darboux and binary darboux transformations for discrete integrable systems i discrete potential kdv equation the hirota miwa equation can be written in nonlinear form in two ways the discrete kp equation and by using a compatible continuous variable the discrete potential kp equation for both systems we consider the darboux and binary darboux transformations expressed in terms of the continuous variable and obtain exact solutions in wronskian and grammian form we discuss reductions of both systems to the discrete kdv and discrete potential kdv equation respectively and exploit this connection to find the darboux and binary darboux transformations and exact solutions of these equations

direct linearizing transform for three dimensional discrete integrable systems the lattice akp bkp and ckp equations a unified framework is presented for the solution structure of three dimensional discrete integrable systems including the lattice akp bkp and ckp equations this is done through the so called direct linearizing transform which establishes a general class of integral transforms between solutions as a particular application novel soliton type solutions for the lattice ckp equation are obtained

discretization of bilinear identities for the noncommutative kp and ckp hierarchies the noncommutative kp hierarchy is known to be expressed by a bilinear identity for wavefunctions although the hierarchy does not seem to have any function it is shown that direct discretization of the bilinear identity gives the noncommutative discrete kp equation the bilinear identity for the noncommutative ckp hierarchy is also discretized and the noncommutative discrete ckp equation is obtained

darboux and binary darboux transformations for the nonautonomous discrete kp equation it is shown how darboux and binary darboux transformations for a nonautonomous discrete kp equation can be obtained from fermion analysis this equation is obtained by considering a generalized miwa transformation it is also shown to be linked to the discrete kp equation by a special gauge transformation the darboux and binary darboux transformations are used to discuss general classes of solutions in the form of casorati and gramm type determinants n soliton solutions are discussed as well

with a cole hopf transformation to solutions of the noncommutative kp hierarchy in terms of wronski matrices in the case of the kp hierarchy where the dependent variable takes values in an arbitrary associative algebra it is known that there are solutions which can be expressed in terms of quasideterminants of a wronski matrix which solves the linear heat hierarchy we obtain these solutions without the help of quasideterminants in a simple way via solutions of matrix kp hierarchies over and by use of a cole hopf transformation for this class of exact solutions we work out a correspondence with weakly nonassociative algebras

uniform convexity of ψ direct sums of banach spaces abstract let x and y be banach spaces and a continuous convex function on the unit interval 0 1 satisfying certain conditions let x y be the direct sum of x and y equipped with the associated norm with we show that x y is uniformly convex if and only if x y are uniformly convex and is strictly convex as a corollary we obtain that the l p q direct sum x p q y 1 q p not p q 1 nor is uniformly convex if and only if x y are where l p q is the lorentz sequence space these results extend the well known fact for the l p sum x p y 1 some other examples are also presented

a note on geometrical properties of banach spaces using ψ direct sums abstract let x be a banach space and a continuous convex function on 0 1 satisfying certain conditions let x x be the direct sum of x in this note we characterize the strict convexity uniform convexity and uniformly non squareness of banach spaces using direct sums which extends the well known characterization of these spaces

absolute norms on cn abstract in this paper we study the absolute normalized norms on c n we show that the set of all absolute normalized norms on c n is in one to one correspondence with the set n of all continuous convex functions on n with some suitable conditions where n s1 sn 1 r n 1 s1 s2 sn 1 1 si 0 1 i n 1 as some applications we show that an absolute norm on c n is strictly convex if and only if the corresponding function on n is strictly convex further we consider the von neumann jordan constant of c n with an absolute normalized norm

orthogonality preserving transformations on indefinite inner product spaces generalization of uhlhorn s version of wigner s theorem abstract we present an analogue of uhlhorn s version of wigner s theorem on symmetry transformations for the case of indefinite inner product spaces this significantly generalizes a result of van den broek the proof is based on our main theorem which describes the form of all bijective transformations on the set of all rank one idempotents of a banach space which preserve zero products in both directions

what are schur complements anyway abstract given a matrix m a b c d the schur complements of a in m are the matrices of the form s d cab where a is a generalized inverse of a we survey several recent characterizations of schur complements and discuss where they arose and how they are related

tetrahedron maps and symmetries of three dimensional integrable discrete equations a relationship between the tetrahedron equation for maps and the consistency property of integrable discrete equations on z3 is investigated our approach is a generalization of a method developed in the context of yang baxter maps based on the invariants of symmetry groups of the lattice equations the method is demonstrated by a case by case analysis of the octahedron type lattice equations classified recently leading to some new examples of tetrahedron maps and integrable coupled lattice equations a relationship between the tetrahedron equation for maps and the consistency property of integrable discrete equations on z3 is investigated our approach is a generalization of a method developed in the context of yang baxter maps based on the invariants of symmetry groups of the lattice equations the method is demonstrated by a case by case analysis of the octahedron type lattice equations classified recently leading to some new examples of tetrahedron maps and integrable coupled lattice equations

a semi discrete kadomtsev petviashvili equation and its coupled integrable system we establish connections between two cascades of integrable systems generated from the continuum limits of the hirota miwa equation and its remarkable nonlinear counterpart under the miwa transformation respectively among these equations we are mainly concerned with the semi discrete bilinear kadomtsev petviashvili kp equation which is seldomly studied in literature we present both of its casorati and grammian determinant solutions through the pfaffianization procedure proposed by hirota and ohta we are able to derive the coupled integrable system for the semi discrete kp equation

on a direct approach to quasideterminant solutions of a noncommutative kp equation a noncommutative version of the modified kp equation and a family of its solutions expressed as quasideterminants are discussed the origin of these solutions is explained by means of darboux transformations and the solutions are verified directly we also verify directly an explicit connection between quasideterminant solutions of the noncommutative mkp equation and the noncommutative kp equation arising from the miura transformation

maps on states preserving the relative entropy let h be a finite dimensional hilbert space the aim of this short note is to prove that every bijective map on the space s h of all density operators on h which preserves the relative entropy is of the form u u with some unitary or antiunitary operator u on h

darboux transformations and the discrete kp equation this paper presents two results first it is shown how the discrete kp equation arises from a superposition principle associated with the darboux transformation of the two dimensional toda system then darboux transformations and binary darboux transformations are derived for the discrete kp equation and it is shown how these may be used to construct exact solutions

noncommutative korteweg de vries and modified korteweg de vries hierarchies via recursion methods here noncommutative hierarchies of nonlinear equations are studied they represent a generalization to the operator level of corresponding hierarchies of scalar equations which can be obtained from the operator ones via a suitable projection a key tool is the application of backlund transformations to relate different operator valued hierarchies indeed in the case when hierarchies in 1 1 dimensions are considered a backlund chart depicts links relating in particular the korteweg de vries kdv to the modified kdv mkdv hierarchy notably analogous links connect the hierarchies of operator equations the main result is the construction of an operator soliton solution depending on an infinite dimensional parameter to start with the potential kdv hierarchy is considered then backlund transformations are exploited to derive solution formulas in the case of kdv and mkdv hierarchies it is remarked that hierarchies of matrix equations of any dimension are also incorporated in the present framework

characterizing sequential isomorphisms on hilbert space effect algebras let be any sequential product on the hilbert space effect algebra with and be a bijective map we show that if satisfies ab a b for then there is either a unitary or an anti unitary operator u such that a uau for every let be a borel function satisfying g 0 0 g 1 1 and let us define a binary operation g on by a gb a1 2 g a bg a a1 2 where t denotes the conjugate of the operator t we also show that a bijective map satisfies a gb a g b for if and only if there is either a unitary or an anti unitary operator u such that a uau for every

monodromy of the matrix schrodinger equations and darboux transformations a schrodinger operator with a matrix valued rational potential u z is said to have trivial monodromy if all the solutions of the corresponding schrodinger equations are single valued in the complex plane for any a local criterion of this property in terms of the laurent coefficients of the potential u near its singularities which are assumed to be regular is found it is proved that any such operator with a potential vanishing at infinity can be obtained by a matrix analogue of the darboux transformation from the schrodinger operator this generalizes the well known duistermaat grunbaum result to the matrix case and gives the explicit description of the schrodinger operators with trivial monodromy in this case

entropy preserving maps on quantum states abstract in the paper we give a characterization of surjective maps on quantum states preserving quantum entropy of convex combinations let s h be the set of all quantum states positive operators with the unit trace on a complex hilbert space h with dim h n and s the quantum entropy of the quantum state for arbitrary a surjective map s h s h we claim that preserves quantum entropy of convex combinations of quantum states i e satisfies s t 1 t s t 1 t for arbitrary s h and t 0 1 if and only if there exists a unitary or anti unitary operator u on h such that u u for all s h

riemann equations in bidifferential calculus we consider equations that formally resemble a matrix riemann or hopf equation in the framework of bidifferential calculus with different choices of a first order bidifferential calculus we obtain a variety of equations including a semi discrete and a fully discrete version of the matrix riemann equation a corresponding universal solution generating method then either yields a continuous or discrete cole hopf transformation or leaves us with the problem of solving riemann equations hence an application of the hodograph method if the bidifferential calculus extends to second order solutions of a system of riemann equations are also solutions of an equation that arises on the universal level of bidifferential calculus as an integrability condition depending on the choice of bidifferential calculus the latter can represent a number of prominent integrable equations like self dual yang mills as well as matrix versions of the two dimensional toda lattice hirota s bilinear difference equation 2 1 dimensional nonlinear schrodinger nls kadomtsev petviashvili kp equation and davey stewartson equations for all of them a recent non isospectral binary darboux transformation result in bidifferential calculus applies which can be specialized to generate solutions of the associated riemann equations for the latter we clarify the relation between these specialized binary darboux transformations and the aforementioned solution generating method from arbitrary size matrix versions of the riemann equations associated with an integrable equation possessing a bidifferential calculus formulation multi soliton type solutions of the latter can be generated this includes breaking multi soliton type solutions of the self dual yang mills and the 2 1 dimensional nls equation which are parametrized by solutions of riemann equations

von neumann jordan constant of absolute normalized norms on c2 abstract we determine and estimate the von neumann jordan constant of absolute normalized norms on c 2 by means of their corresponding continuous convex functions on 0 1 this provides many new interesting examples including those of non lp type as well as some previous ones it is also shown that all such norms are uniformly non square except l1 and l norms

quasideterminant solutions of a non abelian toda lattice and kink solutions of a matrix sine gordon equation two families of solutions of a generalized non abelian toda lattice are considered these solutions are expressed in terms of quasideterminants constructed by means of darboux and binary darboux transformations as an example of the application of these solutions we consider the 2 periodic reduction to a matrix sine gordon equation in particular we investigate the interaction properties of polarized kink solutions

non commutative q painleve vi equation by applying suitable centrality conditions to non commutative non isospectral lattice modified gel fand dikii type systems we obtain the corresponding non autonomous equations then we derive the non commutative q discrete painleve vi equation with full range of parameters as the 2 2 similarity reduction of the non commutative non isospectral and non autonomous lattice modified korteweg de vries equation we also comment on the fact that in making the analogous reduction starting from the schwarzian korteweg de vries equation no such non isospectral generalization is needed

darboux transformations for linear operators on two dimensional regular lattices darboux transformations for linear operators on regular two dimensional lattices are reviewed the six point scheme is considered as the master linear problem whose various specifications reductions and sublattice combinations lead to other linear operators together with the corresponding darboux transformations the second part of the review deals with multidimensional aspects of basic reductions of the four point scheme as well as the three point scheme

von neumann entropy and majorization abstract we consider the properties of the shannon entropy for two probability distributions which stand in the relationship of majorization then we give a generalization of a theorem due to uhlmann extending it to infinite dimensional hilbert spaces finally we show that for any quantum channel one has s s for all quantum states if and only if there exists an isometric operator v such that v v

backlund transformations and the atiyah ward ansatz for non commutative anti self dual yang mills equations we present backlund transformations for the non commutative anti self dual yang mills equation where the gauge group is g gl 2 and use it to generate a series of exact solutions from a simple seed solution the solutions generated by this approach are represented in terms of quasi determinants we also explain the origins of all the ingredients of the backlund transformations within the framework of non commutative twistor theory in particular we show that the generated solutions belong to a non commutative version of the atiyah ward ansatz

the automorphism group of separable states in quantum information theory we show that the linear group of automorphism of hermitian matrices which preserves the set of separable states is generated by natural automorphisms change of an orthonormal basis in each tensor factor partial transpose in each tensor factor and interchanging two tensor factors of the same dimension we apply our results to the preservers of the product numerical range

on solutions to the non abelian hirota miwa equation and its continuum limits in this paper we construct grammian like quasideterminant solutions of a non abelian hirota miwa equation through continuum limits of this non abelian hirota miwa equation and its quasideterminant solutions we construct a cascade of non commutative differential difference equations ending with the non commutative kp equation for each of these systems the quasideterminant solutions are constructed as well

quasideterminant solutions of a non abelian hirota miwa equation a non abelian version of the hirota miwa equation is considered in an earlier paper of nimmo 2006 j phys a math gen 39 5053 65 it was shown how solutions expressed as quasideterminants could be constructed for this system by means of darboux transformations in this paper we discuss these solutions from a different perspective and show that the solutions are quasi plucker coordinates and that the non abelian hirota miwa equation may be written as a quasi plucker relation the special case of the matrix hirota miwa equation is also considered using a more traditional bilinear approach and the techniques are compared

on a non abelian hirota miwa equation a generalization of the hirota miwa equation to an abstract non abelian associative algebra is considered this system is integrable in the sense that it arises as compatibility condition for a linear system and has solutions constructed by means of the application of an arbitrary number of darboux transformations these solutions are in general expressed in terms of quasideterminants

darboux transformations for a twisted derivation and quasideterminant solutions to the super kdv equation this paper is concerned with a generalized type of darboux transformations defined in terms of a twisted derivation d satisfying d ab d a a b where is a homomorphism such twisted derivations include regular derivations difference and q difference operators and superderivatives as special cases remarkably the formulae for the iteration of darboux transformations are identical with those in the standard case of a regular derivation and are expressed in terms of quasideterminants as an example we revisit the darboux transformations for the manin radul super kdv equation studied in liu and manas liu manas 1997 a phys lett b 396 133 140 doi 10 1016 s0370 2693 97 00134 2 the new approach we take enables us to derive a unified expression for solution formulae in terms of quasideterminants covering all cases at once rather than using several subcases then by using a known relationship between quasideterminants and superdeterminants we obtain expressions for these solutions as ratios of superdeterminants this coincides with the results of liu and manas in all the cases they considered but also deals with the one subcase in which they did not obtain such an expression finally we obtain another type of quasideterminant solution to the manin radul super kdv equation constructed from its binary darboux transformations these can also be expressed as ratios of superdeterminants and are a substantial generalization of the solutions constructed using binary darboux transformations in earlier work on this topic

maps on states preserving the relative entropy ii abstract let h be a finite dimensional complex hilbert space the aim of this paper is to prove that every transformation on the space of all density operators on h which preserves the relative entropy is implemented by either a unitary or an antiunitary operator on h

on convexity properties of ψ direct sums of banach spaces the direct sum of banach spaces is strictly convex respectively uniformly convex locally uniformly convex uniformly convex in every direction if each of the banach spaces are strictly convex respectively uniformly convex locally uniformly convex uniformly convex in every direction and the corresponding norm is strictly convex