈЩ†вЂ¬fn ( x) = sinnx, x в€€ [в€’ ПЂ ,ПЂ ], n = 1,2,... вЂ«п»џп»ґп»њп»¦вЂ¬
n
вЂ« Щ€ п»Јпє п»¤п»®п»‹пє” пє—п»Њпє®п»іп»”п»Є п»›пєњп»ґп»’вЂ¬p в€’ 1 >
f n в€€ D(в€‚), n в€€ N
. fn
2
ПЂ
ПЂ
1
= в€« sin x dx = 2 вЂ«п»Јп»®пєїпє¤п»ґп»¦ ШЈЩ†вЂ¬
n
n
в€’ПЂ
2
L2
:вЂ«Щ€ п»§пє®Щ‰ пє‘пєґп»¬п»®п»џпє”вЂ¬
fn в€’ 0
L2
=
fn
ПЂ
=
L2
n
в†’в€ћ
пЈ§ nпЈ§
пЈ§в†’ 0
:вЂ«Щ€ п»џп»њп»¦вЂ¬
ПЂ
2
в€‚f n
L
2
в€«ПЂ
=
в€’
2
d 1
sin nx dx =
dx n
ПЂ
в€«ПЂ cos nx
2
dx = ПЂ в‰ 0
в€’
: вЂ«Щ€ п»Јп»Ёп»Є п»іп»Ёпє�пєћвЂ¬
. в€‚ вЂ« Щ€ п»«пє¬Ш§ п»іп»Њп»Ёп»І п»‹пєЄЩ… Ш§пє—пєјпєЋЩ„ Ш§п»џп»¤пє†пє›пє®вЂ¬.
вЂ«п»—пєЋп»‹пєЄШ© п»“п»ІвЂ¬
{Ei }iв€ћ=1 вЉ‚ H 2 (H )
A, B
в†’в€ћ
в€‚fn в€’0 L2 пЈ§nпЈ§в†’
пЈ§ ПЂ в‰ 0 вЂ« Щ€ п»џп»њп»¦вЂ¬f n пЈ§пЈ§H в†’ 0 , n в†’ в€ћ
ШЊ H вЂ«п»—пєЋп»‹пєЄШ© п»“п»І п»“п»ЂпєЋШЎ п»«п» пє’пє®ШЄвЂ¬
вЂ« п»§п»”пє®Ш¶вЂ¬.вЂ«п»§п»€пє®п»іпє” пє—п»¤п»¬п»ґпєЄп»іпє”вЂ¬
: вЂ« п»‹п»ЁпєЄпє‹пє¬вЂ¬A, B в€€ H 2 (H ) вЂ« Щ€вЂ¬H 2 (H )
= B в€— o A, I B
#
{ei }iв€ћ=1
#
= I A , A* o B
(25)
#
I A : D ( A) в†’ H , I A h = h
0 = Ei , E j
#
1 = Ei , , E i
= E в€—j o Ei , I j
$
#
= I i , Eiв€— o E j , i в‰ j
#
= I i , Eiв€— o Ei
#
= Eiв€— o Ei , I i
#
Ii : D(Ii ) в†’ H, Iih = h, I j : D(I j ) в†’ H, I jh = h,i, j = 1,2,...
:вЂ«пє‘пєЋп»‹пє�пє’пєЋШ±вЂ¬
(26)
(27 )
вЂ«пє‘пєЋп»‹пє�пє’пєЋШ±вЂ¬
: (8) вЂ«( п»§пєґпє�п»Њп»¤п»ћ Ш§п»џпє�п»Њпє®п»іп»’вЂ¬25) вЂ« п»џпє’пє®п»«пєЋЩ†вЂ¬.вЂ«Ш§п»џпє’пє®п»«пєЋЩ†вЂ¬
A, B
в€ћ
#
= C 2 в€‘ fi
2
i =1
в€ћ
= C 2 в€‘ fi
i =1
Aei , Bei
2
H
B в€— o Aei , I B ei
H
18
вЂ«пє—п»„пє’п»ґп»�пєЋШЄ п»“п»І п»“п»ЂпєЋШЎШ§ШЄ п»«п» пє’пє®ШЄ п»џп» п»¤пє†пє›пє®Ш§ШЄ Ш§п»џпєЁп»„п»ґпє” п»Џп»ґпє® Ш§п»џп»¤пє¤пєЄЩ€ШЇШ©вЂ¬
вЂ«п»§п»€пє®п»іпє”вЂЄ .4вЂ¬п»§п»”пє®Ш¶ вЂЄ {ei }i =1вЂ¬п»—пєЋп»‹пєЄШ© п»“п»І п»“п»ЂпєЋШЎ п»«п» пє’пє®ШЄвЂ¬
вЂ«в€ћвЂ¬
вЂ«вЂЄA в€€ H 2 (H ) ШЊ HвЂ¬вЂ¬
вЂ«Щ€ п»«пє¬Ш§ п»іп»Њп»„п»ІвЂЄ :вЂ¬п»іп»®пєџпєЄвЂ¬
вЂ«вЂЄ . в€Ђh в€€ D ( A) ШЊ Ah H в†’ 0, h в†’ 0,в‡’ A # < M ШЊ M > 0вЂ¬Щ€ п»џп»њп»¦ Ш§п»»п»—пє�п»ЂпєЋШЎ Ш§п»џп»Њп»њпєґп»І п»Џп»ґпє®вЂ¬
вЂ«п»Јпє¤п»�п»–вЂЄ.вЂ¬вЂ¬
вЂ«Ш§п»џпє’пє®п»«пєЋЩ†вЂЄ .вЂ¬п»џп»ґп»њп»¦ ) вЂЄ A в€€ H (HвЂ¬Щ€ )вЂЄ h в€€ D( AвЂ¬пє‘пєЋп»‹пє�пє’пєЋШ±вЂЄв†’ 0 , for , h в†’ 0 :вЂ¬вЂ¬
вЂ«вЂЄHвЂ¬вЂ¬
вЂ«вЂЄ AhвЂ¬п»Јп»¦ ШЈпєџп»ћ п»›п»ћ вЂЄ< ОґвЂ¬вЂ¬
вЂ«вЂЄHвЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«п»Јп»¦ ШЈпєџп»ћ п»›п»ћ вЂЄ ШЊ в€€> 0вЂ¬п»іп»®пєџпєЄ вЂЄ Оґ (в€€) > 0вЂ¬пє‘пєЋп»‹пє�пє’пєЋШ±вЂЄ<в€€ :вЂ¬вЂ¬
вЂ«ШЈЩ† п»§п»њпє�пєђвЂЄв‰¤ M f i :вЂ¬вЂ¬
вЂ«вЂЄHвЂ¬вЂ¬
вЂ«вЂЄ. hвЂ¬п»џпє¬Ш§ п»іп»¤п»њп»¦вЂ¬
вЂ«вЂЄ f i AeiвЂ¬Щ€ п»Јп»Ёп»Є п»§п»њпє�пєђвЂЄ:вЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«Щ€ пєЈпєґпєђ пє—п»Њпє®п»іп»’ Ш§п»џп»Ёп»€п»ґп»ўвЂЄ:вЂ¬вЂ¬
вЂ«вЂЄHвЂ¬вЂ¬
вЂ«вЂЄAhвЂ¬вЂ¬
вЂ«Щ€ п»«п»њпє¬Ш§вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄfiвЂ¬вЂ¬
вЂ«вЂЄв‰¤ MвЂ¬вЂ¬
вЂ«в€ћвЂ¬
вЂ«в€‘вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄв‰¤MвЂ¬вЂ¬
вЂ«вЂЄi =1вЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄ#вЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄHвЂ¬вЂ¬
вЂ«вЂЄf i Ae iвЂ¬вЂ¬
вЂ«в€ћвЂ¬
вЂ«в€‘вЂ¬
вЂ«вЂЄi =1вЂ¬вЂ¬
вЂ«вЂЄ. AвЂ¬вЂ¬
вЂ«п»џп»њп»¦ Ш§п»џп»Њп»њпєІ п»Џп»ґпє® пє»пє¤п»ґпєў п»џп»¬пє¬Ш§ п»іп»њп»”п»І пє—п»�пєЄп»іп»ў Ш§п»џп»¤пєњпєЋЩ„ Ш§п»џпє�пєЋп»џп»ІвЂЄ:вЂ¬вЂ¬
вЂ«вЂЄПЂвЂ¬вЂ¬
вЂ«вЂЄпЈ±вЂ¬вЂ¬
вЂ«вЂЄпЈјвЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«п»ЈпєњпєЋЩ„вЂЄ .3вЂ¬п»џп»ґп»њп»¦вЂЄ, в€« f (t ) dt < в€ћпЈЅ :вЂ¬вЂ¬
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вЂ«в€ћвЂ¬
вЂ«в€‘вЂ¬
вЂ«вЂЄi =1вЂ¬вЂ¬
вЂ«вЂЄ=CвЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄpвЂ¬вЂ¬
вЂ«вЂЄTвЂ¬вЂ¬
вЂ«ШҐШ°Ш§ п»‹пє®п»“п»ЁпєЋвЂ¬
вЂ«вЂЄA : l2 в†’ l2 , B : l2 в†’ l2вЂ¬вЂ¬
вЂ«)вЂЄA(x1,x2,..) =(x1,2x2,...nxn,...),B(x1,x2,...) =(0,0,...,0,(n+1)xn+1,(n+2))xn+2,...вЂ¬вЂ¬
вЂ«Щ€ п»Јп»Ёп»Є п»§п»њпє�пєђвЂ¬
вЂ«вЂЄ . T = A вЉ• BвЂ¬ШҐШ°Ш§ п»‹пє®п»“п»ЁпєЋ ШЈп»іп»ЂпєЋ вЂЄ {T i }i = 1вЂ¬пє‘пєЋп»‹пє�пє’пєЋШ±вЂЄ:вЂ¬вЂ¬
вЂ«в€ћвЂ¬
вЂ«п»‹п»ЁпєЄпє‹пє¬ п»Јп»¦ ШЈпєџп»ћ вЂЄi < jвЂ¬вЂ¬
вЂ«вЂЄ+вЂ¬вЂ¬
вЂ«вЂЄHвЂ¬вЂ¬
вЂ«вЂЄi2вЂ¬вЂ¬
вЂ«вЂЄ(0,0,...,xi ,...), i =1,2,..., p = 2вЂ¬вЂ¬
вЂ«вЂЄC2вЂ¬вЂ¬
вЂ«вЂЄTiei TjejвЂ¬вЂ¬
вЂ«вЂЄ,вЂ¬вЂ¬
вЂ«вЂЄip ipвЂ¬вЂ¬
вЂ«вЂЄ+ ... +вЂ¬вЂ¬
вЂ«вЂЄHвЂ¬вЂ¬
вЂ«вЂЄTie2 Tje2вЂ¬вЂ¬
вЂ«вЂЄ,вЂ¬вЂ¬
вЂ«вЂЄ2p 2pвЂ¬вЂ¬
вЂ«вЂЄ12вЂ¬вЂ¬
вЂ«= )вЂЄTi (x1, x2 ,...вЂ¬вЂ¬
вЂ«вЂЄ+вЂ¬вЂ¬
вЂ«вЂЄпЈ« T e Tj e1вЂ¬вЂ¬
вЂ«вЂЄ= Cp2 пЈ¬ i p1 , pвЂ¬вЂ¬
вЂ«вЂЄpвЂ¬вЂ¬
вЂ«вЂЄпЈ¬ 1 1вЂ¬вЂ¬
вЂ«вЂЄпЈвЂ¬вЂ¬
вЂ«вЂЄпЈ¶вЂ¬вЂ¬
вЂ«вЂЄTiej TjejвЂ¬вЂ¬
вЂ«вЂЄ...+ p , pвЂ¬вЂ¬
вЂ«= вЂЄ+ ...пЈ·вЂ¬вЂ¬
вЂ«вЂЄпЈ·вЂ¬вЂ¬
вЂ«вЂЄjвЂ¬вЂ¬
вЂ«вЂЄj HвЂ¬вЂ¬
вЂ«вЂЄпЈёвЂ¬вЂ¬
вЂ«вЂЄTi ,TjвЂ¬вЂ¬
вЂ«вЂЄHвЂ¬вЂ¬
вЂ«пє—п»„пє’п»ґп»�пєЋШЄ п»“п»І п»“п»ЂпєЋШЎШ§ШЄ п»«п» пє’пє®ШЄ п»џп» п»¤пє†пє›пє®Ш§ШЄ Ш§п»џпєЁп»„п»ґпє” п»Џп»ґпє® Ш§п»џп»¤пє¤пєЄЩ€ШЇШ©вЂ¬
:вЂ«п»—пєЋп»‹пєЄШ© п»“п»ґп»ЄвЂ¬
{e i }i в€€ I
вЂ« п»“п»ЂпєЋШЎ п»«п» пє’пє®ШЄ Щ€вЂ¬H вЂ« п»џп»ґп»њп»¦вЂ¬.(2) вЂ«ШЈп»Јпєњп» пє” пє—п»„пє’п»ґп»�п»ґпє”вЂ¬
пЈ± A : D ( A ) вЉ‚ H в†’ H : It is a closed
пЈґ
в€ћ
H p (H ) := пЈІ
Aei
в€ћ
в€—
{
}
вЉ‚
such
that
e
D
A
:
:
в€‘
i i =1
пЈґ
ip
i =1
пЈі
( )
linear operator пЈј
пЈґ
2
пЈЅ (1
1
< в€ћ, p >
пЈґ
2
H
пЈѕ
:вЂ« пє‘пєЋп»џпєёп»њп»ћвЂ¬A, B в€€ H p (H ) вЂ«п»§п»Њпє®ЩЃ Ш§п»џпє пєЄШ§ШЎ Ш§п»џпєЄШ§пє§п» п»І п»џп»њп»ћ п»Јпє†пє›пє®п»іп»¦вЂ¬
A, B
p
в€ћ
пЈ« 1 пЈ¶
= C в€‘пЈ¬ 2 p пЈ· Aei , Bei
пЈё
i =1 пЈ i
2
p
в€’
H
пЈ«в€ћ 1пЈ¶
, Cp = пЈ¬ в€‘ 2 p пЈ·
пЈ i =1 i пЈё
1
2
(20)
пЈ± A : D (H ) вЉ‚ H в†’ H : A is a closed linear
пЈґ
2
в€ћ
H r (H ) := пЈІ
Aei
в€ћ
в€—
{
}
пЈґ such that : ei i =1 вЉ‚ D A : в€‘ r i , r > 1
i =1
H
пЈі
operator ,пЈј
пЈґ
пЈЅ (2
пЈґ
пЈѕ
:вЂ« пєіп»®ЩЃ п»іпєёп»њп»ћ п»“п»ЂпєЋШЎ п»«п» пє’пє®ШЄ п»Јпє®п»“п»�пєЋЩ‹ пє‘пєЋп»џпє пєЄШ§ШЎ Ш§п»џпєЄШ§пє§п» п»ІвЂ¬H r (H ) вЂ«п»‹п»ЁпєЄпє‹пє¬вЂ¬
( )
в€ћ
A, B r = Cr2 в€‘
i =1
Aei Bei
,
ri ri
пЈ«в€ћ 1пЈ¶
, Cr = пЈ¬в€‘ 2 пЈ·
пЈ i=1 r пЈё
H
в€’
1
2
(21)
operator пЈј
пЈ± A : D ( A) вЉ‚ H в†’ H : is a closed linear
пЈґ
пЈґ (3
2
в€ћ
H ! (H ) = пЈІ
пЈЅ
Aei
в€ћ
в€—
<в€ћ
пЈґ such that :{ei }i =1 вЉ‚ D A : в€‘ i!
пЈґ
i =1
H
пЈі
пЈѕ
:вЂ« пєіп»®ЩЃ п»іпєёп»њп»ћ п»“п»ЂпєЋШЎ п»«п» пє’пє®ШЄ п»Јп»�пє®Щ€п»§пєЋЩ‹ пє‘пєЋп»џпє пєЄШ§ШЎ Ш§п»џпєЄШ§пє§п» п»ІвЂ¬H ! (H ) вЂ«п»‹п»ЁпєЄпє‹пє¬вЂ¬
( )
A, B ! = C
2
!
в€ћ
в€‘
i =1
Aei Bei
,
i! i!
в€’1
2
пЈ«в€ћ 1 пЈ¶
,C! = пЈ¬пЈ¬ в€‘ 2 пЈ·пЈ·
H
пЈ i =1 (i!) пЈё
(22)
:вЂ«( пє‘пєЋп»џпєёп»њп»ћвЂ¬21) вЂ«( п»џп»® п»‹пє®п»“п»ЁпєЋ Ш§п»џпє пєЄШ§ШЎ Ш§п»џпєЄШ§пє§п» п»І п»“п»ІвЂ¬I :(4)вЂ«п»Јп»јпєЈп»€пє”вЂ¬
в€ћ
A, B r = в€‘
i =1
Aei Bei
,
ri ri
H
: вЂ«п»‹п»ЁпєЄпє‹пє¬вЂ¬
. H r (H ) = H S вЂ« п»іп»њп»®Щ† п»џпєЄп»іп»ЁпєЋвЂ¬r = 1 вЂ«( п»Јп»¦ ШЈпєџп»ћвЂ¬1
11
вЂ«п»‹пє п»®Ш± Щ€ пє‘п»ґп»њпє®ШЇвЂ¬
вЂ«)вЂЄпЈ± A h , h в€€ D(AвЂ¬вЂ¬
вЂ«вЂЄ:= пЈІ BвЂ¬вЂ¬
вЂ«) вЂЄпЈі0 , h в€€ H / D ( AвЂ¬вЂ¬
вЂ«Щ€ п»«п»њпє¬Ш§ п»§п»њпє�пєђвЂЄ:вЂ¬вЂ¬
вЂ«~вЂ¬
вЂ«вЂЄ= ABвЂ¬вЂ¬
вЂ«)вЂЄD( AвЂ¬вЂ¬
вЂ«вЂЄABвЂ¬вЂ¬
вЂ«``вЂ¬
вЂ«вЂЄABвЂ¬вЂ¬
вЂ«вЂЄ .вЂ¬п»џпє’пє®п»«пєЋЩ† )вЂЄ (14вЂ¬пє‘пєґп»¬п»®п»џпє” п»§пє пєЄ п»Јп»¦ ) вЂЄ (6вЂ¬ШЈЩ†вЂ¬
вЂ«) вЂЄA # в‰¤ A B , в€Ђ в€€ B(HвЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«Щ€ п»«пє¬Ш§ п»іп»Њп»„п»І вЂЄ:вЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄBвЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄA # в€’ AвЉҐ # в‰¤ ABвЂ¬вЂ¬
вЂ«вЂЄ{e i }i в€€ IвЂ¬вЂ¬
вЂ«п»§п»€пє®п»іпє” пє—п»¤п»¬п»ґпєЄп»іпє”вЂЄ .2вЂ¬п»џпє�п»њп»¦вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄ HвЂ¬Щ€)вЂ¬
вЂ«п»—пєЋп»‹пєЄШ© п»“п»ІвЂ¬
вЂ«в‰¤ вЂЄ.0вЂ¬вЂ¬
вЂ«(вЂ¬
вЂ«в‹…вЂЄ H (H ), в‹…,вЂ¬п»Јп»Њпє®п»“пє” п»›п»¤пєЋ пєіпє’п»–вЂЄ ШЊвЂ¬п»‹п»ЁпєЄпє‹пє¬вЂЄ:вЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄ#вЂ¬вЂ¬
вЂ«п»›п»ћ ) вЂЄ A в€€ H 2 (HвЂ¬п»іп»¤п»њп»¦ ШҐЩ† п»іп»Њпє’пє® п»‹п»Ёп»Є пє‘пєёп»њп»ћ Щ€пєЈп»ґпєЄ п»›п»¤пєЋ п»іп» п»ІвЂЄ:вЂ¬вЂ¬
вЂ«) вЂЄ{ai }iв€ћ=1 вЉ‚ l 2 (KвЂ¬вЂ¬
вЂ«)вЂЄ(15вЂ¬вЂ¬
вЂ«)вЂЄ(16вЂ¬вЂ¬
вЂ«вЂЄ= 1, в€Ђi = 1,2,3,...вЂ¬вЂ¬
вЂ«) вЂЄ(17вЂ¬вЂ¬
вЂ«вЂЄ=0вЂ¬вЂ¬
вЂ«вЂЄ#вЂ¬вЂ¬
вЂ«вЂЄ#вЂ¬вЂ¬
вЂ«вЂЄ, {Ai }i =1 вЉ‚ H 2 (H ),вЂ¬вЂ¬
вЂ«в€ћвЂ¬
вЂ«вЂЄiв‰ jвЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄaiвЂ¬вЂ¬
вЂ«)вЂЄ(19вЂ¬вЂ¬
вЂ«вЂЄi =1вЂ¬вЂ¬
вЂ«вЂЄAiвЂ¬вЂ¬
вЂ«вЂЄ: Ai , A jвЂ¬вЂ¬
вЂ«)вЂЄ(18вЂ¬вЂ¬
вЂ«в€ћвЂ¬
вЂ«вЂЄA = в€‘ ai AiвЂ¬вЂ¬
вЂ«в€ћвЂ¬
вЂ«в€‘вЂ¬
вЂ«=вЂ¬
вЂ«вЂЄi =1вЂ¬вЂ¬
вЂ«вЂЄв€Ђ i, j в€€ N ,вЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄ#вЂ¬вЂ¬
вЂ«вЂЄAвЂ¬вЂ¬
вЂ«вЂЄA, Ai # = ai , i = 1,2,3,...вЂ¬вЂ¬
вЂ«ШЈп»›пєњпє® п»Јп»¦ Ш°п»џп»љ ШҐШ°Ш§ п»›пєЋЩ†вЂ¬
вЂ«вЂЄIвЂ¬вЂ¬
вЂ«п»Јпє†пє›пє® пє§п»„п»І вЂЄIh = h, в€Ђ h в€€ H :вЂ¬вЂ¬
вЂ«п»‹п»ЁпєЄпє‹пє¬ вЂЄ= 1вЂ¬вЂ¬
вЂ«вЂЄ#вЂ¬вЂ¬
вЂ«вЂЄ. IвЂ¬вЂ¬
вЂ«Ш§п»џпє’пє®п»«пєЋЩ†вЂЄ .вЂ¬п»Јп»¦ ШЈпєџп»ћ Ш§п»џпє’пє®п»«пєЋЩ† Ш§п»§п»€пє® п»“п»І) ]вЂЄ [2вЂ¬Ш§п»џпєјп»”пє¤пє” ) вЂЄ .( ( p.87вЂ¬п»§пє’пє®п»«п»¦ ШЈЩ†вЂЄ= 1 :вЂ¬вЂ¬
вЂ«вЂЄ=1вЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«в€ћвЂ¬
вЂ«вЂЄ= C 2 в€‘ fiвЂ¬вЂ¬
вЂ«вЂЄi =1вЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«в€ћвЂ¬
вЂ«вЂЄf i Ie iвЂ¬вЂ¬
вЂ«вЂЄHвЂ¬вЂ¬
вЂ«вЂЄ10вЂ¬вЂ¬
вЂ«в€‘вЂ¬
вЂ«вЂЄi =1вЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄ= CвЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄ#вЂ¬вЂ¬
вЂ«вЂЄIвЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄ#вЂ¬вЂ¬
вЂ«вЂЄIвЂ¬вЂ¬
вЂ«пє—п»„пє’п»ґп»�пєЋШЄ п»“п»І п»“п»ЂпєЋШЎШ§ШЄ п»«п» пє’пє®ШЄ п»џп» п»¤пє†пє›пє®Ш§ШЄ Ш§п»џпєЁп»„п»ґпє” п»Џп»ґпє® Ш§п»џп»¤пє¤пєЄЩ€ШЇШ©вЂ¬
вЂ«Ш§п»џпє’пє®п»«пєЋЩ†вЂЄ .вЂ¬п»џпє�п»њп»¦ ) вЂЄ {A}n =1 вЉ‚ B (HвЂ¬п»Јпє�пє�пєЋп»џп»ґпє” п»Јпє�п»�пєЋШ±пє‘пє” п»“п»І ) вЂЄ H 2 (HвЂ¬ШҐп»џп»° вЂЄAвЂ¬вЂ¬
вЂ«в€ћвЂ¬
вЂ«ШЈЩ† ) вЂЄ : A в€€ B (HвЂ¬п»Јп»¦ ШЈпєџп»ћ ШЈЩЉвЂ¬
вЂ«)вЂЄh = в€‘ ai ei в€€ D( AвЂ¬вЂ¬
вЂ«п»“п»І ) вЂЄ(HвЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄ . HвЂ¬Щ€ п»џп»Ёпє’пє®п»«п»¦вЂ¬
вЂ«вЂЄiв€€IвЂ¬вЂ¬
вЂ«вЂЄAh H = ( A в€’ An )h + An h HвЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄв‰¤ ( A в€’ An )h H + Ah HвЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄhвЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄhвЂ¬вЂ¬
вЂ«вЂЄHвЂ¬вЂ¬
вЂ«вЂЄHвЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄBвЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄBвЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄ(A в€’вЂ¬вЂ¬
вЂ«вЂЄAn ei ) H + AnвЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄ+ AnвЂ¬вЂ¬
вЂ«вЂЄHвЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄaiвЂ¬вЂ¬
вЂ«в€‘вЂ¬
вЂ«в‰¤вЂ¬
вЂ«вЂЄiв€€ IвЂ¬вЂ¬
вЂ«) вЂЄв‹… max ( A в€’ An e iвЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄiвЂ¬вЂ¬
вЂ«вЂЄiв€€ IвЂ¬вЂ¬
вЂ«п»џп»њп»ћ ) вЂЄ( A) в€© D( AnвЂ¬вЂ¬
вЂ«вЂЄв€‘aвЂ¬вЂ¬
вЂ«вЂЄiв€€ IвЂ¬вЂ¬
вЂ«вЂЄ . h в€€ DвЂ¬ШҐШ°Щ†вЂЄ:вЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄHвЂ¬вЂ¬
вЂ«вЂЄhвЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄ+ AnвЂ¬вЂ¬
вЂ«вЂЄBвЂ¬вЂ¬
вЂ«вЂЄ.вЂ¬вЂ¬
вЂ«)вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄHвЂ¬вЂ¬
вЂ«вЂЄ3вЂ¬вЂ¬
вЂ«вЂЄiвЂ¬вЂ¬
вЂ«вЂЄ)в€‘ aвЂ¬вЂ¬
вЂ«вЂЄiв€€IвЂ¬вЂ¬
вЂ«в‰¤вЂ¬
вЂ«(вЂ¬
вЂ«) вЂЄв‰¤ max ( Aei в€’ An eiвЂ¬вЂ¬
вЂ«вЂЄiв€€IвЂ¬вЂ¬
вЂ«(вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄHвЂ¬вЂ¬
вЂ«вЂЄAhвЂ¬вЂ¬
вЂ«вЂЄ= h H max( Aei в€’ An ei ) + AnвЂ¬вЂ¬
вЂ«)(вЂ¬
вЂ«)(вЂ¬
вЂ«) (вЂ¬
вЂ«вЂЄiв€€IвЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«Щ€ п»«п»њпє¬Ш§ п»іп»њп»®Щ† Ш§п»џп»¤пє†пє›пє® вЂЄ AвЂ¬п»Јпє¤пєЄЩ€ШЇШ§Щ‹ п»‹п» п»° вЂЄ D A в€©D An = D AвЂ¬п»·Щ† вЂЄ AnвЂ¬п»Јпє¤пєЄЩ€ШЇ Щ€ п»џп»њп»ћ вЂЄ. nвЂ¬вЂ¬
вЂ«пє‘пє®п»«пєЋЩ† )вЂЄ (11) ШЊ (10вЂ¬Щ€ )вЂЄ (12вЂ¬п»Јп»¦ п»§п»€пє®п»іпє” Ш§п»џп»”п»ЂпєЋШЎ Ш§п»џпє пє°пє‹п»І Ш§п»џп»¤п»ђп» п»– п»џп»”п»ЂпєЋШЎ п»«п» пє’пє®ШЄвЂЄ .вЂ¬Ш§п»§п»€пє® Ш§п»џпєјп»”пє¤пє” )вЂЄ (80вЂ¬п»Јп»¦вЂ¬
вЂ«]вЂЄ . [3вЂ¬п»џпє’пє®п»«пєЋЩ† )вЂЄ (13вЂ¬п»§п»њпє�пєђвЂЄ:вЂ¬вЂ¬
вЂ«вЉҐвЂЄAB = A в€’ AвЂ¬вЂ¬
вЂ«п»‹п»ЁпєЄпє‹пє¬вЂ¬
вЂ«))вЂЄA в€’ AвЉҐ в€€ B(D( AвЂ¬вЂ¬
вЂ«Щ€ п»«п»њпє¬Ш§ п»іп»њп»®Щ†вЂ¬
вЂ«)вЂЄв‰¤ M 2 h H , h в€€ D ( AвЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄHвЂ¬вЂ¬
вЂ«вЂЄAB hвЂ¬вЂ¬
вЂ«Щ€ п»«пє¬Ш§ п»іпє†ШЇЩЉ ШҐп»џп»°вЂ¬
вЂ«)вЂЄв‰¤ M 2 h H , h в€€ D ( AвЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«п»§пєґпє�п»„п»ґп»Љ пє—п»¤пєЄп»іпєЄвЂ¬
вЂ«вЂЄABвЂ¬вЂ¬
вЂ«ШҐп»џп»° вЂЄ: HвЂ¬вЂ¬
вЂ«вЂЄ9вЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄHвЂ¬вЂ¬
вЂ«вЂЄ( A в€’ AвЉҐ )hвЂ¬вЂ¬
вЂ«п»‹пє п»®Ш± Щ€ пє‘п»ґп»њпє®ШЇвЂ¬
вЂ«)вЂ¬
вЂ«(вЂ¬
вЂ«вЂЄ1 2вЂ¬вЂ¬
вЂ«вЂЄb + d2вЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄA H 2 = a2 + c2 +вЂ¬вЂ¬
вЂ«пє‘п»ґп»Ёп»¤пєЋвЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄA S = a 2 + c 2 + b2 + d 2вЂ¬вЂ¬
вЂ«Щ€вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄSвЂ¬вЂ¬
вЂ«в‰¤вЂ¬
вЂ«вЂЄAвЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄBвЂ¬вЂ¬
вЂ«вЂЄAвЂ¬вЂ¬
вЂ«вЂЄ-3вЂ¬Ш§п»џп»”п»ЂпєЋШЎШ§ШЄ Ш§п»џпє пє°пє‹п»ґпє” Щ€ Ш§п»џпє�п»ЊпєЋп»ЈпєЄвЂ¬
вЂ«п»“п»ґп»¤пєЋ п»іп» п»І пєіп»®ЩЃ п»§п»Њпє®ЩЃ Ш§п»џпє пєЄШ§ШЎ Ш§п»џпєЄШ§пє§п» п»І п»“п»І ) вЂЄ H 2 (HвЂ¬п»›п»¤пєЋ п»іп» п»І вЂЄ:вЂ¬вЂ¬
вЂ«вЂЄ1вЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄв€’вЂ¬вЂ¬
вЂ«вЂЄпЈ«вЂ¬вЂ¬
вЂ«вЂЄ2пЈ¶вЂ¬вЂ¬
вЂ«вЂЄ, C := пЈ¬в€‘ fi пЈ·вЂ¬вЂ¬
вЂ«)вЂЄ(8вЂ¬вЂ¬
вЂ«вЂЄHвЂ¬вЂ¬
вЂ«вЂЄiв€€IвЂ¬вЂ¬
вЂ«вЂЄпЈ iв€€IвЂ¬вЂ¬
вЂ«вЂЄпЈёвЂ¬вЂ¬
вЂ«п»Јп»¦ ШЈпєџп»ћ п»›п»ћ п»Јпє†пє›пє®п»іп»¦ ) вЂЄ. A, B в€€ H 2 (HвЂ¬вЂ¬
вЂ«п»§п»€пє®п»іпє” пє—п»¤п»¬п»ґпєЄп»іпє”вЂЄ . 1вЂ¬п»џп»ґп»њп»¦ ) вЂЄ H 2 (HвЂ¬Щ€ ) вЂЄ B (HвЂ¬Ш§п»џп»¤п»Њпє®п»“п»ґп»¦ пєіпєЋпє‘п»�пєЋЩ‹ п»‹п»ЁпєЄпє‹пє¬ п»§пє пєЄ ШЈп»§п»ЄвЂЄ:вЂ¬вЂ¬
вЂ«вЉҐвЂ¬
вЂ«) вЂЄH 2 (H ) = B (H ) вЉ• B (HвЂ¬вЂ¬
вЂ«)вЂЄ(9вЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«Щ€ п»«п»њпє¬Ш§ п»“пє€Щ† п»›п»ћ п»Јпє†пє›пє® вЂЄ AвЂ¬п»Јп»¦ ) вЂЄ H (HвЂ¬п»іп»¤п»њп»¦ ШЈЩ† п»§п»Њпє’пє® п»‹п»Ёп»Є пє‘пєЋп»џпєёп»њп»ћ Ш§п»џпє�пєЋп»џп»ІвЂЄ:вЂ¬вЂ¬
вЂ«)вЂЄ(10вЂ¬вЂ¬
вЂ«вЂЄA, B # = C2 в€‘ fi Aei , BejвЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЉҐвЂЄA = AB + AвЂ¬вЂ¬
вЂ«вЂЄAB в€€ B(H ) ШЊвЂ¬вЂ¬
вЂ«вЂЄШЊвЂ¬вЂ¬
вЂ«) вЂЄAвЉҐ в€€ B(HвЂ¬вЂ¬
вЂ«вЉҐвЂ¬
вЂ«Щ€ п»Јп»Ёп»Є п»§пє пєЄвЂЄ:вЂ¬вЂ¬
вЂ«)вЂЄ(11вЂ¬вЂ¬
вЂ«Щ€вЂ¬
вЂ«)вЂЄ(12вЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄ#вЂ¬вЂ¬
вЂ«вЉҐвЂЄ+ AвЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄ#вЂ¬вЂ¬
вЂ«вЂЄ= ABвЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄAвЂ¬вЂ¬
вЂ«вЂЄ#вЂ¬вЂ¬
вЂ«) вЂЄA в€’ AвЉҐ в€€ B(HвЂ¬вЂ¬
вЂ«Щ€вЂ¬
вЂ«)вЂЄ(13вЂ¬вЂ¬
вЂ«вЂЄ( A в€’ AвЉҐ )h H в‰¤ M h H ,вЂ¬вЂ¬
вЂ«вЂЄh в€€ D( A) , M > 0вЂ¬вЂ¬
вЂ«Щ€ п»‹п»јЩ€Ш© п»‹п» п»° Ш°п»џп»љвЂ¬
вЂ«)вЂЄ(14вЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄBвЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄA # в€’ AвЉҐ # в‰¤ ABвЂ¬вЂ¬
вЂ«вЂЄ8вЂ¬вЂ¬
вЂ«в‰¤ вЂЄ.0вЂ¬вЂ¬
вЂ«пє—п»„пє’п»ґп»�пєЋШЄ п»“п»І п»“п»ЂпєЋШЎШ§ШЄ п»«п» пє’пє®ШЄ п»џп» п»¤пє†пє›пє®Ш§ШЄ Ш§п»џпєЁп»„п»ґпє” п»Џп»ґпє® Ш§п»џп»¤пє¤пєЄЩ€ШЇШ©вЂ¬
вЂ«п»џпє¬Ш§ п»§п»њпє�пєђвЂЄ:вЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄHвЂ¬вЂ¬
вЂ«в€—вЂЄ= AвЂ¬вЂ¬
вЂ«в€—вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄHвЂ¬вЂ¬
вЂ«вЂЄAnвЂ¬вЂ¬
вЂ«вЂЄlimвЂ¬вЂ¬
вЂ«=вЂ¬
вЂ«в€ћв†’вЂЄnвЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄHвЂ¬вЂ¬
вЂ«вЂЄAnвЂ¬вЂ¬
вЂ«вЂЄlimвЂ¬вЂ¬
вЂ«=вЂ¬
вЂ«=вЂЄ2вЂ¬вЂ¬
вЂ«в€ћв†’вЂЄnвЂ¬вЂ¬
вЂ«вЂЄHвЂ¬вЂ¬
вЂ«вЂЄAвЂ¬вЂ¬
вЂ«п»Јп»јпєЈп»€пє”)вЂЄ :(2вЂ¬п»іп»¤п»њп»¦ ШҐпє›пє’пєЋШЄ Ш§п»џпєЁп»„п»ґпє” п»џп» п»¤пє†пє›пє® Ш§п»џп»¤п»Њпє®ЩЃ вЂЄ AвЂ¬п»“п»І )в€—( ШЇЩ€Щ† Ш°п»›пє® ШЈЩ† Ш§п»џпє�п»¤пєЄп»іпєЄ пє§п»„п»І Щ€ Ш°п»џп»љ пє‘пє€пєїпєЋп»“пє”вЂ¬
вЂ«Ш§п»џпєЁпєЋпє»пє” Ш§п»џпє�пєЋп»џп»ґпє” )ШҐп»џп»° Ш§п»џпє�п»Њпє®п»іп»’ Ш§п»џп»¤пє¬п»›п»®Ш± п»“п»І )в€—( (вЂЄ:вЂ¬вЂ¬
вЂ«вЂЄmвЂ¬вЂ¬
вЂ«вЂЄmвЂ¬вЂ¬
вЂ«вЂЄmвЂ¬вЂ¬
вЂ«вЂЄnв†’в€ћ i=1вЂ¬вЂ¬
вЂ«вЂЄi=1вЂ¬вЂ¬
вЂ«вЂЄi=1вЂ¬вЂ¬
вЂ«вЂЄmвЂ¬вЂ¬
вЂ«вЂЄв€‘fiО±i Aei = Aв€‘fiaiei = в€‘О±ihi =limв€‘О±i An fiei , в€Ђmв€€I , в€ЂО± i в€€ KвЂ¬вЂ¬
вЂ«вЂЄi=1вЂ¬вЂ¬
вЂ«п»Јп»јпєЈп»€пє”)вЂЄ :(3вЂ¬п»іп»¤п»њЩЂп»¦ ШЈЩ† п»§п»®пєіЩЂп»Љ Ш§п»џп»”ЩЂп»ЂпєЋШЎ ) вЂЄ H 2 (HвЂ¬пє‘ЩЂпє„Щ† п»іп»њЩЂп»®Щ† п»›ЩЂп»ћ п»ЈЩЂпє†пє›пє® вЂЄ AвЂ¬п»ЈЩЂп»¦ ) вЂЄ H (HвЂ¬п»—пєЋпє‘ЩЂп»ћ п»џп»єп»ЏЩЂп»јЩ‚вЂ¬
вЂ«пє‘ЩЂЩЂЩЂпєЄп»» п»ЈЩЂЩЂЩЂп»¦ ШЈЩ† п»іп»њЩЂЩЂЩЂп»®Щ† п»Јп»ђп» п»�ЩЂЩЂпєЋЩ‹вЂЄ .вЂ¬Щ€ п»џп»њЩЂЩЂЩЂп»¦ п»іпє ЩЂЩЂЩЂпєђ ШЈЩ† п»§ЩЂЩЂЩЂпєёп»њп»ћ пє»ЩЂЩЂЩЂп»”п»®п»“пєЋЩ‹ п»Јпє�п»њпєЋп»“пєЊЩЂЩЂЩЂпє” п»ЈЩЂЩЂЩЂп»¦ Ш§п»џп»¤ЩЂЩЂЩЂпє†пє›пє®Ш§ШЄвЂЄ :вЂ¬Щ€ Ш°п»џЩЂЩЂЩЂп»љ пє‘ЩЂЩЂЩЂпєЋЩ† п»§п»�ЩЂЩЂЩЂп»®Щ„ ШЈЩ†вЂ¬
вЂ«Ш§п»џп»¤пє†пє›пє®п»іп»¦вЂЄ:вЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄ B : D (B ) вЉ‚ H в†’ H ШЊ A : D ( A ) вЉ‚ H в†’ HвЂ¬п»“ЩЂЩЂЩЂп»І пє»ЩЂЩЂЩЂп»’ Щ€Ш§пєЈЩЂЩЂЩЂпєЄ )п»Јпє�п»њЩЂЩЂЩЂпєЋп»“пєЊп»ґп»¦( ШҐШ°Ш§вЂ¬
вЂ«пє—пє¤п»�п»– Ш§п»џпєёпє®Ш·вЂЄ:вЂ¬вЂ¬
вЂ«вЂЄA в‰Ў B в‡” Aei = Bei , в€Ђi в€€ IвЂ¬вЂ¬
вЂ«п»ЈпєњпєЋЩ„)вЂЄ :(1вЂ¬пє‘п»”пє®Ш¶ вЂЄ RвЂ¬п»Јпє п»¤п»®п»‹пє” Ш§п»·п»‹пєЄШ§ШЇ Ш§п»џпє¤п»�п»ґп»�п»ґпє”вЂЄ .вЂ¬п»§п»Ђп»Љ вЂЄ H = R 2вЂ¬Щ€ п»§п»Њпє�пє’пє®вЂ¬
вЂ«вЂЄпЈ±вЂ¬вЂ¬
вЂ«{вЂ¬
вЂ«}вЂ¬
вЂ«вЂЄпЈјвЂ¬вЂ¬
вЂ«вЂЄпЈ«a b пЈ¶вЂ¬вЂ¬
вЂ«вЂЄпЈ·пЈ·, a, b, c, d в€€ RпЈЅвЂ¬вЂ¬
вЂ«вЂЄпЈc d пЈёвЂ¬вЂ¬
вЂ«вЂЄпЈівЂ¬вЂ¬
вЂ«вЂЄпЈѕвЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«п»§п»Њпє®ЩЃ Ш§п»џпє пєЄШ§ШЎ Ш§п»џпєЄШ§пє§п» п»І п»‹п» п»° ) вЂЄ H (HвЂ¬п»Јп»¦ ШЈпєџп»ћ ШЈЩЉ п»Јпє†пє›пє®п»іп»¦вЂ¬
вЂ«вЂЄH 2 (H ) = A : R 2 в†’ R 2 , linear = пЈІ A = пЈ¬пЈ¬вЂ¬вЂ¬
вЂ«вЂЄb1 пЈ¶вЂ¬вЂ¬
вЂ«вЂЄпЈ·вЂ¬вЂ¬
вЂ«вЂЄd 1 пЈ·пЈёвЂ¬вЂ¬
вЂ«вЂЄbпЈ¶вЂ¬вЂ¬
вЂ«вЂЄпЈ«aвЂ¬вЂ¬
вЂ«вЂЄпЈ·пЈ· , B = пЈ¬пЈ¬ 1вЂ¬вЂ¬
вЂ«вЂЄdпЈёвЂ¬вЂ¬
вЂ«вЂЄпЈ c1вЂ¬вЂ¬
вЂ«вЂЄпЈ«aвЂ¬вЂ¬
вЂ«вЂЄA = пЈ¬пЈ¬вЂ¬вЂ¬
вЂ«вЂЄпЈcвЂ¬вЂ¬
вЂ«пє‘пєЋп»џпєёп»њп»ћ вЂЄ:вЂ¬вЂ¬
вЂ«вЂЄR2вЂ¬вЂ¬
вЂ«вЂЄ1 пЈ«0пЈ¶ пЈ«0пЈ¶вЂ¬вЂ¬
вЂ«вЂЄAпЈ¬ пЈ·, B пЈ¬ пЈ·вЂ¬вЂ¬
вЂ«вЂЄ2 пЈ¬пЈ 1 пЈ·пЈё пЈ¬пЈ 1 пЈ·пЈёвЂ¬вЂ¬
вЂ«вЂЄпЈ«1пЈ¶ пЈ«1пЈ¶вЂ¬вЂ¬
вЂ«вЂЄ= AпЈ¬пЈ¬ пЈ·пЈ·, B пЈ¬пЈ¬ пЈ·пЈ·вЂ¬вЂ¬
вЂ«вЂЄпЈ0пЈё пЈ0пЈёвЂ¬вЂ¬
вЂ«вЂЄ+вЂ¬вЂ¬
вЂ«вЂЄR2вЂ¬вЂ¬
вЂ«вЂЄH2вЂ¬вЂ¬
вЂ«вЂЄA, BвЂ¬вЂ¬
вЂ«Щ€ п»Јп»Ёп»Є п»§пє пєЄвЂЄ:вЂ¬вЂ¬
вЂ«вЂЄ1вЂ¬вЂ¬
вЂ«вЂЄ(b, d ), (b1, d1) R2вЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄA, B H 2 = (a, c), (a1, c1 ) R2 +вЂ¬вЂ¬
вЂ«вЂЄ1вЂ¬вЂ¬
вЂ«) вЂЄ(bb1 + dd1вЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«Щ€ п»Јп»Ёп»Є п»§пє пєЄвЂЄ:вЂ¬вЂ¬
вЂ«вЂЄ7вЂ¬вЂ¬
вЂ«вЂЄ= aa1 + cc1 +вЂ¬вЂ¬
вЂ«п»‹пє п»®Ш± Щ€ пє‘п»ґп»њпє®ШЇвЂ¬
вЂ«п»§пє’пє®п»«п»¦ ШЈЩ† вЂЄ A : D ( A) в†’ HвЂ¬п»—пєЋпє‘п»ћ п»џп»єп»Џп»јЩ‚ Ш§п»§п»€пє® п»“п»І ]вЂЄ [3вЂ¬Ш§п»џпєјп»”пє¤пє” ) вЂЄ . ( p.22вЂ¬п»џп»¬пє¬Щ‡ Ш§п»џп»ђпєЋп»іпє” п»іп»њп»”п»І пє‘пє®п»«пєЋЩ† ШЈЩ†вЂ¬
вЂ«) (вЂ¬
вЂ«в€—вЂЄ D AвЂ¬п»Јпє п»¤п»®п»‹пє” пє—п»Њпє®п»іп»’ Ш§п»џп»¤пє†пє›пє®вЂЄ Aв€— : D(Aв€— ) в†’ H :вЂ¬п»›пєњп»ґп»”пє” п»“п»І вЂЄ : HвЂ¬п»џпє�п»њп»¦вЂ¬
вЂ«п»Јпє�пє�пєЋп»џп»ґпє” п»›п»®пє·п»І Ш§п»џпєґпєЋпє‘п»�пє”вЂЄ ШЊвЂ¬п»‹п»ЁпєЄпє‹пє¬вЂ¬
вЂ«} вЂЄ{AвЂ¬вЂ¬
вЂ«в€ћ *вЂ¬
вЂ«вЂЄn n=1вЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«Ш§п»‹пє�п»¤пєЋШЇШ§Щ‹ п»‹п» п»° )вЂЄ: (5вЂ¬вЂ¬
вЂ«} вЂЄ{f A eвЂ¬вЂ¬
вЂ«в€ћвЂ¬
вЂ«вЂЄi n =1вЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄH2вЂ¬вЂ¬
вЂ«п»Јпє�пє�пєЋп»џп»ґпє” п»›п»®пє·п»І п»“п»І ) вЂЄ . H 2 (HвЂ¬п»іп»¤п»њп»¦ ШЈЩ† п»§п»њпє�пєђвЂЄ:вЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«в€—вЂЄ= TвЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄHвЂ¬вЂ¬
вЂ«вЂЄnвЂ¬вЂ¬
вЂ«вЂЄHвЂ¬вЂ¬
вЂ«п»Јпє�пє�пєЋп»џп»ґпє” п»›п»®пє·п»І п»“п»ІвЂ¬
вЂ«в€—вЂ¬
вЂ«вЂЄAn в€’ AвЂ¬вЂ¬
вЂ«)вЂЄAn в€’ Am = T в€€H2(HвЂ¬вЂ¬
вЂ«вЂЄ TвЂ¬пє‘пєЋп»‹пє�пє’пєЋШ±вЂ¬
вЂ«в€—вЂ¬
вЂ«вЂЄiвЂ¬вЂ¬
вЂ«вЂЄв€— 2вЂ¬вЂ¬
вЂ«вЂЄmвЂ¬вЂ¬
вЂ«вЂЄH2вЂ¬вЂ¬
вЂ«вЂЄ= An в€’ AmвЂ¬вЂ¬
вЂ«вЂЄH2вЂ¬вЂ¬
вЂ«) вЂЄ{An}в€ћn=1 вЉ‚ H 2 (HвЂ¬вЂ¬
вЂ«Щ€п»«п»њпє¬Ш§ п»іп»њп»®Щ† п»џпєЄп»іп»ЁпєЋвЂ¬
вЂ«вЂЄ i в€€ IвЂ¬Щ€ п»›п»¤пєЋ пєіпє’п»– п»§п»Њпє®ЩЃ Ш§п»џп»¤пє†пє›пє®вЂЄ:вЂ¬вЂ¬
вЂ«Щ€ п»џп»њп»ћвЂ¬
вЂ«вЂЄ A : D пЈ«пЈ¬ A пЈ¶пЈ· вЉ‚ H в†’ HвЂ¬пє‘пє¤п»ґпєљ ШЈЩ†вЂЄ:вЂ¬вЂ¬
вЂ«вЂЄпЈ пЈёвЂ¬вЂ¬
вЂ«~вЂ¬
вЂ«~вЂ¬
вЂ«в€—вЂ¬
вЂ«вЂЄf i Aв€—ei := lim f i An ei , в€Ђ i в€€ IвЂ¬вЂ¬
вЂ«в€ћв†’вЂЄnвЂ¬вЂ¬
вЂ«~вЂ¬
вЂ«~вЂ¬
вЂ«п»›п»¤пєЋ пє‘пє®п»«п»ЁпєЋ пєіпєЋпє‘п»�пєЋЩ‹ п»“п»І )в€— в€—( п»іп»¤п»њп»¦ ШЈЩ† п»§пє’пє®п»«п»¦ в€ћ < вЂЄ . A H 2вЂ¬п»Јп»¦ Ш§п»џпє�п»Њпє®п»іп»’ п»§пє пєЄ ШЈп»іп»ЂпєЋЩ‹ ШЈЩ† вЂЄ D пЈ«пЈ¬ A пЈ¶пЈ·вЂ¬п»›пєњп»ґп»”пє”вЂ¬
вЂ«вЂЄпЈ пЈёвЂ¬вЂ¬
вЂ«~вЂ¬
вЂ«п»“п»І вЂЄ . HвЂ¬п»§пє’пє®п»«п»¦ Ш§п»µЩ† в€—вЂЄ A = AвЂ¬п»‹п» п»°вЂ¬
вЂ«вЂЄ{ei }iв€€IвЂ¬вЂ¬
вЂ«вЂЄ, в€Ђi, j в€€ IвЂ¬вЂ¬
вЂ«п»џпє¬Ш§ п»§п»њпє�пєђвЂЄ:вЂ¬вЂ¬
вЂ«вЂЄHвЂ¬вЂ¬
вЂ«вЂЄ= ei , A*e jвЂ¬вЂ¬
вЂ«~вЂ¬
вЂ«вЂЄ= fi ei , f j A e jвЂ¬вЂ¬
вЂ«вЂЄ, в€Ђi, j в€€ IвЂ¬вЂ¬
вЂ«вЂЄHвЂ¬вЂ¬
вЂ«вЂЄHвЂ¬вЂ¬
вЂ«вЂЄHвЂ¬вЂ¬
вЂ«вЂЄAei , e jвЂ¬вЂ¬
вЂ«вЂЄfi Anei , f j e jвЂ¬вЂ¬
вЂ«вЂЄi, j в€€ IвЂ¬вЂ¬
вЂ«пє‘пє„пє§пє¬ Ш§п»џп»Ёп»¬пєЋп»іпє” п»џп»¤пєЋ в€ћ в†’ вЂЄ nвЂ¬п»§пє пєЄвЂЄ :вЂ¬п»џп»њп»ћвЂ¬
вЂ«~вЂ¬
вЂ«вЂЄ= fiei , f j AejвЂ¬вЂ¬
вЂ«вЂЄHвЂ¬вЂ¬
вЂ«вЂЄfi Aei , f jejвЂ¬вЂ¬
вЂ«вЂЄHвЂ¬вЂ¬
вЂ«Щ€ п»«пє¬Ш§ п»іп»Њп»„п»ІвЂ¬
вЂ«~вЂ¬
вЂ«вЂЄ= ei , AejвЂ¬вЂ¬
вЂ«вЂЄ, в€Ђi, j в€€ IвЂ¬вЂ¬
вЂ«вЂЄHвЂ¬вЂ¬
вЂ«п»·Щ† вЂЄfi в‰ 0, в€Ђi в€€ I :вЂ¬вЂ¬
вЂ«вЂЄв€Ђi в€€ IвЂ¬вЂ¬
вЂ«~вЂ¬
вЂ«вЂЄHвЂ¬вЂ¬
вЂ«вЂЄei , Aв€—e jвЂ¬вЂ¬
вЂ«вЂЄ.вЂ¬Щ€ п»«пє¬Ш§ п»іп»Њп»Ёп»ІвЂЄ:вЂ¬вЂ¬
вЂ«~вЂ¬
вЂ«в€—вЂ¬
вЂ«вЂЄ Aв€—ei = AeiвЂ¬Щ€п»Јп»Ёп»Є п»§п»њпє�пєђвЂЄ A = A :вЂ¬п»‹п» п»°вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄHвЂ¬вЂ¬
вЂ«в€—вЂЄ= AвЂ¬вЂ¬
вЂ«вЂЄ6вЂ¬вЂ¬
вЂ«вЂЄ{ei }iв€€IвЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄHвЂ¬вЂ¬
вЂ«вЂЄAвЂ¬вЂ¬
вЂ«вЂЄ .вЂ¬п»§пєњпє’пє– ШЈЩ†вЂЄ:вЂ¬вЂ¬
вЂ«пє—п»„пє’п»ґп»�пєЋШЄ п»“п»І п»“п»ЂпєЋШЎШ§ШЄ п»«п» пє’пє®ШЄ п»џп» п»¤пє†пє›пє®Ш§ШЄ Ш§п»џпєЁп»„п»ґпє” п»Џп»ґпє® Ш§п»џп»¤пє¤пєЄЩ€ШЇШ©вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄBвЂ¬вЂ¬
вЂ«вЂЄ1вЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄв€’вЂ¬вЂ¬
вЂ«пє‘пєЋп»‹пє�пє’пєЋШ±вЂ¬
вЂ«вЂЄпЈ¶вЂ¬вЂ¬
вЂ«вЂЄпЈ·вЂ¬вЂ¬
вЂ«вЂЄпЈёвЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄiвЂ¬вЂ¬
вЂ«в€‘вЂ¬
вЂ«вЂЄfвЂ¬вЂ¬
вЂ«вЂЄiв€€ IвЂ¬вЂ¬
вЂ«вЂЄпЈ«вЂ¬вЂ¬
вЂ«вЂЄпЈ¬вЂ¬вЂ¬
вЂ«вЂЄпЈвЂ¬вЂ¬
вЂ«вЂЄв‰¤ C в€’2 AвЂ¬вЂ¬
вЂ«=вЂ¬
вЂ«вЂЄ.CвЂ¬вЂ¬
вЂ«вЂЄ-2вЂ¬Ш§п»№пє—п»¤пєЋЩ…вЂЄ :вЂ¬п»§пє’пє®п»«п»¦ ШЈЩ† ) вЂЄ H 2 (HвЂ¬п»«п»® п»“п»ЂпєЋШЎ пє—пєЋЩ… вЂЄ:вЂ¬п»џпє�п»њп»¦ ) вЂЄ {An }n =1 вЉ‚ H (HвЂ¬п»Јпє�пє�пєЋп»џп»ґпє” п»›п»®пє·п»ІвЂЄ .вЂ¬пєіп»Ёпє’пє®п»«п»¦ п»‹п» п»°вЂ¬
вЂ«в€ћвЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«Щ€пєџп»®ШЇ п»§п»¬пєЋп»іпє” п»«п»І Ш§п»џп»¤пє†пє›пє® вЂЄ A : D( A) вЉ‚ H в†’ HвЂ¬Ш§п»џп»¤пє¬п»›п»®Ш± Щ€ п»Јпє®Ш§п»“п»�п»Є п»Јпє п»¤п»®п»‹пє” пє—п»Њпє®п»іп»’ п»›п»ћ п»Јп»Ёп»¬п»¤пєЋ п»›пєњп»ґп»”пє” п»“п»ІвЂ¬
вЂ«вЂЄ HвЂ¬Щ€ пє—пє¤п»®ЩЉ п»‹п»ЁпєЋпє»пє® Ш§п»џп»�пєЋп»‹пєЄШ© Щ€ п»іпє пєђ пє—п»®пєїп»ґпєў пє§п»„п»ґпє” вЂЄ AвЂ¬Щ€ п»—пєЋпє‘п» п»ґпє�п»Є п»џп»єп»Џп»јЩ‚ )п»іп»®пєџпєЄ п»Јпє†пє›пє® пє§п»„п»І п»Јп»ђп» п»–вЂ¬
вЂ«вЂЄAвЂ¬вЂ¬
вЂ«пє‘п»¤пєЋвЂ¬
вЂ«ШЈЩ†вЂ¬
вЂ«п»іпє¤п»�п»–вЂ¬
вЂ«Ш§п»џпєЁпєЋпє»пє”вЂЄ:вЂ¬вЂ¬
вЂ«) вЂЄвЉ‚ H 2 (HвЂ¬вЂ¬
вЂ«вЂЄ<ОµвЂ¬вЂ¬
вЂ«вЂЄ( AвЉ‚ AвЂ¬вЂ¬
вЂ«вЂЄH2вЂ¬вЂ¬
вЂ«Щ€вЂ¬
вЂ«п»«пє¬Ш§вЂ¬
вЂ«Ш§п»џп»¤пє†пє›пє®вЂ¬
вЂ«} вЂЄ {AвЂ¬п»Јпє�пє�пєЋп»џп»ґпє” п»›п»®пє·п»ІвЂЄ ШЊвЂ¬п»“п» п»њп»ћ вЂЄОµ > 0вЂ¬вЂ¬
вЂ«в€ћвЂ¬
вЂ«вЂЄn n =1вЂ¬вЂ¬
вЂ«вЂЄ An в€’ AmвЂ¬п»џп»њп»ћ вЂЄn, m > n0вЂ¬вЂ¬
вЂ«вЂЄ<Оµ2вЂ¬вЂ¬
вЂ«Щ€ п»«п»њпє¬Ш§ п»“пє€Щ†вЂ¬
вЂ«Ш§п»џп»¤п»ђп» п»–вЂ¬
вЂ«вЂЄHвЂ¬вЂ¬
вЂ«вЂЄ{ f i Anei }в€ћn =1вЂ¬вЂ¬
вЂ«п»«п»®вЂ¬
вЂ«п»іп»®пєџпєЄ п»‹пєЄШЇ п»ѓпє’п»ґп»Њп»ІвЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄ= в€‘ fiвЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄH2вЂ¬вЂ¬
вЂ«вЂЄiв€€ IвЂ¬вЂ¬
вЂ«вЂЄAn в€’ AnвЂ¬вЂ¬
вЂ«п»Јпє�пє�пєЋп»џп»ґпє” п»›п»®пє·п»І п»“п»І вЂЄ HвЂ¬Щ€ п»Јп»¦ ШЈпєџп»ћ п»›п»ћ вЂЄ . i в€€ IвЂ¬п»џпє¬Ш§ п»§п»њпє�пєђвЂ¬
вЂ«вЂЄHвЂ¬вЂ¬
вЂ«вЂЄ fi Anei пЈ§пЈ§в†’ hiвЂ¬п»џп»¤пєЋ в€ћв†’ вЂЄnвЂ¬вЂ¬
вЂ«Щ€ п»Јп»¦ ШЈпєџп»ћ п»›п»ћ вЂЄ . i в€€ IвЂ¬Щ€ п»Јп»Ёп»Є п»§п»Њпє®ЩЃ Ш§п»џп»¤пє†пє›пє®вЂЄ:вЂ¬вЂ¬
вЂ«вЂЄA : D(H ) в†’ HвЂ¬вЂ¬
вЂ«пє‘пєЋп»‹пє�пє’пєЋШ±вЂ¬
вЂ«вЂЄfi Aei = lim fi Anei в€Ђ i в€€ IвЂ¬вЂ¬
вЂ«)в€—(вЂ¬
вЂ«в€ћв†’ вЂЄnвЂ¬вЂ¬
вЂ«Щ€ п»«пє¬Ш§ Ш§п»џпє�п»¤пєЄШЇ пє§п»„п»ІвЂЄ .вЂ¬п»‹п»ЁпєЄпє‹пє¬ п»џп»њп»ћвЂ¬
вЂ«вЂЄОµвЂ¬вЂ¬
вЂ«вЂЄiвЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«= вЂЄ< ОµiвЂ¬вЂ¬
вЂ«вЂЄHвЂ¬вЂ¬
вЂ«вЂЄiв€€IвЂ¬вЂ¬
вЂ«вЂЄf i An ei в€’ f i AeiвЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄ<Оµ 2вЂ¬вЂ¬
вЂ«вЂЄHвЂ¬вЂ¬
вЂ«п»іп»¤п»њп»¦ ШЈпє§пє¬вЂ¬
вЂ«вЂЄОµвЂ¬вЂ¬
вЂ«вЂЄ2iвЂ¬вЂ¬
вЂ«= вЂЄ Оµ iвЂ¬пє‘пєЋп»‹пє�пє’пєЋШ± ШЈЩ†вЂ¬
вЂ«п»џп»њп»ћ вЂЄ . n > n 0вЂ¬Щ€ п»«п»њпє¬Ш§ п»іп»њп»®Щ†вЂЄ:вЂ¬вЂ¬
вЂ«вЂЄ= в€‘ f i Aei в€’ f i An eiвЂ¬вЂ¬
вЂ«вЂЄiв€€IвЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄH2вЂ¬вЂ¬
вЂ«вЂЄA в€’ AnвЂ¬вЂ¬
вЂ«п»џпє¬Ш§ п»іп»њп»®Щ† п»џпєЄп»іп»ЁпєЋ п»Јп»¦ ШЈпєџп»ћ вЂЄ n в€€ NвЂ¬п»Јпєњпє’пє– п»џпє¤п»€п»ґпєЋЩ‹ вЂЄ:вЂ¬вЂ¬
вЂ«)в€— в€—(вЂ¬
вЂ«в€ћ<вЂ¬
вЂ«вЂЄH2вЂ¬вЂ¬
вЂ«вЂЄn0вЂ¬вЂ¬
вЂ«вЂЄ.вЂ¬Щ€ п»Јп»Ёп»Є п»§п»њпє�пєђвЂЄ:вЂ¬вЂ¬
вЂ«вЂЄ( An в€’ Am )eiвЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄAвЂ¬вЂ¬
вЂ«Ш§п»џп»Ёп»¬пєЋп»іпє”вЂ¬
вЂ«Ш§п»џп»¤п»„п» п»®пє‘пє”вЂЄ.вЂ¬вЂ¬
вЂ«вЂЄ+ AnвЂ¬вЂ¬
вЂ«вЂЄH2вЂ¬вЂ¬
вЂ«вЂЄA H 2 в‰¤ A в€’ AnвЂ¬вЂ¬
вЂ«п»Јп»¦ пє—п»Њпє®п»іп»’ вЂЄ AвЂ¬п»іп»¤п»њп»¦ п»Јп»јпєЈп»€пє” ШЈЩ† п»џп»Є п»Јпє п»¤п»®п»‹пє” пє—п»Њпє®п»іп»’ пє—пє¤п»®ЩЉ п»‹п»ЁпєЋпє»пє® Ш§п»џп»�пєЋп»‹пєЄШ©вЂЄ.вЂ¬вЂ¬
вЂ«вЂЄ5вЂ¬вЂ¬
вЂ«пє‘пє¤п»ґпєљ ШЈЩ†вЂ¬
вЂ«п»‹пє п»®Ш± Щ€ пє‘п»ґп»њпє®ШЇвЂ¬
= в€‘ f j Bв€—hj , Aв€—hj
2
jв€€I
H
= в€‘в€‘ f j Bв€—hj , gk
2
jв€€I kв€€I
gk , Aв€—hj
H
Agk , hj
H
H
= в€‘в€‘ fk hj , Bgk
2
jв€€I kв€€I
= в€‘ fk
2
k в€€I
Ag k , Bg k
(f
в€‘в€‘
jв€€ I k в€€ I
H
.
H
2
в€’ fk
j
2
)в‹… a
jk
= 0
:вЂ«п»џп»�пєЄ Ш§пєіпє�п»Њп»¤п» п»ЁпєЋ Ш§п»џпєЁпєЋпє»пє”вЂ¬
:вЂ«пє‘пєЋп»‹пє�пє’пєЋШ±вЂ¬
a jk = h j , Bg k
H
Ag k , h j
H
, j, k в€€ I
вЂ«( Щ€ Ш°п»џп»љвЂ¬3 ) вЂ« Щ€ п»›пє¬п»џп»љ п»Јп»¤пєЋ пєіпє’п»– п»§пєґпє�п»Ёпє�пєћ пє—пє¤п»�п»–вЂ¬.вЂ«Щ€ п»Јп»Ёп»Є п»§пє пєЄ ШЈЩ† Ш§п»џпє пєЄШ§ШЎ Ш§п»џпєЄШ§пє§п» п»І п»Јпєґпє�п»�п»ћ п»‹п»¦ Ш§пє§пє�п»ґпєЋШ± Ш§п»џп»�пєЋп»‹пєЄШ©вЂ¬
: A, B в€€ H 2 (H ) вЂ«п»·Щ† п»Јп»¬п»¤пєЋ п»іп»њп»¦вЂ¬
A, B
H
=в€‘ fj
вЂ«п»“пє€Щ†вЂ¬
BA
2
H
2
= в€‘ fi
2
iв€€ I
Bв€—h j , Aв€—h j
вЂ«Щ€вЂ¬
H2
B в€€ B(H ) вЂ« п»Јп»¬п»¤пєЋ п»іп»њп»¦вЂ¬: (4) вЂ«п»џпє’пє®п»«пєЋЩ†вЂ¬
2
2
2
2
Aei
iв€€I
B
H
= A* , B*
H
B ( Aei ) H
2
B
Ae i , Be i
A в€€ H 2 (H )
в‰¤ B B в€‘ fi
=
2
iв€€I
2
jв€€I
:= в€‘ f i
2
A
H
2
H
2
.
вЂ«п»“пє€Щ†вЂ¬
A в€€ H 2 (H ) вЂ« п»Јп»¬п»¤пєЋ п»іп»њп»¦вЂ¬: (4) вЂ«п»џпє’пє®п»«пєЋЩ† Ш§п»џпє пє°ШЎ Ш§п»џпєњпєЋп»§п»І п»Јп»¦вЂ¬
A H 2 = в€‘ fi Aei
2
2
iв€€I
4
2
H
вЂ«пє—п»„пє’п»ґп»�пєЋШЄ п»“п»І п»“п»ЂпєЋШЎШ§ШЄ п»«п» пє’пє®ШЄ п»џп» п»¤пє†пє›пє®Ш§ШЄ Ш§п»џпєЁп»„п»ґпє” п»Џп»ґпє® Ш§п»џп»¤пє¤пєЄЩ€ШЇШ©вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«)вЂЄ(6вЂ¬вЂ¬
вЂ«вЂЄBвЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄBвЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄв‰¤ AвЂ¬вЂ¬
вЂ«вЂЄHвЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄBAвЂ¬вЂ¬
вЂ«вЂЄHвЂ¬вЂ¬
вЂ«) вЂЄв‰¤ c в€’ 2 A B , в€ЂA в€€ B ( HвЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«) вЂЄ(7вЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄAвЂ¬вЂ¬
вЂ«вЂЄHвЂ¬вЂ¬
вЂ«вЂЄв€’1вЂ¬вЂ¬
вЂ«вЂЄпЈ«вЂ¬вЂ¬
вЂ«вЂЄ2 пЈ¶ 2вЂ¬вЂ¬
вЂ«пє‘пєЋп»‹пє�пє’пєЋШ±вЂ¬
вЂ«в€‘ вЂЄ. C := пЈ¬вЂ¬вЂ¬
вЂ«вЂЄfiвЂ¬вЂ¬
вЂ«вЂЄпЈ·вЂ¬вЂ¬
вЂ«вЂЄпЈ iв€€ IвЂ¬вЂ¬
вЂ«вЂЄпЈёвЂ¬вЂ¬
вЂ«вЂЄ (5вЂ¬ШҐЩ† Ш§п»џп»”п»ЂпєЋШЎ ) вЂЄ H 2 (HвЂ¬Ш§п»џп»¤п»�пє�пє®Щ† пє‘пєЋп»џпє пєЄШ§ШЎ Ш§п»џпєЄШ§пє§п» п»І вЂЄ:вЂ¬вЂ¬
вЂ«вЂЄ:= в€‘ f iвЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄAei , BeiвЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄHвЂ¬вЂ¬
вЂ«вЂЄiв€€IвЂ¬вЂ¬
вЂ«вЂЄHвЂ¬вЂ¬
вЂ«вЂЄA, BвЂ¬вЂ¬
вЂ«п»іпєёп»њп»ћ п»“п»ЂпєЋШЎ п»«п» пє’пє®ШЄ п»“п»®Щ‚ Ш§п»џпє¤п»�п»ћ Ш§п»џпє�пєЋЩ… вЂЄ. KвЂ¬вЂ¬
вЂ«Ш§п»џпє’пє®п»«пєЋЩ†вЂЄ .вЂ¬п»§пє’пє®п»«п»¦ ШЈЩ†вЂЄ в€ЂA в€€ H 2 (H ) :вЂ¬Щ€ п»›пєЋЩ† вЂЄ A H 2 = 0вЂ¬п»“пє€Щ† вЂЄ A = 0вЂ¬п»‹п» п»°вЂ¬
вЂ«п»Јпє п»¤п»®п»‹пє” пє—п»Њпє®п»іп»”п»Є )вЂЄ : D( AвЂ¬п»Јп»¬п»¤пєЋ п»іп»њп»¦ )вЂЄ в€Ђh в€€ D ( AвЂ¬п»“пє€Щ†вЂЄШЊвЂ¬вЂ¬
вЂ«вЂЄiвЂ¬вЂ¬
вЂ«вЂЄв€‘a eвЂ¬вЂ¬
вЂ«вЂЄiвЂ¬вЂ¬
вЂ«= вЂЄ hвЂ¬Щ€ п»Јп»¦ пєџп»¬пє” ШЈпє§пє®Щ‰ пєЈпєґпєђвЂ¬
вЂ«вЂЄiв€€IвЂ¬вЂ¬
вЂ«пє—п»Њпє®п»іп»’ Ш§п»џп»Ёп»€п»ґп»ўвЂЄ:вЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄAeiвЂ¬вЂ¬
вЂ«вЂЄ0 = в€‘ fiвЂ¬вЂ¬
вЂ«вЂЄiв€€IвЂ¬вЂ¬
вЂ«Щ€ п»Јп»Ёп»Є п»§пє пєЄвЂЄ:вЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄAeiвЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄ 0 = f iвЂ¬п»Јп»¬п»¤пєЋ п»іп»њп»¦вЂ¬
вЂ«Щ€ пє‘п»¤пєЋ ШЈЩ† вЂЄ f i в‰ 0вЂ¬п»Јп»¬п»¤пєЋ п»іп»њп»¦ вЂЄ i в€€ IвЂ¬п»“пє€Щ† вЂЄAei = 0вЂ¬вЂ¬
вЂ«вЂЄ= 0вЂ¬вЂ¬
вЂ«вЂЄiвЂ¬вЂ¬
вЂ«вЂЄв€‘ a AeвЂ¬вЂ¬
вЂ«вЂЄiвЂ¬вЂ¬
вЂ«пє‘п»”пє®Ш¶вЂ¬
вЂ«вЂЄAei = 0вЂ¬вЂ¬
вЂ«Щ€вЂ¬
вЂ«= вЂЄAhвЂ¬вЂ¬
вЂ«вЂЄiв€€ IвЂ¬вЂ¬
вЂ«} {вЂ¬
вЂ«п»џп»њп»ћвЂ¬
вЂ«вЂЄIвЂ¬вЂ¬
вЂ«в€€ вЂЄ . iвЂ¬ШҐШ°Щ† вЂЄ:вЂ¬вЂ¬
вЂ«вЂЄ.вЂ¬вЂ¬
вЂ«вЂЄ{ei }iв€€ IвЂ¬вЂ¬
вЂ«п»§пє’пє®п»«п»¦ ШЈЩ† Ш§п»џпє пєЄШ§ШЎ Ш§п»џпєЄШ§пє§п» п»І Ш§п»џп»¤п»Њпє®ЩЃ п»Јпєґпє�п»�п»ћ п»‹п»¦ Ш§п»џп»�пєЋп»‹пєЄШ© Ш§п»џп»¤пєЁпє�пєЋШ±Ш©вЂ¬
вЂ«вЂЄjв€€ IвЂ¬вЂ¬
вЂ«вЂЄiв€€IвЂ¬вЂ¬
вЂ«п»“п»І вЂЄ. HвЂ¬вЂ¬
вЂ«вЂЄ {g k }k в€€ I ШЊ h jвЂ¬п»—пєЋп»‹пєЄпє—п»ґп»¦ вЂЄ . HвЂ¬п»‹п»ЁпєЄпє‹пє¬ п»Јп»¦ ШЈпєџп»ћ п»›п»ћ п»Јпє†пє›пє®п»іп»¦ ) вЂЄA, B в€€ H 2 (HвЂ¬вЂ¬
вЂ«вЂЄAe i , Be iвЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄHвЂ¬вЂ¬
вЂ«вЂЄпЈ¶вЂ¬вЂ¬
вЂ«вЂЄhj , Bei пЈ·пЈ·вЂ¬вЂ¬
вЂ«вЂЄHвЂ¬вЂ¬
вЂ«вЂЄпЈёвЂ¬вЂ¬
вЂ«вЂЄHвЂ¬вЂ¬
вЂ«вЂЄHвЂ¬вЂ¬
вЂ«вЂЄBв€—hj , eiвЂ¬вЂ¬
вЂ«вЂЄfiвЂ¬вЂ¬
вЂ«в€‘вЂ¬
вЂ«=вЂЄ:вЂ¬вЂ¬
вЂ«вЂЄiв€€ IвЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄHвЂ¬вЂ¬
вЂ«вЂЄA, BвЂ¬вЂ¬
вЂ«вЂЄ2пЈ«вЂ¬вЂ¬
вЂ«вЂЄ= в€‘ fi пЈ¬пЈ¬ в€‘ Aei , hjвЂ¬вЂ¬
вЂ«вЂЄiв€€IвЂ¬вЂ¬
вЂ«вЂЄпЈ jв€€IвЂ¬вЂ¬
вЂ«вЂЄei , Aв€—hjвЂ¬вЂ¬
вЂ«вЂЄHвЂ¬вЂ¬
вЂ«вЂЄ3вЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄв€‘в€‘ fвЂ¬вЂ¬
вЂ«вЂЄiвЂ¬вЂ¬
вЂ«вЂЄiв€€I jв€€IвЂ¬вЂ¬
вЂ«=вЂ¬
вЂ«п»‹пє п»®Ш± Щ€ пє‘п»ґп»њпє®ШЇвЂ¬
вЂ«)вЂЄ(2вЂ¬вЂ¬
вЂ«в€ћвЂ¬
вЂ«вЂЄA, B S := в€‘ Aei , BeiвЂ¬вЂ¬
вЂ«вЂЄHвЂ¬вЂ¬
вЂ«вЂЄi =1вЂ¬вЂ¬
вЂ«)п»Јп»¦ ШЈпєџп»ћ ШЈЩЉ п»Јпє†пє›пє®п»іп»¦ ) вЂЄ ( A, B в€€ B (HвЂ¬пєџпєЄШ§ШЎ ШЇШ§пє§п» п»І п»“п»І вЂЄ . HSвЂ¬Ш§п»§п»€пє® п»“п»ІвЂЄ [6] ):вЂ¬Ш§п»џпєјп»”пє¤пє” ) вЂЄ ( p.87вЂ¬Щ€вЂ¬
вЂ«]вЂЄ [3вЂ¬Ш§п»џпєјп»”пє¤пє” )вЂЄ.( ( p.123вЂ¬вЂ¬
вЂ«п»§п»Њпє®ЩЃ ШЈп»іп»ЂпєЋ Ш§п»џп»¤пє п»¤п»®п»‹пє”вЂЄ:вЂ¬вЂ¬
вЂ«)вЂЄ(3вЂ¬вЂ¬
вЂ«вЂЄпЈ±A : D( A) вЉ‚ H в†’ H : A: is a closed linear operatorпЈјвЂ¬вЂ¬
вЂ«вЂЄпЈґвЂ¬вЂ¬
вЂ«вЂЄ H (H ) := пЈґвЂ¬п»Јп»¦ ШЈпєџп»ћ п»Јпє�пє�пєЋп»џп»ґпє”вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«в€—вЂ¬
вЂ«вЂЄпЈЅвЂ¬вЂ¬
вЂ«=вЂЄпЈІsuch that: A 2 :вЂ¬вЂ¬
вЂ«{вЂ¬
вЂ«}вЂ¬
вЂ«вЂЄ,вЂ¬вЂ¬
вЂ«вЂЄfвЂ¬вЂ¬
вЂ«вЂЄAeвЂ¬вЂ¬
вЂ«вЂЄeвЂ¬вЂ¬
вЂ«вЂЄDвЂ¬вЂ¬
вЂ«вЂЄAвЂ¬вЂ¬
вЂ«<вЂ¬
вЂ«в€ћвЂ¬
вЂ«вЉ‚вЂ¬
вЂ«в€‘вЂ¬
вЂ«вЂЄiвЂ¬вЂ¬
вЂ«вЂЄi HвЂ¬вЂ¬
вЂ«вЂЄi iв€€IвЂ¬вЂ¬
вЂ«вЂЄпЈґпЈѕвЂ¬вЂ¬
вЂ«вЂЄпЈґпЈівЂ¬вЂ¬
вЂ«вЂЄiв€€IвЂ¬вЂ¬
вЂ«) (вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«ШЈп»‹пєЄШ§ШЇ пєЈп»�п»ґп»�п»ґпє” п»ЈпєЋ вЂЄ { fi }iв€€I вЉ‚ KвЂ¬пє‘пєЋп»‹пє�пє’пєЋШ± вЂЄfi в‰ 0,в€Ђi в€€IвЂ¬вЂ¬
вЂ«Щ€вЂ¬
вЂ«в€ћ<вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄiвЂ¬вЂ¬
вЂ«вЂЄв€‘fвЂ¬вЂ¬
вЂ«вЂЄiв€€IвЂ¬вЂ¬
вЂ«) (вЂ¬
вЂ«вЂЄ .вЂ¬пєЈп»ґпєљ п»—пєјпєЄп»§пєЋ ШЈЩ† )вЂЄ D( AвЂ¬Щ€ в€—вЂЄ D AвЂ¬п»«п»¤пєЋ п»Јпє п»¤п»®п»‹пє” пє—п»Њпє®п»іп»’ п»›п»ћ п»Јп»¦ Ш§п»џп»¤пє†пє›пє® вЂЄ AвЂ¬Щ€вЂ¬
вЂ«п»Јпє®Ш§п»“п»�п»Є в€—вЂЄ. AвЂ¬вЂ¬
вЂ«п»Јп»јпєЈп»€пє”)вЂЄ :(1вЂ¬п»Јп»¦ Ш§п»џпєґп»¬п»ћ п»Јп»јпєЈп»€пє” ШЈЩ† )вЂЄ D( AвЂ¬п»›пєњп»ґп»”ЩЂпє” п»“ЩЂп»І вЂЄ . HвЂ¬Щ€ пє‘п»¤ЩЂпєЋ ШЈЩ† вЂЄ AвЂ¬пє§п»„ЩЂп»І п»“ЩЂпє€Щ† п»Јпє®Ш§п»“п»�ЩЂп»Є п»Јп»®пєџЩЂп»®ШЇвЂЄШЊвЂ¬Щ€вЂ¬
вЂ«п»Јпє п»¤п»®п»‹ЩЂЩЂпє” пє—п»Њпє®п»іЩЂЩЂп»’ Ш§п»џп»¤пє®Ш§п»“ЩЂЩЂп»– в€—вЂЄ AвЂ¬пє—п»њЩЂЩЂп»®Щ† п»›пєњп»ґп»”ЩЂЩЂпє” ШҐШ°Ш§ Щ€ п»“п»�ЩЂЩЂп»‚ ШҐШ°Ш§ п»›ЩЂЩЂпєЋЩ† вЂЄ AвЂ¬п»—пєЋпє‘ЩЂЩЂп»ћ п»џп»єп»ЏЩЂЩЂп»јЩ‚вЂЄ.вЂ¬Ш§п»§п»€ЩЂЩЂпє® п»“ЩЂЩЂп»І ]вЂЄ [3вЂ¬Ш§п»џЩЂЩЂпєјп»”пє¤пє”вЂ¬
вЂ«) вЂЄ. ( p.22вЂ¬вЂ¬
вЂ«п»§пє�пєЋпє‹пєћвЂЄ (1 :вЂ¬ШҐШ°Ш§ п»‹пє®п»“п»ЁпєЋ п»‹п»¤п» п»ґпє” Ш§п»џпє п»¤п»Љ п»Јп»¦ ШЈпєџп»ћ ШЈЩЉ п»Јпє†пє›пє®п»іп»¦ ) вЂЄA , B в€€ H (HвЂ¬вЂ¬
вЂ«) вЂЄ( A + B )h = Ah + Bh, в€Ђh в€€ D( A) в€© D(BвЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«Щ€ п»‹п»¤п» п»ґпє” Ш§п»џп»Ђпє®ШЁ пє‘п»ЊпєЄШЇ п»Јп»¦ Ш§п»џпє¤п»�п»ћвЂ¬
вЂ«)вЂЄ(О»A)h = О» ( Ah ), в€ЂО» в€€ K , h в€€ D( AвЂ¬вЂ¬
вЂ«п»“пє€Щ† ) вЂЄ H 2 (HвЂ¬п»іпєёп»њп»ћ п»“п»ЂпєЋШЎ пє·п»ЊпєЋп»‹п»І п»“п»®Щ‚ Ш§п»џпє¤п»�п»ћ вЂЄ. KвЂ¬вЂ¬
вЂ«вЂЄ (2вЂ¬п»§п»Њпє®ЩЃ п»‹п» п»° ) вЂЄ H 2 (HвЂ¬Ш§п»џпє пєЄШ§ШЎвЂЄ:вЂ¬вЂ¬
вЂ«)вЂЄ(4вЂ¬вЂ¬
вЂ«вЂЄAei , BeiвЂ¬вЂ¬
вЂ«вЂЄHвЂ¬вЂ¬
вЂ«п»Јп»¦ ШЈпєџп»ћ ШЈЩЉ п»Јпє†пє›пє®п»іп»¦ ) вЂЄ(HвЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄ:= в€‘ f iвЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄ . A, B в€€ HвЂ¬п»‹п»ЁпєЄпє‹пє¬ )вЂ¬
вЂ«вЂЄiв€€IвЂ¬вЂ¬
вЂ«в‹…вЂЄв‹…,вЂ¬вЂ¬
вЂ«)вЂЄ. (unitary spaceвЂ¬вЂ¬
вЂ«вЂЄ (3вЂ¬п»Јп»¬п»¤пєЋ п»іп»њп»¦ ) вЂЄ(HвЂ¬вЂ¬
вЂ«)вЂЄ(5вЂ¬вЂ¬
вЂ«вЂЄ (4вЂ¬п»Јп»¬п»¤пєЋ п»іп»њп»¦вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄAв€€ HвЂ¬вЂ¬
вЂ«п»“пє€Щ†вЂ¬
вЂ«вЂЄв€— 2вЂ¬вЂ¬
вЂ«вЂЄH2вЂ¬вЂ¬
вЂ«) вЂЄB в€€ B(HвЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄA H2 = AвЂ¬вЂ¬
вЂ«Щ€ ) вЂЄ A в€€ H (HвЂ¬п»“пє€Щ†вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄH2вЂ¬вЂ¬
вЂ«вЂЄA, BвЂ¬вЂ¬
вЂ«вЂЄ(H (H ),вЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«п»іпєёп»њп»ћ п»“п»ЂпєЋШЎШ§Щ‹ ШЈпєЈпєЋШЇп»іпєЋЩ‹вЂ¬
вЂ«Ш§п»џп»¤пє п» пєЄ вЂЄ ШЊ16вЂ¬Ш§п»џп»ЊпєЄШЇ вЂЄ ШЊ2007 ШЊ1вЂ¬Шµ ШµвЂЄ24-1 .вЂ¬вЂ¬
вЂ«Ш§пє‘пє¤пєЋШ« Ш§п»џп»ґпє®п»Јп»®Щѓ "пєіп» пєґп» пє” Ш§п»џп»Њп» п»®Щ… Ш§п»·пєіпєЋпєіп»ґпє” Щ€Ш§п»џп»¬п»ЁпєЄпєіп»ґпє”"вЂ¬
вЂ«пє—п»„пє’п»ґп»�пєЋШЄ п»“п»І п»“п»ЂпєЋШЎШ§ШЄ п»«п» п±ЄШЄ п»џп» п»¤пє†пє›пє®Ш§ШЄ Ш§п»џпєЁп»„п»ґпє” п»ЏпІ‘ Ш§піЊпє¤пєЄЩ€ШЇШ©вЂ¬
вЂ«ШЈпєЈп»¤пєЄ п»Јпє¤п»¤пєЄ п»‹пє п»®Ш±* Щ€ Ш±п»іп»Ёпє®вЂ¬
вЂ«пє‘п»ґп»њпє®ШЇ**вЂ¬
вЂ«пє—пєЋШ±п»іпє¦ п»—пє’п»®п»џп»ЄвЂЄ2006/10/19 :вЂ¬вЂ¬
вЂ«пє—пєЋШ±п»іпє¦ Ш§пєіпє�п»јЩ… Ш§п»џпє’пє¤пєљвЂЄ2004/8/30 :вЂ¬вЂ¬
вЂ«п»Јп» пєЁпєєвЂ¬
вЂ«п»“п»І п»«пє¬Щ‡ Ш§п»џп»¤п»�пєЋп»џпє” п»§п»Њпє®ЩЃ ) вЂЄ H 2 (HвЂ¬п»›пєјп»’ п»ЈЩЂп»¦ Ш§п»џп»¤ЩЂпє†пє›пє®Ш§ШЄ Ш§п»џпєЁп»„п»ґЩЂпє” п»Џп»ґЩЂпє® Ш§п»џп»¤пє¤ЩЂпєЄЩ€ШЇШ© Щ€ Ш§п»џп»¤п»ђп» п»�ЩЂпє”вЂЄ ШЊвЂ¬п»Јпє п»¤п»®п»‹ЩЂпє”вЂ¬
вЂ«пє—п»Њпє®п»іп»’ п»›п»ћ п»Јпє†пє›пє® п»Јп»¦ п»«пє¬Ш§ Ш§п»џпєјп»’ Щ€ п»Јпє п»¤п»®п»‹пє” пє—п»Њпє®п»іп»’ п»Јпє®Ш§п»“п»�п»Є п»›п»ћ п»Јп»Ёп»¬п»¤пєЋ п»›пєњп»ґп»”ЩЂпє” п»“ЩЂп»І п»“ЩЂп»ЂпєЋШЎ п»«п» пє’ЩЂпє®ШЄ вЂЄ HвЂ¬Щ€ пє—пє¤ЩЂп»®ЩЉвЂ¬
вЂ«п»‹п»ЁпєЋпє»ЩЂЩЂпє® Ш§п»џп»�пєЋп»‹ЩЂЩЂпєЄШ© п»“ЩЂЩЂп»І вЂЄ . HвЂ¬пє›ЩЂЩЂп»ў п»§ЩЂЩЂпєЄШ±Ші Ш§п»џпє�п»ЊпєЋп»ЈЩЂЩЂпєЄ Щ€ Ш§п»џп»”ЩЂЩЂп»ЂпєЋШЎШ§ШЄ Ш§п»џпє пє°пє‹п»ґЩЂЩЂпє” п»џп»”ЩЂЩЂп»ЂпєЋШЎ п»«п» пє’ЩЂЩЂпє®ШЄ ) вЂЄ H 2 (HвЂ¬Ш§п»џп»¤пє†п»џЩЂЩЂп»’ п»ЈЩЂЩЂп»¦вЂ¬
вЂ«п»Јпє п»¤п»®п»‹ЩЂЩЂпє” Ш§п»џп»¤ЩЂЩЂпє†пє›пє®Ш§ШЄ Ш§п»џпєЁп»„п»ґЩЂЩЂпє” п»Џп»ґЩЂЩЂпє® Ш§п»џп»¤пє¤ЩЂЩЂпєЄЩ€ШЇШ© )ШЈЩ€ Ш§п»џп»¤пє¤ЩЂЩЂпєЄЩ€ШЇШ© пє‘ЩЂЩЂпєёп»њп»ћ п»‹ЩЂЩЂпєЋЩ…(вЂЄ..вЂ¬пє›ЩЂЩЂп»ў п»§пє¤ЩЂЩЂпєґпєђ п»§п»€ЩЂЩЂп»ґп»ў п»«ЩЂЩЂпє¬Ш§ Ш§п»џп»¤ЩЂЩЂпє†пє›пє® п»ЈЩЂЩЂп»¦вЂ¬
вЂ«) вЂЄ . H 2 (HвЂ¬п»§п»Њпє’пє® п»‹п»¦ п»›п»ћ п»Јпє†пє›пє® п»ЈЩЂп»¦ ) вЂЄ H 2 (HвЂ¬п»›п»¤пє п»¤ЩЂп»®Ш№ пє§п»„ЩЂп»І п»џп»¤ЩЂпє†пє›пє®п»іп»¦ п»ЈЩЂп»¦ ) вЂЄ H 2 (HвЂ¬ШҐпєЈЩЂпєЄШ§п»«п»¤пєЋ п»Јпє¤ЩЂпєЄЩ€ШЇвЂ¬
вЂ«вЂЄ.вЂ¬пє›п»ў п»§пє’п»ґп»¦ ШЈЩ† ШЈЩЉ п»ЈЩЂпє†пє›пє® п»ЈЩЂп»¦ ) вЂЄ H 2 (HвЂ¬п»іп»Њпє’ЩЂпє® п»‹п»ЁЩЂпє” п»›п»¤пє п»¤ЩЂп»®Ш№ пє§п»„ЩЂп»І п»џп»Њп»ЁпєЋпє»ЩЂпє® Ш§п»џп»�пєЋп»‹ЩЂпєЄШ© п»“ЩЂп»І ) вЂЄ. H 2 (HвЂ¬Щ€ ШЈпє§п»ґЩЂпє®Ш§Щ‹вЂ¬
вЂ«п»§п»®пєџпєЄ п»‹п»јп»—пє” пє‘п»ґп»¦ Ш§п»џп»„п»ґп»’ Ш§п»џп»¤п»Ёп»”пєјп»ћ п»џп» п»¤пє†пє›пє® п»Јп»¦ ) вЂЄ H 2 (HвЂ¬Щ€ п»§п»€п»ґп»¤п»ЄвЂЄ .вЂ¬п»Јп»Љ пє‘п»Њпєѕ Ш§п»·п»Јпєњп» пє” ШЈп»іп»ЂпєЋЩ‹вЂЄ.вЂ¬вЂ¬
вЂ«п»Јп»�пєЄп»Јпє”вЂЄ :вЂ¬ШҐЩ† п»§п»€п»ґп»ў Ш§п»џп»¤пє†пє›пє® Ш§п»џпєЁп»„п»І Ш§п»џп»¤пє¤пєЄЩ€ШЇ п»» п»іп»¤п»њп»¦ пє—п»®пєіп»ґп»Њп»Є п»џп»Ёп»®пєџпєЄ п»Јп»Њп»ґпєЋШ±Ш§Щ‹ п»џп»¤пє†пє›пє® пє§п»„п»І п»Џп»ґпє® п»Јпє¤ЩЂпєЄЩ€ШЇ Щ€ п»›ЩЂпє¬п»џп»љвЂ¬
вЂ«ШҐЩ† Ш§п»џп»Ёп»€ЩЂЩЂп»ґп»ў Ш§п»џп»¤п»ЊЩЂЩЂпє®ЩЃ п»‹п» ЩЂЩЂп»° п»“ЩЂЩЂп»ЂпєЋШЎ п»«п» пє’ЩЂЩЂпє®ШЄ пє·ЩЂЩЂп»¤пє– Ш§п»џп»¤п»ЊЩЂЩЂпє®ЩЃ п»»пєЈп»�ЩЂпєЋЩ‹ п»» п»іЩЂЩЂпє†ШЇЩЉ Ш§п»џп»ђЩЂЩЂпє®Ш¶ ШЈп»іЩЂЩЂп»ЂпєЋЩ‹ пєЈпє�ЩЂЩЂп»° ШЈЩ† п»—п»ґп»¤ЩЂЩЂпє” п»§п»€ЩЂЩЂп»ґп»ўвЂ¬
вЂ«Ш§п»џп»¤п»®пє›пє® Ш§п»џпє¤п»ґпєЋШЇЩЉ п»“п»ґп»Є п»Џп»ґпє® п»Јп»Ёпє�п»¬п»ґпє”вЂЄ.вЂ¬вЂ¬
вЂ«п»џпє¬Ш§ ШЈЩ€пєџпєЄп»§пєЋ п»“п»ЂпєЋШЎ п»«п» пє’пє®ШЄ п»Јп»¦ Ш§п»џп»¤пє†пє›пє®Ш§ШЄ Ш§п»џпєЁп»„п»ґЩЂпє” п»Џп»ґЩЂпє® Ш§п»џп»¤пє¤ЩЂпєЄЩ€ШЇШ© пєїЩЂп»¤п»¦ пє·ЩЂпє®Щ€Ш· пєіпє�п»”пє®пєїЩЂп»¬пєЋ п»џЩЂп»Ђпє®Щ€Ш±Ш© Щ€ пєџЩЂп»®ШЇ Щ€вЂ¬
вЂ«ШҐпє—п»¤пєЋЩ… Ш§п»џп»”п»ЂпєЋШЎвЂЄ.вЂ¬вЂ¬
вЂ«вЂЄ - 1вЂ¬пє—п»¤п»¬п»ґпєЄвЂЄ :вЂ¬п»џп»ґп»њп»¦ вЂЄ HвЂ¬п»“п»ЂпєЋШЎ п»«п» пє’пє®ШЄ п»—пєЋпє‘п»ћ п»џп» п»”пєјп»ћ п»Јп»�пє®Щ€п»§пєЋЩ‹ пє‘пє¤п»�п»ћ пє—ЩЂпєЋЩ… вЂЄ KвЂ¬Щ€ вЂЄ {ei }iв€€I вЉ‚ HвЂ¬п»—пєЋп»‹ЩЂпєЄШ© п»“п»ґЩЂп»Є Щ€ вЂЄIвЂ¬вЂ¬
вЂ«п»Јпє п»¤п»®п»‹ЩЂЩЂпє” ШЈШЇп»џЩЂЩЂпє” п»—пєЋпє‘п» ЩЂЩЂпє” п»џп» п»ЊЩЂЩЂпєЄ п»‹п» ЩЂЩЂп»° Ш§п»·п»›пєњЩЂЩЂпє®вЂЄ .вЂ¬п»§пє®п»ЈЩЂЩЂпє° п»џп»”ЩЂЩЂп»ЂпєЋШЎ Ш§п»џп»¤ЩЂЩЂпє†пє›пє®Ш§ШЄ Ш§п»џпєЁп»„п»ґЩЂЩЂпє” Ш§п»џп»¤пє¤ЩЂЩЂпєЄЩ€ШЇШ© пє‘ЩЂЩЂпєЋп»џпє®п»Јпє°вЂЄ . B (H ) :вЂ¬п»§ЩЂЩЂпє¬п»›пє®вЂ¬
вЂ«пє‘п»”п»ЂпєЋШЎ п»«п» пє’пє®ШЄ пє·п»¤пє–вЂЄ:вЂ¬вЂ¬
вЂ«)вЂЄ(1вЂ¬вЂ¬
вЂ«пє‘пєЋп»‹пє�пє’пєЋШ±вЂ¬
вЂ«вЂЄ{ei }iв€ћ=1вЂ¬вЂ¬
вЂ«вЂЄпЈјвЂ¬вЂ¬
вЂ«вЂЄ< в€ћпЈЅвЂ¬вЂ¬
вЂ«вЂЄпЈѕвЂ¬вЂ¬
вЂ«вЂЄ2вЂ¬вЂ¬
вЂ«вЂЄHвЂ¬вЂ¬
вЂ«вЂЄпЈ±вЂ¬вЂ¬
вЂ«вЂЄHS := пЈІ A в€€ B(H ) : в€‘ AeiвЂ¬вЂ¬
вЂ«вЂЄi =1вЂ¬вЂ¬
вЂ«вЂЄпЈівЂ¬вЂ¬
вЂ«в€ћвЂ¬
вЂ«п»—пєЋп»‹пєЄШ© п»“п»І вЂЄ HвЂ¬Щ€вЂ¬
вЂ«В© пєџп»¤п»ґп»Љ Ш§п»џпє¤п»�п»®Щ‚ п»Јпє¤п»”п»®п»‡пє” п»џпє пєЋп»Јп»Њпє” Ш§п»џп»ґпє®п»Јп»®Щѓ вЂЄ.2007вЂ¬вЂ¬
вЂ«* п»›п» п»ґпє” Ш§п»џп»¬п»ЁпєЄпєіпє” Ш§п»џп»њп»¬пє®пє‘пєЋпє‹п»ґпє” Щ€Ш§п»№п»џп»њпє�пє®Щ€п»§п»ґпє”вЂЄ ШЊвЂ¬пєџпєЋп»Јп»Њпє” пєЈп» пєђвЂЄ ШЊвЂ¬пєіп»®Ш±п»іпєЋвЂЄ.вЂ¬вЂ¬
вЂ«** п»‹п»¤п»ґпєЄ п»›п» п»ґпє” Ш§п»џпє�пє¤п» п»ґп»ћ п»“п»І Ш§п»џпє пєЋп»Јп»Њпє” Ш§п»џпє�п»�п»Ёп»ґпє” пє‘п»¤пєЄп»іп»Ёпє” ШЇШ±ШІШЇЩ†вЂЄ.вЂ¬вЂ¬
вЂ«п»Јпє¤пє�п»®п»іпєЋШЄ Ш§п»џп»ЊпєЄШЇвЂ¬
1
вЂ«ШЈпєЈп»¤пєЄ п»Јпє¤п»¤пєЄ п»‹пє п»®Ш± Щ€ Ш±п»іп»Ёпє® пє‘п»ґп»њпє®ШЇвЂ¬
вЂ«Ш§п»џпє’пє¤п»®Ш« пє‘пєЋп»џп» п»ђпє” Ш§п»џп»Њпє®пє‘п»ґпє”вЂ¬
вЂ«пє—п»„пє’п»ґп»�пєЋШЄ п»“п»І п»“п»ЂпєЋШЎШ§ШЄ п»«п» пє’пє®ШЄ п»џп» п»¤пє†пє›пє®Ш§ШЄ Ш§п»џпєЁп»„п»ґпє” п»Џп»ґпє®вЂ¬
*
вЂ«Ш§п»џп»¤пє¤пєЄЩ€ШЇШ©вЂ¬
вЂ«Ш§п»џпє’пє¤п»®Ш« пє‘пєЋп»џп» п»ђпє” Ш§п»»п»§пє п» п»ґпє°п»іпє”вЂ¬
*
Use of X-ray Diffraction and Mud Master
Program in the Evaluation of Crystallite Size and
Size Distributions of Illite and Kaolinite at Various
Depths of Rock Formations in Jordan
Ibrahim M. Odeh, Waleed A.
Saqqa and Denis Eberl
1
*
Assessment of Background Radiation Dose due to
Radon in Kharja Village, Irbid Governorate,
Jordan
Khitam M. Khasawinah and
Khalid M. Abumurad
19
*
Determination of the Specific Activities of
Radioactive Nuclides of Surface Soils of Some
HadhramoutвЂ™s Valleys in Yemen Using GammaRay Spectrometry
Awad Ali Bin- Jaza and
Abdulazize Omer Bazohair
27
*
Origin and Migration of Primordial Germ Cells in
Mosquitofish, Gambusia affinis
Janti S. Qar and Mohammad
A. Al-Adhami
39
*
The Role of Selenium and Lycopene in the
Protection of Mice Against Microcystin Toxicity
Saad Al-Jassabi and Ahmad
Khalil
49
*
Geoeletrical and Hydrochemical Investigations in
the Hallabat Area, North Jordan
Ghunaim S. Nasher, Hakam
A. Mustafa and Nabil
Abderahman
59
*
Environmental Impact Assessment of Hiswa Clay
Deposits, South Jordan
Talal Al-Momani
93
*
Performance Evaluation of One UMTS Cell by the
New Modelling Specification and Evaluation
Language MOSEL-2
Aymen Issa Zreikat
105
*
A Generic Parallel Map: Extending the List Map
Higher Order Function to Arbitrary Data
Structures
Mohammad M. Hamdan
123
*
Effect of Access Point Moving on the Wireless
Network Performance Using Opnet Modeler
Sameh H. Ghwanmeh
135
*
New Techniques for Improving H.263
Videoconferencing
Ammar M. Kamel and Moh'd
Belal. Al-Zoubi
147
вЂ«Ш§п»џпє�п»®пє›п»ґп»–вЂЄ:вЂ¬вЂ¬
вЂ«ШЈвЂЄ -вЂ¬пє—п»®пє›п»ґп»– Ш§п»џп»¤пє®Ш§пєџп»Љ Щ€Ш§п»џп»¤пєјпєЋШЇШ± Ш§п»џп»¤п»Ёпєёп»®Ш±Ш©вЂЄ :вЂ¬п»іпє�п»ў Ш°п»џп»љ ШЇШ§пє§п»ћ Ш§п»џп»¤пє�п»¦ пє‘пєЋпєіпє�пєЁпєЄШ§Щ… п»§п»€пєЋЩ… Ш§п»џпє�пє®п»—п»ґп»ў ]вЂЄ[ 1вЂ¬вЂ¬
вЂ«пє‘п»ґп»¦ п»—п»®пєіп»ґп»¦вЂЄ.вЂ¬вЂ¬
вЂ«вЂў пє—п»ЊпєЄ п»—пєЋпє‹п»¤пє” пє‘пєЋп»џп»¤ЩЂпєјпєЋШЇШ± Щ€Ш§п»џп»¤пє®Ш§пєџЩЂп»Љ Ш§п»џп»¤п»ЁЩЂпєёп»®Ш±Ш© п»“ЩЂп»І п»§п»¬пєЋп»іЩЂпє” Ш§п»џпє’пє¤ЩЂпєљ пєЈЩЂпєґпєђ Ш§п»џпє�ЩЂпє®п»—п»ґп»ў Ш§п»џЩЂп»®Ш§Ш±ШЇ п»“ЩЂп»ІвЂ¬
вЂ«Ш§п»џп»ЁпєєвЂЄ.вЂ¬вЂ¬
вЂ«ШҐШ°Ш§ п»›пєЋЩ† Ш§п»џп»¤пє®пєџп»Љ п»›пє�пєЋпє‘пєЋЩ‹ п»іп»њпє�пєђ п»«п»њпє¬Ш§вЂЄ:вЂ¬вЂ¬
вЂ«] [ п»‹ЩЂЩЂпє°Щ€ШІвЂЄ ШЊвЂ¬ШЇШ±п»іЩЂЩЂпєЄвЂЄ .вЂ¬п»Јп»ґп»њпєЋп»§п»ґЩЂЩЂп»љ Ш§п»џп»¤п»®Ш§пє‹ЩЂЩЂп»ЉвЂЄ ШЊвЂ¬п»ЈпєЄп»іпє®п»іЩЂЩЂпє” Ш§п»џп»њпє�ЩЂЩЂпєђ Щ€Ш§п»џп»¤п»„пє’п»®п»‹ЩЂЩЂпєЋШЄвЂЄ ШЊвЂ¬пєџпєЋп»Јп»ЊЩЂЩЂпє” пєЈп» ЩЂЩЂпєђвЂЄ ШЊвЂ¬пєЈп» ЩЂЩЂпєђвЂЄШЊвЂ¬вЂ¬
вЂ«вЂЄ.1979вЂ¬вЂ¬
вЂ«Щ€Ш§Ш°Ш§ п»›пєЋЩ† Ш§п»џп»¤пє®пєџп»Љ пє‘пє¤пєњпєЋЩ‹ п»“п»І ШЇЩ€Ш±п»іпє” п»іп»њпє�пєђ п»«п»њпє¬Ш§вЂЄ:вЂ¬вЂ¬
вЂ«] [ п»Јпє¤п»¤пєЄвЂЄ ШЊвЂ¬п»§п»®Ш±ЩЉ пє»пєЋпє‘пє®вЂЄ ШЊвЂ¬п»—п»ґпєЋШі пєЈЩЂпєґпєЋпєіп»ґпє” Ш§п»џпє�пєЁпє®п»іЩЂпє° ШЈпє›п»ЁЩЂпєЋШЎ п»›ЩЂпєґпє® Ш§п»џп»”ЩЂп»®п»»Ш° Ш§п»џп»њпє®пє‘ЩЂп»®п»§п»ІвЂЄ ШЊвЂ¬ШЇШ±Ш§пєіЩЂпєЋШЄвЂЄ:вЂ¬вЂ¬
вЂ«Ш§п»џп»Њп» п»®Щ… Ш§п»џп»¬п»ЁпєЄпєіп»ґпє” Щ€Ш§п»џпє�п»њп»Ёп»®п»џп»®пєџп»ґпє”вЂЄ ШЊвЂ¬п»ЈпєћвЂЄ ШЊ12вЂ¬Ш№вЂЄ ШЊ1985 ШЊ9вЂ¬Шµ вЂЄ.24вЂ¬вЂ¬
вЂ«Щ€Ш§Ш°Ш§ п»›пєЋЩ† Ш§п»џп»¤пє®пєџп»Љ п»Јп»�пєЋп»џпє” ШЈЩ€ п»“пєјп»јЩ‹ п»“п»І п»›пє�пєЋШЁ п»іп»њп»®Щ† п»›пєЋп»џпє�пєЋп»џп»ІвЂЄ:вЂ¬вЂ¬
вЂ«] [ п»‹пє’ЩЂпєЄ Ш§п»џпє¤ЩЂпєЋп»“п»†вЂЄ ШЊвЂ¬пєіЩЂпєЋп»Јп»І пє§ЩЂп»Ђпє®вЂЄ .вЂ¬Ш§п»џпє�ЩЂпєёпєЁп»ґпєє Ш§п»џп»¤пєЁпє’ЩЂпє®ЩЉ п»џп» п»„п»”п»ґп» п»ґЩЂпєЋШЄвЂЄ ШЊвЂ¬п»ѓЩЂпє®Щ‚ ШІШ±Ш№ Ш§п»џп»„п»”п»ґп» п»ґЩЂпєЋШЄвЂЄШЊвЂ¬вЂ¬
вЂ«Ш§Ш±пє‘пєЄвЂЄ ШЊвЂ¬пєџпєЋп»Јп»Њпє” Ш§п»џп»ґпє®п»Јп»®ЩѓвЂЄ ШЊ1986 ШЊвЂ¬Шµ вЂЄ.181-175вЂ¬вЂ¬
вЂ«ШЁвЂЄ -вЂ¬пє—п»®пє›п»ґЩЂЩЂп»– Ш§п»џп»¬ЩЂЩЂп»®Ш§п»Јпє¶ Щ€Ш§п»џп»¤ЩЂЩЂпєјпєЋШЇШ± п»Џп»ґЩЂЩЂпє® Ш§п»џп»¤п»ЁЩЂЩЂпєёп»®Ш±Ш©вЂЄ :вЂ¬п»іЩЂЩЂпє�п»ў Ш°п»џЩЂЩЂп»љ п»“ЩЂЩЂп»І Ш§п»џп»¤ЩЂЩЂпє�п»¦ пє‘пє€пє›пє’ЩЂЩЂпєЋШЄ п»›п» п»¤ЩЂЩЂпє” )п»«ЩЂЩЂпєЋп»Јпє¶(вЂ¬
вЂ«п»Јпє�пє’п»®п»‹ЩЂЩЂпє” пє‘ЩЂЩЂпєЋп»џпє®п»—п»ў Ш§п»џп»¤пє�пєґп» ЩЂЩЂпєґп»ћ п»џп» п»¬ЩЂЩЂпєЋп»Јпє¶ ШЇШ§пє§ЩЂЩЂп»ћ п»—п»®пєіЩЂЩЂп»ґп»¦вЂЄ ШЊвЂ¬п»«п»њЩЂЩЂпє¬Ш§вЂЄ) :вЂ¬п»«ЩЂЩЂпєЋп»Јпє¶вЂЄ .(1вЂ¬Щ€пє—ЩЂЩЂпє¬п»›пє® Ш§п»џп»¤п»Њп» п»®п»ЈЩЂЩЂпєЋШЄвЂ¬
вЂ«Ш§п»џпє�п»”пєјп»ґп» п»ґпє” п»џп»њп»ћ п»«пєЋп»Јпє¶ п»“п»І п»§п»¬пєЋп»іпє” Ш§п»џпє’пє¤пєљ пє—пє¤пє– п»‹п»Ёп»®Ш§Щ† Ш§п»џп»¬п»®Ш§п»Јпє¶ Щ€п»—пє’п»ћ п»—пєЋпє‹п»¤пє” Ш§п»џп»¤пє®Ш§пєџп»ЉвЂЄ:вЂ¬вЂ¬
вЂ«п»«пєЋп»Јпє¶вЂЄ :1вЂ¬пє§п»Ђпє®вЂЄ ШЊвЂ¬пє‘пєґпєЋЩ… Ш±Ш§пє·ЩЂпєЄвЂЄ ШЊвЂ¬п»—п»ґЩЂпєЋШі пє—пє®п»›п»ґЩЂпє° Ш§п»џЩЂпє®Ш§ШЇЩ€Щ†вЂЄ 222вЂ¬п»“ЩЂп»І п»«ЩЂп»®Ш§ШЎ п»Јп»ЁЩЂпєЋШІЩ„ п»ЈпєЄп»іп»ЁЩЂпє” Ш§Ш±пє‘ЩЂпєЄвЂЄ ШЊвЂ¬Ш§п»·Ш±ШЇЩ†вЂЄШЊвЂ¬вЂ¬
вЂ«Ш±пєіпєЋп»џпє” п»ЈпєЋпєџпєґпє�п»ґпє® п»Џп»ґпє® п»Јп»Ёпєёп»®Ш±Ш©вЂЄ ШЊвЂ¬пєџпєЋп»Јп»Њпє” Ш§п»џп»ґпє®п»Јп»®Щѓ вЂЄ ШЊ1990вЂ¬Шµ вЂЄ.11-8вЂ¬вЂ¬
вЂ«п»Јпє®Ш§пєџп»ЊпєЋШЄ Ш§п»џп»њпє�пєђвЂЄ :вЂ¬пє—п»�пє’п»ћ п»џп» п»Ёпєёпє® п»“п»І Ш§п»џп»¤пє п» пє” п»Јпє®Ш§пєџп»ЊпєЋШЄ Ш§п»џп»њпє�пєђ Ш§п»џпє¤пєЄп»іпєњпє” Ш§п»џп»�п»ґп»¤пє”вЂЄ.вЂ¬вЂ¬
вЂ«Ш§п»џпє�пєјпє®ЩЃвЂЄ :вЂ¬п»іпє¤п»– п»џпє®пє‹п»ґпєІ Ш§п»џпє�пє¤пє®п»іпє® ШҐпєџпє®Ш§ШЎ Ш§п»џпє�п»ђп»ґп»ґпє®Ш§ШЄ Ш§п»џпє�п»І п»іпє®Ш§п»«пєЋ пєїпє®Щ€Ш±п»іпє” п»·п»Џпє®Ш§Ш¶ Ш§п»џпєјп»ґпєЋп»Џпє”вЂЄ.вЂ¬вЂ¬
вЂ«Ш§п»џп»¤пєґпє�п»јШЄвЂЄ :вЂ¬п»іп»¤п»Ёпєў п»›п»ћ п»Јп»¦ п»іп»ЁЩЂпєёпє® пє‘пє¤пєњЩЂп»Є п»§ЩЂпєґпєЁпє” Щ€Ш§пєЈЩЂпєЄШ© п»ЈЩЂп»¦ п»‹ЩЂпєЄШЇ Ш§п»џп»¤пє п» ЩЂпє” Ш§п»џЩЂпє¬ЩЉ п»іп»ЁЩЂпєёпє® п»“п»ґЩЂп»Є Ш§п»џпє’пє¤ЩЂпєљвЂ¬
вЂ«пє‘пєЋп»№пєїпєЋп»“пє” ШҐп»џп»° п»‹пєёпє®п»іп»¦ п»Јпєґпє�п» пє” п»Јп»Ёп»ЄвЂЄ.вЂ¬вЂ¬
вЂ«пє—пє®пєіп»ћ Ш§п»џпє’пє¤п»®Ш« Щ€Ш§п»џп»¤пє®Ш§пєіп»јШЄ ШҐп»џп»°вЂ¬
вЂ«Ш±пє‹п»ґпєІ пє—пє¤пє®п»іпє® п»Јпє п» пє”вЂ¬
вЂ«ШЈпє‘пє¤пєЋШ« Ш§п»џп»ґпє®п»Јп»®Щѓ "пєіп» пєґп» пє” Ш§п»џп»Њп» п»®Щ… Ш§п»·пєіпєЋпєіп»ґпє” Щ€Ш§п»џп»¬п»ЁпєЄпєіп»ґпє”"вЂ¬
вЂ«п»‹п»¤пєЋШЇШ© Ш§п»џпє’пє¤пєљ Ш§п»џп»Њп» п»¤п»І Щ€Ш§п»џпєЄШ±Ш§пєіпєЋШЄ Ш§п»џп»Њп» п»ґпєЋвЂ¬
вЂ«пєџпєЋп»Јп»Њпє” Ш§п»џп»ґпє®п»Јп»®ЩѓвЂ¬
вЂ«Ш§Ш±пє‘пєЄвЂЄ-вЂ¬Ш§п»џп»¤п»¤п» п»њпє” Ш§п»·Ш±ШЇп»§п»ґпє” Ш§п»џп»¬пєЋпє·п»¤п»ґпє”вЂ¬
вЂ«п»іп»¤п»њп»¦ Ш§п»џпє¤ЩЂпєјп»®Щ„ п»‹п» ЩЂп»° ШЈпє‘пє¤ЩЂпєЋШ« Ш§п»џп»ґпє®п»ЈЩЂп»®Щѓ п»ЈЩЂп»¦ п»—ЩЂпєґп»ў Ш§п»џпє�пє’ЩЂпєЋШЇЩ„ п»“ЩЂп»І п»Јп»њпє�пє’ЩЂпє” пєџпєЋп»Јп»ЊЩЂпє” Ш§п»џп»ґпє®п»ЈЩЂп»®ЩѓвЂЄ ШЊвЂ¬ШЈЩ€ п»‹п»¤ЩЂпєЋШЇШ©вЂ¬
вЂ«Ш§п»џпє’пє¤пєљ Ш§п»џп»Њп» п»¤п»І Щ€Ш§п»џпєЄШ±Ш§пєіпєЋШЄ Ш§п»џп»Њп» п»ґпєЋ п»џп»�пєЋШЎ ШЇп»іп»ЁпєЋШ± п»џп» п»ЁпєґпєЁпє” Ш§п»џп»®Ш§пєЈпєЄШ©вЂЄ.вЂ¬вЂ¬
вЂ«Ш§п»»пє·ЩЂЩЂпє�пє®Ш§Щѓ Ш§п»џЩЂЩЂпєґп»Ёп»®ЩЉвЂЄ :вЂ¬ШЇп»іп»ЁЩЂЩЂпєЋШ±Ш§Щ† Щ€п»§ЩЂЩЂпєјп»’ п»“ЩЂЩЂп»І Ш§п»·Ш±ШЇЩ†вЂЄ ШЊвЂ¬Щ€пє›п»¤пєЋп»§п»ґЩЂЩЂпє” ШЇп»§ЩЂЩЂпєЋп»§п»ґпє® ШЈЩ€ Ш§пє›п»ЁЩЂЩЂп»І п»‹ЩЂЩЂпєёпє® ШЇЩ€п»»Ш±Ш§Щ‹ ШЈп»Јпє®п»іп»њп»ґ ЩЂпєЋЩ‹ п»“ЩЂЩЂп»ІвЂ¬
вЂ«Ш§п»џп»®п»ѓп»¦ Ш§п»џп»Њпє®пє‘п»ІвЂЄ ШЊвЂ¬Щ€пє›п»¤пєЋп»§п»ґпє” п»‹пєёпє® ШЇЩ€п»»Ш±Ш§Щ‹ ШЈп»Јпє®п»іп»њп»ґпєЋЩ‹ ШЈЩ€ п»ЈпєЋ п»іп»ЊпєЋШЇп»џп»¬пєЋ п»“п»І Ш§п»џпє’п» пєЄШ§Щ† Ш§п»·пє§пє®Щ‰вЂЄ.вЂ¬вЂ¬
вЂ«пє‘Щђпєґп»ў пєЌп»џп» п± п»Є пєЌп»џпє®п№јпєЈп№ѕп»¤п»¦п№ѕ пєЌп»џпє®п№јпєЈп№єп»ґп»ўп№ѕвЂ¬
вЂ«ШЈпє‘пє¤пєЋШ« Ш§п»џп»ґпє®п»Јп»®ЩѓвЂ¬
вЂ«пєіп» пєґп» пє” Ш§п»џп»Њп» п»®Щ… Ш§п»·пєіпєЋпєіп»ґпє” Щ€Ш§п»џп»¬п»ЁпєЄпєіп»ґпє”вЂ¬
вЂ«п»Јпє п» пє” п»‹п» п»¤п»ґпє” п»Јпє¤п»њЩ‘п»¤пє”вЂ¬
вЂ«пє—п»Ёпєёпє® Ш§п»џп»¤пє п» пє” Ш§п»џпє’пє¤п»®Ш« Ш§п»·пє»п» п»ґпє” Ш§п»џпє�п»І пє—пє�п»®Ш§п»“пє® п»“п»ґп»¬пєЋвЂ¬
вЂ«Ш§п»џп»¤п»Ёп»¬пє п»ґпє” Ш§п»џпєґп» п»ґп»¤пє”вЂЄ ШЊвЂ¬Щ€Ш§п»џпє�п»І п»џп»ў пє—п»�пєЄЩ… п»џп» п»Ёпєёпє® п»“п»І ШЈЩЉ п»Јп»њпєЋЩ† Шўпє§пє®вЂ¬
вЂ«п»—п»®Ш§п»‹пєЄ Щ€ШҐпєџпє®Ш§ШЎШ§ШЄ Ш§п»џп»Ёпєёпє®вЂ¬
вЂ«вЂў Ш§п»џп» п»ђЩЂЩЂпє”вЂЄ :вЂ¬пє—п»њпє�ЩЂЩЂпєђ Ш§п»џпє’пє¤ЩЂЩЂп»®Ш« пє‘пєЋп»џп» п»ђЩЂЩЂпє” Ш§п»џп»Њпє®пє‘п»ґЩЂЩЂпє” ШЈЩ€ пє‘пєЋп»џп» п»ђЩЂЩЂпє” Ш§п»№п»§пє п» п»ґпє°п»іЩЂЩЂпє” Щ€п»» пє—ЩЂЩЂпєґпє�п» п»ў Ш§п»џпє’пє¤ЩЂЩЂп»®Ш« пє‘п»ђп»ґЩЂЩЂпє®вЂ¬
вЂ«п»«пєЋпє—п»ґп»¦ Ш§п»џп» п»ђпє�п»ґп»¦вЂЄ.вЂ¬вЂ¬
вЂ«вЂў пє—п»�пєЄп»іп»ў Ш§п»џпє’пє¤п»®Ш«вЂЄ :вЂ¬пє—п»�пєЄЩ… Ш§п»џпє’пє¤п»®Ш« п»“ЩЂп»І ШЈШ±пє‘ЩЂп»Љ п»§ЩЂпєґпє¦ п»Јп»„пє’п»®п»‹ЩЂпє” пє‘п»”пє®Ш§п»ЏЩЂпєЋШЄ п»Јпє°ШЇЩ€пєџЩЂпє” Щ€п»‹п» ЩЂп»° Щ€пєџЩЂп»ЄвЂ¬
вЂ«Щ€Ш§пєЈпєЄвЂЄ ШЊвЂ¬Щ€п»«п»®Ш§п»Јпє¶ пєЈпє п»ў Ш§п»џп»®Ш§пєЈпєЄ п»Јп»Ёп»¬пєЋ вЂЄ2.5вЂ¬пєіп»ўвЂЄ.вЂ¬вЂ¬
вЂ« п»іпє ЩЂЩЂпєђ ШЈЩ† п»» п»іпє°п»іЩЂЩЂпєЄ п»‹ЩЂЩЂпєЄШЇ пє»ЩЂЩЂп»”пє¤пєЋШЄ Ш§п»џпє’пє¤ЩЂЩЂпєљ пє‘п»¤ЩЂЩЂпєЋ п»“ЩЂЩЂп»І Ш°п»џЩЂЩЂп»љ Ш§п»»пє·ЩЂЩЂп»њпєЋЩ„ Щ€Ш§п»џпє®пєіЩЂЩЂп»®Щ… Щ€Ш§п»џп»¤пє®Ш§пєџЩЂЩЂп»ЉвЂ¬вЂ«Щ€Ш§п»џпє пєЄШ§Щ€Щ„ Щ€Ш§п»џп»¤п»јпєЈп»– п»‹п»¦ )вЂЄ (30вЂ¬пє»п»”пє¤пє”вЂЄ.вЂ¬вЂ¬
вЂ« п»іп»�пєЄЩ… Ш§п»џпє’пєЋпєЈпєљ п»Јп» пєЁпєјп»ґп»¦ п»џпє’пє¤пєњп»Є п»“п»І пє»п»”пє¤пє�п»ґп»¦ п»Јп»Ёп»”пєјп» пє�п»ґп»¦ ШЈпєЈпєЄп»«п»¤пєЋ пє‘пєЋп»џп» п»ђпє” Ш§п»џп»Њпє®пє‘п»ґпє” Щ€Ш§п»µпє§пє®вЂ¬вЂ«пє‘пєЋп»џп» п»ђпє” Ш§п»№п»§пє п» п»ґпє°п»іпє” п»“п»І п»ЈпєЋ п»» п»іпє°п»іпєЄ п»‹п» п»° вЂЄ 200вЂ¬п»›п» п»¤пє” п»џп»њп»ћ п»Јп»Ёп»¬п»¤пєЋвЂЄ.вЂ¬вЂ¬
вЂ« п»іп»њпє�пєђ п»‹п»Ёп»®Ш§Щ† Ш§п»џпє’пє¤пєљ Щ€Ш§пєіп»ў Ш§п»џп»¤пє†п»џп»’ Щ€Ш±пє—пє’пє�ЩЂп»Є Ш§п»џп»Њп» п»¤п»ґЩЂпє” Щ€Ш§п»џп»¤пє†пєіЩЂпєґпє” Ш§п»џпє�ЩЂп»І п»іп»Њп»¤ЩЂп»ћ пє‘п»¬ЩЂпєЋ п»‹п» ЩЂп»°вЂ¬вЂ«пє»п»”пє¤пє” п»Јп»Ёп»”пєјп» пє”вЂЄ ШЊвЂ¬пє›п»ў п»іп»њпє�пєђ п»‹п»Ёп»®Ш§Щ† Ш§п»џпє’пє¤пєљ п»Јпє®Ш© ШЈпє§пє®Щ‰ п»‹п» ЩЂп»° Ш§п»џЩЂпєјп»”пє¤пє” Ш§п»·Щ€п»џЩЂп»° п»ЈЩЂп»¦ Ш§п»џпє’пє¤ЩЂпєљвЂ¬
вЂ«Щ€п»‹п» п»° пє»п»”пє¤пє” п»›п»ћ п»Јп» пєЁпєєвЂЄ.вЂ¬вЂ¬
вЂ« п»іп»�пєЄЩ… Ш§п»џпє’пє¤пєљ пє‘п»ЊпєЄ Ш§п»џп»¤п»®Ш§п»“п»�пє” п»‹п» п»° п»§пєёпє®Щ‡ п»Јп»„пє’п»®п»‹пєЋЩ‹ Щ€п»Јпє¤п»”п»®п»‡пєЋЩ‹ п»‹п» п»° п»—ЩЂпє®Шµ п»›п»¤пє’п»ґЩЂп»®пє—пє® п»—п»ґЩЂпєЋШівЂ¬вЂ«вЂЄ 3.5вЂ¬ШҐп»§пє¶ п»Јпє�п»®Ш§п»“п»– п»Јп»Љ ШЈп»§п»€п»¤пє” )вЂЄIBM (Microsoft Word 2000, XPвЂ¬вЂ¬
вЂ«Ш§п»»пє·п»њпєЋЩ„ Щ€Ш§п»џпє®пєіп»®п»ЈпєЋШЄвЂЄ:вЂ¬вЂ¬
вЂ«вЂў пє—п»®пєїп»Љ Ш§п»·пє·п»њпєЋЩ„ Щ€Ш§п»џпє®пєіп»®п»ЈпєЋШЄ Щ€Ш§п»џпє пєЄШ§Щ€Щ„ п»“п»І п»§п»¬пєЋп»іпє” Ш§п»џпє’пє¤пєљ п»Јп»Љ Ш§п»№пє·пєЋШ±Ш© ШҐп»џп»° ШЈп»ЈпєЋп»›п»Ёп»¬пєЋ Ш§п»џп»¤п»ЁпєЋпєіЩЂпє’пє” п»“ЩЂп»ІвЂ¬
вЂ«Ш§п»џп»¤пє�п»¦вЂЄ ШЊвЂ¬Щ€Щ€пє»п»’ п»џп»¬пєЋ пє»п»”пє¤пє” п»Јп»Ёп»”пєјп» пє”вЂЄ.вЂ¬вЂ¬
вЂ«вЂў пє—п»�пєЄЩ… Ш§п»·пє·п»њпєЋЩ„ Щ€Ш§п»џпє®пєіп»®п»ЈпєЋШЄ Щ€Ш§п»џпє пєЄШ§Щ€Щ„ п»Јпє®пєіп»®п»Јпє” пє‘пєЋп»џпє¤пє’пє® Ш§п»·пєіп»®ШЇ п»‹п» п»° Щ€Ш±Щ‚ пє·ЩЂп»”пєЋЩЃ вЂЄ(TracingвЂ¬вЂ¬
вЂ«)вЂЄ ШЊPaperвЂ¬Щ€п»“ЩЂЩЂЩЂЩЂп»І пє»ЩЂЩЂЩЂЩЂп»”пє¤пєЋШЄ п»Јп»Ёп»”ЩЂЩЂЩЂЩЂпєјп» пє” пє‘пє¤п»ґЩЂЩЂЩЂЩЂпєљ п»» пє—пє�пє ЩЂЩЂЩЂЩЂпєЋЩ€ШІ ШЈпє‘п»ЊЩЂЩЂЩЂЩЂпєЋШЇ п»«ЩЂЩЂЩЂЩЂпє¬Щ‡ Ш§п»»пє·ЩЂЩЂЩЂЩЂп»њпєЋЩ„ Щ€Ш§п»џпє®пєіЩЂЩЂЩЂЩЂп»®п»ЈпєЋШЄвЂ¬
вЂ«)вЂЄ19вЂ¬пєіп»ўГ—вЂЄ12вЂ¬пєіп»ў(вЂЄ.вЂ¬вЂ¬
вЂ«вЂў ШЈпєіп»¤пєЋШЎ Ш§п»·п»‹п»јЩ… Ш§п»·пєџп»Ёпє’п»ґпє”вЂЄ :вЂ¬п»‹п»ЁпєЄ Щ€Ш±Щ€ШЇ ШЈпєіп»¤пєЋШЎ ШЈп»‹п»јЩ… ШЈпєџп»Ёпє’п»ґпє” п»“п»І Ш§п»џпє’пє¤п»®Ш« Ш§п»џп»¤п»�пєЄп»Јпє” пє‘пєЋп»џп»Њпє®пє‘п»ґпє”вЂ¬
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вЂ«Ш§п»џпєЄп»›пє�ЩЂЩЂЩЂЩЂЩЂЩЂЩЂЩЂЩЂЩЂЩЂЩЂЩЂЩЂЩЂЩЂЩЂЩЂЩЂЩЂЩЂЩЂп»®Ш± п»Јпє¤п»¤пєЄ пє‘п»јЩ„ Ш§п»џпє°п»‹пє’п»ІвЂ¬
вЂ«пєіп»њпє®пє—пІ‘ Ш§п»џпє�пє¤пє®п»іпє®вЂ¬
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вЂ«В© пєџп»¤п»ґп»Љ пєЈп»�п»®Щ‚ Ш§п»џп»„пє’п»Љ п»Јпє¤п»”п»®п»‡пє” п»џпє пєЋп»Јп»Њпє” Ш§п»џп»ґпє®п»Јп»®Щѓ вЂЄ2007вЂ¬вЂ¬
вЂ«п»» п»іпє п»®ШІ п»§пєёпє® ШЈЩЉ пєџпє°ШЎ п»Јп»¦ п»«пє¬Щ‡ Ш§п»џп»¤пє п» пє” ШЈЩ€ Ш§п»—пє�пє’пєЋпєіп»Є ШЇЩ€Щ† Ш§п»џпє¤пєјп»®Щ„ п»‹п» п»°вЂ¬
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вЂ«Ш§п»џп»¤пє®Ш§пєџп»Њп»Є Ш§п»џп» п»ђп»®п»іпє”вЂ¬
вЂ«Ш§п»»пєіпє�пєЋШ° Ш§п»џпєЄп»›пє�п»®Ш± Ш§пє‘пє®Ш§п»«п»ґп»ў пєџпє’пє®п»іп»ћвЂ¬
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вЂ«ШЈпєЈп»¤пєЄ ШЈпє‘п»®п»«п»¤пєЋЩ… Щ€п»Јпє пєЄЩЉ Ш§п»џпєёп»ЁпєЋЩ‚вЂ¬
вЂ«вЂЄISSN 1023-0149вЂ¬вЂ¬
вЂ«ШЈпє‘пє¤ЩЋЩЂпєЋШ« Ш§п»џп»ґЩЋпє®п»ЈЩЂп»®ЩѓвЂ¬
вЂ«пєіп» пєґп» пє” Ш§п»џп»Њп» п»®Щ… Ш§п»·пєіпєЋпєіп»ґпє” Щ€Ш§п»џп»¬п»ЁпєЄпєіп»ґпє”вЂ¬
вЂ«п»Јпє п» пє” п»‹п» п»¤п»ґпє” п»Јпє¤п»њЩ‘п»¤пє”вЂ¬
вЂ«Ш§п»џп»¤пє п» пєЄ Ш§п»џпєґпєЋШЇШі п»‹пєёпє®вЂ¬
вЂ«Ш§п»џп»ЊпєЄШЇ Ш§п»·Щ€Щ„вЂ¬
вЂ«вЂЄ1428вЂ¬п»«ЩЂвЂЄ2007/вЂ¬Щ…вЂ¬

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