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

a note on recent lie parasuperstructures the authors discuss two recently published triple products seen as generalizations in parasupersymmetric quantum mechanics of the usual anticommutator between odd operators in the supersymmetric context they lead to lie parasuperalgebras which are not isomorphic in general

on an interconnection between the lipschitz continuity of the solution map and the positive principal minor property in linear complementarity problems over euclidean jordan algebras in the setting of euclidean jordan algebras we study the lipschitz continuity of the solution map of linear complementarity problems we show that if the solution map is lipschitz continuous and if the linear transformation has the q property then the linear transformation has the positive principal minor property

on the irreducibility lyapunov rank and automorphisms of special bishop phelps cones abstract motivated by optimization considerations we consider cones in r n to be called special bishop phelps cones of the form t x t x where is a norm on r n 1 we show that when n 3 such cones are always irreducible defining the lyapunov rank of a proper cone k as the dimension of the lie algebra of the automorphism group of k we show that the lyapunov rank of any special bishop phelps polyhedral cone is one extending an earlier known result for the l 1 cone which is a special bishop phelps cone with 1 norm we show that any l p cone for 1 p p 2 has lyapunov rank one we also study automorphisms of special bishop phelps cones in particular giving a complete description of the automorphisms of the l 1 cone

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

exact evaluation of the collapse phase boundary for two dimensional directed polymers the phase boundary of the collapsed phase for a directed polymer on a square lattice in the presence of an attractive wall and monomer monomer interactions is calculated exactly using transfer matrix techniques the position of the multicritical point is also identified

on the p 2 and p 2 properties in the semidefinite linear complementarity problem abstract motivated by the so called p 2 property in the semidefinite linear complementarity problems in this article we introduce the concept of p 2 property for a linear transformation on the space of real n n symmetric matrices while these two properties turn out to be different we show that they are equivalent for the lyapunov transformation l a double sided multiplicative transformation m a and a particular class of stein transformations we also show that p 2 implies the ssm and q properties

characterizing q linear transformations for semidefinite linear complementarity problems abstract in this paper we introduce a new class called f of linear transformations defined from the space of real n n symmetric matrices into itself within this new class we show the equivalence between q and q b transformations we also provide conditions under which a linear transformation belongs to f moreover this class when specialized to square matrices of size n turns out to be the largest class of matrices for which such equivalence holds true in the context of standard linear complementary problems

merit functions for semi definite complementarity problems merit functions such as the gap function the regularized gap function the implicit lagrangian and the norm squared of the fischer burmeister function have played an important role in the solution of complementarity problems defined over the cone of nonnegative real vectors we study the extension of these merit functions to complementarity problems defined over the cone of block diagonal symmetric positive semi definite real matrices the extension suggests new solution methods for the latter problems

on the lipschitz continuity of the solution map in linear complementarity problems over second order cone let k irn denote the second order cone given an n n real matrix m and a vector q irn the second order cone linear complementarity problem solcp m q is to find a vector x irn such that x k y mx q kandytx 0 we say that m q if solcp m q has a solution for all q irn an n n real matrix a is said to be a z matrix with respect to k iff x k y kandxty 0 xtmy 0 let m q denote the set of all solutions to solcp m q the following results are shown in this paper if m z q then m is lipschitz continuous if and only if m is positive definite on the boundary of k if m is symmetric then m is lipschitz continuous if and only if m is positive definite

nonnegative generalized inverses and certain subclasses of singular q matrices abstract the notion of q matrices is quite well understood in the theory of linear complementarity problems in this article the author considers three variations of q matrices typically applicable for singular matrices the main result presents a relationship of these notions for a z matrix with the nonnegativity of the moore penrose inverse of the matrix concerned

jordan quadratic ssm property and its relation to copositive linear transformations on euclidean jordan algebras in this paper we introduce jordan quadratic ssm property and study its relation to copositive linear transformations on euclidean jordan algebras in particular we study this relationship for normal z transformations lyapunov like transformations and cone invariant transformations

existence and uniqueness of search directions in interior point algorithms for the sdp and the monotone sdlcp various search directions used in interior point algorithms for the semidefinite program sdp and the monotone semidefinite linear complementarity problem sdlcp are characterized by the intersection of a maximal monotone affine subspace and a maximal and strictly antitone affine subspace this observation provides a unified geometric view over the existence of those search directions

applications of geometric means on symmetric cones let v be a euclidean jordan algebra and let omega be the corresponding symmetric cone the geometric mean a b of two elements a and b in omega is defined as a unique solution which belongs to omega of the quadratic equation p x a 1 b where p is the quadratic representation of v in this paper we show that for any a in omega the sequence of n mathrm th iterate f a n x of the function f a omega to omega defined by f a x 1 over 2 x p a x 1 converges to a as applications we obtain that the geometric mean a b of omega can be represented as a limit of successive iteration of arithmetic means and harmonic means and we derive the lowner heinz inequality on the symmetric cone omega 0 leq a leq b mathrm implies a p leq b p mathrm for 0 leq p leq 1 furthermore we obtain a formula mathrm exp x y lim n to infty mathrm exp frac 2 n x exp frac 2 n y n which leads a golden thompson type inequality mathrm exp x y leq p mathrm exp x over 2 mathrm exp y for the spectral norm on v

positive and doubly stochastic maps and majorization in euclidean jordan algebras abstract a positive map between euclidean jordan algebras is a symmetric cone order preserving linear map we show that the norm of such a map is attained at the unit element thus obtaining an analog of the operator matrix theoretic russo dye theorem a doubly stochastic map between euclidean jordan algebras is a positive unital and trace preserving map we relate such maps to jordan algebra automorphisms and majorization

on the lipschitz continuity of the solution map in semidefinite linear complementarity problems in this paper we investigate the lipschitz continuity of the solution map in semidefinite linear complementarity problems for a monotone linear transformation defined on the space of real symmetricn xn matrices we show that the lipschitz continuity of the solution map implies the globally uniquely solvable gus property for lyapunov transformations with theq property we prove that the lipschitz continuity of the solution map is equivalent to the strong monotonicity property for the double sided multiplicative transformations we show that the lipschitz continuity of the solution map implies the gus property

relationship between strong monotonicity property p2 property and the gus property in semidefinite linear complementarity problems in a recent paper on semidefinite linear complementarity problems gowda and song 2000 introduced and studied thep property p2 property gus property and strong monotonicity property for linear transformationl s n s n wheres nis the space of all symmetric and realn x n matrices in an attempt to characterize thep2 property they raised the following two questions i does the strong monotonicity imply thep2 property ii does the gus property imply thep2 property in this paper we show that the strong monotonicity property implies thep2 property for any linear transformation and describe an equivalence between these two properties for lyapunov and other transformations we show by means of an example that the gus property need not imply thep2 property even for lyapunov transformations

the gus property of second order cone linear complementarity problems the globally uniquely solvable gus property of the linear transformation of the linear complementarity problems over symmetric cones has been studied recently by gowda et al via the approach of euclidean jordan algebra in this paper we contribute a new approach to characterizing the gus property of the linear transformation of the second order cone linear complementarity problems soclcp via some basic linear algebra properties of the involved matrix of soclcp some more concrete and checkable sufficient and necessary conditions for the gus property are thus derived

cubulating graphs of free groups with cyclic edge groups we prove that if g is a group that splits as a finite graph of finitely generated free groups with cyclic edge groups and g has no non euclidean baumslag solitar subgroups then g is the fundamental group of a compact nonpositively curved cube complex in addition if g is also word hyperbolic i e if g contains no baumslag solitar subgroups of any type we show that g is linear in fact is a subgroup of sl n z

a uniqueness result for linear complementarity problems over the jordan spin algebra abstract given a euclidean jordan algebra v with the corresponding symmetric cone k a linear transformation l v v and q v the linear complementarity problem lcp l q is to find a vector x v such that x k y l x q k and x y 0 to investigate the global uniqueness of solutions in the setting of euclidean jordan algebras the p property and its variants of a linear transformation were introduced in gowda et al 2004 3 and it is shown that if lcp l q has a unique solution for all q v then l has the p property but the converse is not true in general in the present paper when v is the jordan spin algebra we show that lcp l q has a unique solution for all q v if and only if l has the p property and l is positive semidefinite on the boundary of k

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

the quantum group suq 2 and a q analogue of the boson operators a new realisation of the quantum group suq 2 is constructed by means of a q analogue to the jordan schwinger mapping determining thereby both the complete representation structure and q analogues to the wigner and racah operators to achieve this realisation a new elementary object is defined a q analogue to the harmonic oscillator the uncertainty relation for position and momentum in a q harmonic oscillator is quite unusual

commensurability of groups quasi isometric to raags let g be a right angled artin group with defining graph gamma and let h be a finitely generated group quasi isometric to g we show if g satisfies that 1 its outer automorphism group is finite 2 gamma does not contain any induced 4 cycles 3 gamma is star rigid then h is commensurable to g we show condition 2 is sharp in the sense that if gamma contains an induced 4 cycle then there exists an h quasi isometric to g but not commensurable to g moreover one can drop condition 1 if h is a uniform lattice acting on the universal cover of the salvetti complex of g as a consequence we obtain a conjugation theorem for such uniform lattices the ingredients of the proof include a blow up building construction in huang and kleiner duke math j 167 3 537 602 2018 https doi org 10 1215 00127094 2017 0042 and a haglund wise style combination theorem for certain class of special cube complexes however in most of our cases relative hyperbolicity is absent so we need new ingredients for the combination theorem

compact matrix pseudogroups the compact matrix pseudogroup is a non commutative compact space endowed with a group structure the precise definition is given and a number of examples is presented among them we have compact group of matrices duals of discrete groups and twisted deformed su n groups the representation theory is developed it turns out that the tensor product of representations depends essentially on their order the existence and the uniqueness of the haar measure is proved and the orthonormality relations for matrix elements of irreducible representations are derived the form of these relations differs from that in the group case this is due to the fact that the haar measure on pseudogroups is not central in general the corresponding modular properties are discussed the haar measures on the twistedsu 2 group and on the finite matrix pseudogroup are found

complementarity forms of theorems of lyapunov and stein and related results abstract the well known lyapunov s theorem in matrix theory continuous dynamical systems asserts that a complex square matrix a is positive stable i e all eigenvalues lie in the open right half plane if and only if there exists a positive definite matrix x such that ax xa is positive definite in this paper we prove a complementarity form of this theorem a is positive stable if and only if for any hermitian matrix q there exists a positive semidefinite matrix x such that ax xa q is positive semidefinite and x ax xa q 0 by considering cone complementarity problems corresponding to linear transformations of the form i s we show that a complex matrix a has all eigenvalues in the open unit disk of the complex plane if and only if for every hermitian matrix q there exists a positive semidefinite matrix x such that x axa q is positive semidefinite and x x axa q 0 by specializing q to i we deduce the well known stein s theorem in discrete linear dynamical systems a has all eigenvalues in the open unit disk if and only if there exists a positive definite matrix x such that x axa is positive definite

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

the linear order complementarity problem the classical complementarity problem in euclidean space can be viewed alternatively as a variational inequality or as a lattice orthogonality problem generalizations of the former have been extensively studied but infinite dimensional analogues of the latter have been largely ignored moreover as we show many well known results about the classical complementarity problem are more appropriately viewed order theoretically this is particularly true of least element solutions which are central to the present study of order complementarity in vector lattices we emphasize that the lattice theoretic descriptions we employ are very useful even in the standard finite dimensional setting

lowest rank solutions of continuous and discrete lyapunov equations over symmetric cone abstract the low rank solutions of continuous and discrete lyapunov equations are of great importance but generally difficult to compute in control system analysis and design fortunately mesbahi and papavassilopoulos 1997 30 showed that with the semidefinite cone constraint the lowest rank solutions of the discrete lyapunov inequality can be efficiently solved by a linear semidefinite programming in this paper we further show that the lowest rank solutions of both the continuous and discrete lyapunov equations over symmetric cone are unique and can be exactly solved by their convex relaxations the symmetric cone linear programming problems therefore they are polynomial time solvable since the underlying symmetric cone is a more general algebraic setting which contains the semidefinite cone as a special case our results also answer an open question proposed by recht fazel and parrilo 2010 36

on the bilinearity rank of a proper cone and lyapunov like transformations a real square matrix q q is a bilinear complementarity relation on a proper cone k k in mathbb r n r n if begin aligned x in k s in k text and langle x s rangle 0 rightarrow x t qs 0 end aligned x k s k and x s 0 x t q s 0 where k k is the dual of k k the bilinearity rank of k k is the dimension of the linear space of all bilinear complementarity relations on k k in this article we continue the study initiated by rudolf et al math prog ser b 129 5 31 2011 we show that bilinear complementarity relations are related to lyapunov like transformations that appear in dynamical systems and in complementarity theory and further show that the bilinearity rank of k k is the dimension of the lie algebra of the automorphism group of k k in addition we correct a result of rudolf et al compute the bilinearity ranks of symmetric and completely positive cones and state some schur type results for lyapunov like transformations

the strong atiyah conjecture for virtually cocompact special groups we provide new conditions for the strong atiyah conjecture to lift to finite group extensions in particular we show that fundamental groups of compact special cube complexes satisfy these conditions so the conjecture holds for finite extensions of these groups

on the algebraic riccati inequality arising in cone preserving time delay systems abstract this paper studies the problem of riccati stability of a pair of matrices for a matrix pair a b it was recently shown that if its corresponding time delay system is internally positive meaning that a is metzler and b is nonnegative then the pair a b is diagonally riccati stable if and only if a b is hurwitz we extend this to the case when the pair a b corresponds to a time delay system with a more general cone preserving property we show that if the time delay system relating to the pair a b is invariant on a symmetric cone the corresponding algebraic riccati inequality admits positive definite solutions which can be constructed via the scaling transformation on the euclidean jordan algebra associated with the symmetric cone for the special case when the symmetric cone is the positive semi definite cone an application to a class of stochastic systems is discussed

extended m matrices and subtangentiality the concept of a singular m matrix a with respect to a proper cone x is extended by replacing the usual regularity condition a al b for a xnonnegative matrix b with the weaker condition exponential nonnegativity of a as in earlier work which dealt with the nonsingular case in the present characterizations the lack of regularity is overcome by employing subtangentiality

on the game theoretic value of a linear transformation relative to a self dual cone this paper is concerned with a generalization of the concept of value of a zero sum matrix

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

interior point methods for the monotone semidefinite linear complementarity problem in symmetric matrices the sdlcp semidefinite linear complementarity problem in symmetric matrices introduced in this paper provides a unified mathematical model for various problems arising from systems and control theory and combinatorial optimization it is defined as the problem of finding a pair x y of n times n symmetric positive semidefinite matrices which lies in a given n n 1 2 dimensional affine subspace fc of sc 2 and satisfies the complementarity condition x bullet y 0 where sc denotes the n n 1 2 dimensional linear space of symmetric matrices and x bullet y the inner product of x and y the problem enjoys a close analogy with the lcp in the euclidean space in particular the central trajectory leading to a solution of the problem exists under the nonemptiness of the interior of the feasible region and a monotonicity assumption on the affine subspace fc the aim of this paper is to establish a theoretical basis of interior point methods with the use of newton directions toward the central trajectory for the monotone sdlcp

groups generated by reflections and aspherical manifolds not covered by euclidean space a coxeter system r v is a group r a coxeter group together with a set of generators v such that each element of v has order two and such that all relations in r are consequences of relations of the form vw m v w 1 where v w e v and m v w denotes the order of vw the m v w s which are positive integers or oo obviously determine the coxeter system up to isomorphism the list of finite coxeter groups is short and well known however general coxeter groups are much more flexible in fact the set v and the m v w s can be specified arbitrarily at least if v is finite subject only to the conditions m v v 1 and m v w m w v 2 if v w suppose that r v is a coxeter system that x is a hausdorff space and that xv v v is a locally finite family of closed subspaces indexed by v the xv are called the panels of x 2 there is a classical method for constructing a transformation group from these data for each x e x let v x denote the set of v in v such that x e xv for each subset s of v let rs be the subgroup generated by s and let xs be the face of x defined by

a cone programming approach to the bilinear matrix inequality problem and its geometry we discuss an approach for solving the bilinear matrix inequality bmi based on its connections with certain problems defined over matrix cones these problems are among others the cone generalization of the linear programming lp and the linear complementarity problem lcp referred to as the cone lp and the cone lcp respectively specifically we show that solving a given bmi is equivalent to examining the solution set of a suitably constructed cone lp or cone lcp this approach facilitates our understanding of the geometry of the bmi and opens up new avenues for the development of the computational procedures for its solution

a holder type inequality and an interpolation theorem in euclidean jordan algebras abstract in a euclidean jordan algebra v of rank n which carries the trace inner product to each element x we associate the eigenvalue vector x whose components are the eigenvalues of x written in the decreasing order for any p 1 we define the spectral p norm of x to be the p norm of x in r n in this paper we show that x y 1 x p y q where x y denotes the jordan product of two elements x and y in v and q is the conjugate of p for a linear transformation on v we state and prove an interpolation theorem relative to these spectral norms in addition we compute estimate the norms of lyapunov transformations quadratic representations and positive transformations on v

normal maps inducted by linear transformations we study a certain piecewise linear manifold which we call the normal manifold associated with a polyhedral convex set and a family of continuous functions called normal maps that are induced on this manifold by continuous functions from rn to rn these normal maps occur frequently in optimization and equilibrium problems and the subclass of normal maps induced by linear transformations plays a key role our main result is that the normal map induced by a linear transformation is a lipschitzian homeomorphism if and only if the determinant of the map in each n cell of the normal manifold has the same nonzero sign

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

on the p property of z and lyapunov like transformations on euclidean jordan algebras abstract the p z and s properties of a linear transformation on a euclidean jordan algebra are generalizations of the corresponding properties of a square matrix on rn motivated by the equivalence of p and s properties for a z matrix 2 and a similar result for lyapunov and stein transformations on the space of real symmetric matrices 6 5 in this paper we present two results supporting the conjecture that p and s properties are equivalent for a z transformation on a euclidean jordan algebra we show that the conjecture holds for lyapunov like transformations and z transformations satisfying an additional condition

residual finiteness growths of virtually special groups let g be a virtually special group then the residual finiteness growth of g is at most linear this result cannot be found by embedding g into a special linear group indeed the special linear group mathrm sl k mathbb z for k 2 has residual finiteness growth n k 1

weakly homogeneous variational inequalities and solvability of nonlinear equations over cones given a closed convex cone c in a finite dimensional real hilbert space h a weakly homogeneous map f c rightarrow h is a sum of two continuous maps h and g where h is positively homogeneous of degree gamma ge 0 on c and g x o x gamma as x rightarrow infty in c given such a map f a nonempty closed convex subset k of c and a q in h we consider the variational inequality problem text vi f k q of finding an x in k such that langle f x q x x rangle ge 0 for all x in k in this paper we establish some results connecting the variational inequality problem text vi f k q and the cone complementarity problem text cp f infty k infty 0 where f infty h is the homogeneous part of f and k infty is the recession cone of k we show for example that text vi f k q has a nonempty compact solution set for every q when zero is the only solution of text cp f infty k infty 0 and the topological index of the map x mapsto x pi k infty x g x at the origin is nonzero where g is a continuous extension of f infty to h as a consequence we generalize a complementarity result of karamardian j optim theory appl 19 227 232 1976 formulated for homogeneous maps on proper cones to variational inequalities the results above extend some similar results proved for affine variational inequalities and for polynomial complementarity problems over the nonnegative orthant in mathcal r n as an application we discuss the solvability of nonlinear equations corresponding to weakly homogeneous maps over closed convex cones in particular we extend a result of hillar and johnson proc am math soc 132 945 953 2004 on the solvability of symmetric word equations to euclidean jordan algebras

cross positive matrices revisited abstract for a closed pointed n dimensional convex cone k in r n let k denote the set of all n n real matrices a which as linear operators map k into itself let k denote the set of all n n matrices that are cross positive on k and l k k k the lineality space of k let r i the set of all real multiples of the n n identity matrix i then k k l k cl k k the final equality was proved in 1970 by schneider and vidyasagar who showed also that k k when k is polyhedral but not when k is a three dimensional circular cone they asked for a general characterization of those k for which the equality holds it is shown here that if n 3 and the cone k is strictly convex or smooth then k k hence for n 3 the equality fails for almost all k in the sense of baire category however the equality does hold for some nonpolyhedral k as was shown by a construction that appeared in the third author s 1977 dissertation and is explained here in more detail in 1994 it was shown by stern and wolkowicz that the weaker equality k l k k holds for all ellipsoidal as well as for all polyhedral k and they wondered whether this equality holds for all k however their equality certainly fails for all strictly convex or smooth k such that l k and it is shown here that this also includes almost all k when n 3

hall littlewood symmetric functions and the bkp equation the connection between solutions of the bkp equation and hall littlewood symmetric functions is utilised in a unified approach to soliton and polynomial solutions this is analogous to the wronskian formulation of the solution of the kp equation as a by product two novel expressions for certain hall littlewood functions in terms of pfaffians are derived

some extensions on the bounded real lemma for positive systems recent advances have shown that the l 2 gain of positive systems is fully determined by the static gain matrix and hence the linear matrix inequality lmi characterizing their h infty performance admits a diagonal lyapunov matrix variable in this technical note we aim to extend this property for systems with input state and output defined on general cones although it is known that the l 2 gain is no longer governed by the static gain matrix for systems defined on general proper cones we show that this property is still valid when the cone in the state space is symmetric and the cones in the input and output spaces are self dual meanwhile it is proved that the lyapunov matrix appearing in the lmi characterization for the h infty performance can be constructed by the quadratic representation of the jordan algebra asso ciated with the symmetric cone in the state space finally the established results are illustrated via a linear system defined on second order cones

some p properties for nonlinear transformations on euclidean jordan algebras in this article we introduce the concepts of p and p0 properties for a nonlinear transformation defined on a euclidean jordan algebra and study existence of solution in the associated complementarity problems in particular we show in this general setting that if a transformation has the p0 and r0 properties then all associated complementarity problems have solutions we also describe a necessary condition for a transformation to have the global uniqueness of solution property

mixed 3 manifolds are virtually special letm be a compact oriented irreducible 3 manifold which is neither a graph manifold nor a hyperbolic manifold we prove that 1m is virtually special

limit groups are subgroup conjugacy separable abstract a group g is called subgroup conjugacy separable if for every pair of non conjugate finitely generated subgroups of g there exists a finite quotient of g where the images of these subgroups are not conjugate we prove that limit groups are subgroup conjugacy separable we also prove this property for one relator groups of the form r a 1 a n w n with n w the property is also proved for infinite virtual retracts equivalently for infinite quasiconvex subgroups of hyperbolic virtually special groups

the bianchi groups are separable on geometrically finite subgroups let d be a square free positive integer and od the ring of integers in q p id the main result of this paper is that the groups psl 2 od are

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 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

on some interconnections between strict monotonicity globally uniquely solvable and p properties in semidefinite linear complementarity problems in the setting of semidefinite linear complementarity problems on sn the implications strict monotonicity p2 gus p are known here p and p2 properties for a linear transformation l sn sn are respectively defined by x sn xl x l x x 0 x 0 and x 0 y 0 x y l x l y x y 0 x y gus refers to the global unique solvability in semidefinite linear complementarity problems correspond ing to l in this article we show that the reverse implications hold for any self adjoint linear transformation and for normal lyapunov and stein transformations by introducing the concept of a principal subtransformation of a linear transformation we show that l sn sn has the p2 property if and only if for every n n real invertible matrix q every principal subtransformation of l has the p property where l x qtl qxqt q based on this we show that p2 gus and p properties coincide for the two sided multiplication transfor mation

semismooth matrix valued functions matrix valued functions play an important role in the development of algorithms for semidefinite programming problems this paper studies generalized differential properties of such functions related to nonsmooth smoothing newton methods the first part of this paper discusses basic properties such as the generalized derivative rademacher s theorem b derivative directional derivative and semismoothness the second part shows that the matrix absolute value function the matrix semidefinite projection function and the matrix projective residual function are strongly semismooth

coxeter groups are virtually special abstract in this paper we prove that every finitely generated coxeter group has a finite index subgroup that is the fundamental group of a special cube complex some consequences include every f g coxeter group is virtually a subgroup of a right angled coxeter group every word hyperbolic coxeter group has separable quasiconvex subgroups

on quantum deformations of the simplest lie parasuperalgebra quantum deformations of the simplest lie parasuperalgebra psqm 2 are constructed by taking advantage of the well known representations of suq 2 c as a particular result the authors show that the fundamental witten superalgebra sqm 2 cannot be deformed

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

on the lipschitzian property in linear complementarity problems over symmetric cones abstract let v be a euclidean jordan algebra with symmetric cone k we show that if a linear transformation l on v has the lipschitzian property and the linear complementarity problem lcp l q over k has a solution for every invertible q v then l c c 0 for all primitive idempotents c in v we show that the converse holds for lyapunov like transformations stein transformations and quadratic representations we also show that the lipschitzian q property of the relaxation transformation r a on v implies that a is a p matrix

on the block norm p property a real n n matrix m is said to be a p matrix if all its principal minors are positive in a recent paper chua and yi 2 described this property in terms of norm there exists a 0 such that for all nonnegative diagonal matrices d and vectors x mx dx x in this paper we introduce a block version of this property for a linear transformation defined on a product of normed or inner product spaces in addition to relating this to real positive stability and positive principal minor properties we study the invariance of this property by principal subtransformations and schur complements we also specialize this property to z transformations and to euclidean jordan algebras

the automorphism group of a completely positive cone and its lie algebra abstract given a closed cone c in r n we consider the corresponding completely positive convex cone k generated by uu t u c in s n under certain conditions on c we describe the automorphism group of k and its corresponding lie algebra in terms of those of c c and or c in particular we show that when c is a closed convex proper cone the automorphism groups of c and k are isomorphic and their corresponding lie algebras are isomorphic

a combination theorem for special cube complexes we prove that certain compact cube complexes have special nite covers this means they have nite covers whose fundamental groups are quasiconvex subgroups of right angled artin groups as a result we obtain linearity and the separability of quasiconvex subgroups for the groups we consider our result applies in particular to a compact negatively curved cube complex whose hyperplanes do not self intersect for a cube complex with word hyperbolic fundamental group we show that it is virtually special if and only if its hyperplane stabilizers are separable in a nal application we show that the fundamental groups of every simple type uniform arithmetic hyperbolic manifolds are cubical and virtually special

a note on the ostrowski schneider type inertia theorem in euclidean jordan algebras abstract in a recent paper 7 gowda et al extended ostrowski schneider type inertia results to certain linear transformations on euclidean jordan algebras in particular they showed that in a in x whenever a x 0 by the min max theorem of hirzebruch where the inertia of an element x in a euclidean jordan algebra is defined by in x x x x with x x and x denoting respectively the number of positive negative and zero eigenvalues counting multiplicities in this paper we present a peirce decomposition version of wimmer s result 13 and show that it is equivalent to the above result in addition we extend higham and cheng s result 8 lemma 4 2 to the setting of euclidean jordan algebras

tits alternatives for graph products we discuss various types of tits alternative for subgroups of graph products of groups and prove that under some natural conditions a graph product of groups satisfies a given form of tits alternative if and only if each vertex group satisfies this alternative as a corollary we show that every finitely generated subgroup of a graph product of virtually solvable groups is either virtually solvable or large as another corollary we prove that every non abelian subgroup of a right angled artin group has an epimorphism onto the free group of rank 2 in the course of the paper we develop the theory of parabolic subgroups which allows to describe the structure of subgroups of graph products that contain no non abelian free subgroups we also obtain a number of results regarding the stability of some group properties under taking graph products

the q property of composite transformations and the p property of stein type transformations on self dual and symmetric cones motivated by the q property of nonsingular m matrices lyapunov and stein transformations corresponding to positive stable and schur stable matrices respectively and their products a berman r j plemmons nonnegative matrices in the mathematical sciences siam philadelphia 1994 m s gowda y song math prog 88 2000 575 587 m s gowda t parthasarathy linear algebra appl 320 2000 131 144 m s gowda y song siam j matrix anal appl 24 2002 25 39 in the first part of the paper we present a unifying result on the product of q transformations defined on self dual closed convex cones the second part deals with the p property of the linear transformation i s on a euclidean jordan algebra where s leaves the corresponding symmetric cone invariant and s 1 we prove the p property for the lorentz cone and present some partial results in the general case

generalized m matrices abstract let r n be a proper cone from the theory of m matrices see e g 1 it is known that if there exist 0 and a matrix b such that a b i then the following conditions are equivalent i a is monotone ii a is seminegative iii re spectrum a e ta t 0 is more general than the structural assumption a b i conditions i iii are nevertheless all equivalent to iv x ax 0 when e holds