some results on the apostol bernoulli and apostol euler polynomials the main object of this paper is to investigate the apostol bernoulli polynomials and the apostol euler polynomials we first establish two relationships between the generalized apostol bernoulli and apostol euler polynomials it can be found that many results obtained before are special cases of these two relationships moreover we have a study on the sums of products of the apostol bernoulli polynomials and of the apostol euler polynomials

some new results on products of apostol bernoulli and apostol euler polynomials abstract we perform a further investigation for the apostol bernoulli and apostol euler polynomials and numbers by making use of an elementary idea used by euler in the discovery of his famous pentagonal number theorem we establish some new formulae for products of an arbitrary number of apostol bernoulli and apostol euler polynomials and numbers these results are the corresponding generalizations of some known formulae including the higher order convolution ones discovered by agoh and dilcher 2014 5 on the classical bernoulli and euler polynomials

small on large geometric anelasticity in this paper we are concerned with finding exact solutions for the stress fields of nonlinear solids with non symmetric distributions of defects or more generally finite eigenstrains that are small perturbations of symmetric distributions of defects with known exact solutions in the language of geometric mechanics this corresponds to finding a deformation that is a result of a perturbation of the metric of the riemannian material manifold we present a general framework that can be used for a systematic analysis of this class of anelasticity problems this geometric formulation can be thought of as a material analogue of the classical small on large theory in nonlinear elasticity we use the present small on large anelasticity theory to find exact solutions for the stress fields of some non symmetric distributions of screw dislocations in incompressible isotropic solids

small deformations superposed on large deformations of an elastic plastic material abstract in the context of a purely mechanical rate type theory of plasticity and utilizing a strain space formulation an infinitesimal theory is developed for motions superposed on any given motion of an elastic plastic material the given motion may involve all forms of elastic plastic deformation including both loading and unloading and in addition the loading conditions are allowed to change due to the superposed motion the infinitesimal theory is properly invariant under arbitrary finite superposed rigid motions

on the dynamical axisymmetric stress field in a finite pre stretched bilayered slab resting on a rigid foundation the dynamic axisymmetric stress field in an initially finite pre stretched bilayered slab resting on a rigid foundation is studied within the framework of the piecewise homogeneous body model the three dimensional linearized theory of elastic waves in initially stressed bodies is used it is assumed that a time harmonic point located normal force acts on the free face plane of the slab the considered problem is solved by employing the hankel integral transformation the materials of the layers are assumed to be incompressible and the elastic relations are given through the treloar potential the formulation and solution to the problem coincide with the corresponding ones of classical linear theory of elasticity for an incompressible body in the case where there is no initial stretching in the layers numerical results are presented and these results involve stresses acting on the interface planes in particular it is established that stresses on the interface planes decrease as the pre stretching is increased

stability of discrete linear inclusion let m ai be a set of linear operators on rn the discrete linear inclusion dli m is the set of possible trajectories xi i 0 such that xn ainain 1 ai1x0 where x0 rn and aij m we study several notions of stability for dli m including absolute asymptotic stability aas which is that all products ain ai1 0 as n we mainly study the case that m is a finite set we give criteria for the various forms of stability two new approaches are taken one relates the question of aas of dli m to formal language theory and finite automata while the second connects the aas property to the structure of a lie algebra associated to the elements of m more generally the discrete linear inclusion dli m makes sense for m contained in a banach algebra b we prove some results for aas in this case and give counterexamples showing that some results valid for finite sets of operators on rn are not true for finite sets m in a general banach algebras b

inequalities for numerical invariants of sets of matrices we prove three inequalities relating some invariants of sets of matrices such as the joint spectral radius one of the inequalities in which proof we use geometric invariant theory has the generalized spectral radius theorem of berger and wang as an immediate corollary

a chebyshev collocation method for solving two phase flow stability problems this paper describes a chebyshev collocation method for solving the eigenvalue problem that governs the stability of parallel two phase flow the method is based on the expansion of the eigenfunctions in terms of chebyshev polynomials point collocation and the subsequent solution of the resulting generalized eigenvalue problem with the qz algorithm we concentrate on the question how to handle difficulties that arise when these standard techniques are applied to the stability problem of a thin film of liquid that is sheared by a gas after discussing this specific problem in detail it is argued that the method of solution can readily be applied to other two phase flow configurations as well

use of geometric algebra compound matrices and the determinant of the sum of two matrices in this paper we demonstrate the capabilities of geometric algebra by the derivation of a formula for the determinant of the sum of two matrices in which both matrices are separated in the sense that the resulting expression consists of a sum of traces of products of their compound matrices for the derivation we introduce a vector of grassmann elements associated with an arbitrary square matrix we recall the concept of compound matrices and summarize some of their properties this paper introduces a new derivation and interpretation of the relationship between p forms and the pth compound matrix and it demonstrates the use of geometric algebra which has the potential to be applied to a wide range of problems

pade approximation and apostol bernoulli and apostol euler polynomials using the pade approximation of the exponential function we obtain recurrence relations between apostol bernoulli and between apostol euler polynomials as applications we derive some new lacunary recurrence relations for bernoulli and euler polynomials with gap of length 4 and lacunary relations for bernoulli and euler numbers with gap of length 6

ruga formation instabilities of a graded stiffness boundary layer in a neo hookean solid we present an analysis of ruga formation instabilities arising in a graded stiffness boundary layer of a neo hookean half space caused by lateral plane strain compression in this study we repres

from local to global triangularization abstract it is shown that a semigroup of shatten p class operators is simultaneously triangularizable if each pair of operators in the semigroup is triangularizable several sufficient conditions for triangularizability of semigroups are obtained as corollaries a block triangularization theorem for algebras of compact operators is established consequences of which include a number of necessary and sufficient conditions for triangularization of such algebras

the haruki rassias and related integral representations of the bernoulli and euler polynomials abstract haruki and rassias h haruki t m rassias new integral representations for bernoulli and euler polynomials j math anal appl 175 1993 81 90 found the integral representations of the classical bernoulli and euler polynomials and proved them by making use of the properties of certain functional equation in this sequel we rederive in a completely different way the results of haruki and rassias and deduce related and new integral representations our proofs are quite simple and remarkably elementary

the compound matrix method for ordinary differential systems abstract a generalization of the compound matrix method is presented to deal with eigenvalue and boundary value problems involving unstable systems of ordinary differential equations details are given for fourth and sixth order problems next is shown that a simple equivalence relation exists between the compounds of the solution matrices of the eigenvalue problems and their adjoints and how this relation can be exploited to simplify the calculation of the adjoint eigenfunctions is discussed using the orr sommerfeld problem as an example it is also shown how the techniques used in the derivation of certain auxiliary systems which play a crucial role in the generalization of the compound matrix method can also provide an alternate method for the direct computation of certain quantities defined in terms of the eigenfunctions finally there is a brief discussion of the relationship between the compound matrix method anf the riccati method

on products of unbounded collections of matrices abstract families a and b of n n complex matrices with bounded products are studied in particular it is shown that if a b k b 1 b 2 b k b i b i 1 2 k is bounded for some k then b m is bounded for all m n the latter result is used to extend the relation lim sup k x k b 1 k k b 1 k due to i daubechies and j c lagarias for unbounded families b where g9 k b sup t b 1 b k b i b i 1 k and k 1 2

evaluation of eigenfunctions from compound matrix variables in non linear elasticity ii sixth order systems we show how the compound matrix method can be extended to give eigenfunctions as well as eigenvalues to bifurcation problems in non linear elasticity the non trivial boundary conditions create some difficulties and we find that sixth order systems for elasticity problems will require a shooting method for two functions of two unknown parameters over and above the calculations required for comparable problems in fluids

the finite step realizability of the joint spectral radius of a pair of d d matrices one of which being rank one abstract we study the finite step realizability of the joint generalized spectral radius of a pair of real square matrices s 1 and s 2 one of which has rank 1 where 2 d let a denote the spectral radius of a square matrix a then we prove that there always exists a finite length word i 1 i m 1 2 m for some finite m 1 such that s i 1 s i m m sup n 1 max i 1 i n 1 2 n s i 1 s i n n in other words there holds the spectral finiteness property for s 1 s 2 explicit formula for computation of the joint spectral radius is derived this implies that the stability of the switched system induced by s 1 s 2 is algorithmically decidable in this case

dynamics of a system comprising an orthotropic layer and orthotropic half plane under the action of an oscillating moving load this paper investigates the dynamic response to a time harmonic oscillating moving load of a system comprising a covering layer and half plane within the scope of the piecewise homogeneous body model utilizing of the exact equations of the linear theory of elastodynamics it is assumed that the materials of the layer and half plane are anisotropic orthotropic and that the velocity of the line located time harmonic oscillating moving load is constant as it acts on the free face of the covering layer our investigations were carried out for a two dimensional problem plane strain state under subsonic velocity for a moving load in complete and incomplete contact conditions the corresponding numerical results were obtained for the stiffer layer and soft half plane system in which the modulus of elasticity of the covering layer material for the moving direction of the load is greater than that of the half plane material numerical results are presented and discussed for the critical velocity displacement and stress distribution for various values of the problem parameters in particular it is established that the critical velocity of the moving load is controlled mainly with a rayleigh wave speed of a half plane material and the existence of the oscillation of the moving load causes two types of critical velocity to appear one of which is less but the other one is greater than that attained for the case where the mentioned oscillation is absent

on the dynamics of a finite pre strained bi layered slab resting on a rigid foundation under the action of an oscillating moving load this paper investigates the dynamic response of a finitely pre strained bi layered slab resting on a rigid foundation to a time harmonic oscillating moving load within the scope of the piecewise homogeneous body model utilizing the 3d linearized wave propagation theory in the initially stressed body it is assumed that the materials of the layers are highly elastic ones and their elasticity relations are given in terms of the harmonic potential moreover it is also assumed that the velocity of the line located time harmonic oscillating moving load is constant as it acts on the free face of the upper layer of the slab our investigations were carried out for a 2d problem plane strain state under subsonic velocity for a moving load in complete contact conditions the corresponding numerical results were obtained for the stiff upper layer and soft lower layer system in which the constants of elasticity for the upper layer material are greater than those of the lower layer material numerical results are presented and discussed for the critical velocity and stress distribution for various values of the problem parameters in particular it is established that with the oscillating frequency of the moving load the values of the critical velocity decrease moreover it is established that the initial stretching of the upper layer of the slab causes the critical velocity to increase and the absolute values of the normal stresses acting on the mid planes of the layers of the slab to decrease

some relationships between the apostol bernoulli and apostol euler polynomials recently srivastava and pinter 1 investigated several interesting properties and relationships involving the classical as well as the generalized or higher order bernoulli and euler polynomials they also showed among other things that the main relationship proven earlier by cheon 2 can easily be put in a much more general setting the main object of the present sequel to these earlier works is to derive several general properties and relationships involving the apostol bernoulli and apostol euler polynomials some of these general results can indeed be suitably specialized in order to deduce the corresponding properties and relationships involving the generalized bernoulli and generalized euler polynomials other relationships associated with the stirling numbers of the second kind are also considered

axisymmetric bifurcations of thick spherical shells under inflation and compression incremental equilibrium equations and corresponding boundary conditions for an isotropic hyperelastic and incompressible material are summarized and then specialized to a form suitable for the analysis of a spherical shell subject to an internal or an external pressure a thick walled spherical shell during inflation is analyzed using four different material models specifically one and two terms in the ogden energy formulation the gent model and an i1 formulation recently proposed by lopez pamies we investigate the existence of local pressure maxima and minima and the dependence of the corresponding stretches on the material model and on shell thickness these results are then used to investigate axisymmetric bifurcations of the inflated shell the analysis is extended to determine the behavior of a thick walled spherical shell subject to an external pressure we find that the results of the two terms ogden formulation the gent and the lopez pamies models are very similar for the one term ogden material we identify additional critical stretches which have not been reported in the literature before

certain results for the 2 variable apostol type and related polynomials in this article the 2 variable general polynomials are taken as base with apostol type polynomials to introduce a family of 2 variable apostol type polynomials these polynomials are framed within the context of monomiality principle and their properties are established certain summation formulae for these polynomials are also derived examples of some members belonging to this family are considered and numbers related to some mixed special polynomials are also explored

some new identities for the apostol bernoulli polynomials and the apostol genocchi polynomials in this paper we present a further investigation for the apostol bernoulli polynomials and the apostol genocchi polynomials by making use of the generating function methods and summation transform techniques we establish some new identities involving the products of the apostol bernoulli polynomials and the apostol genocchi polynomials many of the results presented here are the corresponding generalizations of some known formulas on the classical bernoulli polynomials and the classical genocchi polynomials

the boundedness of all products of a pair of matrices is undecidable we show that the boundedness of the set of all products of a given pair sigma of rational matrices is undecidable furthermore we show that the joint or generalized spectral radius rho sigma is not computable because testing whether rho sigma less than or equal to1 is an undecidable problem as a consequence the robust stability of linear systems under time varying perturbations is undecidable and the same is true for the stability of a simple class of hybrid systems we also discuss some connections with the so called finiteness conjecture our results are based on a simple reduction from the emptiness problem for probabilistic finite automata which is known to be undecidable c 2000 elsevier science b v all rights reserved

on the eversion of compressible elastic cylinders we consider the eversion problem for compressible hyperelastic isotropic thick walled cylinders we give a new strain energy function for a highly compressible material that admits an exact solution to the problem with point wise end conditions previously only averaged loads could be accounted for secondly we investigate the bifurcation problem for a wide range of material models we find qualitively similar results to those obtained for incompressible materials while thin walled cylinders can be everted into other right circular cylinders if the wall thickness exceeds some critical value bifurcation into non cylindrical shapes is possible

sets of matrices all infinite products of which converge an infinite product iit lmi of matrices converges on the right if limi m mi exists a set z ai i l of n x n matrices is called an rcp set rightconvergent product set if all infinite products with each element drawn from z converge such sets of matrices arise in constructing self similar objects like von koch s snowflake curve in various interpolation schemes in constructing wavelets of compact support and in studying nonhomogeneous markov chains this paper gives necessary conditions and also some sufficient conditions for a set x to be an rcp set these are conditions on the eigenvalues and left eigenspaces of matrices in 2 and finite products of these matrices necessary and sufficient conditions are given for a finite set z to be an rcp set having a limit function m d rit lad where d d d which is a continuous function on the space of all sequences d with the sequence topology finite rcp sets of column stochastic matrices are completely characterized some results are given on the problem of algorithmically deciding if a given set x is an rcp set

instability of the hocking stewartson pulse and its implications for three dimensional poiseuille flow the linear stability problem for the hocking stewartson pulse obtained by linearizing the complex ginzburg landau cgl equation is formulated in terms of the evans function a complex analytic function whose zeros correspond to stability exponents a numerical algorithm based on the compound matrix method is developed for computing the evans function using values in the cgl equation associated with spanwise modulation of plane poiseuille flow we show that the hocking stewartson pulse associated with points along the neutral curve is always linearly unstable due to a real positive eigenvalue implications for the spanwise structure of nonlinear poiseuille problem between parallel plates are also discussed

corrigendum addendum to sets of matrices all infinite products of which converge abstract this corrigendum addendum supplies corrected statements and proofs of some results in our paper appearing in linear algebra appl 161 1992 227 263 these results concern special kinds of bounded semigroups of matrices it also reports on progress on the topics of this paper made in the last eight years

matrix algebras and radjavi s trace condition abstract the minimal projections of a transitive algebra of n n matrices are studied the result is then applied to find new criteria for such an algebra to be equal to the whole algebra of n n matrices also triangularizability of an algebra satisfying tr abc tr bac for all a b c in the algebra is examined in terms of its minimal projections

general theory of small elastic deformations superposed on finite elastic deformations using tensor notations a general theory is developed for small elastic deformations of either a compressible or incompressible isotropic elastic body superposed on a known finite deformation without assuming special forms for the strain energy function the theory is specialized to the case when the finite deformation is pure homogeneous when two of the principal extension ratios are equal the changes in displacement and stress due to the small superposed deformation are expressed in terms of two potential functions in a manner which is analogous to that used in the infinitesimal deformation of hexagonally aeolotropic materials the potential functions are used to solve the problem of the infinitesimally small indentation by a spherical punch of the plane surface of a semi infinite body of incompressible isotropic elastic material which is first subjected to a finite pure homogeneous deformation symmetrical about the normal to the force free plane surface

black box learning of multigrid parameters abstract this paper studies the optimality of the restriction and prolongation operators in the geometric multigrid method gmg gmg is used in solving discretized partial differential equation pde and it relies greatly on the restriction and prolongation operators many methods to find these operators were proposed but most of them have limited optimality proofs to study their optimality we introduce stochastic convergence functional which estimates the spectral radius of the iteration matrix for given gmg parameters we implement the gmg method in a modern machine learning framework that can automatically compute the gradients of the introduced convergence functional with respect to restriction and prolongation operators therefore we can minimize the proposed functional starting from some initial parameters and get better ones after some iterations of stochastic gradient descent to illustrate the performance of the proposed approach we carry out experiments on the discretized poisson equation helmholtz equation and singularly perturbed convection diffusion equation and demonstrate that proposed approach gives operators which lead to faster convergence

convergence of inhomogenous products of matrices and coefficients of ergodicity abstract given a square matrix a and a norm the coefficient of ergodicity of a with respect to is defined as max x t all x r n x 1 xtf 0 with f as a matrix satisfying af 0 we demonstrate that for a bounded set of such matrices with all coefficients of ergodicity of the matrices in the set below 1 all sequences constructed through inhomogenous products of matrices from the set converge geometrically

characterization of joint spectral radius via trace the joint spectral radius for a bounded collection of the square matrices with complex entries and of the same size is characterized by the trace of matrices this characterization allows us to give some estimates concerning the computation of the joint spectral radius

algebraic properties and fourier expansions of two dimensional apostol bernoulli and apostol euler polynomials we define the two dimensional 2d apostol bernoulli and the 2d apostol euler polynomials respectively via the generating functions t e x t y t m e t 1 n 0 b n x y t n n 2 e x t y t m e t 1 n 0 e n x y t n n as parametrized polynomial families they are essentially the same we study their basic algebraic properties generalizing some well known formulas and relations for apostol bernoulli and bernoulli polynomials we determine the fourier series of x x b n x y y x b n x y and x y x b n x y for x y 0 1 0 1 these contain as a special case the fourier series of the one dimensional apostol bernoulli and apostol euler polynomials

numerical solution of two point boundary value problems abstract this paper presents an original formulation of two point boundary value and eigenvalue problems expressed as a system of first order equations the fundamental difference between the new method and other methods based on a first order approach is the introduction of conditions of an integral character to supplement the simultaneous set of first order equations which are hence never regarded as an initial value problem the consideration of integral conditions leads to establish a class of linear multipoint schemes for the numerical solution of boundary value problems for ordinary differential equations furthermore the global character of the integral conditions nonlocality combined with the block structure of the system of algebraic equations allow dealing with stiff problems by means of the classical procedure of iterative refinement introduced by wilkinson the properties of the numerical schemes are illustrated by the solution of linear and nonlinear problems and by the accurate and efficient determination of some eigensolutions of a difficult problem of hydrodynamic stability the proposed method is conceptually simpler and numerically more convenient than existing initial value methods while still retaining all the advantages of a formulation based on a first order system

new integral representations for bernoulli and euler polynomials abstract the purpose of this paper is to prove new integral represenlations for bernoulli and euler polynomials and to give two applications of our result to prove our main result we use the functional equation formula although similar functional equations have been studied before in the literature the one above is new

some generalizations of the apostol bernoulli and apostol euler polynomials abstract the main object of this paper is to give analogous definitions of apostol type see t m apostol on the lerch zeta function pacific j math 1 1951 161 167 and h m srivastava some formulas for the bernoulli and euler polynomials at rational arguments math proc cambridge philos soc 129 2000 77 84 for the so called apostol bernoulli numbers and polynomials of higher order we establish their elementary properties derive several explicit representations for them in terms of the gaussian hypergeometric function and the hurwitz or generalized zeta function and deduce their special cases and applications which are shown here to lead to the corresponding results for the classical bernoulli numbers and polynomials of higher order

some generalizations of the apostol genocchi polynomials and the stirling numbers of the second kind abstract recently the authors introduced some generalizations of the apostol bernoulli polynomials and the apostol euler polynomials see q m luo h m srivastava j math anal appl 308 2005 290 302 and q m luo taiwanese j math 10 2006 917 925 the main object of this paper is to investigate an analogous generalization of the genocchi polynomials of higher order that is the so called apostol genocchi polynomials of higher order for these generalized apostol genocchi polynomials we establish several elementary properties provide some explicit relationships with the apostol bernoulli polynomials and the apostol euler polynomials and derive various explicit series representations in terms of the gaussian hypergeometric function and the hurwitz or generalized zeta function we also deduce their special cases and applications which are shown here to lead to the corresponding results for the genocchi and euler polynomials of higher order by introducing an analogue of the stirling numbers of the second kind that is the so called stirling numbers of the second kind we derive some basic properties and formulas and consider some interesting applications to the family of the apostol type polynomials furthermore we also correct an error in a previous paper q m luo h m srivastava comput math appl 51 2006 631 642 and pose two open problems on the subject of our investigation

elliptic extensions of the apostol bernoulli and apostol euler polynomials in this paper we investigate the elliptic analogues of the apostol bernoulli and apostol euler polynomials and obtain the closed expressions of sums of products for these elliptic type polynomials some interesting special cases are also shown

the generalized spectral radius theorem an analytic geometric proof abstract let be a bounded set of complex matrices m a 1 a m a i the generalized spectral radius theorem states that where and are defined as follows lim sup m 1 m where m sup a a m with a the spectral radius lim sup m 1 m where m sup a a m with any matrix norm we give an elementary proof based on analytic and geometric tools which is in some ways simpler than the first proof by berger and wang

numerical solution of eigenvalue problems using spectral techniques abstract two algorithms based on spectral chebyshev and pseudospectral chebyshev methods are presented for solving difficult eigenvalue problems that are valid over connected domains coupled through interfacial conditions to demonstrate the applicability of these methods we have examined the eigenvalue problems that describe the linear stability of two superposed newtonian and inelastic power law fluids in plane poiseuille flow for a selected range of parameters both algorithms provide accurate results and the pseudospectral code appears to be more efficient in handling linear stability problems

approximation of the joint spectral radius using sum of squares we provide an asymptotically tight computationally efficient approximation of the joint spectral radius of a set of matrices using sum of squares sos programming the approach is based on a search for an sos polynomial that proves simultaneous contractibility of a finite set of matrices we provide a bound on the quality of the approximation that unifies several earlier results and is independent of the number of matrices additionally we present a comparison between our approximation scheme and earlier techniques including the use of common quadratic lyapunov functions and a method based on matrix liftings theoretical results and numerical investigations show that our approach yields tighter approximations

the orr sommerfeld equation on a manifold the most effective and widely used methods for integrating the orr sommerfeld equation by shooting are the continuous orthogonalization method and the compound matrix method in this paper we consider this problem from a differential geometric point of view a new definition of orthogonalization is presented restriction of the orr sommerfeld to a complex stiefel manifold and this definition leads to a new formulation of continuous orthogonalization which differs in a precise and interesting geometric way from existing orthogonalization routines present orthogonalization methods based on davey9s algorithm are shown to have a different differential geometric interpretation restriction of the orr sommerfeld equation to a complex grassmanian manifold this leads us to introduce the concept of a grassmanian integrator which preserves linear independence and not necessarily orthogonality using properties of grassmanian manifolds and their tangent spaces a new grassmanian integrator is introduced that generalizes davey9s algorithm furthermore it is shown that the compound matrix method is a dual grassmanian integrator it uses plucker coordinates for integrating on a grassmanian manifold and this characterization suggests a new algorithm for constructing the compound matrices extension of the differential geometric framework to general systems of linear ordinary differential equations is also discussed

an efficient shooting algorithm for evans function calculations in large systems abstract in evans function computations of the spectra of asymptotically constant coefficient linear operators a basic issue is the efficient and numerically stable computation of subspaces evolving according to the associated eigenvalue ode for small systems a fast shooting algorithm may be obtained by representing subspaces as single exterior products j c alexander r sachs linear instability of solitary waves of a boussinesq type equation a computer assisted computation nonlinear world 2 4 1995 471 507 l q brin numerical testing of the stability of viscous shock waves ph d thesis indiana university bloomington 1998 l q brin numerical testing of the stability of viscous shock waves math comp 70 235 2001 1071 1088 l q brin k zumbrun analytically varying eigenvectors and the stability of viscous shock waves in seventh workshop on partial differential equations part i 2001 rio de janeiro mat contemp 22 2002 19 32 t j bridges g derks g gottwald stability and instability of solitary waves of the fifth order kdv equation a numerical framework physica d 172 1 4 2002 190 216 for large systems however the dimension of the exterior product space quickly becomes prohibitive growing as n k where n is the dimension of the system written as a first order ode and k typically n 2 is the dimension of the subspace we resolve this difficulty by the introduction of a simple polar coordinate algorithm representing pure monomial products as scalar multiples of orthonormal bases for which the angular equation is a numerically optimized version of the continuous orthogonalization method of drury davey a davey an automatic orthonormalization method for solving stiff boundary value problems j comput phys 51 2 1983 343 356 l o drury numerical solution of orr sommerfeld type equations j comput phys 37 1 1980 133 139 and the radial equation is evaluable by quadrature notably the polar coordinate method preserves the important property of analyticity with respect to parameters

asymptotic estimates for apostol bernoulli and apostol euler polynomials we analyze the asymptotic behavior of the apostol bernoulli polynomials bn x in detail the starting point is their fourier series on 0 1 which it is shown remains valid as an asymptotic expansion over compact subsets of the complex plane this is used to determine explicit estimates on the constants in the approximation and also to analyze oscillatory phenomena which arise in certain cases these results are transferred to the apostol euler polynomials en x via a simple relation linking them to the apostol bernoulli polynomials

efficient computation of analytic bases in evans function analysis of large systems in evans function computations of the spectra of asymptotically constant coefficient linearized operators of large systems a problem that becomes important is the efficient computation of global analytically varying bases for invariant subspaces of the limiting coefficient matrices in the case that the invariant subspace is spectrally separated from its complementary invariant subspace we propose an efficient numerical implementation of a standard projection based algorithm of kato for which the key step is the solution of an associated sylvester problem this may be recognized as the analytic cousin of a c k algorithm developed by dieci and collaborators based on orthogonal projection rather than eigenprojection as in our case for a one dimensional subspace it reduces essentially to an algorithm of bridges derks and gottwald based on path finding and continuation methods

on accuracy of approximation of the spectral radius by the gelfand formula abstract the famous gelfand formula a limsup n a n 1 n for the spectral radius of a matrix is of great importance in various mathematical constructions unfortunately the range of applicability of this formula is substantially restricted by a lack of estimates for the rate of convergence of the quantities a n 1 n to a in the paper this deficiency is made up to some extent by using the bochi inequalities we establish explicit computable estimates for the rate of convergence of the quantities a n 1 n to a the obtained estimates are then extended for evaluation of the joint spectral radius of matrix sets

fourier expansions and integral representations for the apostol bernoulli and apostol euler polynomials we investigate fourier expansions for the apostol bernoulli and apostol euler polynomials using the lipschitz summation formula and obtain their integral representations we give some explicit formulas at rational arguments for these polynomials in terms of the hurwitz zeta function we also derive the integral representations for the classical bernoulli and euler polynomials and related known results

norm conditions for convergence of infinite products abstract we give a finite number of necessary and sufficient conditions on two matrices so that any infinite left product in them is convergent

numerical solution of eigenvalue problem using the compound matrix method abstract an algorithm based on the compound matrix method is presented for solving difficult eigen value problems details are given for systems of ordinary differential equations of fourth order that are valid over connected domains coupled through interfacial conditions as an example we examine the linear stability of two superposed fluids in plane poiseuille flow and consider both interfacial and shear modes the orr sommerfeld system that describes linear stability is solved for a selected range of parameters

incremental elastic modulus for ventricles in diastole abstract utilizing the formulation of so called small deformations superimposed on a large initial deformation the incremental pressure modulus of a ventricle in diastole is studied and the explicit expression of it is obtained as a function of intraventricular pressure in the analysis the ventricular wall material is assumed to be homogeneous incompressible isotropic and the stress strain relation is exponential the numerical results for a dog left ventricle indicate that above a critical value of inner pressure the incremental pressure modulus increases with increasing intra ventricular pressure furthermore the relationship between the stiffness and pressure is seen to be curvilinear particularly for low pressure level but for large values of inner pressure the behavior of the curve may be approximated by a set of straight lines

bounded semigroups of matrices in this note are proved two conjectures of daubechies and lagarias the first asserts that if z is a bounded set of matrices such that all left infinite products converge then 8 generates a bounded semigroup the second asserts the equality of two differently defined joint spectral radii for a bounded set of matrices one definition involves the conventional spectral radius and one involves the operator norm

some q extensions of the apostol bernoulli and the apostol euler polynomials of order n and the multiple hurwitz zeta function abstract in this paper we first investigate several further interesting properties of the multiple hurwitz lerch zeta function n z s a which was introduced recently by choi et al j choi d s jang h m srivastava a generalization of the hurwitz lerch zeta function integral transform spec funct 19 2008 we then introduce and investigate some q extensions of the multiple hurwitz lerch zeta function n z s a the apostol bernoulli polynomials b k n x of order n and the apostol euler polynomials e k n x of order n relevant connections of the results presented here with those obtained in earlier works are also indicated precisely

asymptotic stability and generalized gelfand spectral radius formula let be a set of n n complex matrices for m 1 2 let m be the set of all products of matrices in of length m denote by the multiplicative semigroup generated by is said to be asymptotically stable in the sense of dynamical systems if there is 0 1 such that there are bounded neighborhoods u v cn of the origin for which av mu for all a m m 1 2 for a bounded set of n n complex matrices it is shown that the following conditions are mutually equivalent i is asymptotically stable ii g9 lim supm supa m a 1 m 1 iii lim supm supa m a 1 m 1 where a stands for the spectral radius of a and iv there exists a positive number such that a 1 for all a this fact answers an open question raised by brayton and tong the generalized gelfand spectral radius formula that is g9 conjectured by daubechies and lagarias and proved by berger and wang using advanced tools from ring theory and then by elsner using analytic geometric tools follows immediately form the above asymptotic stability theorem

asymptotic height optimization for topical ifs tetris heaps and the finiteness conjecture given an iterated function system ifs of topical maps verifying some conditions we prove that the asymptotic height optimization problems are equivalent to finding the extrema of a continuous functional the average height on some compact space of measures we give general results to determine these extrema and then apply them to two concrete problems first we give a new proof of the theorem that the densest heaps of two tetris pieces are sturmian second we construct an explicit counterexample to the lagarias wang finiteness conjecture

numerical exterior algebra and the compound matrix method the compound matrix method which was first proposed for numerically integrating systems of differential equations in hydrodynamic stability on k 2 3 dimensional subspaces of mathbb c n by using compound matrices as coordinates is reformulated in a coordinate free way using exterior algebra spaces bigwedge k mathbb c n