1 NEW TIME SERIES PREDICTABILITY METRICS FOR ... - CiteSeerX

NEW TIME SERIES PREDICTABILITY METRICS FOR NEAREST NEIGHBOR BASED FORECASTING Syed Rahat Abbas, Muhammad Arif Department of Computer and Information Sciences Pakistan Institute of Engineering and Applied Sciences (PIEAS) Islamabad-Pakistan Time series prediction or forecasting is an important area of research in various fields of science and engineering. Predictability refers to the degree that a correct forecasting of a time series can be made. Prediction can be erroneous; so it is important to know the predictability of the time series before going for prediction. In this paper six new predictability metrics have been presented. The proposed predictability metrics are evaluated by using single-step-ahead and multistep-ahead forecasting of time series by nearest neighbor method. The variation of predictability metrics with signal to noise ratio is determined. The proposed metrics found stable. The predictability metrics for three benchmark time series is calculated and compared with already proposed metrics in literatures. It is found that proposed metrics are more useful for analyzing the time series. Keywords: Predictability, signal processing, time series forecasting, nearest neighbor method. A CONCRETE EXAMPLE TO SHOW IMPORTANCE OF IDENTITY ELEMENT IN MATRIX NEAR RING Sarwar Jahan Abbasi University of Karachi Karachi-Pakistan Matrix near rings were constructed in 1985. Meldrum, Van Der Walt, Abbasi and Meyer are actively involved in studying their structure. It turns out that Identity element plays an important role in the study of ideals of matrix near ring. In this paper we present a concrete example for our claim. THE BASIC CASE OF FERMAT LAST THEOREM Nur Azman Abu, Shahrin Sahib, Nanna Suryana Faculty of Information and Communication Technology 1

Universiti Teknikal Melaka Ayer Keroh, Melaka-Malaysia Fermat Last Theorem An + Bn ≠ Cn has been a pure number theory problem. It has also been analyzed in literature on Fermat last theorem. It is popularly reduced to the basic elliptic curve of y2 = a·x3 + b·x + c. This paper will illustrate for the basic case n = 3, A3 + B3 ≠ C3 instead, it shall be reduced to the case of (x –k )3 = m·x + h. The result shall illustrate that there is no solution for natural number x. We also discuss a few related relations. L2-APPROXIMATION THEORY ON COMPACT GROUP AND THEIR REALIZATION FOR THE GROUP SU(2) AND SO(3) E. Aghdassi Faculty of Math. Sciences Tabriz-Iran S. F. Razev Khazar University Baku-Azerbaijan In this article we use results work c18 we have analogous results for the group SU(2) and prove specific integral formulas for the matrix elements representations of group SU(2) (In particularly for the spherical functions) and given some results concerning to classical orthogonal polynomials. AMS Subject Classification: 41XX, 42A10, 62L20. Keywords. Approximation theory, Harmonic analysis. TOWARDS MATHEMATICAL THEORY OF BENDING OSCILLATION: NON-STATIONARY SELF-GRAVITATING DISK M. Iqbal Ahmad Federal Urdu University Karachi-Pakistan Mathematical theory of bending oscillation is first constructed for stationary Model by Hunter and Toomre (Ap.J, 1969). However, real gravitational disks are non-linearly non-stationary. That’s why today it is interesting studying non- stationary disk models for analysis of different bending instabilities. The fact is that disk models are always unstable and nonstationary in nature. So we have an opportunity to construct a new 2

anisotropic disk model and investigate its vertical instabilities. The disk is studied by us relatively to main bending oscillation modes. Critic values of basic disk parameters are found. FLOW AND HEAT TRANSFER OF A THIRD-GRADE NONNEWTONIAN FLUID Rana Iftikhar Ahmad Department of Mathematics GC University, Lahore-Pakistan I have considered an unsteady unidirectional third grade fluid in a porous medium with porous boundaries. Modified Darcy's law is introduced to incorporate the effects of pores in a porous media. A strong magnetic field is applied, which acts perpendicular to the velocity field, usually known as the Hall effects. I further assumed that there is a slip between the boundary and the third grade fluid in contact with the plate boundary. The heat transfer analysis with regard to the modified Darcy's law is also discussed. The nonlinear governing equation with superimposed suction/blowing and velocity amplitude at the boundary is solved by applying perturbation technique in the third grade parameter. In this way three linear systems are obtained and solved by separating the variables. In order to obtain the physical interpretation the graphs are plotted in which the effects at the boundary and in the flow region are examined. Mainly the effects of magnetic field, Hall parameter, second and third grade parameters, porosity parameter and suction/blowing parameter are observed and then compared to the results already available in the literature. A C6 APPROXIMATING SUBDIVISION SCHEME Nadeem Ahmad, Shahid S. Siddiqi University of the Punjab Lahore-Pakistan Developing new subdivision schemes for curve designing has its own importance. Subdivision is becoming an important subject with many applications in fields including Computer Graphics, Computer Aided Geometric Design and Computer Animation due to its simplicity and efficiency. The approximating subdivision scheme, developed by Siddiqi and Ahmad [2006], is extended. It is proved that the new scheme generates C6 curves. 3

The support of the limit function is determined. The usefulness of the scheme is illustrated in different examples. MODIFIED KAPITZA METHOD AND STABLE POINTS OF THE OPEN-LOOP CONTROLLED OSCILLATOR Babar Ahmad School of Mathematical Sciences, GC University Lahore-Pakistan In this report we discuss the modified Kapitza method of averaging in a rapidly oscillating field. In the place of harmonic external force we put the arbitrary periodic function extended in Fourier series. This method has been applied to the open loop controlled oscillator to investigate the stable points. GRAVITATIONAL DUST COLLAPSE WITH COSMOLOGICAL CONSTANT Zahid Ahmad, M. Sharif University of the Punjab Lahore-Pakistan In this paper, we study the effect of a positive cosmological constant on spherically symmetric dust collapse. We take the FRW metric in the interior region whereas Schwarzschild-de Sitter in the exterior region. The matching conditions between exterior and interior spacetimes are given in the presence of a cosmological constant. We also discuss the apparent horizons and their physical significance. MONOMIAL IDEALS IN A POLYNOMIAL ALGEBRA OF FOUR VARIABLES DEFINING COHEN-MACAULAY QUOTIENTS Sarfraz Ahmad School of Mathematical Sciences, GC University Lahore-Pakistan

(

) (

) (

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Let K be a field, S = K [ x1 ,..., x4 ] and I = x1a1 , x 2a2 ∩ x1b1 , x3b3 ∩ x3c3 , x 4c4 for some a1 , a 2 , b1 , b3 , c3 , c 4 ∈ N. We describe all numbers a1 , a2 , b1 , b3 , c3 , c4 for which S / I is Cohen-Macaulay, which happens exactly when S / I is pretty clean.

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Key words: Cohen-Macaulay Algebras, Pretty Clean Modules. POLAR CREMONA TRANSFORMATIONS AND MILNOR ALGEBRA Imran Ahmed School of Mathematical Sciences, GC University Lahore-Pakistan Consider the gradient map associated to any non-constant homogeneous polynomial f ∈ [ x0 ,… , xn ] of degree d, defined by

φ f = grad ( f ) : D( f ) → Ρ n , ( x0 : … : xn ) → ( f 0 ( x) : … : f n ( x)) where D( f ) = {x ∈ Ρ n ; f ( x) ≠ 0} is the principal open set associated to f and ∂f . This map corresponds to polar Cremona transformations. Now, fi = ∂xi consider the Milnor algebra given by M(f) = [ x0 ,… , xn ] /( f 0 ,… , f n ) as the artinian -algebra. I have to establish the link between the algebraic invariant and topological invariant in the fore-mentioned scenario.

PROJECTIVE LINE ARRANGEMENT AND POINCARÉ SERIES OF MILNOR ALGEBRA Shahid Ahmed School of Mathematical Sciences, GC University Lahore-Pakistan In this note we explore useful information about the Poincaré series of the Milnor algebra associated to a hypersurface with isolated singularities. We also discuss the particular case of Projective line arrangement. NON POLYNOMIAL SPLINE SOLUTION OF EIGHTH ORDER BOUNDARY VALUE PROBLEMS Ghazala Akram, Shahid S. Siddiqi University of the Punjab Lahore-Pakistan Non polynomial spline is used for the numerical solutions of the eighth order linear special case boundary value problems. The method presented in this 5

paper has also been proved to be second order convergent. To compare the method developed in this paper with that developed by Inc and Evans [2004], two examples are considered and is observed that our method is more efficient. ON SMOOTHING OF NON-SMOOTH FUNCTIONS IN UNDERWATER SOUND PROBLEMS Alexander Alenitsyn St. Petersburg State University St. Petersburg-Russia In certain problems of sound propagation in the sea, a proper restoration of functions given by a table is required. Let the sound waves be generated by a source situated in water at some depth under the sea surface and received in water at a distance from the source. The intensity of sound pressure can be v( z )dz calculated with the help of the “phase integral” F ( µ ) = ∫ , 2 2 (µ ) 1 − µ v ( z) where v(z) is the sound velocity at the depth z, µ > 0, and ∆(µ) is z-interval such that 1−µv(z) ≥ 0. The function v(z) is usually given by a table and needs to be restored as a continuous function. If the simplest restoration with a broken line is applied, each corner point produces a nonphysical singularity of F(µ). In the present report a “running means” procedure for smoothing the broken line is proposed to remove the singularities. ON THE RAMSEY NUMBERS OF PATHS AND JAHANGIR GRAPHS J3,m Kashif Ali School of Mathematical Sciences, GC University Lahore-Pakistan Edy Tri Baskoro Institut Teknologi Bandung Bandung-Indonesia Ioan Tomescu University of Bucharest Bucharest-Romania For given graphs G and H, the Ramsey number R (G, H) is the least natural number n such that for every graph F of order n the following condition 6

holds: either F contains G or complement of F contains H. In this paper, we determine the Ramsey number of path versus Jahangir graphs J3, m .We also determine the Ramsey number R (tPn, H), where H is general Jahangir graph Js,m , t ≥ 2, s ≥ 2 is even and m ≥ 3. Keywords: Ramsey number, paths, general Jahangir graph ACTION OF THE GROUP ON A PROJECTIVE LINE OVER INFINITE IMAGINARY FIELD Q(I) Iftikhar Ali Quaid-i-Azam University Islamabad-Pakistan The main purpose of this article is to study the behavior of the group through its action on projective line over the field i.e., first we have found the linear fractional transformations and satisfying the relations and then performed the action of on by constructing coset diagrams for this action and using the information obtained from these, it is shown that the action of on gives us two orbits and hence the action is intransitive. AMBIGUITY-FREE PRESENTATION OF MBn AND ITS APPLICATION TO THE SOLUTION OF THE WORD PROBLEM IN Bn (n=3,4) Usman Ali, Zaffar Iqbal School of Mathematical Sciences, GC University Lahore-Pakistan Emil Artin was the first who gave his solution to the word problem by introducing a normal form of elements in Bn. In 1969, Garside gave a new normal form with the help of a special element (Garside braid ∆n) and thus solved the word and conjugacy problems in Bn. The algorithm to get this new normal form has six steps. The most difficult one is to find the diagram of a positive word. This step can be eliminated if we have ambiguity-free presentation for MBn. This will also be helpful to know, how many times a positive word is divisible by ∆n. In this paper we only consider the case when n = 3,4.

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TELEPARARLLEL ENERGY-MOMENTUM DISTRIBUTION OF SPATIALLY HOMOGENEOUS ROTATING SPACETIMES Muhammad Jamil Amir, M. Sharif Department of Mathematics University of the Punjab Lahore-Pakistan The energy-momentum distribution of spatially homogeneous and rotating spacetimes in the context of teleparallel theory of gravity is investigated. For this purpose, we use the teleparallel version of Moller prescription. It is found that the components of energy-momentum density are finite and welldefined. We compare the results with those already found in General Relativity. Some special cases are also discussed. AN UPPER BOUND FOR THE REGULARITY OF IDEALS OF BOREL TYPE Imran Anwar, Sarfraz Ahmad School of Mathematical Sciences, GC University Lahore-Pakistan We show that the regularity of monomial ideals of K [ x1,…,xn] (K being a field), whose associated prime ideals are totally ordered by inclusion is bounded above by a linear function in n. Key words: Stable Ideals, Borel Ideals, Regularity STOCHASTIC TRANSPORT AND FRACTIONAL CALCULUS V. E. Arkhincheev Federal Urdu University Karachi-Pakistan & Buryat Science Center Ulan-Ude-Russia It is shown that to describe the "anomalous" stochastic transport - random wals in disordered systems the fractional calculus is necessary to apply. The new generalized diffusion equations of fractional order are deduced from microscopic models with anomalous diffusion as Comb model, Continuous Random Wals and Levy flights. It is shown that three types of equations are 8

possible: with fractional temporal, with fractional spatial derivatives and with mixed derivatives. The solutions of these equations are obtained and the physical sense of these fractional equations is discussed. The relation between diffusion andconductivity is studied and the well-known Einstein relation is generalized for the anomalous diffusion case. It is shown that for Levy flight diffusion the Ohm's law is not applied. The nonlinear response instead of Ohm's law is established. The exponent of nonlinearity is founded, it is connected with index of anomalous super-diffusion. DATA, TRAFFIC AND CONGESTION D. K. Arrowsmith, R. J. Mondragon University of London, U.K The talk assembles information on the nature of traffic on a network. It addresses the complex synthesis of the topological nature of a network, the dynamics of packet traffic in communications, and the way in which emergent behaviour can be obtained through mean field and numerical approaches. SPECTRAL FUNCTION AND KINETIC EQUATION FOR NORMAL FERMI LIQUID M. Arshad, I. Siddique School of Mathematical Sciences, GC University Lahore-Pakistan A. S. Kondratyev Department of physics, Herzen State Pedagogical University St. Petersburg-Russia We offer a new ansatz for the approximation of the spectral function in the Kadanoff-Baym version of Green’s function method in quantum statistical mechanics. This ansatz satisfies the equation for spectral function in the case of non-equilibrium systems with slowly varying in space and time disturbances. The ansatz provides a new approach to the microscopic derivation of the Landau-Silin kinetic equation for normal Fermi-liquid and increase the temperature range of this equation’s validity.

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INTRODUCE SUMUDU TRANSFORM TO DIFFERENCE EQUATIONS Muhammad Ashfaq National College of Business Administration & Economics Lahore-Pakistan The Sumudu Transform was introduced by Watugala in 1993, is little known and not widely used. However being the theoretical dual to the Laplace transform, the Sumudu transform rivals it in problem solving. Having scale and unit preserving properties, the Sumudu transform may be used to solve problems without restoring to a new frequency domain. These unit and scale preserving properties make visualization easier and application has been demonstrated in the solution of partial differential equations, integral equations, integro-differential equations and dynamic systems. In this paper, application of Sumudu transform to solution of Difference equation is described. ON A MEASURE OF NON-COMPACTNESS FOR SOME CLASSICAL OPERATORS Usman Ashraf, Muhammad Asif School of Mathematical Sciences, GC University Lahore-Pakistan Alexander Meskhi A. Razmadze Mathematical Institute, Georgian Academy of Sciences Tbilisi-Georgia A measure of non-compactness (essential norm) for maximal functions and Riesz potentials acting in weighted Lebesgue spaces defined on homogeneous groups are estimated from below. The similar problem for one-sided potentials, Poisson integrals, partial sums of the Fourier series, identity mapping and Sobolev embeddings are studied. In particular, in the most cases we conclude that there is no a weight pair for which these operators acting between two weighted Lebesgue spaces are compact. Our investigation, in some sense, is a natural continuation of the works done by D. E. Edmunds and A. Meskhi; D. E. Edmunds, A. Fiorenza and A. Meskhi.

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SOME GENERALIZATIONS OF LITTLEWOOD-PALEY INEQUALITY IN THE POLYDISK K. L. Avetisyan Yerevan State University Yerevan, Armenia R. F. Shamoyan Bryans Pedagogical University Bryans, Russia The paper generalizes the well-known inequality of Littlewood-Paley in the polydisk. We establish a family of inequalities which are analogues and extensions of Littlewood-Paley type inequalities proved by Sh. Yamashita and D. Luecing in the unit dis. Some other generalizations of the LittlewoodPaley inequality are stated in terms of anisotropic Triebel-Lizorin spaces. With the help of an extension of Hardy-Stein identity, we also obtain area inequalities and representations for quasi-norms inweighted spaces of holomorphic functions in the polydisk. PROJECTIVE LIMITS OF PSEUDO RESOLVENTS Abdul Sami Awan School of Mathematical Sciences, GC University Lahore-Pakistan The present paper is divided in three sections. In the first section the notions of projective family of topological vector spaces and projective family of operators are given. In the second section, projective families of pseudo resolvents acting on locally convex spaces are considered. The last section is devoted to pseudo resolvents of multiplicative operators on locally convex spaces. We have introduced a new concept of L∞-type Pseudo Resolvent which on Banach Spaces generalizes the similar concept introduced by F.Hirsch in 1972.We have pointed out the connection between the class of L∞-type Pseudo Resolvents and the class of C0, equicontinuous semi groups on locally convex spaces

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GENERALIZED NO GO THEOREM FOR KINEMATIC SELFSIMILARITY Sehar Aziz, M. Sharif University of the Punjab Lahore-Pakistan We investigate plane and cylindrical symmetric spacetimes with perfect fluid obeying the polytropic equation of state. These admit a kinematic selfsimilar vector of the second kind which is tilted to the fluid flow. It is found that the corresponding self-similar solutions (if exist) must be vacuum. This provides the generalization of No Go Theorem already available for spherical symmetric spacetimes. MATRIX LIE RINGS THAT CONTAIN A ONE-DIMENSIONAL LIE ALGEBRA OF SEMI-SIMPLE MATRICES Evgeni L. Bashkirov Belorussian State University of Informatics and Radio electronics Minsk-Belarus Let k be a field and k an algebraic closure of . Suppose that contains more than five elements if char k ≠ 2. Let h be a one-dimensional subalgebra of the Lie-algebra sl2 ( k ) consisting of semi-simple matrices. In this paper, it is

proved that if g is a subring of the Lie ring sl2 ( k ) containing h, then g is either solvable or there exists a quaternion algebra A over a subfield F of k such that F ⊇ k and g is isomorphic to the Lie F-algebra of all elements in A that are sew-symmetric with respect to a symplectic type involution defined on A.

ON THE EXISTENCE OF ALMOST MOORE DIGRAPHS Edy Tri Baskoro Institut Teknologi Bandung Bandung-Indonesia One of the interesting problems in the design of interconnection networks is to find digraphs with the number of vertices n as large as possible, given the values of maximum out-degree d and diameter k (see [4], [6], [7]). It is easy to see that 12

n ≤ M d ,k =1+ d + d 2 +…+ d k . This upper bound is called the Moore bound. It is well known that this bound is attained only for d = 1 by the cycle digraph of order k + 1, and for k = 1 by the complete digraph Kd+1 (for the proof see [5] or [12]). This motivates us to study the existence of digraphs with order 'close' to the Moore bound, for d ≥ 2, k ≥ 2 . A (d;k )-digraph is a diregular digraph of degree d ≥ 2, diameter k ≥ 2 and the number of vertices n is one less than the Moore bound, that is, n = d + d 2 + … + d k . Such a (d;k )-digraph is also called an almost Moore digraph. Since the order is one less than the Moore bound then every (d;k )digraph G has property that for every vertex u ∈ V (G) there exists exactly one vertex v such that there are two walks of length ≤ k from u to v. Such a vertex v is called the repeat of u, denoted by r(u) = v. In case r(u) = u, vertex u is called a selfrepeat (the two walks, in this case, have lengths 0 and k). The study of the existence of an almost Moore digraphs of degree d and diameter has received much attention. Fiol, Allegre and Yebra [6] showed the existence of (d; 2)-digraphs for d ≥ 2. In particular, for d = 2 and k = 2, Miller and Fris [10] enumerated the exact number of (2; 2)-digraphs. Further, Gimbert [9] showed that there is only one (d; 2)-digraph, namely the line digraph L(Kd+1) of the complete digraph Kd+1 for d ≥ 3. For diameter k ≥ 3, it is known that there are no (2;k)-digraphs [11]. Recently, it was proved that there are no (3;k )-digraphs with k ≥ 3 [3]. Thus, the remaining case still open is the existence of (d;k )-digraphs with d ≥ 4 and k ≥ 3. Several necessary conditions for the existence of (d;k )-digraphs, in general, have been obtained (see [1],[2],[8]). One such condition is that any (d;k )digraph contains at most one selfrepeat, k ≥ 3 [2]. This talk will give a survey on the existence study of an almost Moore digraph. Open problems related to this study are also listed. References 1. E. T. Baskoro, M. Miller, J. Plesnik, Further results on almost Moore digraph, Ars Combinatoria 56, (2000), 43-63. 2. E. T. Baskoro, M. Miller, J. Plesnik, On the structure of digraphs with order close to the Moore bound, Graphs and Combinatorics 14 (1998), 109119. 3. E.T. Baskoro, M. Miller, J. ·Širáň dan M. Sutton, Complete 13

characterization of almost Moore digraphs of degree three, Journal of Graph Theory 48:2 (2005), 112-126. 4. J.C. Bermond, C. Delorme, J.J. Quisquater, Strategies for interconnection networks: Some methods from graph theory, Journal of Parallel and Distributed Computing 3, (1986) 433-449. 5. W. G. Bridges, S. Toueg, On impossibility of directed Moore graphs, J. Combinatorial Theory Series B29 (1980), 339-341. 6. M. A. Fiol, I. Alegre and J. L. A. Yebra, Line digraph iteration and the (d,k) problem for directed graphs, Proc. 10th Symp. Comp. Architecture, Stocholn (1983) 174-177. 7. M. A. Fiol and J. L. A. Yebra, Dense bipartite digraphs, J. Graph Theory, 14 (1990) 687-700. 8. J. Gimbert, On the existence of (d;k)-digraphs, Discrete Mathematics 197/198 (1999), 375-391. 9. J. Gimbert, Enumeration of almost Moore digraphs of diameter 2, Discrete Mathematics 231 (2001), 177-190. 10. M. Miller, I. Fris, Minimum diameter of direguler digraphs of degree 2, Computer Journal 31 (1988) 71-75. 11. M. Miller, I. Fris, Maximum order digraphs for diameter 2 or degree 2, Pullman Volume of Graphs and Matrices, Lecture Note in Pure and Applied Mathematics139 (1992), 269-298. 12. J. Plesnik, ·Š. Znám, Strongly geodetic directed graphs, Acta F.R.N. Univ. Comen. Mathematica XXIX (1974), 29-34. RECURSIVE CLASSES AND REDUCTIONS OF GRAPHS Sheng Bau Fuzhou University Fuzhou-China A recursive class is a class of objects that can be constructed from a basic set of objects by means of a set of well defined operations. Recursive classes are encountered everywhere in mathematics, especially in logic, computation, combinatorics and graph theory. This article provides a review of classical and recent results on recursive and inductive classes and reductions of graphs, with an attention to their relationship with concepts such as connectivity, cyclability and hamiltonicity. Open problems are reviewed and some new problems are proposed.

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ON THE SUMS OF COMPLEMENTARY DIVISORS OF INTEGERS Mircea Becheanu University of Bucharest Bucharest-Romania Florian Luca Univesidad Nacional Autonoma de Mexico Michoacan-Mexico Igor Shparlinski Macquarie University Sydney-Australia It is known that the arithmetic properties of the divisors d of a positive integer n are heavily influenced by the arithmetic properties of n itself. The various sums d + n/d of all complementary divisors of n exhibits some interesting properties connected with shifted primes, factoring, cryptography, etc. In this paper, we study various arithmetic properties of d + n/d, where d runs over all τ ( n) positive divisors d of the positive integer n. For example, denoting by ω ( n) the number of prime values among these sums, we study how often ω ( n) > o and ω ( n) = τ ( n) and we also evaluate the average value of ω ( n) . We estimate some character sums with d + n/d and study the distribution of quadratic nonresidues and primitive roots among these sums on average over n ≤ x . Main results are presented in theorems 1-4. ITERATED TSCHIRNHAUSEN OPERATORS Barbu Berceanu Institute of Mathematics “Simion Stoilow” Bucharest-Romania In order to understand Aut(C[X, Y ]), we study an associated system of 2n − 1 differential equations. We prove that this system is completely integrable and we express the solutions as an affine algebraic variety in C2n-1×C[X]n-1. The irreducible components of this variety are related to log-partitions of the number n (for instance, if n = 6, log 6, log 2 + log 3, log 3 + log 2).

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ON SOME CHROMATIC PROPERTIES OF JAHANGIR GRAPH Akhlaq Ahmad Bhatti School of Mathematical Sciences GC University Lahore-Pakistan In this note we compute the chromatic polynomial of the Jahangir graph J 2 p , and we prove that it is chromatically unique for p = 3 . AMS Subject classification: 05C15 Keywords: Jahangir graph, Chromatic polynomial, Chromatic uniqueness. ALMOST PERIODICITY OF WEAKLY ALMOST PERIODIC FUNCTIONS DEFINED ON An WITH VALUES IN A TOPOLOGICAL LINEAR SPACE Muhammad Iqbal Bhatti Department of Mathematics University of Engineering & Technology Lahore. Muhammad Amer Latif Pakistan International School, English Section Riyadh-Saudi Arabia Amerio’s in his paper describes functions from Å into a Banch space which are weakly almost periodic in a natural sense. Afterwards Gaston studied weakly almost periodic functions from Å to topological linear spaces. In this paper we extend the study and in particular explore the condition under which a weakly almost periodic function defined on Ån with values in a topological linear space, is almost periodic. ON THE EIGENVALUES AND EIGENVECTORS OF COMBINATION INVARIANT OF CHARACTERISTIC GRAPH OF PLANER HONEYCOMB GRAPH Faqir M. Bhatti Dept of Mathematics, SSE Lahore University of Management Sciences(LUMS) Lahore-Pakistan

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Aisha Perveen Dept of Computer Science COMSATS Institute of Information Technology Defense Road off Raiwind Road, Lahore-Pakistan The planer Honeycomb graphs consist of equal regular hexagons. Using the concept of He-matrix corresponding to characteristic graph of Honeycomb graph, we study various properties of eigenvalues and eigenvectors. The product of He-matrix of two different characteristic graphs of planer honeycomb graphs is calculated and their Eigenvalues, Eigenvectors, and a few properties of resulting graphs are also given. ORTHOGONAL 0-INTERPOLANTS WITH APPLICATIONS IN APPROXIMATION THEORY M. A. Bokhari Dept. of Math. Science FUPM, Dhahran- Saudi Arabia The Orthogonal 0-interpolants “OZI” are orthogonal polynomials which interpolate the “zero-function” at pre-assigned nodes x i , i = 1, 2,… , n , up to derivatives of a pre-assigned order m i . OZI can be determined by the 3-term recurrence relation. These interpolants have significant applications in constrained approximation problems. We shall present theoretical and computational aspects of OZI on a bounded interval [a,b] and discuss their applications. With a similar approach, this concept will be extended to orthogonal exponential 0-interpolants (OEZI) on the semi-infinite interval [0, ∞) . ANALYSIS OF NAVIGATION OF POPULATIONS OF MOBILE ROBOTS USING AUTOMATA AND GRAPH THEORY Banhan Bootsingha, Pongbundit Nongpromma, Kanokorn Jeenmueang, Elvin J Moore, Utomporn Phalavonk Department of Mathematics King Mongut’s Institute of Technology North Bangkok-Thailand Mobile robots have been proposed as a replacement for humans in operations in dangerous environments, e.g., buildings that are on fire or contaminated by radiation or in military operations. The model considered in 17

this paper is a heterogeneous population of robots whose task is to reach a set of target rooms in a building. Because of the different properties of the rescue robots, any given robot might be able to reach certain target rooms and not others and in addition the “costs” of given robots reaching target rooms might be different. The rescue problem considered here is to assign robots to target rooms so that, if possible, each target room is reached by some robot and so that the assignment corresponds to a minimum total cost. This paper describes some automata and graph theory models and some computer programs which can be used to solve this rescue robot problem for problems of arbitrary size. The problem is solved in two main stages. Graph theory methods (the FloydWarshall algorithm [1]) are first used to determine allowed minimum-cost paths for each robot between all pairs of rooms, a minimum edge-weight bipartite matching algorithm (the Hungarian algorithm [2]) is then used to allocate a target room for each robot to give a minimum total cost. A Matlab program has been developed to solve this rescue-robot problem. Results obtained from the Matlab program are described. ANALYTICAL SOLUTION FOR THE ENERGY BALANCE EQUATION IN DEVELOPED TURBULENCE Sergei Borisenok Herzen State Pedagogical University of Russia St. Petersburg-Russia Here we present the modified Heisenberg model of the spectrum transfer function for homogeneous developed decay turbulence and solve analytically PDE of energy balance. The result is proved by experimental data. DUAL INTEGRAL CHARACTERIZATIONS FOR NONUNIFORMEXPONENTIAL STABILITY OF EVOLUTION FAMILIES ON BANACH SPACES C. Buşe Department of Mathematics, West University of Timisoara, Timisoara, Romania A. D. R. Choudary School of Mathematical Sciences, GC University Lahore-Pakistan 18

In 1960, Mircea Reghiş introduced a non-uniform stability concept and he formulated theorems of connection between this concept and Perron conditions related to well-chosen pairs of Banach function spaces. Later, in1971, Mircea Reghiş introduced a class of spaces which proved useful in the study of the non-uniform behavior of solutions of abstract linear differential equations. Here we make a study about non-uniform stability by the perspective of Barbashin's theorem of stability. Let J be one of the intervals R; R+ : = [0, ∞ ) or R- = (−∞, ] . We shall prove that an evolution family U = {U (t , s )}t ≥ s satisfying some natural assumptions is non-uniformly exponentially stable if there exists a positive real number α and a nondecreasing function φ : R+ → R+ with φ (t ) > 0 for all t > 0 and such that t −s

sup t >s

∫ φ (e

αu

U ( s + u , u ) x du = M φ ( s ) < ∞,

∀s ∈ J , ∀x ∈ X , x ≤ 1

0

The symmetric theorem reaches the same line under the assumption that t

sup ∫ φ (eα t e −α T U (t , T )* x* ) dT = Lφ ( s ) < ∞,

∀s ∈ J , ∀x* ∈ X * , x* ≤ 1

t >s s

If φ is a convex function then the converse statements are obviously true. ON SELF MAPS OF BCC ALGEBRAS Ms. Bushra University of the Punjab Lahore-Pakistan In this paper, we define implicative BCC algebra, self maps, and left regular maps of BCC algebras. We also study their properties. Further we prove that there exist left regular maps of proper BCC algebras which are different from right maps as well as BCC homomorphism QUALITY SURFACE CONSTRUCTION R. J. Cripps University of Birmingham Birmingham-U.K Current surface construction methods in CADCAM use parametric polynomial equations in the form of a NURBS given by: S( u,v ) =

m

n

∑ ∑ N i, p (u)N j,q (v )Pi , j i= 0 j= 0

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where

Ni ,p ( u ) & N j ,q ( u )

vectors [U ] = {u0 ,… ,um + p +1 }

are the basis functions defined by not &

vq ≤ v ≤ vn +1 and Pi , j = Vi , j wi , j

[V ] = {v0 ,… ,vn+ q +1}

with

u p ≤ u ≤ um +1

are the homogeneous control points,

0 ≤ i ≤ m , 0 ≤ j ≤ n . The surface consists of ( m + 1 − p ) × ( n + 1 − q ) subpatches, which are Cp-1 and Cq-1 continuous in the u- and v-directions respectively, assuming there are no repeated interior knots. This representation is ideal for computer-based implementations, allowing efficient interrogation. However, issues exist in constructing and manipulating such surfaces. When constructing a NURBS surface there are difficulties in determining constraints such as parameterisation, tangent magnitudes and twist vectors. Controlling the geometric features lie curvature profiles of section/longitudinal of NURBS surface is problematical as is joining several such surfaces together. A cause of the difficulties in controlling the NURBS is that the control points do not lie on the surface itself.

Fig 1. NURBS surface representation. An alternative approach is to specify the curvature and construct the surface to satisfy the curvature constraints. Since NURBS does not directly allow this, a fundamentally different approach is required. The key is to adopt a point-based approach where the surface is defined by a small number of points lying on the surface. Intermediate points are then constructed using a recursive approach which is defined to ensure that the curvature profile between adjacent points is of a very high quality. A drawback is that the mathematics becomes less ‘nice’ and less suited to computer implementations. However, with increasing computer speed and cheaper memory and storage, the recursive approach is an attractive alternative.

20

Fig 2. (a) Equivalent point-based surface (b) Intermediate points on the surface The recursive scheme is based on a GCS which has a rational linear curvature profile, parameterised with respect to arc length, s, and is given by: ( κ1 − κ0 + rκ1 ) s + κ0S , r>-1 κ (s) rs + S where κ 0 and κ 1 are the curvatures at the start and end of the curve segment, S is the total arc length and r is the shape factor which controls the fullness of the curve.

=

The algorithms that are required to construct a 3-d surface are outlined and the results are illustrated. Consideration is also given to approximating point-based surfaces in a NURBS representation to enable existing CAE systems to tae advantage of the quality surfaces constructed using the recursive scheme. The article is concluded by given consideration to some outstanding issues and suggestions for future research. References [1] Ali, J. M., Tookey, R.M., Ball, J.V., Ball, A.A. The generalised Cornu spiral and its application to span generation, Journal of Computational and Applied Mathematics, 102, 37-47, 1999. [2] R.J. Cripps and A. A. Ball, Visualisation and Quality assessment of freeform surfaces. Proceedings of the Institute of Mechanical Engineers, Part B, 212, 207--214, 1998. [3] R.J. Cripps and P.R. Coo. Point-based CADCAM. In it Advances in Manufacturing Technology XIII, Eds. A. N. Bramley, A. R. Mileham, L. B. Newnes, G. W. Owen, 149-153. Professional Engineering Publishing Ltd., 1999. [4] R.J. Cripps. Algorithms to support point-based CADCAM. The International Journal of Machine Tools and Manufacture. 43,425-432, 2002.

21

MATHEMATICAL MODELS OF CODING B. K . Dass Department of Mathematics University of Delhi Delhi-India It is now known that coding and decoding play a significant role in almost all communication systems. This talk shall begin by proposing several mathematical and algebraic models of coding, their limitations and possible remedies. The models shall include a linear code as a subspace of the vector space of n-tuples followed by the construction of a generator matrix and a parity-check matrix of the code. Applications of such codes in ISBN, telecommunications, military communication, computers and other fields lie credit card systems etc. will be discussed. ZIGZAG AND CENTRAL CIRCUIT STRUCTURE OF TWO-FACED PLANE GRAPHS Michel-Marie Deza Ecole Normale Superieure Paris-France A zigzag in a k-valent plane graph G is a circuit of edges, such that any two, but not three consecutive edges belong to the same face. A railroad in G is a circuit of evengonal faces, such that any face is adjacent to its neighbors on opposite edges. Boundary circuits of a railroad are two "parallel" zigzags if k = 3, or, in a 4-valent graph, two such central circuits. We consider the zigzag and railroad structure of two-faced 3- and 4-valent plane graphs (generalizations of Platonic polyhedra) and their connections with other problems. A FIXED POINT THEOREM FOR MULTIVALUED NONEXPANSIVE MAPPINGS ON A SPACE HAVING THE UNIFORM KLEE-KADEC PROPERTY S. Dhompongsa Faculty of Sciences, Chiang Mai University Chiang Mai-Thailand Let X be a Banach space, C a weakly compact convex subset of X and T : C → KC (C ) a nonexpansive mapping with convex compact values. In 22

2004, T. Dominguez and P. Lorenzo solved an open problem by proving that T has a fixed point if X is nearly uniformly convex. We improve this result by removing the reflexivity condition from the space. FINITELY GENERATED-FRAGMENTED DOMAINS Tiberiu Dumitrescu Faculty of Mathematics and Informatics University of Bucharest Bucharest-Romania (Joint work with Jim Coykendall.) An integral domain D is called a fragmented domain (concept introduced by David Dobbs), if every proper principal ideal aD of D is contained in all powers of bD for some proper principal ideal bD. In this talk we introduce a related concept. We say that an integral domain D is a finitely generated-fragmented domain (FGF domain), if every proper finitely generated ideal I of D is contained in all powers of J for some proper finitely generated ideal J. We prove the following results. A semi-quasi-local fragmented domain is an FGF domain. An FGF domain which is not a field is infinite-dimensional. A domain D is FGF iff it is locally FGF, provided for each maximal ideal M of D there exists a finitely generated ideal I such that M is the only maximal ideal containing I. We construct a fragmented non-FGF domain D which has a finitely generated maximal ideal M such that the localization of D in M is a Noetherian domain. We show that not all FGF domains are fragmented by constructing an FGF domain satisfying the ascending chain condition for the principal ideals. SINGULAR-PERTURBED BOUNDARY-VALUE PROBLEMS WITH A MULTIPLE PURE IMAGINARY SPECTRUM Abubakir Dzhuraev Osh State University Kyrgyzstan The creation of mathematical models of real processes is the important direction of modern applied mathematics. For the analysis of these models often are used asymptotical methods. Now asymptotical methods have the 23

advanced theory in ordinary area of stability, i.e. when the own meanings are negative. In the present work we develop the new theory of singularperturbed problems with boundary value for the differential equations with a multiple pure imaginary spectrum in the extended area of stability. THE PERFECT NUMBERS AND THEIR APPLICATIONS Ahmed Ali Elwakshi 7th October University Libya As we know that the perfect numbers are not increasing than 38 numbers and perfect from the others are not increasing than 10 numbers. Up to now we do not know that these numbers are finite or infinite. In the present study we explained the presence of the upper perfect from any order k where k is a natural number. TWO NEW FORMULAS FOR GENERATING INFINITE SETS OF INTEGER NUMBERS BY THE SUM OF TWO AND THREE SIGNED CUBIC NUMBERS Rand Al-Faris, Mohamad Rushdan Md Said University Putra Malaysia Selangor Darul Ehsan-Malaysia In this paper, we introduce new study for the cubic numbers from the view point of representing the integer numbers as a sum of cubic numbers. We put two new formulas for generating infinite set of integer numbers as the sum of signed cubic numbers. The latest formula is representing an infant set of integers as sum of four signed cubic. These two new formulas reduce the number that needs to represent an integer numbers as a sum of cubic numbers from four cubic to two and three signed cubic numbers. These two formulas are depending on quadratic equations. The first formula we call it ( JR - 2CN ), generate infinite set of integer numbers by the sum of only two signed cubic numbers. And the second formula we call it ( JR - 3CN ) generate infinite set of integer numbers by the sum of three signed cubic numbers and this formula proved that there exist an infinite set of integer numbers can generated by the sum of three signed cubic numbers and it extends the conjecture that says only for specific integers of the set N 1, 0 ≤ α qj < t . Extending a result of Aramova and Herzog we showed under mild assumptions that β ij ( I ) do not depend on p and all the Koszul homology modules H i ( x; S / I ) have monomial cycle basis. ON THE CONTINUOUS AUTOMORPHISMS OF AN ALGEBRAIC EXTENSION OF VALUED FIELDS Sever Angel Popescu Technical University of Civil Engineering of Bucharest Bucharest-Romania In this paper we describe the subgroup of all continuous automorphisms of the Galois group Gal(L/K) of a Galois extension of valued fields L/K. We present six equivalent characterizations of this subgroup in terms of valuations and topological structure Galois groups. IMPRIMITIVE, LINE-TRANSITIVE LINEAR SPACES: THEORY AND SEARCH TOOLS Cheryl E. Praeger University of Western Australia Australia (Joint work with Anton Betten, Anne Delandtsheer, Maska Law, Alice Niemeyer, Shenglin Zhou) A finite linear space S = (P,L) consists of a finite set P of points and a set L of distinguished subsets of P called lines such that any two points lie in exactly one line and each line contains at least two points. If a group G of automorphisms of S is line-transitive, then it is also transitive on points, and in this case all lines have the same size. If G leaves invariant a non-trivial point-partition, then S is called an imprimitive, line-transitive linear space. In 1991, Weidong Fang and Huiling Li proved that there are only finitely many non-trivial imprimitive, line-transitive linear spaces for a given value of gcd(k, r), where k is the line size and r is the number of lines on a point. 54

The research team above has just completed a long-running project to make this result of Fang and Li effective. We collected together existing theory, and proved additional theoretical restrictions of both a combinatorial and group theoretic nature. We then organized this information into a series of algorithms (our search tools) that, for gcd(k, r) up to a given maximum value, return a list of candidate parameter values and candidate groups. As a decisive test of these search tools we rank these algorithms for values of gcd(k, r) up to 8. We examined in detail each of the possibilities returned, and completed the classification in this case. Theorem Suppose that S is a non-trivial, imprimitive, line-transitive linear space with v points, line size , and r lines on each point, and that gcd(k, r) ≤ 8. Then, up to isomorphism, S is one of (a) the Desarguesian projective plane of order 4 or 7, (b) the Mills design or Colbourn-McCalla design, both with (v,k ) = (91, 6), (c) one of 467 designs with (v,k ) = (729, 8) constructed by Nickel, Niemeyer, O’Keefe, Penttila and Praeger. HAMILTONIAN PROPERTIES OF GENERALIZED HALIN GRAPH Ahmad Mahmood Qureshi, Shabnam Malik School of Mathematical Sciences, GC University Lahore-Pakistan A Halin graph is a graph G = T ∪ C , where T is a tree with no vertex of degree two, and C is a cycle connecting the end vertices of T in the cyclic order determined by a plane embedding of T . In this paper, we define classes of generalized Halin graph, called k -Halin graphs, and investigate their Hamiltonian properties. GENETIC ALGORITHM HYBRID WITH SIMULATED ANNEALING FOR BINARY VALUED INVERSE RADON TRANSFORM APPROXIMATION THROUGH PARALLEL-RAY TRANSMISSION TOMOGRAPHY Shahzad Ahmad Qureshi Department of Computer & Information Sciences, Pakistan Institute of Engineering & Applied Sciences (PIEAS) Islamabad-Pakistan Sikander M. Mirza Department of Physics and Applied Mathematics, Pakistan Institute of Engineering & Applied Sciences (PIEAS) Islamabad-Pakistan 55

M. Arif Department of Electrical Engineering, Pakistan Institute of Engineering & Applied Sciences (PIEAS) Islamabad-Pakistan The Genetic Algorithm (GA) and Simulated Annealing (SA) have been used in collaboration for approximating the Inverse Radon transform through parallel ray transmission tomography for the binary head and lung phantoms of various sizes. This hybrid methodology is developed by combining GA in the initial part of reconstruction with SA in the posterior part where the convergence slope is relatively less sheer. The fitness function is based on minimizing the weighted least-squares distance between the measured projection data and the re-projections of a postulated image. The parameters lie population size, selection schemes, crossover schemes and mutation variations have been analyzed. Various selection schemes have been utilized with respect to their specific efficacy for image reconstruction. Crossover operators lie single point, multipoint and uniform crossover are compared based on the resultant image quality. The mutation operation with slow and fast probability profiles has also been attempted. Influence of the number of projections on image quality is investigated using Mean-Squared Error (MSE), Pea-Signal-to-Noise ratios (PSNR), Euclidean Error (EE) and Universal Image Quality Index (UIQI) as numerical measures. The comparison of convergence time for this hybrid with simulated annealing is endeavored. Same image quality demanded relatively more time from SA in comparison to this hybrid technique. In comparison with the standard deterministic techniques including filtered-bac-projection and the algebraicreconstruction techniques, the proposed hybrid genetic algorithm has been found superior yielding PSNR values of 32.37 and 30.45 for 64 × 64 head and lung phantoms respectively. The parallel ray transmission tomography consists of passing radiations through an object and detecting the transmitted intensities, forming the projections, at various angles. For an object described by the function f(x,y) in xy-space, a projection along (θ,t)−line is given by: Pθ ( t ) =

∞ ∞

∫ ∫ f ( x, y ) δ ( x cos θ + y sin θ − t ) dx dy,

−∞ −∞

where the right hand side forms the Radon transform of the function f(x,y). The Sinogram Pa consisting of actual measurements is: Pa ( t , θ ) =

RMax −1 CMax −1 θ n

∑ ∑ ∑ f ( x , y ) z (θ ) , x =0

y =0 θ =0

56

(2)

where ⎧1 ⎩0

iff

z (θ ) = ⎨

t = x cos θ + y sin θ otherwise

The fitness F function is based on the relative difference between the computed and the measured projections for Nθ [0,π] views and ls parallel rays per angular position as follows: −1 F = (1 + ERMSE ) , (3) ERMSE =

1 Nθ ls

Nθ

ls

∑∑{P ( j, i ) − P ( j, i )} , 2

FBP

i =0 j = 0

a

(4)

where ERMSE is the root-mean squared error (RMSE) of the system of equations. For genetic algorithm, the th string of population P is initialized by: P ( k ,ψ ) = f FBP (η ,ν ) ; η ∈ [1, ls ] , ν ∈ [1, ls ] ,

(5)

where ψ = (η-1)ls + ν. The F is calculated by equation (3). The selection operation provides a means for the chromosomes with better fitness to form the mating pool (MP). Various selection schemes analyzed in this work are briefly given below: In proportionate or roulette wheel scheme the sectors of roulette wheel are assigned to each individual according to fitness. The selection probability for ith chromosome is based on the size m of MP as follows: (6) F (i ) p (i) =

m

∑F(k )

.

k =1

The sector S on roulette wheel for ith string is allotted as: i

(7)

S ( i ) = ∑ p ( j ). j =1

A random number RNo is generated in the range [0, 1]. The smallest index, for which S is equal to or greater than RNo, is used to identify the string to inhibit MP. In truncation scheme the selected chromosomes inhibit MP and the mating chromosomes are then selected only from this pool depending on the fraction φ of the best chromosomes. M len = φ m, (8) M p ( i,:) = P ( i,:) ; i = 1, 2,..., M len , where Mp is the matrix for MP, Mlen is the length of the mating pool and m is the total number of chromosomes. In tournament scheme a group of chromosomes is randomly selected from P. The best individual out of this group inhibits MP. This process is used repeatedly to fill MP. The algorithm used is given by: 57

T ( i,:) = P ( RNo ,:) ; i = 1, 2,..., Tlen , ⎧ Swap Fq and Fr of T ⎫ if Fq < Fr ⎨ ⎬ ; r = q + 1, q + 2,..., Tlen , th th ⎩ Swap q and r strings ⎭ q = 1, 2,..., Tlen − 1,

M p ( j ,:) = T (1,:) ; j = 1, 2,..., m,

where Mp represents the mating pool, T represents the matrix containing the group of randomly selected strings from existing population P, Tlen is number of individuals in the team and m is number of strings in the population. In Crossover operation (CO) the probability of forming offspring is Pc with typical value Pc = 0.8. Various crossover schemes used are briefly given below: In Single-Point CO, a random integer P1, is generated in the range [1, n], where n represents the length of the chromosome. The crossover between ith and (i+1)th strings taes place when RI ≤ Pc, where RI is a random number in the range [0, 1], and swapping of strings at P1 taes place. The offspring possesses features from both parents. In Multi-Point CO, q random positions are chosen as CO points and each chromosome is split into q+1 portions. The alleles between successive points are exchanged if RI ≤ Pc. In Uniform CO, each gene is crossed with some probability typically onehalf so that each pair of genes exchanges values independently and avoids position biasing in forming the new population Np. If RI ≤ Pc then CO for th position, where = 1, 2,…, n, is carried out between the mating chromosomes. The mutation operation allows changing slightly or completely the allele with a small probability of occurrence and the resulting small variation in fitness induces diversity in the population. Various mutation schemes are given below: In Standard scheme a random number RNo is generated in the range [0, 1]. The gene is mutated if RNo is less than mutation probability Pm. Another random value Rval is then assigned to this gene with the allele lying within the variation domain. The algorithm for standard mutation is: (10) if RNo < Pm N p ( i, j ) = RVal ; j = 1, 2,..., n, i = 1, 2,..., m, where n is the total number of genes in a string and m is the total number of chromosomes. In Gradient-based scheme some researchers introduced decreasing law for mutation probability Pm as a function of generation Pm(i)..The falling Pm.results in wider exploration of search space in the initial portion of the algorithm and fast convergence in the final portion of the program. The 58

algorithm for this scheme is given by: Pm ( i ) = Pm0 e

(

− ( i / G ) log10 Pm0 ×103

) ; i = 1, 2,..., G ,

(11)

where i represents the generation number and G is the total number of generations. In Off-set based scheme a fixed offset gray level Oset in either direction is used to define the range for random value finding as the possible mutate. The offset value depends on the degree of complexity of the reconstruction. The algorithm used is given by: a = N p ( i, j ) − Oset if a < 0.0 a = 0.0, (12) b = N p ( i, j ) + Oset if b > 1.0 b = 1.0, if RNo < Pm N p ( i, j ) = Rval ( b − a ) + a ; j = 1, 2,..., n, i = 1, 2,..., m, where Rval is the random number generated in the range [0, 1], RNo is another number generated to chec the occurrence of mutation. a or b are set to 0 or 1 if the range [0, 1] is exceeded at either end. The algorithm for Simulated Annealing (SA) is engaged when the rate of change of RMSE of projections has fallen below a certain predefined threshold. This limiting value has been found different for both the head phantom and the lung phantom. The methodology for SA starts with a template f ( x, y ) and by inserting a random change in the absorption k

density ∆f , the kth supposed solution

f k +1 ( x, y ) is

generated: (13)

f k +1 ( x, y ) = f k ( x, y ) + ∆f

The change in absorption density with location is carried out with random pixel-selection and value-selection in the grey level range [0, 255] for the estimated cross-section that may or may not result in minimizing the energy. Let C and C+1 represent the cost (or energy) of f k ( x, y ) and f k +1 ( x, y ) respectively, calculated through forward problem. For minimization type cost function, if ∆C = Ck +1 − Ck < 0 (i.e., energy of C+1 is favorable), then f k +1 ( x, y ) is accepted; otherwise, the acceptance probability is given by: ⎛ ∆C ⎞ Pr ( k ) = exp ⎜ − ⎟ ⎜ T ⎟ k ⎠ ⎝

(14)

where T is the temperature corresponding to the kth iteration. Eq. (14) shows that the probability of acceptance for worst case decreases as the temperature is reduced. The increase in energy is accepted which prevents the algorithm from getting trapped into local minima. An annealing profile is given below that modifies T as: (T − T ) (15) Tk = T0 − k 0 N N

where T0 and TN are the initial and final temperatures and N represents the 59

number of iterations selected as the termination criterion. Keywords: Genetic Algorithm, Simulated Annealing, Transmission Tomography, Inverse Radon Transform, Filtered Back-Projection. PROPERTIES OF (THETA-PRE, S)-CONTINUOUS FUNCTIONS Muhammad Rafiq Department of Computer Science Raja Sarfraz Khan Study Center Allama Iqbal Open University Chakwal-Pakistan Raja Mohammad Latif Department of Mathematical Sciences King Fahd University of Petroleum and Minerals Dhahran-Saudi Arabia We introduce (theta-pre, s)-continuous functions and study basic characterizations and investigate several properties concerning these new types of functions. The general cases for the composition of functions under specific conditions which yield (theta-pre, s)-continuous functions are also studied and some results are obtained. VERTEX-MAGIC TOTAL LABELING OF THE UNION OF SUNS M.T. Rahim School of Mathematical Sciences, GC University Lahore-Pakistan Slamin Universitas Jember Jember -Indonesia Let G be a graph with vertex set V = V (G) and edge set E = E(G) and let e = |E(G)| and v = |V (G)|. A one-to-one map λ from V ∪ E onto the integers {1, 2, ..., v +e} is called vertex magic total labeling if there is a constant so that for every vertex x, λ ( x) + ∑ λ ( xy ) = k where the sum is over all vertices y adjacent to x. Let us call the sum of labels at vertex x the weight wλ ( x) of the vertex under labeling λ ; we require wλ ( x) = for all x. The constant is called the magic constant for λ . 60

A sun Sn is a cycle on n vertices Cn, for n ≥ 3 , with an edge terminating in a vertex of degree 1 attached to each vertex. In this paper, we present the vertex magic total labeling of the union of suns, including the union of m non-isomorphic suns for any positive integer m ≥ 3 , proving the conjecture given in [8]. References [1] M. Bača, M. Miller and Slamin, Vertex-magic total labelings of generalized Petersen graphs, Int. J. of Computer Mathematics 79, Issue 12, (2002) pp.1259–1264. [2] R. Bodendie and G. Walther, On number theoretical methods in graph labelings, Res. Exp. Math. 21 (1995) 3–25. [3] N. Hartsfield and G. Ringel, Pearls in Graph Theory, Academic Press, Boston-San Diego-New York-London, 1990. [4] J. A. Gallian, A dynamic survey of graph labeling, The Electronic Journal of Combinatorics, 5 (2005) #DS6. [5] I.D. Gray, J. MacDougall, J.P. McSorley and W.D. Wallis, Vertex-magic labeling of trees and forests, Discrete Mathematics 261 (2003) 285-298 [6] J. MacDougall, M. Miller, Slamin and W. D. Wallis, Vertex-magic total labelings, Utilitas Math., 61 (2002) 3–21 [7] Slamin and M. Miller, On two conjectures concerning vertex-magic total labelings of generalized Petersen graphs, Bulletin of ICA, 32 (2001) 9–16. [8] Slamin, A.C. Prihandoko, T.B. Setiawan, V. Rosita and B. Shaleh, Vertex-magic total labelings of disconnected graphs, Journal of Prime Research in Mathematics, to appear. [9] W. D. Wallis, Magic Graph, Birhäuser, (2001). [10] M. E. Watins, A Theorem on Tait Colorings with an Application to the Generalized Petersen Graphs, J. Combin. Theory 6 (1969) 152–164. [11] D. B. West, An Introduction to Graph Theory, Prentice-Hall, (1996). ON THE LOCALIZATION OF SPECTRAL EXPANSIONS Abdumalik A. Rakhimov Federal Urdu University of Arts, Science and Technology Karachi-Pakistan & National University of Uzbekistan Tashkent-Uzbekistan Present paper is an introduction of the localization and sum-ability problems of spectral expansions connected with partial differential operators of mathematic physics and it is also given some theorems regarding these 61

problems recently obtained. Here we mostly pay attention to the problem of convergence and sumability of spectral expansions of distributions connected with the development of mathematical tools for modern physics. Especially simple and important example of spectral expansion of distributions is Fourier series of Dirac’s delta function, partial sum of which is well known Dirichlet’s ernel. From the classic theory of trigonometric series it is known that Dirichlet’s kernel is not uniformly approximation of delta function. So spectral expansions of Dirac’s delta function is not convergent in any compact set out of the support of the distribution. But arithmetic means of the partial sum of Fourier series of Dirac’s delta function coincides with Fejer’s kernel and in one dimensional case it uniformly convergent to zero in any compact set where delta function is equal to zero. In multidimensional case the problem become more complicated. SELF-SIMILARITY IN GEOMETRY, ALGEBRA AND ARITHMETIC Arash Rastegar Department of Mathematical Sciences Sharif University of Technology (SUT) Tehran-Iran We define the concept of self-similarity of an object by considering endomorphisms of the object as ‘similarity’ maps. A variety of interesting examples of self-similar objects in geometry, algebra and arithmetic are introduced. Selfsimilar objects provide a framework in which, one can unite some results and conjectures in different mathematical frameworks. In some general cases, one can define a well-behaved notion of dimension for selfsimilar objects. Morphisms between self-similar objects are also defined and a categorical treatment of this concept is provided. TOPOLOGY OF COMPLEX POLYNOMIAL FUNCTIONS USING DISCRIMINANT Zahid Raza School of Mathematical Sciences, GC University Lahore-Pakistan The description of the topology of the corresponding level curves Ct : P(x; y) = t is directly related to the vanishing of the leading coefficients cj(t) of 62

the discriminant of the polynomial P(x,y)-t, regarded as polynomials in t. In this paper, we investigate the topology of complex polynomials P(x; y) in two variables using the discriminant of the polynomial of two variables. In particular, we give the condition such that Ct is smooth and contractible, since this implies that the corresponding polynomial is a coordinate on 2 . We give detailed description including the homotopy types of the curves Ct and the associated geometry up to the degree 3 and 4. APPLICATIONS OF δ - OPEN SETS Muhammad Razaq Department of Computer Science Raja Sarfraz han Study Center Allama Iqbal Open University Chakwal-Pakistan Raja Mohammad Latif Department of Mathematical Sciences King Fahd University of Petroleum and Minerals Dhahran-Saudi Arabia Velico [1968] introduced the concepts of δ -closure and δ -interior operations. We introduce and study topological properties of δ -derived; δ border; δ - frontier and δ - exterior of a set using the concept of δ -open sets and study also other properties of the well-known notions of δ - closure and δ -interior. THE DYNAMICS OF A VERTICALLY TRANSMITTED DISEASE M. R. Razvan Department of Mathematical Sciences Sharif University of Technology Tehran-Iran An SIRS epidemiological model for a vertically transmitted disease is discussed. We give a complete global analysis in terms of three explicit threshold parameters which respectively govern the existence and stability of an endemic proportion equilibrium, the increase of the total population and the growth of the infective population. This paper generalizes the results of Busenberg and van den Driessche.

63

EFFECT OF REORDERING ON PRECONDITIONERS FOR THE NAVIER STOKES EQUATIONS M. ur Rehman, C. Vuik, G. Segal Delft Institute of Applied Mathematics Delft University of Technology Delft-The Netherlands In this paper, we discuss some recently published preconditioners for saddlepoint problems. In combination with Krylov subspace methods, they give a fast convergence for the solution of the Navier Stokes equations. We have modified some of these preconditioners, and compared this with the original ones. Besides that, we have applied some reordering of nodes. The effect of this reordering has been tested for some bench mark problems. Our results show that modification of the preconditioner, in combination with the reordering technique, has a nice effect on the convergence of the Krylov subspace method. THE AMERICAN-STYLE FOREIGN EXCHANGE OPTIONS IN GENERAL ONE-DIMENSIONAL DIFFUSION MODEL FOR EXCHANGE RATE Nasir Rehman School of Mathematical Sciences, GC University Lahore-Pakistan The classical Garman-Kohlhagen model for the currency exchange assumes that the domestic and foreign currency risk-free interest rates are constant and the exchange rate follows a log-normal diffusion process. We consider the general case, when exchange rate evolves according to arbitrary one-dimensional diffusion process with local volatility that is the function of time and the current exchange rate and where the domestic and foreign currency risk-free interest rates may depend on time. We give a rigorous proof of the early exercise premium representation of the value function of the American put option as the sum of the European put option value and the early exercise premium. The proof essentially relies on the properties of the stochastic integral with respect to arbitrary continuous semi-martingale over the predictable subsets of its zeros.

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SOME STRINGY INVARIANTS OF ADE-SINGULARITIES Marko Roczen Humboldt University Berlin-Germany The theory of motivic integration was invented about 1995 by ONTSEVICH as a tool to study Hodge numbers of algebraic varieties. Meanwhile, a whole theory on the subject has developed. Using motivic integration, BATYREV succeeded to introduce new invariants of (at most) log-terminal singularities: He showed independence of the string-theoretic E-function Estr(X; u, v) (which encodes a series of invariants, the string-theoretic Hodge numbers) on the choice of a resolution X for a given singularity. Estr gives rise to the string-theoretic Euler numbers estr(X) = limu ,v→1 Estr (X; u, v). Explicit results for given singularities were obtained by DAIS in the case of simple A-singularities, by DAIS, Roczen for the 3-dimensional ADE-singularities and later completed by Schepers for simple singularities in arbitrary dimensions ≥ 3. Those formulas can be applied to compute stringy Euler numbers of several compact complex 3folds with ADE-Singularities. Mathematics Subject Classification (MSC 2000): 14Q15, 32S35, 32S45, 14B05, 14E15 A CONTOUR METHOD ON CAYLEY TREE U.A. Rozikov Institute of Mathematics Tashkent-Uzbekistan The lattice spin systems are large class of systems considered in statistical mechanics. Some of them have a real physical meaning, others are studied as suitable simplified models of more complicated systems. One of the key problems related to lattice spin systems is the description of the set of Gibbs measures. The structure of the lattice plays an important role in the investigations of spin systems. For example in order to study the phase transition problem (non-uniqueness of Gibbs measure) for a system on Z d and on Cayley tree, respectively, there are two different methods: contour method (Pirogov-Sinai theory) on Z d (see e.g. [5]) and Markov random field theory on Cayley tree (see e.g. [1]). Note, that Pirogov-Sinai theory on Cayley tree is not simply applicable and not much work has been done to develop contour methods on trees ([2]-[4]). 65

As a continuation of our previous papers [2]-[4] devoted to the introduction of a contour method on Cayley tree we consider a finite range lattice models on Cayley tree with two basic properties: the existence of only a finite number of ground states and with Peierls type condition. We define notion of a contour for the model on the Cayley tree. By a contour argument we show the existence of s different (where s is the number of ground states) Gibbs measures. References. 1. Bleher, P.M., Ganihodjaev, N.N.: On pure phases of the Ising model on the Bethe lattice. Theor. Probab. Appl. 35, 216-227 (1990) 2. Rozikov, U.A.: An example of one-dimensional phase transition. Siber. Adv. Math. 16, 121-125 (2006) 3. Rozikov, U.A.: On q¡ component models on Cayley tree: contour method. Lett. Math. Phys. 71, 27-38 (2005) 4. Rozikov, U. A.: A constructive description of ground states and Gibbs measures for Ising model with two-step interactions on Cayley tree. J. Stat. Phys. 122, 217-235 (2006) 5. Sinai, Ya. G.: Theory of phase transitions: Rigorous Results, Oxford: Pergamon, 1982. LIE SYMMETRIES OF TENSORS OF SECOND RANK K. Saifullah Center for advanced Mathematics and Physics National University of Science and Technology Rawalpindi-Pakistan Application of Lie derivative on various geometrical and physical quantities define symmetries in mathematical physics [1]. Thus for a four dimensional Larentzian manifold symmetries of the metric, Ricci and matter tensors are known as motions, Ricci and matter collineations [2]. In Riemannian geometry these symmetries are used not only to obtain new physically important spaces but to classify them also [3]. Here we restrict our study to tensors of second ran only and provide different applications. Some classification schemes are also discussed here. Reference: 1.Hall, G. S., Symmetries and curvature structure in general relativity, World Scientific Publishing Co., 2004. 2.Qadir A., Saifullah ., Ziad M., Classification of cylindrically symmetric static spacetimes according to their Ricci collineations, Gen. Rel. Grav., 66

2003, V. 35, 1927-1975; Saifullah ., Lie symmetries of the energymomentum tensor for plane symmetric static spacetimes, Int. J. Mod. Phys. D, 2005, V. 14, 797-816. 3.Stephani H., ramer D., MacCallum M.A.H., Hoenselaers C., Herly E., Exact solutions of Einstein’s Field equations, Cambidge University Press, 2003. SHELL MODEL OF BEAM SPLITTING IN MOMENTUM SPACE Saifullah School of Mathematical Sciences, GC University Lahore-Pakistan In this paper we describe a shell model for the quantum open-loop controlled beam splitter in momentum space. The splitting effect has been demonstrated in both asymptotic of the small and large number of shells. ON TRANSNORMAL EMBEDDINGS B. A. Saleemi Air University Islamabad-Pakistan Let M n be a compact, connected, smooth n-manifold which is smoothly embedded in E n+ N , N ≥ 1. Let p ∈ M n and π ( p ) be the normal N-plane to M n at p. Then M n is said to be transnormal in E n+ N if, for each pair (p, q) ∈ M n , q ∈ π ( p ) if and only if π ( p ) = π (q ) . The map π from a transnormal manifold M n to the space of normal N-plane of M n is a covering map. We have proved: Theorem: Every transnormally embedded compact manifold M n in E n+ N is minimally embedded in the sense of the theory of total absolute curvature. AN ULTRAPARABOLIC MODEL OF TWO-PHASE FILTRATION WITH DEGENERATE HYDRODYNAMIC DISPERSION Sergey Sazhenkov NUST–CAMP Rawalpindi- Pakistan 67

& Lavrentyev Institute of Hydrodynamics Novosibirsk-Russia A Cauchy problem for a two-dimensional ultra-parabolic model of filtration through a porous ground of a viscous incompressible fluid containing a solute (tracer) is considered. The fluid is driven by the buoyancy force. The phenomenon of molecular diffusion of the tracer into the porous ground is taken into account. The porous ground consists of one-dimensional filaments oriented along some smooth non-degenerate vector field. Two cases are distinguished depending on spatial orientation of the filaments, and existence of generalized entropy solutions is proved for the both. In the first case, all filaments are parallel to the buoyancy (gravitational) force and, except for this, the equations of the model have rather general forms. In the second case, the filaments can be nonparallel to the buoyancy force and to each other, in general, but their geometric structure must be genuinely nonlinear. The proofs rely on the method of kinetic equation and the theory of Young measures and H-measures. Keywords: Ultraparabolic equation, genuine nonlinearity, non-isotropic porous medium, nonlinear diffusion-convection. AMS subject classification: 3565, 76S05, 76R50, 35D05. MILNOR ALGEBRAS AND BRIESKORN MODULES OF HOMOGENEOUS POLYNOMIALS Khurram Shabbir School of Mathematical Sciences, GC University Lahore-Pakistan Let f∈ [ X 1 ,… , X n ] be the homogeneous polynomial and M(f) be the corresponding Milnor algebra which is the quotient of polynomial ring by jacobian ideal. This algebra M(f) is a graded -algebra and its dimension is Milnor number denoted as µ(f) when f has an isolated singularity at the origin. Let B(f) be the Brieskorn module which is the quotient of the polynomial ring by some specific -vector space. The ring of polynomials [t] acts on B(f) in the following way tk[p] =[fk.p]. It is known that B(f) is a free C[t] module of rank µ(f) when f has an isolated singularity at the origin. The torsion of Brieskorn module B(f) is known only for two variables. The goal of my research is to find ways that allow us to study the torsion part of B(f) for more than two variables, using Milnor algebra.

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AVANT-GRADE SPLITTING FOR THE SOLUTION OF SPARSE NON-SYMMETRIC LINEAR SYSTEMS Ahmed Ali Shah Institute of Business Administration Karachi-Pakistan A separate splitting is presented in all investigations, old and new, for the solution of certain sparse linear systems with a non-symmetric matrix. We present the comparison and the convergence performance of these investigations with previous and the contemporary methods and include explicit comments. It is also studied that the methods that may work efficiently with a symmetric matrix but show insufficiencies lie numerical instability and non-scaling invariant with non-symmetric matrix. A splitting is proposed which can overcome these deficiencies. MONOGENIC BEHAVIOUR OF QUARTIC ABELIAN NUMBER FIELDS Syed Inayat Ali Shah, Muhammad Iqbal Islamia College University of Peshawar Peshawar-Pakistan Let K be an algebraic number field and Z K be the ring of integers in K . It is called that K is monogenic or K has a power integral basis if the ring Z K = Z [α ], α ∈ K .When K is a full cyclotomic field kn or its maximal real subfield, K is monogenic. On the contrary K is a quartic field with prime conductor over the rationals, K is non-monogenic except for k5 [Mh. Math. 94(1984)125-132]. We endevour to determine the monogenic behaviour of several prototypes applying previous works [Nagoya Math.J.168(2002),8592,Arch. Math (2004), 309-316]. MATHEMATICAL DEVELOPMENT IN THE INDUS VALLEY CIVILIZATION Mohibullah Shaikh Ex-Professor, St. Mary's Academy Rawalpindi-Pakistan About 700 wells of varying diameters in the ruins of Moen-jo-daro invite our attention towards the fact that: To establish relation “between 69

CIRCUMFERENCE and DIAMETER of a circle” was their necessity, which helped them in preparing the WOODEN-RING called GARH to be used as basement for erecting and sliding the pucca wall of each well upto the waterstream level. This assertion is further strengthened by the WEDGE-SHAPED bricks used in constructing the wells. Furthermore, the supply of hundreds of thousands of baked-bricks from the KILN to the site of construction reminds that ‘ even the illiterate labourers and workers of the kiln were highly efficient in counting orally the large number of bricks supplied thereat by making piles of certain number of bricks. So the survivors after the catastrophe were able to transmit these number-words to their posterity. TOPOLOGY AND FACTORIZATION OF POLYNOMIALS Hani Shaker School of Mathematical Sciences, GC University Lahore-Pakistan Let P be a polynomial in two variables with complex coefficients. The question under investigations is, how many irreducible components the plane curve C:P(x,y)=0 has?. The answer to this question is directly related to the study of the topology of the complement of C in the complex plane 2 using de Rham cohomology. The main problem is to extend this result for more variables and to obtain other related results on algebraic affine hypersurfaces. SYMMETRIES OF THE ENERGY-MOMENTUM TENSOR M. Sharif University of the Punjab Lahore-Pakistan The symmetries of the energy-momentum tensor for the static spacetimes with maximal symmetric transverse spaces are analyzed by taking three nonzero components of the vector xi a . This gives three, four and five independent matter collineations for the non-degenerate as well as for the degenerate cases and generalizes the degenerate case of the static spherically symmetric spacetimes. Some constraint equations are obtained which, on solving, may provide some new exact solutions of the Einstein field equations.

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THE AMERICAN PUT OPTION IN A ONE-DIMENSIONAL DIFFUSION MODEL WITH LEVEL-DEPENDENT VOLATILITY Malkhaz Shashiashvili Tbilisi State University Tbilisi -Georgia We generalize the well-known results on the American put option problem derived for the classical Black-Sholesmodel to the case of diffusion model with level-dependent volatility. We establish the early exercise premium representation of a value function of the American put option. Then we derive the nonlinear integral equation for the optimal exercise boundary. The uniqueness of the solution of this integral equation is proved by introducing the corresponding Cauchy problem with discontinuous right-hand side and with a unique solution in the second order Sobolev space of functions. References 1. C. Berge, “Hypergraphes” , Bordas, Paris, 1987. 2. F.M.Dong, K.L.Teo, C.H.C Little, M. Hendy, K. M.oh, “Chromatically Unique Multibridge Graphs", The Electronic Journal of Combinatorics 11 (2004),R12. 3. F.M.Dong, K.M.oh, K.L.Teo, “Chromatic polynomilas and Chromaticity of Graphs”, first edition, World Scientific Publishing Co. Pte. Ltd. Singapore, 2005. 4. Ioan Tomescu, “Chromatic coeffcients of Linera uniform Hypergraphs", Journal of Combinatorial Theory, Series B, Vol. 72, No. 2, 1998, (229-235). 5. J. H. van Lint and R. M. Wilson, “A course in combinatorics", second edition, Cambridge Univ. Press United Kingdom, 2001. 6. R. L. Graham, M. Grotschel and L. Lovaasz, “Handbook of combinatorics",Vol. I and II North-Holan, 1995. 7. R. C. Read, “An introduction to chromatic polynomials", J. Combin. Theory 4(1968), 52-71. WAVE PROPERTIES IN NON-MAGNETIZED, ISOTHERMAL SCHWARZSCHILD GEOMETRY Umber Sheikh, M. Sharif University of the Punjab Lahore-Pakistan The behavior of waves near the non-rotating black hole horizon is analyzed. 71

We consider non-magnetized isothermal Schwarzschild geometry and form the GRMHD equations. The 3+1 formalism and Fourier analysis methods are applied to reduce these equations to ordinary differential equations. The dispersion relations are then found to obtain the wave numbers. Finally, we obtain information like phase and group velocities etc. which help to discuss the nature and characteristics of waves. COHOMOLOGY OF CONFIGURATION SPACE OF CPm Tanweer Sohail School of Mathematical Sciences, GC University Lahore- Pakistan In this paper we compute topological invariants such as Poincare polinomial, cohomological algebra for configuration spaces of some of very important algebraic manifolds: Complex projective spaces. The method we shall use is based on the results of Fulton-MacPherson and Kriez. We express our topological invariants using cyclotomic polynomials. We present complete results for configuration spaces of two points in CPm. Keywords and phrases: Poincare polynomials, cohomological algebra, cyclotomic polynomials. APPLIED STABILITY METHOD OF PERIODIC SOLUTION: RING MODES IN NON-LINEAR GALAXY MODEL Mariam Sultana Federal Urdu University Karachi-Pakistan Mathematical modeling problems for ring-like structure formation are considered. Till day, many authors have studied the problem in the frame of stationary model. In this work, we investigate a possibility of construction of new pulsating model for dissipation less collision less astrophysical system. To achieve the same, a stability method of period oscillation is used. It is found that critic value of pulsation amplitude is connected with virial ratio at initial moment of collapse. Marginal state for ring-like oscillation mode is studied.

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A GENERALIZATION OF KRASNER’S LEMMA Sobia Sultana School of Mathematical Sciences, GC University Lahore-Pakistan Let

p

be the p -adic number field and let

Let G p be the absolute Galois group of C = Cont (G p ,

p

) be the

p

p

be an algebraic closure of it. p

, G p = Gal (

p

/

p

) . Let

-Banach algebra of all continuous functions

defined on the compact space G p , with values in the complex p -adic numbers field p . In this paper, we present a general form of the classical Krasner’s lemma for the

p

-algebra C . We also prove that this last result is

a” natural” generalization of Krasner’s lemma. UIR MATRIX ELEMENTS OF FINITE ROTATIONS OF SO(2, 1) DECOMPOSED ACCORDING TO THE SUBGROUP T1 Ansaruddin Syed Department of Mathematics BUITMS, Quetta-Pakistan Using a technique of Kalnins, UIR of principle series of SO(2,1), decomposed according to the subgroup T1 , are realized on the space of homogeneous functions on the cone ξ 02 - ξ 12 - ξ 22 = 0 as the carrier space. It is then shown that the matrix elements of an arbitrary finite rotation of SO (2. 1) are determined by those of two specific type of rotations, each depending on a single parameter; matrix elements of these two specific types of finite rotations are then explicitly computed. ECENT DEVELOPMENT IN SCIENTIFIC COMPUTING Tao Tang Dept of Mathematics Hong Kong Baptist University Hong Kong

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Scientific Computing deals with the development of efficient and reliable algorithms for computations with floating-point numbers on computers, as well as the underlying theory related to the accuracy and stability of the algorithms. With the advance of modern high-performance computers, and in particular computers with parallel architecture, the implementation aspects have come to play a very useful role. Moreover, as the computational problems increase in size and complexity, the interplay between the mathematical modeling, the numerical algorithms, and the computer implementation becomes increasingly important. In this talk, the history of the scientific computing research will be reviewed and its current development will be introduced. The applications of scientific computing in several important areas such as computational fluid dynamics, scientific visualization and simulation of physical processes will be also discussed. ON THE CHROMATICITY OF SUNFLOWER HYPERGRAPHS Ioan Tomescu University of Bucharest Bucharest-Romania A sunflower hypergraph SH (n,p,h) is an h-hypergraph of order n = h+(k-1)p and size k(0