Browsing by Author "Aywa, Shem."

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Advances in composite integer factorization
(2013) Wanambisi, Aldrin W.; Aywa, Shem.; Maende, Cleophas.; Muketha, Geoffrey Muchiri.
In this research we propose a new method of integer factorization. Prime numbers are the building blocks of arithmetic. At the moment there are no efficient methods (algorithms) known that will determine whether a given integer is prime or and its prime factors [1]. This fact is the basis behind many of the cryptosystems currently in use.
Advances in composite integer factorization
(Mathematical theory and modelling, 2013) Wanambisi, Aldrin.; Aywa, Shem.; Maende, Cleophas.; Muketha, Geoffrey Muchiri.
In this research we propose a new method of integer factorization. Prime numbers are the building blocks of arithmetic. At the moment there are no efficient methods (algorithms) known that will determine whether a given integer is prime or and its prime factors [1]. This fact is the basis behind many of the cryptosystems currently in use.
Algebraic approach to composite integer factorization
(International journal of mathematics and statistics, 2013) Wanambisi, W.; Aywa, Shem.; Maende, C.; Muketha, MG.
There various algorithms that can factor large integers but very few of these algorithms run in polynomial time. This fact makes them inefficient. The apparent difficulty of factoring large integers is the basis of some modern cryptographic algorithms. In this paper we propose an algebraic approach to factoring composite integer. This approach reduces the number of steps to a finite number of possible differences between two primes.
Aspects of (p.q) summing multipliers
(Vdm verlag Dr. Muller aktiengeselischaft & co. kg., 2010) Aywa, Shem.; Agure, J.O.; Rao, W.
A sequence (uj )j∈N of operators in L(X, Y ) is a (p, q)-summing multiplier (or (p, q)-summing sequence of operators), in short (uj ) ∈ `πp,q (X, Y ), if there exists a constant C > 0 such that, for any finite collection of vectors x1, x2, . . . xn in X, it holds that ³Xn j=1 kujxjk p ´1/p ≤ C sup n³Xn j=1 |x ∗ xj | q ´1/q ; x ∗ ∈ BX∗ o . Some examples of these operators, inclusions between the spaces and connections with spaces of multipliers are presented.∗ Mathematics Subject Classification (2000): 47B10. Key words: Summing operators, vector-valued multipliers. 1. Introduction. Let X and Y be two real or complex Banach spaces and let E(X) and F(Y ) be two Banach spaces whose elements are defined by sequences of vectors in X and Y (containing any eventually null sequence in X or Y ). A sequence of operators (un) ∈ L(X, Y ) is called a multiplier sequence from E(X) to F(Y ) if there exists a constant C > 0 such that ° °(ujxj ) n j=1 ° ° F (Y ) ≤ C ° °(xj ) n j=1 ° ° E(X) for all finite families x1, . . . , xn in X. The set of all of multiplier sequences is denoted by (E(X), F(Y )). For the study of such multipliers for the cases of E(X) and F(Y ) corresponding to vector-valued Hardy spaces, vector-valued Bergman spaces, vector-valued BMOA or spaces of vector valued Bloch functions the reader is referred to [AB1, Bl1, Bl2, Bl3, Bl4].
Characterization of codes of ideals of the polynomial ring f30 2 [x] mod ô€€€ x30 ô€€€ 1 for error control in computer applicatons
(2016-06-30) Olege, Fanuel.; Oduor, Oduor M.; Aywa, Shem.; Okaka, Colleta A.
The study of ideals in algebraic number system has contributed immensely in preserving the notion of unique factorization in rings of algebraic integers and in proving Fermat's last Theorem. Recent research has revealed that ideals in Noethe-rian rings are closed in polynomial addition and multiplication.This property has been used to characterize the polynomial ring Fn 2 [x] mod (xn 1) for error control. In this research we generate ideals of the polynomial ring using GAP software and characterize the polycodewords using Shannon's Code region and Manin's bound.
Conecture of banach space operator ideals in nuclear spaces
(Asian journal of current engineering and maths, 2013) Musundi, W.; Ombaka, S.O.; Njogu, C.M.; Muthengi, S.F.; Mugambi, D.; Aywa, Shem.
We apply the notion of Banach space operator ideals in nuclear spaces through topological vector spaces. The motivation for this study came from attempts to generalize the structure of nuclear spaces as a result of nuclear maps from functional analysis context. The compact closed structure associated with the category of relations results to nuclear ideals. Basic properties of Banach space operator ideals in relation to the structure of nuclear spaces will be demonstrated. We therefore establish a close correspondence between Banach space operator ideals and nuclear ideals through topological vector.
Equivalent banach operator ideal norms
(International journal of mathematical analysis, 2012) Musundi, S.; Aywa, Shem.; Fourie, J.
We continue the investigation of coorbit spaces which can be attached to every integrable, irreducible, unitary representation of a locally compact groupG and every reasonable function space onG. Whereas Part I was devoted to atomic decompositions of such spaces, Part II deals with general properties of these spaces as Banach spaces. Among other things we show that inclusions, the quality of embeddings, reflexivity and minimality and maximality of coorbit spaces can be completely characterized by the same properties of the corresponding sequence spaces. In concrete examples (cf. Part III) one recovers several and often difficult theorems with ease.
Equivalent Banach operator ideal norms1
(Hikari: International Journal of mathematical analysis, 2012-12-01) Musundi, Sammy.; Aywa, Shem.; Jan, Fourie.
We present some results concerning the general theory of Banach ideals of operators and give several applications to Banach space theory. We give, in Section 3, new proofs of several recent results, as well as new operator characterizations of the p-spaces of Lindenstrauss and Pelczynski. In Section 4 we prove that the space of absolutely summing operators from E to F is reflexive if both E and F are reflexive and E has the approximation property. Section 5 concerns Hilbert spaces. In particular, we compute the relative projection constant of Hilbert spaces in Lp(μ)-spaces.
Factorization of large composite integer
(International journal of mathematics and studies, 2013) Wanambisi, Aldrin.; Aywa, Shem.; Maende, Cleophas.; Muketha, Geofrey Muchiri.
This paper is concerned with the problem of obtaining theoretical estimates for the number of arithmetical operations required to factorize a large integer n or test it for primality. One way of making these problems precise uses a multi-tape Turing machine (e.g. (1), although we require a version with an input tape). At the start of the calculation n is written in radix notation on one of the tapes, and the machine is to stop after writing out the factors in radix notation or after writing one of two symbols denoting ‘prime’ or ‘composite’. There are, of course, other definitions which could be used; but the differences between these are unimportant for our purpose.
Finite difference solution othird order viscous wave equation
(International journal of engineering, science and mathematics, 2014) Oganga, Duncan.; Okoya, Michael O.; Aywa, Shem.; Lawi, George O.; Nthiiri, Joyce .
A new numerical method for Nwogu’s (ASCE Journal of Waterway, Port, Coastal and Ocean En-gineering1993;119:618) two-dimensional extended Boussinesq equations is presented using a lineartriangular nite element spatial discretization coupled with a sophisticated adaptive time integrationpackage. The authors have previously presented a nite element method for the one-dimensional formof these equations (M. Walkley and M. Berzins (International Journal for Numerical Methods inFluids1999;29(2):143)) and this paper describes the extension of these ideas to the two-dimensionalequations and the application of the method to complex geometries using unstructured triangular grids.Computational results are presented for two standard test problems and a realistic harbour model. Copy-right?2002 John Wiley & Sons, Ltd
Generation for measurable covers for measurable sets using multiple integral of functions
(Science sign post publishing, 2015) Otanga, o.; Levi, O.; Aywa, Shem.; Owino, M.
The point-objective problem and the Weber problem are two well-known formulations for locating a new facility with respect to a set of fixed facilities. When locations are represented as points on a plane, the point-objective problem is a multiple objective formulation of minimizing the distance from a variable point to each of the fixed points. Similarly, the Weber problem is a single objective formulation of minimizing the sum of transportation costs between the variable point and the fixed points, where transportation cost is a function of distance. Generalizing solution properties for these problems from distance measures given by the Euclidean, rectilinear, Ip, and one-infinity norms; this paper develops solution properties under the broad classes of distance measures given by block and round norms. For the point-objective problem, we show that (i) the efficient set for all round norms is the convex hull of the set of fixed points and (ii) the efficient set under a block norm tends to the convex hull for a sequence of block norms approaching a round norm. For the Weber problem, we prove that (i) an optimal location for any block norm may be found in a finite set of intersection points belonging to the convex hull and (ii) this set tends to the convex hull for a sequence of block norms approaching a round norm. Finally, we use these results to propose a synthesis of some of the main properties in continuous and network location theory.
Ideals if polynomial ring got error control in computer applications
(Mathematical theory and modelling, 2013) Olege, F.; Aywa, Shem.; Rao, R.D.K.; Wanambisi, A.
We generalize the classical Knuth-Schonhage algorithm computing GCD of two polynomials for solving arbitrary linear Diophantine systems over polynomials in time, quasi-linear in the maximal degree. As an application, we consider the following weighted curve fitting problem: given a set of points in the plain, find an algebraic curve (satisfying certain degree conditions) that goes through each point the prescribed number of times. The main motivation for this problem comes from coding theory, namely it is ultimately related to the list decoding of Reed-Solomon codes. We present a new fast algorithm for the weighted curve fitting problem, based on the explicit construction of Groebner basis. This gives another fast algorithm for soft-decoding of Reed-Solomon codes different from the procedure proposed by Feng (1999), which works in time (w/r)/sup O(1)/ n log/sup 2/ n loglogn, where r is the rate of the code, and w is the maximal weight assigned to a vertical line.
Income source diversification and financial performance of commercial banks in Kenya
(Mku Journals, 2011-01-21) Teimet, Paul Rotich.; Ochieng, Damianus Okaka.; Aywa, Shem.
The profitability of commercial banks depends heavily on the net of income generating activities and the related activities’ expense. Due to the problem of profitability and stiff competition in the industry, commercial banks have changed their behavior of income sources, by increasingly diversifying into non-intermediation income generating activities as opposed to the traditional inter-mediation income generating activities. The objective of this paper was to establish the impact of income source diversification on financial performance of commercial banks in Kenya. This has been achieved through: establishing the level of income source diversification of commercial banks in Kenya and establish whether income source diversification improves financial position of commercial banks. This was a census study of all registered 44 commercial banks in Kenya and relied heavily on documentary secondary data for 5 year study period (2005-2009) and validated by primary data achieved through keyinformant method. Herfindahl-Hirschman Index, Correlations and Regression analysis were mainly used and revealed on aggregate that all commercial banks in Kenya are diversified with large banks in lead while Islamic banks trail. Further, diversification level has a positive influence on financial performance of commercial banks in Kenyan and the two main revenue streams are positively related.
Measurable feller semigroups on rn
(2007) Oduor, Maurice.; Owino; Aywa, Shem.
Fractional derivatives are used to model anomalous diffusion, which occurs when the particles spread in a different manner than the prediction of the classical diffusion equation ∂ ∂tu(x, t) = D ∂ 2 ∂x2 u(x, t), u(x, 0) = f(x). The solution u (x, t) depends on location x ∈ R and time t ≥ 0 and models the dispersion. A known model for an anomalous diffusion (see [6]) is the fractional diffusion equation, where the usual second derivative in space is replaced by a fractional derivative of order α, 0 < α < 2, ∂ ∂tu(x, t) = D ∂ α ∂xα u(x, t), u(x, 0) = f(x). We observe that ∂ α ∂xα is a pseudodifferential operator. Thus, we can extend this equation to ∂u ∂t (x, t) = (Au(· , t))(x), u(x, 0) = u0(x), where A is a pseudodifferential operator. A study of the solutions of a generalized reaction-diffusion equation of the form ∂u ∂t (x, t) = (Au (· , t)) (x) + f (x, u (x, t)), u (x, 0) = u0 (x), MATH. REPORTS 12(62), 2 (2010), 181–188 where A is a pseudodifferential operator which generates a Feller semigroup, was given in [9]. In this paper we consider the fractional Cauchy problem ∂ β ∂tβ u(x, t) = (Au(·, t))(x), u(x, 0) = f(x), where ∂ β ∂tβ u (x, t) is the Caputo fractional derivative in time and A is a pseudodifferential operator which generates a Feller semigroup. In [1] and [2] was shown that the solution of fractional Cauchy problem, where 0 < β < 1, t ≥ 0 and A is the generator of bounded continuous semigroup {T (t)}t≥0 on the Banach space X, can be expressed as an integral transform of the solution to the initial Cauchy problem ∂ ∂tu(x, t) = (Au(·, t))(x), u(x, 0) = f(x). Starting from this integral transform, we give a formula for the solution u(x, t) = S(t)f(x) of the fractional Cauchy problem. We show that {S(t)}t≥0 is a family of pseudodifferential operators. Their symbols are obtained by transformation of the symbols of the semigroup {T(t)}t≥0, where u(x, t) = T(t)f(x) is the solution to the initial Cauchy problem.
Mixed galerkin inite element solution of the homogenous burgers equation
(International journal of engineering & scientific research, 2015) Adenya, K.; Okoya, RO.; Aywa, Shem.; Oganga, OD.
Nonlinear partial differential equations arise in a large number of mathematical and engineering problems. Systems of nonlinear partial differential equations have attracted much attention in studying solid state physics, fluid mechanics, chemical, propagation of undular bores in shallow water waves [1], propagation of waves in elastic tube filled with a viscous fluid [2], and plasma physics [3]. Burgers equation is one of the well-known equations in mathematics and physics. This equation has been found to describe various kinds of phenomena such as the mathematical model of turbulence [4] and the approximate theory of flow through a shock wave traveling in a viscous fluid [5]. The Korteweg–de Vries–Burgers (KdV–Burgers) equation is a 1-D generalization of the model description of the density and velocity fields that takes into account pressure forces as well as the viscosity and the dispersion. Several numerical methods are used such as Chebyshev spectral collocation method [6], meshfree interpolation method [7], modified extended backward differentiation formula [8], direct variational methods [9], and so on to solve these equations [10, 11]. In this paper, mixed finite difference [12] and Galerkin methods are used to solve the 1-D, KdV [13], and coupled Burgers equations with interpolating scaling functions (ISFs). Burgers equation in this paper is represented in three types as .E1/ 1-D Burgers equation ut C ˛uux uxx D 0, .x, t/ 2 Œa, b Œ0, T, (1.1) with the initial and boundary conditions u.x, 0/ D f.x/, x 2 Œa, b, (1.2) u.x, t/ D g.t/, .x, t/ 2 Œa, b Œ0, T, (1.3) respectively, where ˛ and are arbitrary constants
On characterization of u-ideals determined by sequences
(2012) Matuya, Wanyonyi John.; Makila, Patrick.; Achiles, Simiyu.; Aywa, Shem.; Sammy, Musundi.
The area of ideals is important in the study of Analysis, algebra, Geometry and Computer science. The various types of ideals have been studied, for example m ideals and h ideals. The m ideals defined on real Banach spaces are referred to as u - ideals. The natural examples of u - ideals with respect to their biduals, are order continuous Banach lattices. Using the approximation property, we shall study properties of u - ideals and their characterization. We define the set of compact operators K X( ) on X to be u - ideals given that X is a separable reflexive Banach space with approximation property if and only if there is a sequence (Tn ) of finite rank of operators with lim 2 1 n n →∞ I T − = and lim n n →∞Tx x = . We shall show that u -ideals containing no copies of sequences 1  are strict u - ideals.
On convergence of sections of sequences in Banach spaces
(Springer-verlag, 2000-02-01) Aywa, Shem.; Fourie, Jan H.
An elementary proof of the (known) fact that each element of the Banach spaceℓ w p (X) of weakly absolutelyp-summable sequences (if 1≤p<∞) in the Banach spaceX is the norm limit of its sections if and only if each element ofℓ w p (X) is a norm null sequence inX, is given. Little modification to this proof leads to a similar result for a family of Orlicz sequence spaces. Some applications to spaces of compact operators on Banach sequence spaces are considered.
On generation of measurable covers for measurable sets using multiple integral of functions
(200-09-20) Olwamba, Otanga Levi.; Aywa, Shem.; Oduor, Owino Maurice.
A fundamental notion in many areas of mathematics, including optimization, probability, variational problems, functional analysis and operator theory, is that of an integral functional. By this is meant an expression of the form If(x) = ~ f(s,x(s))p(ds), x E X, where X is a linear space of measurable functions defined on a measure space (S,A,~) and having values in a linear space E. The function f: S x E § R is the associated Inte~rand. Classically, only finite integrands on S x R n were studied, usually under the assumption that f(s,x) was continuous in x and measurable in s (the Carathgodory condition). However, from the modern point of view it is essential to admit possibly infinite values for f and If, since it is in this way that important kinds of constraints can most efficiently be represented. Such integrands require a distinctly new theoretical approach, where questions of measurability and the existence of measurable selections are prominent and are reflected in a concept of "normality". The purpose of these notes is to provide a relatively thorough treatment of the most common case in applications, that where E = R n. While many of the results have extensions in one way or another beyond this case, as indicated to some extent in the text, these are often more complicated technically and may require further restrictions. For example, it is only for R n that one presently knows how to develop a complete theory without assuming that the measurable space is complete, an assumption which appears to be awkward in some situations. In treating Inflnite-dimenslonal spaces E, there are the usual problems of the multiplicity of topologies and dualities which must be ironed out. It is desirable, therefore, to have available a full and consistent exposition of the details in the basic case of E = R n, freeing one from the need to search for auxilliary results through sequences of papers wlth varying frameworks.
On redemacher boundedness in operator – norm
(Journal of agriculture, pure and applied science and technology, 2012) Wanyonyi, W.; Simiyu, J.A.; Aywa, Shem.; Kirui, C.S.
In these lecture notes we report on recent breakthroughs in the functional analytic approach to maximal regularity for parabolic evolution equations, which set off a wave of activity in the last years and allowed to establish maximal L p -regularity for large classes of classical partial differential operators and systems. In the first chapter (Sections 2-8) we concentrate on the singular integral approach to maximal regularity. In particular we present effective Mihlin multiplier theorems for operator-valued multiplier functions in UMD-spaces as an interesting blend of ideas from the geometry of Banach spaces and harmonic analysis with R-boundedness at its center. As a corollary of this result we obtain a characterization of maximal regularity in terms of R-boundedness. We also show how the multiplier theorems “bootstrap” to give the R-boundedness of large classes of classical operators. Then we apply the theory to systems of elliptic differential operators on Rn or with some common boundary conditions and to elliptic operators in divergence form. In Chapter II (Sections 9-15) we construct the H∞ -calculus, give various characterizations for its boundedness, and explain its connection with the “operator-sum” method and R-boundedness. In particular, we extend McIntosh’s square function method form the Hilbert space to the Banach space setting. With this tool we prove, e.g., a theorem on the closedness of sums of operators which is general enough to yield the characterization theorem of maximal L p -regularity. We also prove perturbation theorems that allow us to show boundedness of the H∞ -calculus for various classes of differential operators we studied before. In an appendix we provide the necessary background on fractional powers of sectorial operators.
On summing multipliers and applications
(Academic press, 2001-01-01) Aywa, Shem.; Fourie, Jan H.
A scalar sequence (αi) is said to be a p-summing multiplier of a Banach space E, if ∑∞i = 1‖αixi‖p < ∞ for all weakly p-summable sequences in E. We study some important properties of the space mp(E) of all p-summing multipliers of E, consider applications to E-valued operators on the sequence space lp, and extend this work to general “summing multipliers.” The case p = 1 shows close resemblance to the work of B. Marchena and C. Piñeiro (Quaestiones Math., to appear), where the results originated from the authors' interest in sequences in the ranges of vector measures.