on some matrix inequalities the arithmetic geometric mean inequality for singular values due to bhatia and kittaneh says that 2sj ab sj a a b b j 1 2 for any matrices a b we first give new proofs of this inequality and its equivalent form then we use it to prove the following trace inequality let a0 be a positive definite matrix and a1 ak be positive semidefinite matrices then tr j 1k i 0jai 2aj tra0 1

structure preserving parallel algorithms for solving the bethe salpeter eigenvalue problem abstract the bethe salpeter eigenvalue problem is a dense structured eigenvalue problem arising from discretized bethe salpeter equation in the context of computing exciton energies and states a computational challenge is that at least half of the eigenvalues and the associated eigenvectors are desired in practice we establish the equivalence between bethe salpeter eigenvalue problems and real hamiltonian eigenvalue problems based on theoretical analysis structure preserving algorithms for a class of bethe salpeter eigenvalue problems are proposed we also show that for this class of problems all eigenvalues obtained from the tamm dancoff approximation are overestimated in order to solve large scale problems of practical interest we discuss parallel implementations of our algorithms targeting distributed memory systems several numerical examples are presented to demonstrate the efficiency and accuracy of our algorithms

solving satisfiability problems by fluctuations the dynamics of stochastic local search algorithms stochastic local search algorithms are frequently used to numerically solve hard combinatorial optimization or decision problems we give numerical and approximate analytical descriptions of the dynamics of such algorithms applied to random satisfiability problems we find two different dynamical regimes depending on the number of constraints per variable for low constraintness the problems are solved efficiently i e in linear time for higher constraintness the solution times become exponential we observe that the dynamical behavior is characterized by a fast equilibration and fluctuations around this equilibrium if the algorithm runs long enough an exponentially rare fluctuation towards a solution appears

field theoretic approach to the counting problem of hamiltonian cycles of graphs a hamiltonian cycle of a graph is a closed path that visits each site once and only once i study a field theoretic representation for the number of hamiltonian cycles for arbitrary graphs by integrating out quadratic fluctuations around the saddle point one obtains an estimate for the number which reflects characteristics of graphs well the accuracy of the estimate is verified by applying it to 2d square lattices with various boundary conditions this is the first example of extracting meaningful information from the quadratic approximation to the field theory representation

singular value inequalities for compact operators abstract a singular value inequality due to bhatia and kittaneh says that if a and b are compact operators on a complex separable hilbert space such that a is self adjoint b 0 and a b then s j a s j b b for j 1 2 we give an equivalent inequality which states that if a b and c are compact operators such that a b b c 0 then s j b s j a c for j 1 2 moreover we give a sharper inequality and we prove that this inequality is equivalent to three equivalent inequalities considered by tao in particular we show that if a and b are compact operators such that a is self adjoint b 0 and a b then 2 s j a s j b a b a for j 1 2 some applications of these results will be given

commutators and accretive operators abstract in this paper singular values of commutators of hilbert space operators are estimated to this aim the accretivity of a transform of the operators is applied some recent results of kittaneh f kittaneh singular value inequalities for commutators of hilbert space operators linear algebra appl 430 2009 2362 2367 are extended

mixed mean inequalities for several positive definite matrices on basis of the geometric mean proposed recently by t ando chi kwong li and roy mathias in this paper we present several kinds of mixed means for three or more positive definite matrices and prove some related mixed mean inequalities

matrix young inequalities for the hilbert schmidt norm abstract let a b and x be n n complex matrices such that a and b are positive semidefinite if p q 1 with 1 p 1 q 1 it is shown that 1 p a p x 1 q xb q 2 2 1 r 2 a p x xb q 2 2 axb 2 2 where r max p q and 2 is the hilbert schmidt norm generalizations and applications of this inequality are also considered

relaxation in graph coloring and satisfiability problems using t 0 monte carlo simulation we study the relaxation of graph coloring k col and satisfiability k sat two hard problems that have recently been shown to possess a phase transition in solvability as a parameter is varied a change from exponentially fast to power law relaxation and a transition to freezing behavior are found these changes take place for smaller values of the parameter than the solvability transition results for the coloring problem for colorable and clustered graphs and for the fraction of persistent spins for satisfiability are also presented

graph partitioning and dilute spin glasses the minimum cost solution the authors calculate the ground state cost function energy for graph bipartitioning and spin glasses on networks with fixed finite valences allowing a continuous distribution for the effective field compared with previous calculations using a field distribution in integral multiples of the coupling strength j the ground state energy is higher by less than one per cent although the distribution itself is altered drastically

on drury s solution of bhatia kittaneh s question abstract let a b be n n positive semidefinite matrices bhatia and kittaneh asked whether it is true j a b 1 2 j a b j 1 n where j j is the j th largest singular value eigenvalue respectively the question was recently solved by drury in the affirmative this article revisits drury s solution in particular we simplify the proof for a key auxiliary result in his solution

on some inequalities for unitarily invariant norms and singular values abstract in this note we improve and generalize some existing inequalities for unitarily invariant norms and singular values and present some other inequalities for unitarily invariant norms

unification of the arithmetic geometric mean and holder inequalities for unitarily invariant norms abstract in this paper we obtain an inequality for unitarily invariant norms which unifies the arithmetic geometric mean inequality and the holder inequality for unitarily invariant norms

young s inequality in trace class operators if a and b are trace class operators and if u is a partial isometry then where 1 denotes the norm in the trace class the present paper characterises the cases of equality in this young inequality and the characterisation is examined in the context of both the operator and the hilbert schmidt forms of young s inequality

eigenvalues pseudospectrum and structured perturbations we investigate the behavior of eigenvalues under structured perturbations we show that for many common structures such as complex symmetric toeplitz symmetric toeplitz circulant and others the structured condition number is equal to the unstructured condition number for normwise perturbations and prove similar results for real perturbations an exception are complex skewsymmetric matrices we also investigate componentwise complex and real perturbations here hermitian and skew hermitian matrices are exceptional for real perturbations furthermore we characterize the structured complex and real pseudospectrum for a number of structures and show that often there is little or no significant difference to the usual unstructured pseudospectrum

inequalities for sums and direct sums of hilbert space operators abstract we prove several singular value inequalities and norm inequalities involving sums and direct sums of hilbert space operators it is shown among other inequalities that if x and y are compact operators then the singular values of x y 2 are dominated by those of x y applications of these inequalities are also given

the hua matrix and inequalities related to contractive matrices abstract we first deny a conjecture raised in xu et al 2011 14 and then we present some eigenvalue or singular value inequalities related to contractive matrices

matrix power means and the information monotonicity abstract lim and palfia established the notion of the matrix power means for k positive definite matrices k 3 and showed that the matrix power means have the information monotonicity for a unital positive linear mapping in this note by virtue of the generalized kantorovich constant we show counterparts to the information monotonicity of the matrix power means

novel matrix singular value inequalities and their applications to uncertain mimo channels novel matrix singular value inequalities are established for a sum product of three matrices their application to the uncertain compound multiple input multiple output mimo channel subject to normed additive uncertainty establishes the saddle point property for a wide range of performance metrics monotonic in the channel singular values including among others the mutual information mmse error exponent and pairwise error probability this in turn implies that the transmission on the eigenmodes of the nominal or worst case channel is also optimal for the whole set of channels under a general power constraint and hence achieves the compound channel capacity the worst case channel turns out to be antiparallel of the nominal one for all these performance metrics an application of these results to beamforming over compound mimo channels is discussed an optimal robust precoder for the uncertain mimo channel is obtained in a closed form under the sum mse criterion and the total power constraint the saddle point property is shown to hold and the optimal strategy is to diagonalize the nominal or worst case channel

computation of pseudospectra by continuation the concept of pseudospectrum was introduced by trefethen to explain the behavior of nonnormal operators many phenomena for example hydrodynamic instability and convergence of iterative methods for linear systems cannot be accounted for by eigenvalue analysis but are more understandable by examining the pseudospectra the straightforward way to compute pseudospectra involves many applications of the singular value decomposition svd this paper presents several fast continuation methods to calculate pseudospectra of different types of matrices

a note on the arithmetic geometric mean inequality for matrices abstract we give a simple proof of the inequality aa x xbb 2 a ab where a b and x are arbitrary n n matrices and is any unitarily invariant norm for the schatten p norms p with 1 p aa x xbb p 2 a xb p if and only if aa x xbb

comparison of various means for operators for hilbert space operatorsh k xwithh k 0 the norm inequality h1 2xk1 2 12 hx xk is known where is an arbitrary unitarily invariant norm a refinement of this arithmetic geometric mean inequality is studied similar norm inequalities are indeed established for various natural means for operators such as the logarithmic mean

statistical mechanics of the random k satisfiability model the random k satisfiability problem consisting in verifying the existence of an assignment of n boolean variables that satisfy a set of m ensuremath alpha n random logical clauses containing k variables each is studied using the replica symmetric framework of diluted disordered systems we present an exact iterative scheme for the replica symmetric functional order parameter together for the different cases of interest k 2 kg 3 and k ensuremath gg 1 the calculation of the number of solutions which allowed us phys rev lett 76 3881 1996 to predict a first order jump at the threshold where the boolean expressions become unsatisfiable with probability one is thoroughly displayed in the case k 2 the rigorously known critical value ensuremath alpha 1 of the number of clauses per boolean variable is recovered while for kg 3 we show that the system exhibits a replica symmetry breaking transition the annealed approximation is proven to be exact for large k

phase transition in the number partitioning problem universit at magdeburg institut fu r theoretische physik universit atsplatz 2 d 39106 magdeburg germany february 1 2008 number partitioning is an np complete problem of com binatorial optimization a statistical mechanics analysis re veals the existence of a phase transition that separates theeasy from the hard to solve instances and that re ects thepseudo polynomiality of number partitioning the phase di agram and the value of the typical ground state energy arecalculated 64 60 cn 02 60 pn 02 70 lq 89 80 h

entropy barriers and slow relaxation in some random walk models we study the zero temperature limit of a simple model of slow relaxation without energy barriers proposed by ritort 1995 as well as two other closely related models with a much faster relaxation these models can be mapped onto random walk problems which allows for their analytic study we analyse in particular a specific aspect of the former model namely the existence of a bias leading to entropy barriers and to a very slow relaxation

more results on singular value inequalities of matrices abstract the arithmetic geometric mean inequality for singular values due to bhatia and kittaneh says that 2 s j ab s j a a b b j 1 2 for any matrices a b we give a new equivalent form and some relevant generalizations of this inequality in particular we show that s j a 1 4 b 3 4 a 3 4 b 1 4 s j a b j 1 n for any n n positive semidefinite matrices a b which proves a special case of zhan s conjecture posed in 2000

singular value inequalities for commutators of hilbert space operators abstract we prove several singular value inequalities for commutators of hilbert space operators it is shown among other inequalities that if a b and x are operators on a complex separable hilbert space such that a and b are positive and x is compact then the singular values of ax xb are dominated by those of max a b x x where is the usual operator norm

an operator inequality abstract for hilbert space operators with s invertible hermitian it is proved that sts 1 s 1 ts 2 t

a cauchy schwarz inequality for operators with applications abstract for any unitarily invariant norm on hilbert space operators it is shown that for all operators a b x and positive real numbers r we have a xb r 2 aa x r xbb r some consequences are then discussed a simple proof is given for the fact that for positive operators a b the function spr a t b t 1 t is monotone in t on the positive half line

generalized epsilon pseudospectra normal matrices have complete sets of orthonormal eigenvectors and therefore the spectral decomposition is useful in studying the properties of normal operators in contrast the eigenvectors of nonnormal matrices can be nearly linearly dependent and the eigenvalue problem may by highly ill conditioned thus additional concepts and analysis techniques are useful in examining nonnormal operators the concepts of epsilon pseudospectra introduced by l n trefethen has proven to be a powerful tool in the analysis of nonnormal operators the author generalizes epsilon pseudospectra and the associated computational algorithms to the generalized eigenvalue problem rank 1 perturbations are used to determine the epsilon pseudospectra

inequalities for commutators of positive operators abstract it is shown that if a b and x are operators on a complex separable hilbert space such that a and b are compact and positive then the singular values of the generalized commutator a x x b are dominated by those of x a b where is the usual operator norm consequently for every unitarily invariant norm we have a x x b x a b it is also shown that if a and b are positive and x is compact then a x x b max a b x for every unitarily invariant norm moreover if x is positive then the singular values of the commutator a x x a are dominated by those of 1 2 a x x consequently a x x a 1 2 a x x for every unitarily invariant norm for the usual operator norm these norm inequalities hold without the compactness conditions and in this case the first two norm inequalities are the same our inequalities include and improve upon earlier inequalities proved in this context and they seem natural enough and applicable to be widely useful

new inequalities for the generalized karcher mean abstract recently palfia introduced a generalized karcher mean as a solution of an operator equation in this article we present several relations for this new mean in particular we investigate the behavior of this generalized mean when filtered through operator monotone functions and positive linear maps revealing its information monotonicity

d eigenvalues of diffusion kurtosis tensors diffusion kurtosis imaging dki is a new model in medical engineering where a diffusion kurtosis dk tensor is involved a dk tensor is a fourth order three dimensional fully symmetric tensor in this paper we introduce d eigenvalues for a dk tensor the largest the smallest and the average d eigenvalues of a dk tensor correspond with the largest the smallest and the average apparent kurtosis coefficients akc of a water molecule in the space respectively we present their computational methods and discuss related anisotropy value of akc

an arithmetic geometric mean inequality for products of three matrices abstract consider the following noncommutative arithmetic geometric mean inequality given positive semidefinite matrices a 1 a n the following holds for each integer m n 1 n m j 1 j 2 j m 1 n a j 1 a j 2 a j m n m n j 1 j 2 j m 1 all distinct n a j 1 a j 2 a j m where denotes a unitarily invariant norm including the operator norm and schatten p norms as special cases while this inequality in full generality remains a conjecture we prove that the inequality holds for products of up to three matrices m 3 the proofs for m 1 2 are straightforward to derive the proof for m 3 we appeal to a variant of the classic araki lieb thirring inequality for permutations of matrix products

more on pseudospectra for polynomial eigenvalue problems and applications in control theory definitions and characterizations of pseudospectra are given for rectangular matrix poly nomials expressed in homogeneous form p da d d 1 a d 1 da 0 it is shown that problems with infinite pseudo eigenvalues are elegantly treated in this framework for such problems stereographic projection onto the riemann sphere is shown to provide a convenient way to visualize pseudospectra lower bounds for the distance to the nearest nonregular polynomial and the nearest uncontrollable dth order system with equality for standard state space systems are obtained in terms of pseudospectra showing that pseudospectra are a fundamental tool for reasoning about matrix polynomials in areas such as control theory how and why to incorporate linear structure into pseudospectra is also discussed by example

generalizing eigenvalue theorems to pseudospectra theorems analysis of nonsymmetric matrix iterations based on eigenvalues can be misleading in this paper we discuss sixteen theorems involving varepsilon def lam lambda def lam lambda pseudospectra that each generalize a familiar eigenvalue theorem and may provide more descriptive information in some cases our organizing principle is that each pseudospectral theorem reduces precisely to the corresponding eigenvalue theorem when varepsilon def lam lambda def lam lambda 0

structured pseudospectra and structured sensitivity of eigenvalues pseudospectra and structured pseudospectra have been investigated widely in this paper for the matrix spectral norm we give a new equivalent definition of structured pseudospectra of a matrix triplet a b c via the so called restricted singular value decomposition rsvd according to this definition we provide a solution of structured wilkinson s problem which concerns structured sensitivity of eigenvalues of square matrices several examples are given to illustrate our results also we extend the new definition of structured pseudospectra to polynomial eigenvalue problems

some inequalities for unitarily invariant norm abstract we shall prove the inequalities a b a b aa bb 2 ab a b a b 4 ab for all n n complex matrices a b and all unitarily invariant norms if further a b are positive definite it is proved that j 1 k j a b j 1 k j a 1 b 1 k n 0 1 where denotes the operator means considered by kubo and ando and j x 1 j n denote the eigenvalues of x arranged in the decreasing order whenever these all are real a number of inequalities are obtained as applications

on eigenvalue problems of real symmetric tensors abstract we use variational methods to give a positive answer to a conjecture posed by liqun qi l qi eigenvalues of a real supersymmetric tensor j symbolic comput 40 2005 1302 1324 regarding the real eigenvalues of certain higher order tensors

two combinatorial models with identical statics yet different dynamics motivated by the problem of sorting we introduce two simple combinatorial models with distinct hamiltonians yet identical spectra and hence partition function and show that the local dynamics of these models is very different after a deep quench one model slowly relaxes to the sorted state whereas the other model becomes blocked by the presence of stable local minima

contractive maps on operator ideals and norm inequalities ii abstract let i be a norm ideal of operators equipped with a unitarily invariant norm we exploit integral representations of certain functions to prove that certain ratios of linear operators acting on operators in i are contractive this leads to some new and old norm inequalities we also lift a variety of inequalities to the operator setting which were proved in the matrix setting earlier

application of statistical mechanics to np complete problems in combinatorial optimisation recently developed techniques of the statistical mechanics of random systems are applied to the graph partitioning problem the averaged cost function is calculated and agrees well with numerical results the problem bears close resemblance to that of spin glasses the authors find a spin glass transition in the system and the low temperature phase space has an ultrametric structure this sheds light on the nature of hard computation problems

norm bounds for hadamard products and an arithmetic geometric mean inequality for unitarily invariant norms abstract an arithmetic geometric mean inequality for unitarily invariant norms and matrices 2 a xb aa x xbb is an immediate consequence of a basic inequality for singular values of hadamard products

relaxation and metastability in a local search procedure for the random satisfiability problem an analysis of the average properties of a local search procedure randomwalksat for the satisfaction of random boolean constraints is presented depending on the ratio alpha of constraints per variable reaching a solution takes a time t res growing linearly t res approximately tau res alpha n alpha alpha d with the size n of the instance the relaxation time tau res alpha in the linear phase is calculated through a systematic expansion scheme based on a quantum formulation of the evolution operator for alpha alpha d the system is trapped in some metastable state and resolution occurs from escape from this state through crossing of a large barrier an annealed calculation of the height zeta alpha of this barrier is proposed the polynomial to exponential cross over alpha d approximately 2 7 is not related to the onset of clustering among solutions occurring at alpha approximately 3 86

spectra of uniform hypergraphs abstract we present a spectral theory of uniform hypergraphs that closely parallels spectral graph theory a number of recent developments building upon classical work has led to a rich understanding of symmetric hyperdeterminants of hypermatrices a k a multidimensional arrays symmetric hyperdeterminants share many properties with determinants but the context of multilinear algebra is substantially more complicated than the linear algebra required to address spectral graph theory i e ordinary matrices nonetheless it is possible to define eigenvalues of a hypermatrix via its characteristic polynomial as well as variationally we apply this notion to the adjacency hypermatrix of a uniform hypergraph and prove a number of natural analogs of basic results in spectral graph theory open problems abound and we present a number of directions for further study

arithmetic geometric mean and related inequalities for operators abstract in recent years certain arithmetic geometric mean and related inequalities for operators and unitarily invariant norms have been obtained by many authors based on majorization technique and so on we first point out that they are direct consequences of integral expressions of relevant operators furthermore we obtain related new inequalities theorems 4 5 and 6 based on our current approach

better bounds on the numerical radii of hilbert space operators abstract kittaneh proved that if a is a bounded linear operator on a complex hilbert space then 1 4 a 2 a 2 2 a where and are the numerical radius and the usual operator norm and a a a 1 2 in this paper we show that 1 4 a 2 a 2 1 2 2 a 1 8 a a a a 2 a meanwhile we give an improvement of the norm inequality presented by bhatia and kittaneh for the positive operators

an arithmetic geometric harmonic mean inequality involving hadamard products abstract given matrices of the same size a a ij and b b ij we define their hadamard product to be a b a ij b ij we show that if x i 0 and q p 0 then the n n matrices x i x j x i x j x i 1 x j 1 x i x j and x i p x j p x i q x j q are positive definite and we relate these facts to some matrix valued arithmetic geometric harmonic mean inequalities some of which involve hadamard products and others unitarily invariant norms it is known that if a is positive semidefinite then max a b b 1 max a ij where denotes the spectral norm we show that the converse of this statement is false and give a useful partial converse

existence of real eigenvalues of real tensors abstract we use the brouwer degree to establish the existence of real eigenpairs of higher order real tensors in various settings also we provide some finer criteria for the existence of real eigenpairs of two dimensional real tensors and give a complete classification of the brouwer degree zero and 2 maps induced by general third order two dimensional real tensors

componentwise pseudospectrum of a matrix abstract given a nonnegative weight matrix e defining componentwise structured perturbations e e we introduce a componentwise pseudospectrum of a square matrix a as contour sets of the function c a i 1 e where m is the spectral radius of a square matrix m and discuss how to compute it

the singular values of a b and a ib we obtain several inequalities relating the singular values of a b and those of a ib when a and b are hermitian and when one or both of them are positive semidefinite

majorizations and inequalities in matrix theory abstract in matrix theory majorization plays a significant role for instance majorization relations among eigenvalues and singular values of matrices produce a lot of norm inequalities and even matrix inequalities this survey article is intended as a review of recent results in matrix theory related to majorization

more inequalities for positive semidefinite 2 2 block matrices and their blocks abstract it is obtained that for positive semidefinite block matrix h m k k n if a b m n c such that m a x a 2 b 2 2 and f is a nonnegative increasing convex function on 0 satisfying f 0 0 then 2 s j f a k b m a x a 2 b 2 s j f h for j 1 2 n also it is shown that if h m k k n is a positive semidefinite 2 2 block matrix with square matrices m and n of the same size then h r 2 a b r 1 m n for r 1 where a and b are the largest eigenvalues of the matrices m and n respectively in addition if r 3 2 then h r 2 4 m 2 n 2 m 0 n 0 2 r 2 for every unitarily invariant norm

the centred travelling salesman at high temperature a recently formulated statistical mechanics method is used to study the phase transition occurring in a generalization of the travelling salesman problem tsp known as the centred tsp the method shows that the problem has clear signs of a crossover but is only able to access unscaled finite temperatures above the transition point the solution of the problem using this method displays a curious duality

an arithmetic geometric mean inequality for singular values and its applications abstract in this short note we give a new equivalent form of the arithmetic geometric mean inequality for singular values as applications of our result we give a new proof of an inequality due to bhatia and davis 1993 4 and we obtain a singular value inequality for matrix means which is similar to one proved by drury 2012 9 finally we present a log majorization inequality for singular values

on singular value inequalities for matrix means abstract for positive semidefinite n n matrices a and b the singular value inequality 2 t s j a r b 2 r a 2 r b r 2 s j a 2 t a b b 2 is shown to hold for r 1 2 1 3 2 and all 2 t 2

notes on matrix arithmetic geometric mean inequalities abstract for positive semi definite n n matrices the inequality 4 ab a b 2 isshown to hold for every unitarily invariant norm the connection of this with some other matrix arithmetic geometric mean inequalities and trace inequalities is discussed

some bounds for the spectral radius of nonnegative tensors in this paper we extend the well known column sum bound of the spectral radius for nonnegative matrices to the tensor case an upper bound of the spectral radius for a nonnegative tensor is given via the largest eigenvalue of a symmetric tensor also we show some bounds of spectral radius of nonnegative tensors based on the sum of the entries in the other indices of tensors we demonstrate that our new results improve existing results the other main results of this paper is to provide a sharper ky fan type theorem and a comparison theorem for nonnegative tensors finally we make use of our bounds to give a perturbation bound for the spectral radius of symmetric nonnegative tensors this result is similar to the weyl theorem for the matrix case

on a question of bhatia and kittaneh abstract we settle in the affirmative a question of bhatia and kittaneh for p and q positive semidefinite n n matrices the inequality r pq 1 2 r p q holds for r 1 2 n

a singular value inequality for heinz means we prove a matrix inequality for matrix monotone functions and apply it to prove a singular value inequality for heinz means recently conjectured by zhan

semismoothness of the maximum eigenvalue function of a symmetric tensor and its application abstract in this paper we examine the maximum eigenvalue function of an even order real symmetric tensor by using the variational analysis techniques we first show that the maximum eigenvalue function is a continuous and convex function on the symmetric tensor space in particular we obtain the convex subdifferential formula for the maximum eigenvalue function next for an mth order n dimensional symmetric tensor a we show that the maximum eigenvalue function is always th order semismooth at a for some rational number 0 in the special case when the geometric multiplicity is one we show that can be set as 1 2 m 1 n sufficient condition ensuring the strong semismoothness of the maximum eigenvalue function is also provided as an application we propose a generalized newton method to solve the space tensor conic linear programming problem which arises in medical imaging area local convergence rate of this method is established by using the semismooth property of the maximum eigenvalue function

eigenvalue inequalities for convex and log convex functions abstract we give a matrix version of the scalar inequality f a b f a f b for positive concave functions f on 0 we show that choi s inequality for positive unital maps and operator convex functions remains valid for monotone convex functions at the cost of unitary congruences some inequalities for log convex functions are presented and a new arithmetic geometric mean inequality for positive matrices is given we also point out a simple proof of the bhatia kittaneh arithmetic geometric mean inequality

tensor decompositions and applications this survey provides an overview of higher order tensor decompositions their applications and available software a tensor is a multidimensional or n way array decompositions of higher order tensors i e n way arrays with n geq 3 have applications in psycho metrics chemometrics signal processing numerical linear algebra computer vision numerical analysis data mining neuroscience graph analysis and elsewhere two particular tensor decompositions can be considered to be higher order extensions of the matrix singular value decomposition candecomp parafac cp decomposes a tensor as a sum of rank one tensors and the tucker decomposition is a higher order form of principal component analysis there are many other tensor decompositions including indscal parafac2 candelinc dedicom and paratuck2 as well as nonnegative variants of all of the above the n way toolbox tensor toolbox and multilinear engine are examples of software packages for working with tensors

on an operator kantorovich inequality for positive linear maps abstract we improve the operator kantorovich inequality as follows let a be a positive operator on a hilbert space with 0 m a m then for every unital positive linear map a 1 2 m m 2 4 m m 2 a 2 as a consequence a 1 a a a 1 m m 2 2 m m

large scale computation of pseudospectra using arpack and eigs arpack and its sc matlab counterpart tt eigs are software packages that calculate some eigenvalues of a large nonsymmetric matrix by arnoldi iteration with implicit restarts we show that at a small additional cost which diminishes relatively as the matrix dimension increases good estimates of pseudospectra in addition to eigenvalues can be obtained as a by product thus in large scale eigenvalue calculations it is feasible to obtain routinely not just eigenvalue approximations but also information as to whether or not the eigenvalues are likely to be physically significant examples are presented for matrices with dimension up to 200 000

pseudospectra of linear operators if a matrix or linear operator a is far from normal its eigenvalues or more generally its spectrum may have little to do with its behavior as measured by quantities such as an or exp ta more may be learned by examining the sets in the complex plane known as the pseudospectra of a defined by level curves of the norm of the resolvent zi a 1 five years ago the author published a paper that presented computed pseudospectra of thirteen highly nonnormal matrices arising in various applications since that time analogous computations have been carried out for differential and integral operators this paper a companion to the earlier one presents ten examples each chosen to illustrate one or more mathematical or physical principles

some inequalities for positive linear maps it has long been known that an analogue of jensen s inequality holds for positive unital linear maps on matrix algebras provided that instead of ordinary convex functions one restricts to matrix convex functions we show that this restriction is not necessary in the case of 2 2 matrices a noncommutative analogue of the variance is studied and a basic inequality with several applications is established

the hardest constraint problems a double phase transition abstract the distribution of hard graph coloring problems as a function of graph connectivity is shown to have two distinct transition behaviors the first previously recognized is a peak in the median search cost near the connectivity at which half the graphs have solutions this region contains a high proportion of relatively hard problem instances however the hardest instances are in fact concentrated at a second lower transition point near this point most problems are quite easy but there are also a few very hard cases this region of exceptionally hard problems corresponds to the transition between polynomial and exponential scaling of the average search cost whose location we also estimate theoretically these behaviors also appear to arise in other constraint problems this work also shows the limitations of simple measures of the cost distribution such as mean or median for identifying outlying cases

squaring operator polya szego and diaz metcalf type inequalities abstract we square operator polya szego and diaz metcalf type inequalities as follows if operator inequalities 0 m 1 2 a m 1 2 and 0 m 2 2 b m 2 2 hold for some positive real numbers m 1 m 1 and m 2 m 2 then for every unital positive linear map the following inequalities hold a b 2 m 1 m 2 m 1 m 2 2 m 1 m 2 m 1 m 2 4 a b 2 and m 2 m 2 m 1 m 1 a b 2 m 1 m 1 m 2 2 m 2 2 m 2 m 2 m 1 2 m 1 2 2 8 m 2 m 1 m 1 m 2 m 1 2 m 1 2 m 2 m 2 2 a b 2

statistical mechanics of the multi constraint continuous knapsack problem we apply the replica analysis established by gardner to the multi constraint continuous knapsack problem which is one of the linear programming problems and a most fundamental problem in the field of operations research or for a large problem size we analyse the space of solution and its volume and estimate the optimal number of items to go into the knapsack as a function of the number of constraints we study the stability of the replica symmetric rs solution and find that the rs calculation cannot estimate the optimal number of items in the knapsack correctly if many constraints are required in order to obtain a consistent solution in the rs region we try the zero entropy approximation for this continuous solution space and get a stable solution within the rs ansatz on the other hand in the replica symmetry breaking rsb region the one step rsb solution is found by parisi s scheme it turns out that this problem is closely related to the problem of optimal storage capacity and of generalization by maximum stability rule of a spherical perceptron

norm inequalities for commutators of positive operators and applications let x y and z be operators on a hilbert space such that x and z are positive it is shown that left xy yz right leqslant rm max left left x right left z right right left y right applications of this commutator inequality are given

pseudo spectra theory of tensors and tensor polynomial eigenvalue problems abstract this paper is devoted to the extension of the pseudo spectra theory from matrices to tensors based on the definition of an eigenpair of real symmetric tensors and results on the pseudo spectrum of square matrices we first introduce the pseudo spectrum of a complex tensor and investigate its fundamental properties such as its computational interpretations and the link with the stability of its related homogeneous dynamical system we then extend the pseudo spectrum to the setting of tensor polynomial eigenvalue problems we further derive basic structure of the pseudo spectrum for tensor polynomial eigenvalue problems including the symmetry boundedness and number of connected components under suitable mild assumptions finally we discuss the implication of the pseudo spectrum for computing the backward errors and the distance from a regular tensor polynomial to the nearest irregular tensor polynomial

an extension of a matrix inequality of thompson abstract a matrix inequality of thompson is extended as an application a partial affirmative answer to a recent question of lin is provided

phase transitions and the search problem we describe how techniques that were originally developed in statistical mechanics can be applied to search problems that arise commonly in artificial intelligence this approach is useful for understanding the typical behavior of classes of problems in particular these techniques predict that abrupt changes in computational cost analogous to physical phase transitions should occur universally as heuristic effectiveness or search space topology is varied we also present a number of open questions raised by these studies

pseudospectra localization sets of tensors with applications abstract a localization set and an exclusion set for the pseudospectrum of tensors are given as applications we first use the pseudospectra localization sets to locate asymptotically stable sets for homogeneous dynamical systems and then two upper bounds for the largest modulus of the asymptotically stable set are given to estimate the asymptotic decay rate for homogeneous dynamical systems we also provide two lower bounds for the preservation measure to the positive definiteness of the homogeneous polynomial form in addition some sufficient conditions for the global uniqueness and solvability of tensor complementarity problems are presented the last application is to test copositivity of some potential fields in particle physics

on some operator inequalities abstract for hilbert space operators s t with s invertible and self adjoint corach porta and recht recently proved that sts 1 s 1 ts 2 t a generalization of this inequality to larger classes of operators and norms is obtained as an immediate consequence of the operator form of the arithmetic geometric mean inequality some related inequalities are also discussed

geometric operator inequalities abstract the geometrical meaning of several well known inequalities is discussed they include the so called loewner heinz mcintosh and segal inequalities it is shown that some of them can be deduced from the others even for unitarily invariant forms some spectral properties of the elementary operators associated to the inequalities are studied

new properties for certain positive semidefinite matrices abstract we bring in some new notions associated with 2 2 block positive semidefinite matrices these notions concern the inequalities between the singular values of the off diagonal blocks and the eigenvalues of the arithmetic mean or geometric mean of the diagonal blocks we investigate some relations between them many examples are included to illustrate these relations