a generalization of the craig sakamoto theorem to euclidean jordan algebras abstract letac and massam 10 extended the craig sakamoto theorem to euclidean jordan algebras in this paper we give another proof of this generalization by reformulating the result in terms of rank and determinant equalities and by proving the result in each of the simple euclidean jordan algebras

on the completely q property for linear transformations on euclidean jordan algebras abstract in this article we introduce and study ultra and super ssm cone gus properties for linear transformations on euclidean jordan algebras we show that the jordan quadratic strict semimonotonicity jqssm property is equivalent to the completely q property for linear transformations on any euclidean jordan algebra also we investigate some interconnections between the ssm property the jordan linear ssm jlssm property and the completely q property on a euclidean jordan algebra moreover we give a characterization of the completely q property on the jordan spin algebra and study the completely q property for some special transformations on a euclidean jordan algebra

some majorization inequalities in euclidean jordan algebras abstract in this paper we prove various eigenvalue and trace inequalities of objects in the setting of simple euclidean jordan algebras via majorization techniques

convexity and differentiability properties of spectral functions and spectral mappings on euclidean jordan algebras abstract we study in this paper several properties of the eigenvalues function of a euclidean jordan algebra extending several known results in the framework of symmetric matrices in particular we give a concise form for the directional differential of a single eigenvalue we especially focus on spectral functions f on euclidean jordan algebras which are the composition of a symmetric real valued function f with the eigenvalues function we explore several properties of f that are transferred to f in particular convexity strong convexity and differentiability spectral mappings are also considered a special case of which is the gradient mapping of a spectral function answering a problem proposed by h sendov we give a formula for the jacobian of these functions

some p properties for linear transformations on euclidean jordan algebras a real square matrix is said to be a p matrix if all its principal minors are positive it is well known that this property is equivalent to the nonsign reversal property based on the componentwise product of vectors the order p property based on the minimum and maximum of vectors uniqueness property in the standard linear complementarity problem lipschitzian homeomorphism property of the normal map corresponding to the nonnegative orthant in this article we extend these notions to a linear transformation defined on a euclidean jordan algebra we study some interconnections between these extended concepts and specialize them to the space s n of all n n real symmetric matrices with the semidefinite cone s n and to the space r n with the lorentz cone

extension of primal dual interior point algorithms to symmetric cones in this paper we show that the so called commutative class of primal dual interior point algorithms which were designed by monteiro and zhang for semidefinite programming extends word for word to optimization problems over all symmetric cones the machinery of euclidean jordan algebras is used to carry out this extension unlike some non commutative algorithms such as the xs sx method this class of extensions does not use concepts outside of the euclidean jordan algebras in particular no assumption is made about representability of the underlying jordan algebra as a special case we prove polynomial iteration complexities for variants of the short semi long and long step path following algorithms using the nesterov todd xs or sx directions

on the finiteness of the cone spectrum of certain linear transformations on euclidean jordan algebras abstract let l be a linear transformation on a finite dimensional real hilbert space h and k be a closed convex cone with dual k in h the cone spectrum of l relative to k is the set of all real for which the linear complementarity problem x k y l x x k and x y 0 admits a nonzero solution x in the setting of a euclidean jordan algebra h and the corresponding symmetric cone k we discuss the finiteness of the cone spectrum for z transformations and quadratic representations on h

on eigenvalues induced by a cone constraint abstract let a be an n n real matrix and k r n be a closed convex cone the spectrum of a relative to k denoted by a k is the set of all r for which the linear complementarity problem x k ax x k x ax x 0 admitsa nonzero solution x r n the notation k stands for the positive dual cone of k the purpose of this work is to study the main properties of the mapping k and discuss some structural differences existing between the polyhedral case i e k is finitely generated and the nonpolyhedral case

some global uniqueness and solvability results for linear complementarity problems over symmetric cones the purpose of this paper is to give a numerical treatment for a class of quasi linear elliptic equations under nonlinear boundary conditions including the three basic types of linear boundary conditions the quasi linear equation is discretized by the finite difference method and the method of upper lower solutions and its associated monotone iteration are used to compute the solutions of the finite difference system this method leads to monotone iterative schemes for the computation of numerical solutions as well as some comparison results among the monotone iterative schemes it also leads to the existence of a maximal and a minimal finite difference solution including the uniqueness of the solution and the convergence of the finite difference solution to the corresponding continuous solution applications are given to two physical problems in heat conduction and combustion theory and numerical results for the heat conduction problem are given and are compared with the known true continuous solution

an inertia theorem for lyapunov s equation and the dimension of a controllability space abstract let a be an n n complex matrix with inertia in a a a a and let h be an n n hermitian matrix with inertia in a h h h let k be an n n positive semidefinite matrix such that k ah ha suppose that l is the dimension of the controllability space of the pair a k lerer and rodman conjectured that a h n l and a h n l it is our purpose to prove this conjecture

automorphism invariance of p and gus properties of linear transformations on euclidean jordan algebras generalizing the p property of a matrix gowda et al gowda m s r sznajder j tao 2004 some p properties for linear transformations on euclidean jordan algebras linear algebra appl 393 203232 recently introduced and studied p and globally uniquely solvable gus properties for linear transformations defined on euclidean jordan algebras in this paper we study the invariance of these properties under automorphisms of the algebra and of the symmetric cone by means of these automorphisms and the concept of a principal subtransformation we introduce and study ultra and super p gus properties for a linear transformation on a euclidean jordan algebra

z transformations on proper and symmetric cones z transformations motivated by the similarities between the properties of z matrices on r n and lyapunov and stein transformations on the semidefinite cone mathcal s n we introduce and study z transformations on proper cones we show that many properties of z matrices extend to z transformations we describe the diagonal stability of such a transformation on a symmetric cone by means of quadratic representations finally we study the equivalence of q and p properties of z transformations on symmetric cones in particular we prove such an equivalence on the lorentz cone

bubbles in modularity we provide a global technique called neatening for the study of modularity of left linear term rewriting systems objects called bubbles are identified as the responsibles of most of the problems occurring in modularity and the concept of well behaved from the modularity point of view reduction called neat reduction is introduced neatening consists of two steps the first is proving a property is modular when only neat reductions are considered the second is to neaten a generic reduction so to obtain a neat one thus showing that restricting to neat reductions is not limitative this general technique is used to provide a unique uniform method able to elegantly prove all the existing results on the modularity of every basic property of left linear term rewriting systems and also to provide new results on the modularity of termination

lyapunov type least squares problems over symmetric cones abstract the lyapunov type least squares problem over symmetric cone is to find the least squares solution of the lyapunov equation with a constraint of symmetric cone in the euclidean jordan algebra and it contains the lyapunov type least squares problem over cone of semidefinite matrices as a special case in this paper we first give a detailed analysis for the image of lyapunov operator in the euclidean jordan algebra relying on these properties together with some characterizations of symmetric cone we then establish some necessary and or sufficient conditions for solution existence of the lyapunov type least squares problem finally we study uniqueness of the least squares solution

some inertia theorems in euclidean jordan algebras abstract this paper deals with some inertia theorems in euclidean jordan algebras first based on the continuity of eigenvalues we give an alternate proof of kaneyuki s generalization of sylvester s law of inertia in simple euclidean jordan algebras as a consequence we show that the cone spectrum of any quadratic representation with respect to a symmetric cone is finite second we present ostrowski schneider type inertia results in euclidean jordan algebras in particular we relate the inertias of objects a and x in a euclidean jordan algebra when l a x 0 or s a x 0 where l a and s a denote lyapunov and stein transformations respectively

positive groups on hn are completely positive we prove that an operator generates a positive group on the real space of real or complex hermitian matrices if and only if it is a lyapunov operator in particular it follows that every group of positive operators in fact is a group of completely positive operators

stability radii of positive linear volterra stieltjes equations abstract we study stability radii of linear volterra stieltjes equations under multi perturbations and affine perturbations a lower and upper bound for the complex stability radius with respect to multi perturbations are given furthermore in some special cases concerning the structure matrices the complex stability radius can precisely be computed via the associated transfer functions then the class of positive linear volterra stieltjes equations is studied in detail it is shown that for this class complex real and positive stability radius under multi perturbations or multi affine perturbations coincide and can be computed by simple formulae expressed in terms of the system matrices as direct consequences of the obtained results we get some results on robust stability of positive linear integro differential equations and of positive linear functional differential equations to the best of our knowledge most of the results of this paper are new

stability and inertia abstract the purpose of this paper is to present a brief overview of matrix stability and inertia theory a few applications of inertia and stability theorems and a nonspectral implicit matrix equation method for determining stability and inertia of a nonhermitian matrix are also presented inter relationships between different theorems are explicitly stated whenever appropriate the paper concludes with some problems for future research in this area

minimum eigenvalue inequalities for z transformations on proper and symmetric cones abstract a linear transformation l defined on a finite dimensional real hilbert space is said to be a z transformation on a proper cone k if x k y k and x y 0 l x y 0 where k is the dual of k examples of such transformations include z matrices on r n lyapunov and stein transformations on the semidefinite cone for a z transformation l l min l is an eigenvalue of l with a corresponding eigenvector in k in this article when k is a product cone we relate the z property positive stable property minimum real eigenvalue of l with those of a subtransformation of l and its schur complement

the minimax principle and related topics in the jordan setting abstract we prove a minimax principle for weakly compact jb triples characterizing geometrically the singular values of an element among the consequences of this principle we present a weyl inequality on the perturbation of the singular values and a cauchy poincare interlacing theorem we also obtain a version of the ky fan maximum principle in the setting of weakly compact jb triples we study perturbations of the spectral resolutions showing that small perturbations of an element produces small perturbations of the corresponding spectral resolutions as a consequence we obtain that weakly compact jb triples satisfy the property that perturbations of a convex combination of elements in the closed unit ball coincide with a convex combination of perturbations of the elements also in the closed unit ball all these results hold true when particularized to weakly compact jb algebras

strict diagonal dominance and a gersgorin type theorem in euclidean jordan algebras abstract for complex square matrices the levy desplanques theorem asserts that a strictly diagonally dominant matrix is invertible the well known gersgorin theorem on the location of eigenvalues is equivalent to this in this article we extend the levy desplanques theorem to an object in a euclidean jordan algebra when its peirce decomposition with respect to a jordan frame is given as a consequence we prove a gersgorin type theorem for the spectral eigenvalues of an object in a euclidean jordan algebra