maps completely preserving commutativity and maps completely preserving jordan zero product abstract let x y be real or complex banach spaces with infinite dimension and let a b be standard operator algebras on x and y respectively in this paper we show that every map completely preserving commutativity from a onto b is a scalar multiple of either an isomorphism or in the complex case a conjugate isomorphism every map completely preserving jordan zero product from a onto b is a scalar multiple of either an isomorphism or in the complex case a conjugate isomorphism

commutativity preserving linear maps and lie automorphisms of triangular matrix algebras abstract in this article we classify linear maps from the algebra t n of n upper triangular matrices into itself satisfying ab ba 0 if and only if ab ba 0 in particular we show that for n 2 any such map is either the sum of an algebra automorphism and a map into the centre or the sum of the negative of an algebra anti automorphism and a map into the centre as a consequence lie automorphisms of the algebra t n are also classified

linear transformations on symmetric matrices that preserve commutativity it is shown that if a nonsingular linear transformation t on the space of n square real symmetric matrices preserves the commutativity where n 3 then t a qaqt q a in for all symmetric matricesa for some scalar orthogonal matrix q and linear functional q

transformations on positive definite matrices preserving generalized distance measures abstract we substantially extend and unify former results on the structure of surjective isometries of spaces of positive definite matrices obtained in the paper 14 the isometries there correspond to certain geodesic distances in finsler type structures and to a recently defined interesting metric which also follows a non euclidean geometry the novelty in our present paper is that here we consider not only true metrics but also so called generalized distance measures which are parameterized by unitarily invariant norms and continuous real functions satisfying certain conditions among the many possible applications we shall see that using our new result it is easy to describe the surjective maps of the set of positive definite matrices that preserve the stein loss or several other types of divergences we also present results concerning similar preserver transformations defined on the subset of all complex positive definite matrices with unit determinant

linear preserver problems a brief introduction and some special techniques abstract linear preserver problems concern the characterization of linear operators on matrix spaces that leave certain functions subsets relations etc invariant the earliest papers on linear preserver problems date back to 1897 and a great deal of effort has been devoted to the study of this type of question since then we present a brief picture of the subject aiming at giving a gentle introduction to the reader then we describe some techniques used in our recent papers on this type of problem

linear operators that strongly preserve commuting pairs of fuzzy matrices abstract for each n 3 we characterize the linear operators t on the n n fuzzy matrices that preserved commutativity strongly that is t x commutes with t y if and only if x commutes with y for all x and y characterizations of the corresponding linear operators on the fuzzy symmetric matrices are also obtained

on the t method for delay differential equations with infinite lag abstract this paper discusses the method for the numerical solution of delay differential equations with infinite lag our analysis is based on the test equation y t ay t by t where a b c and 0 1 in order to solve the storage problem involved in studying the long time behaviour of the solution we use a grid with increased stepsizes monotonicity uniform boundedness asymptotic stability algebraic decay and global discretisation error of the numerical solution are investigated

commuting traces and commutativity preserving maps on triangular algebras abstract let a be a triangular algebra the problem of describing the form of a bilinear map b a a a satisfying b x x x x b x x for all x a is considered as an application commutativity preserving maps and lie isomorphisms of certain triangular algebras e g upper triangular matrix algebras and nest algebras are determined

commuting traces of biadditive mappings commutativity preserving mappings and lie mappings biadditive mappings b r r r where r is a prime ring with certain additional properties satisfying b x x x xb x x for all x r are characterized as an application we determine the structures of commutativity preserving mappings lie isomorphisms and lie derivations of certain prime rings

linear operators that strongly preserve commuting pairs of boolean matrices abstract we characterize those linear operators t on the class m of square boolean matrices respectively on the set l of symmetric boolean matrices for which t x commutes with t y if and only if x commutes with y for all x y in m respectively in l

linear transformations on self adjoint matrices the preservation of rank one plus scalar abstract let m be the space of self adjoint linear maps for a nondegenerate quadratic form on a finite dimensional vector space characteristic 2 dimension at least 3 let t m m be an invertible linear transformation if t preserves the set of rank one plus scalar maps then t is of the form t x rpxp x i with one exception over z 3 z any t preserving commuting pairs also has this form with no exceptions the proof of this latter result involves computing the dimension of the space of self adjoint maps commuting with any given one

strong commutativity preserving maps in prime rings with involution abstract let a be a prime ring of characteristic not 2 with center z a and with involution let s be the set of symmetric elements of a suppose that f s a is an additive map such that f x f y x y for all x y s then unless a is an order in a 4 dimensional central simple algebra there exists an additive map s z a such that f x x x for all x s or f x x x for all x s

linear maps preserving commutativity abstract commutativity preserving maps on the real space of all real symmetric or complex self adjoint matrices are characterized related results are given for adjoint preserving maps defined on all n n matrices these results are extended to infinite dimensions in the case of invertible maps

stability analysis of theta methods for neutral functional differential equations this paper deals with the subject of numerical stability for the neutral functional differential equation y t ay t by qt cy pt qquad t 0 it is proved that numerical solutions generated by theta methods are convergent if c b if and only if theta geq 1 2 and that the one leg theta method is lambda stable if theta 1 we also find out that inappropriate stepsize causes spurious solution in the marginal case where rm re a 0 and a b

non spectrality of self affine measures on the spatial sierpinski gasket abstract let m d be the self affine measure corresponding to a diagonal matrix m with entries p 1 p 2 p 3 z 0 1 and d 0 e 1 e 2 e 3 in the space r 3 where e 1 e 2 e 3 are the standard basis of unit column vectors in r 3 such a measure is supported on the spatial sierpinski gasket in this paper we prove the non spectrality of m d by characterizing the zero set z m d of the fourier transform m d we obtain that if p 1 2 z and p 2 p 3 2 z 1 then m d is a non spectral measure and there are at most a finite number of orthogonal exponential functions in l 2 m d this completely solves the problem on the finiteness or infiniteness of orthogonal exponentials in the hilbert space l 2 m d

stability analysis of numerical methods for delay differential equations this paper deals with the stability analysis of step by step methods for the numerical solution of delay differential equations we focus on the behaviour of such methods when they are applied to the linear testproblemu t u t u t with 0 and complex a general theorem is presented which can be used to obtain complete characterizations of the stability regions of these methods

similarity preserving linear maps on matrices abstract we determine all similarity preserving linear maps on the space of n x n complex matrices and all unitary equivalence preserving linear maps on the space of n x n hermitian matrices sub majorization preserving linear maps on hermitian matrices are also determined

numerical stability of the t methods for systems of differential equations with several delay terms abstract consider the following two dimensional delay differential equation dde u t a 1 u t b 1 v t d 2 v t a 2 v t b 2 u t d 1 t t 0 with the initial conditions u t u 0 t t 0 d 1 t t 0 v t v 0 t t 0 d 2 t t 0 where a j b j c d j 0 and u 0 t 0 t are continuous and complex valued the author give the sufficient conditions for the asymptotic stability of equation 0 1 the numerical stabilities of the linear method and the one leg method are investigated by analysing their stabilities behaviour in the solutions of 0 1 it is shown that the linear method is gp stable if and only if 1 2 1 and that the one leg method is gp stable if and only if 1

linear additive preservers of rank 2 on spaces of alternate matrices over fields abstract let kn f be the linear space of all n n alternate matrices over a field f and let k n 2 f be its subset consisting of all rank 2 matrices an operator kn f kn f is said to be additive if a b a b for any a b kn f linear if is additive and aa af a for every a f and a kn f and a preserver of rank 2 on kn f if k n 2 f k n 2 f when n 4 we characterize all linear respectively additive preservers of rank 2 on kn f over any field respectively any field that is not isomorphic to a proper subfield of itself

commutativity preserving operators on symmetric matrices abstract if is a nonsingular linear operator on n n symmetric matrices over a formally real field n 3 and if preserves commutativity then a cu 1 au f a i where u t u and c are scalar and is a linear functional this is an extension of a result of chan and lim

stability of the discretized pantograph differential equation in this paper we study discretizations of the general pantograph equation y t ay t by 6 t cy o t t 0 y o yo where a b c and yo are complex numbers and where 0 and 0 are strictly increasing functions on the nonnegative reals with 6 0 q o 0 and 0 t t 0 t t for positive t our purpose is an analysis of the stability of the numerical solution with trapezoidal rule discretizations and we will identify conditions on a b c and the stepsize which imply that the solution sequence yn i is bounded or that it tends to zero algebraically as a negative power of n

on operators preserving commutativity abstract let l x be the algebra of all bounded operators on a non trivial complex banach space x and f l x l x a bijective linear operator such that f and f 1 both send commuting pairs of operators into commuting pairs then either f a uau 1 p a i or f a ua u 1 p a i where p is a linear functional on l x u is a bounded linear bijective operator between the appropriate two spaces is a complex constant and a is the adjoint of a the form of an operator f for which f and f 1 both send projections of rank one into projections of rank one is also determined

additive rank one preserving mappings on triangular matrix algebras abstract we classify surjective additive maps on the space of block upper triangular matrices that preserve matrices of rank one as well as linear maps preserving matrices of rank one on fairly general subspaces of matrices

stability of runge kutta methods for the generalized pantograph equation this paper deals with stability properties of runge kutta rk methods applied to a non autonomous delay differential equation dde with a constant delay which is obtained from the so called generalized pantograph equation an autonomous dde with a variable delay by a change of the independent variable it is shown that in the case where the rk matrix is regular stability properties of the rk method for the dde are derived from those for a difference equation which are examined by similar techniques to those in the case of autonomous ddes with a constant delay as a result it is shown that some rk methods based on classical quadrature have a superior stability property with respect to the generalized pantograph equation stability of algebraically stable natural rk methods is also considered

h stability of runge kutta methods with general variable stepsize for pantograph equation this paper deals with h stability of runge kutta methods with variable stepsize for pantograph equations it is shown that the runge kutta methods with a regular matrix a are h stable if and only if the modulus of stability function at infinity is less than 1 furthermore the same conclusion can be obtained if the runge kutta methods are stiffly accurate

commutativity preserving maps we briefly describe several techniques that have been developed to solve the problem of characterizing commutativity preserving maps we prove a new result describing the general form of injective continuous commutativity preserving maps on full matrix algebras

commutativity preserving maps on lie ideals of prime algebras abstract one of the most studied linear preserver problems is the problem of characterizing maps preserving commutativity in this paper we study maps preserving commutativity on a lie ideal l of a prime algebra a we show by an example that in general we cannot expect the same standard conclusion as in the case when l a therefore we confine ourselves to some special classes of algebras where the usual result can be proved

p stability properties of runge kutta methods for delay differential equations recently the author and others have studied a class of runge kutta methods for ddes these methods make use of certain constrained meshes which allow to get optimal order results in this paper we study their asymptotic stability properties using the test equation begin gathered y t ay t by t tau t 0 hfill y t g t for tau leqq t leqq 0 hfill end gathered wherea b 0 andg t is continuous and complex valued it is known thaty t 0 ast if b re a in particular we show that any astable one step collocation method for odes inherits the same property when it is applied to ddes with a constrained mesh i e it is p stable

commutativity preserving linear maps and lie automorphisms of strictly triangular matrix space abstract in this paper we classify linear maps preserving commutativity in both directions on the space n f of strictly upper triangular n 1 n 1 matrices over a field f we show that for n 3 a linear map on n f preserves commutativity in both directions if and only if f where is a product of standard maps on n f and f is a linear map of n f into its center

stability of a class of runge kutta methods for a family of pantograph equations of neutral type this paper deals with the stability of runge kutta methods for a class of neutral infinite delay differential equations with different proportional delays under suitable conditions the asymptotic stability of some runge kutta methods with variable stepsize are considered by the stability function at infinity it is proved that the even stage gauss legendre methods are not asymptotically stable but the radau ia methods radau iia methods and lobatto iiic methods are all asymptotically stable furthermore some numerical experiments are given to demonstrate the main conclusions