iterative solution of a 3 d scattering problem from arbitrary shaped multidielectric and multiconducting bodies we present an iterative algorithm used for the computation of the scattering from arbitrary shaped bodies made of dielectric and conducting material the harmonic problem is discretized by a hybrid boundary integral method finite element method to solve the discretized linear system a modified version of a minimal residual algorithm has been developed for complex matrices it takes into account the lagrange multiplier constraint in a special way this modified version also allows the linear system to be solved for many right hand sides in an efficient manner

linear preservers of weak majorization on l1 i when i is an infinite set abstract the necessary and sufficient conditions that a bounded linear map on the banach space l 1 i may be considered as a linear preserver of weak majorization on l 1 i where i is an arbitrary infinite set are given also we prove that the set of all linear preservers of weak majorization is closed under the norm topology

a weak majorization involving the matrices a b and ab abstract let a and b be n n positive definite matrices and let the eigenvalues of a b and ab be arranged in decreasing order then for all r 0 i k n i r ab i k n i r a b for k 1 n furthermore it is shown that i k n i r a b i k n i r ab for k 1 n is a limiting case of the weak majorization above which settles a conjecture of bapat and johnson

an extension of a result of hwang and pyo abstract in this paper we extend a majorization result of hwang and pyo laa 332 334 2001 pp 15 21 from the ordinary majorization ordering to the class of group induced cone orderings induced by non effective groups the case of effective groups is also investigated a particular attention is paid to positive operators

log majorization related to renyi divergences abstract for z 0 with 1 motivated by comparison between different kinds of renyi divergences in quantum information we consider log majorization between the matrix functions p a b b 1 2 b 1 2 a b 1 2 b 1 2 q z a b b 1 2 z a z b 1 2 z z of two positive semi definite matrices a b we precisely determine the parameters z for which p a b log q z a b and q z a b log p a b hold respectively

some majorization inequalities induced by schur products in euclidean jordan algebras abstract in a euclidean jordan algebra v of rank n an element x is said to be majorized by an element y symbolically x y if the corresponding eigenvalue vector x is majorized by y in r n in this article we describe pointwise majorization inequalities of the form t x s x where t and s are linear transformations induced by schur products specializing we recover analogs of majorization inequalities of schur hadamard and oppenheimer stated in the setting of euclidean jordan algebras as well as majorization inequalities connecting quadratic and lyapunov transformations on v we also show how schur products induced by certain scalar means such as arithmetic geometric harmonic and logarithmic means naturally lead to majorization inequalities

linear operators preserving directional majorization abstract a characterization of linear operators preserving directional majorization is given the result extends a theorem of t ando linear algebra appl 118 1989 163 and the proof also gives the characterization of linear operators preserving multivariate majorization recently obtained by beasley and lee

group entropies correlation laws and zeta functions the notion of group entropy is proposed it enables the unification and generaliztion of many different definitions of entropy known in the literature such as those of boltzmann gibbs tsallis abe and kaniadakis other entropic functionals are introduced related to nontrivial correlation laws characterizing universality classes of systems out of equilibrium when the dynamics is weakly chaotic the associated thermostatistics are discussed the mathematical structure underlying our construction is that of formal group theory which provides the general structure of the correlations among particles and dictates the associated entropic functionals as an example of application the role of group entropies in information theory is illustrated and generalizations of the kullback leibler divergence are proposed a new connection between statistical mechanics and zeta functions is established in particular tsallis entropy is related to the classical riemann zeta function

a quantitative version of the observation that the hadamard product is a principal submatrix of the kronecker product abstract the hadamard and kronecker products of two n m matrices a b are related by a b p t 1 a b p 2 where p 1 p 2 are partial permutation matrices after establishing several properties of the p matrices this relationship is employed to demonstrate how a simplified theory of the hadamard product can be developed during this process the well known result a b a b aa bb is extended to a b a b 1 2 aa bb ab ba aa bb showing an inherent link between the hadamard product and conventional product of two matrices this leads to a sharper bound on the spectral norm of a b a b 1 2 a 2 b 2 ab 2 1 2 a b and an improvement on the weak majorization of a b 2 a b w 1 2 2 a 2 b 2 ab w 2 a 2 b for a real non singular matrix x and invertible diagonal matrices d e the spectral condition number is shown to be if scaled bounded below as follows dxe 2 x x t 2 x x t 2 1 2 x x t 1 for a 0 we have i a 2 1 2 i a 2 a a i a 2 and a 1 2 a 1 2 2 1 2 i a a 1 a a 1 when a 0 the latter inequality is compared to styan s inequality a a 1 1 2 i a a 1 when a is a correlation matrix and is shown to possess stronger properties of ordering finally the relationship a b p t 1 a b p 2 is applied to determine conditions of singularity of certain orderings of the hadamard products of matrices

illustrating the geometry of coherently controlled unital open quantum systems we extend standard markovian open quantum systems quantum channels by allowing for hamiltonian controls and elucidate their geometry in terms of lie semigroups for standard dissipative interactions with the environment and different coherent controls we particularly specify the tangent cones lie wedges of the respective lie semigroups of quantum channels these cones are the counterpart of the infinitesimal generator of a single one parameter semigroup they comprise all directions the underlying open quantum system can be steered to and thus give insight into the geometry of controlled open quantum dynamics such a differential characterization is highly valuable for approximating reachable sets of given initial quantum states in a plethora of experimental implementations

majorization doubly stochastic matrices and comparison of eigenvalues abstract 1 basic properties of majorization 2 isotone maps and algebraic operations 3 double sub and superstochasticity 4 doubly stochastic matrices 5 doubly stochastic matrices with minimum permanent 6 comparison of eigenvalues 7 doubly stochastic maps

generalized statistics yet another generalization we provide a unifying axiomatics for renyi s entropy and non extensive entropy of tsallis it is shown that the resulting entropy coincides with csiszar s measure of directed divergence known from communication theory

remarks on comments on on q non extensive statistics with non tsallisian entropy physica a 466 2017 160 abstract recently in ilic and stankovic 2017 have suggested that there may be problem for the class of hybrid entropies introduced in jizba and arimitsu 2004 in this comment we point out that the problem can be traced down to the q additive entropic chain rule and to a peculiar behavior of the de finetti kolmogorov relation for escort distributions however despite this one can still safely use the proposed hybrid entropies in most of the statistical thermodynamics considerations

multifractal diffusion entropy analysis optimal bin width of probability histograms abstract in the framework of multifractal diffusion entropy analysis we propose a method for choosing an optimal bin width in histograms generated from underlying probability distributions of interest the method presented uses techniques of renyi s entropy and the mean squared error analysis to discuss the conditions under which the error in the multifractal spectrum estimation is minimal we illustrate the utility of our approach by focusing on a scaling behavior of financial time series in particular we analyze the s p500 stock index as sampled at a daily rate in the time period 1950 2013 in order to demonstrate a strength of the method proposed we compare the multifractal spectrum for various bin widths and show the robustness of the method especially for large values of q for such values other methods in use e g those based on moment estimation tend to fail for heavy tailed data or data with long correlations connection between the spectrum and renyi s q parameter is also discussed and elucidated on a simple example of multiscale time series

permutation invariant proper polyhedral cones and their lyapunov rank abstract the lyapunov rank of a proper cone k in a finite dimensional real hilbert space is defined as the dimension of the space of all lyapunov like transformations on k or equivalently the dimension of the lie algebra of the automorphism group of k this rank measures the number of linearly independent bilinear relations needed to express a complementarity system on k that arises for example from a linear program or a complementarity problem on the cone motivated by the problem of describing spectral proper cones where the complementarity system can be expressed as a square system that is where the lyapunov rank is greater than equal to the dimension of the ambient space we consider proper polyhedral cones in r n that are permutation invariant for such cones we show that the lyapunov rank is either 1 in which case the cone is irreducible or n in which case the cone is isomorphic to r n in the latter case we show that the corresponding spectral cone is isomorphic to a symmetric cone

convex majorization on discrete lp spaces abstract in this paper for p 1 and a nonempty set i we consider a natural preorder on l p i which is said to be convex majorization then some interesting properties of all bounded linear operators t l p i l p i preserving the convex majorization are given we also characterize these operators and assert one of the important properties of them that is the rows of matrix form of such an operator belongs to l 1 i

extremal majorizing and anti majorizing matrices in this note we study the convex set of all real n x n matrices which either increase or decrease mi xednlnrss when applied to any monotonously ordered probability n vector and we compute a complete list of its extremals in both cases 0 i998 published by

the solution of van der waerden s problem for permanents in this paper a solution to van der waerden s problem is presented it is hoped that the solution of this problem will stimulate new and vigorous interest in the theory of permanents and related topics permanents are ideal objects of study having both practical applications and abstract mathematical problems problems involving permanents tend to be easily stated but only solvable using deep methods the theory of permanents as an independent branch of mathematics which has developed parallel to the theory of determinants is far from exhausted many difficult questions must still be resolved in particular the problem of calculation and estimation of permanents for specific classes of matrices is of prime importance the solution of van der waerden s problem makes use of the penetrating force of the method of mixed volumes from combinatorial analysis even hilbert noted that there were connections to be uncovered between the theory of mixed volumes and other areas of mathematics the method of mixed volumes is ideally suited to solving extremal problems and problems of uniqueness and obtaining deep new inequalities it is reasonable to assume that in the future the method of mixed volumes will stand with that of generating functions as one of the basic analytic tools of computational combinatorial analysis

linear operators preserving multivariate majorization abstract in this paper we characterize linear operators on real matrices that preserve multivariate majorization we place no additional restrictions on the linear operator

notes on egoritsjev s proof of the van der waerden conjecture abstract a solution to the van der waerden conjecture on the minimum of the permanent of doubly stochastic matrices has recently been found by g p ergoritsjev in russian because of the wide interest in this conjecture it seems useful to make the solution more accessible professor van lint has kindly consented to have his notes including a direct proof of an inequality available in the literature only in a more general form published in this journal ed r a b

directional quasi convexity asymmetric schur convexity and optimality of consecutive partitions the current paper has the following distinct goals 1 to generalize standard quasi convexity to obtain a weaker property of functions that suffices for optimality of extreme points specifically it is required that intersections of the level sets of the function with line segments having direction in a prescribed set are convex 2 to characterize schur convexity for symmetric functions through the above generalization of quasi convexity thereby obtaining a previously unknown relationship of schur convexity and standard convexity the new characterization is used to extend the definition of schur convexity to functions which are not symmetric 3 to obtain new sufficient conditions for the optimality of consecutive partitions by using the new definition of schur convexity for functions that are not necessarily symmetric the conclusion that schur convexity is an instance of a convexity property that implies the optimality of extreme points unifies two approaches that were used in the literature to prove optimality of subsets of the domain of real valued functions the two tools quasi convexity and schur convexity were previously considered as distinct techniques

g majorization inequalities for linear maps abstract we give a unified approach to majorization inequalities for linear maps via g majorization preorderings a particular attention is paid to positive linear maps some classical and recently obtained inequalities are derived from the results

on the monotonicity of scalar curvature in classical and quantum information geometry we study the monotonicity under mixing of the scalar curvature for the geometries on the simplex of probability vectors from the results obtained and from numerical data we are led to some conjectures about quantum geometries and wigner yanase dyson information finally we show that this last conjecture implies the truth of the petz conjecture about the monotonicity of the scalar curvature of the bogoliubov kubo mori monotone metric

reach a nonlinear consensus for mas via doubly stochastic quadratic operators abstractthis technical note addresses the new nonlinear protocol class of doubly stochastic quadratic operators dsqos for coordination of consensus problem in multi agent systems mas we derive the conditions for ensuring that every agent reaches consensus on a desired rate of the group s decision where the group decision value in its agent s initial statuses varies besides that we investigate a nonlinear protocol sub class of extreme dsqo edsqo to reach a consensus for mas to a common value with nonlinear low complexity rules and fast time convergence if the interactions for each agent are not selfish in addition to extend the results to reach a consensus and to avoid the selfish case we specify a general class of dsqo for reaching a consensus under any given case of initial states the case that mas reach a consensus by dsqo is if each member of the agent group has positive interactions of dsqo pdsqo with the others the convergence of both edsqo and pdsqo classes is found to be directed tow

new bounds on the lieb thirring constants improved estimates on the constants l d for 1 2 3 2 d n in the inequalities for the eigenvalue moments of schrodinger operators are established

majorizations of hadamard products of matrix powers abstract this paper presents a number of intervening terms for the weak majorization i k n i a b i k n i ab for k 1 2 n where a b 0 more specifically it proves for r 0 1 that i k n i a b i k n i 1 r a r b r i k n e i l log a log b i k n e i log a log b i k n i 1 r a r b r i k n i ab for k 1 n it also shows that behind many of these inequalities are stronger ones

bounds for sums of eigenvalues and applications let a be a matrix of order n n with real spectrum 1 2 n let 1 k n 2 if n or 1 is known then we find an upper bound respectively lower bound for the sum of the k largest respectively k smallest remaining eigenvalues of a then we obtain a majorization vector for 1 2 n 1 when n is known and a majorization vector for 2 3 n when 1 is known we apply these results to the eigenvalues of the laplacian matrix of a graph and in particular a sufficient condition for a graph to be connected is given also we derive an upper bound for the coefficient of ergodicity of a nonnegative matrix with real spectrum

on sub and superadditive vector valued maps with applications to group majorization in this paper motivated by results in s s dragomir quasilinearity of some functionals associated with monotonic convex functions rgmia res rep coll 14 2011 artcle 24 13 pp and m niezgoda cone orderings group majorizations and similarly separable sequences linear algebra appl 436 3 2012 pp 579 594 we investigate maps involving quotient p p with an 1 subadditive operator and a positive superadditive functional we pay special attention to vector valued operators we study the subadditivity of the map p p f p p with 1 increasing convex f next we give applications for group majorization preorders we establish some preorder preservation results for the map p p p with a group majorization g and a combination 1 2 of preorders 1 and 2 in consequence we show that the function p f p p is g increasing

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

inequalities involving unitarily invariant norms and operator monotone functions let be a unitarily invariant norm on matrices for matrices a b x with a b positive semidefinite and x arbitrary we prove that the function t atxb1 t r a1 txbt r is convex on 0 1 for each r 0 this convexity result interpolates the matrix cauchy schwarz inequality a1 2xb1 2 r 2 ax r xb r due to r bhatia and c davis linear algebra appl 223 224 1995 119 and also it generalizes a w marshall and i olkin s pacific j math 15 1965 241 result that the condition number as a s is increasing in s 0 we prove that if f t is a nonnegative operator monotone function on 0 and is a normalized unitarily invariant norm then f x f x for every matrix x the special case when f t tr 0 r 1 is used to consider the monotonicity of p ap bp 1 p as well as p ap bp 1 p furthermore we obtain some norm inequalities of holder and minkowski types related to the expression a p b p 1 p for example comparisons are made between c a d b and a p b p 1 p c q d q 1 q where p 1 q 1 1

the biconjugate gradient method for electromagnetic scattering the biconjugate gradient bcg method for solving linear systems is shown to be more efficient than the conjugate gradient cg method for several examples from electromagnetic scattering a remedy for the occasional stagnation of the algorithm is proposed the potential flaw in the bcg algorithm may be avoided when encountered by restarting the algorithm with a perturbed estimate of the solution

classical and quantum probability we follow the development of probability theory from the beginning of the last century emphasizing that quantum theory is really a generalization of this theory the great achievements of probability theory such as the theory of processes generalized random fields estimation theory and information geometry are reviewed their quantum versions are then described

on q non extensive statistics with non tsallisian entropy we combine an axiomatics of renyi with the qq deformed version of khinchin axioms to obtain a measure of information i e entropy which accounts both for systems with embedded self similarity and non extensivity we show that the entropy thus obtained is uniquely solved in terms of a one parameter family of information measures the ensuing maximal entropy distribution is phrased in terms of a special function known as the lambert w function we analyze the corresponding high and low temperature asymptotics and reveal a non trivial structure of the parameter space salient issues such as concavity and schur concavity of the new entropy are also discussed

random unistochastic matrices an ensemble of random unistochastic orthostochastic matrices is defined by taking squared moduli of elements of random unitary orthogonal matrices distributed according to the haar measure on u n or o n an ensemble of symmetric unistochastic matrices is obtained with use of unitary symmetric matrices pertaining to the circular orthogonal ensemble we study the distribution of complex eigenvalues of bistochastic unistochastic and orthostochastic matrices in the complex plane we compute averages entropy traces over the ensembles of unistochastic matrices and present inequalities concerning the entropies of products of bistochastic matrices

majorization and singular values iii abstract we obtain a majorization inequality which relates the singular values of a complex square matrix a and those of t a where t is a linear map the inequality takes a particularly simple form if t is positive the results generalize several known majorization inequalities for eigenvalues and singular values

tsallis thermostatics as a statistical physics of random chains in this paper we point out that the generalized statistics of tsallis havrda charvat can be conveniently used as a conceptual framework for statistical treatment of random chains in particular we use the path integral approach to show that the ensuing partition function can be identified with the partition function of a fluctuating oriented random loop of arbitrary length and shape in a background scalar potential to put some meat on the bare bones we illustrate this with two statistical systems schultz zimm polymer and relativistic particle further salient issues such as the projective special linear group psl 2 r transformation properties of tsallis inverse temperature parameter and a grand canonical ensemble of fluctuating random loops related to the tsallis havrda charvat statistics are also briefly discussed

a note on the lyapunov equation some results on the sensitivity of the solution of the stable lyapunov equation are shown to follow easily from well known theorems of functional analysis

convergence of the conjugate gradient method when applied to matrix equations representing electromagnetic scattering problems an iterative procedure based on the conjugate gradient method is used to solve a variety of matrix equations representing electromagnetic scattering problems in an attempt to characterize the typical rate of convergence of that method it is found that this rate depends on the cell density per wavelength used in the discretization the presence of symmetries in the solution and the degree to which mixed cell sizes are used in the models assuming cell densities used in the discretization are in the range of ten per linear wavelength the iterative algorithm typically requires n 4 to n 2 steps to converge to necessary accuracy where n is the order of the matrix under consideration

observability of renyi s entropy despite recent claims we argue that renyi s entropy is an observable quantity it is shown that contrary to popular belief the reported domain of instability for renyi entropies has zero measure bhattacharyya measure in addition we show that the instabilities can be easily emended by introducing a coarse graining into an actual measurement we also clear up any doubts regarding the observability of renyi s entropy in multi fractal systems and in systems with absolutely continuous probability density functions

comments on on q non extensive statistics with non tsallisian entropy abstract recently in jizba and korbel 2016 four generalized shannon khinchin gsk axioms have been proposed and a generalized entropy which uniquely satisfies the gsk axioms has been derived in this comment we show that the unique class of the entropies derived in the aforementioned paper is not correct as it violates the fourth gsk axiom and we derive the correct one nevertheless the class of entropies proposed in the commented paper still can serve as a basis for generalized statistical mechanics we propose a new axiomatic system which characterizes the class of entropies

lanczos type variants of the cocr method for complex nonsymmetric linear systems motivated by the celebrated extending applications of the well established complex biconjugate gradient cbicg method to deal with large three dimensional electromagnetic scattering problems by pocock and walker m d pocock s p walker the complex bi conjugate gradient solver applied to large electromagnetic scattering problems computational costs and cost scalings ieee trans antennas propagat 45 1997 140 146 three lanczos type variants of the recent conjugate a orthogonal conjugate residual cocr method of sogabe and zhang t sogabe s l zhang a cocr method for solving complex symmetric linear systems j comput appl math 199 2007 297 303 are explored for the solution of complex nonsymmetric linear systems the first two can be respectively considered as mathematically equivalent but numerically improved popularizing versions of the bicr and crs methods for complex systems presented in sogabe s ph d dissertation and the last one is somewhat new and is a stabilized and more smoothly converging variant of the first two in some circumstances the presented algorithms are with the hope of obtaining smoother and hopefully faster convergence behavior in comparison with the cbicg method as well as its two corresponding variants this motivation is demonstrated by numerical experiments performed on some selective matrices borrowed from the university of florida sparse matrix collection by davis

log majorization and complementary golden thompson type inequalities abstract we obtain a log majorization result for power means of positive semidefinite matrices this implies matrix norm inequalities for unitarily invariant norms which are considered as complementary to the golden thompson one other log majorization results are also obtained we give logarithmic trace inequalities and determinant inequalities as applications of our log majorizations

linear preservers of majorization on lp i abstract in this paper we use doubly stochastic operators to extend the notion of majorization on l p where i is assumed to be an infinite set and p 1 and then characterize the structure of all bounded linear maps on this space which preserve majorization we will see that the two cases p 1 and p 1 lead to different structures

an analytical characterization of effective and of irreducible groups inducing cone orderings abstract an analytical characterization of effective and of irreducible groups inducing cone orderings is given the characterization implies easily a necessary condition for a group majorization to be a group induced cone ordering

linear preservers of circulant majorization on rn abstract let m n m be the set of all n m real matrices an n n real matrix a is called circulant doubly stochastic if it is a convex combination of circulant permutation matrices for x y r n x is said to be circulant majorized by y written as x c y if there exists a circulant doubly stochastic matrix d such that x d y in this paper the concept of circulant majorization is investigated and then the linear preservers and strong linear preservers of this concept are characterized on r n

free point processes and free extreme values we continue here the study of free extreme values begun in ben arous and voiculescu ann probab 34 2037 2059 2006 we study the convergence of the free point processes associated with free extreme values to a free poisson random measure voiculescu in lecture notes in mathematics springer heidelberg pp 279 349 1998 barndorff nielsen and thorbjornsen in probab theory relat fields 131 197 228 2005 we relate this convergence to the free extremal laws introduced in ben arous and voiculescu ann probab 34 2037 2059 2006 and give the limit laws for free order statistics

generalized shannon khinchin axioms and uniqueness theorem for pseudo additive entropies abstract we consider the shannon khinchin axiomatic systems for the characterization of generalized entropies such as the sharma mittal and frank daffertshofer entropies we provide the generalization of the shannon khinchin axioms and give the corresponding uniqueness theorem the previous attempts at such axiomatizations are also discussed

new entropy formula with fluctuating reservoir abstract finite heat reservoir capacity c and temperature fluctuation t t lead to modifications of the well known canonical exponential weight factor requiring that the corrections least depend on the one particle energy we derive a deformed entropy k s the resultingformula contains the boltzmann gibbs renyi and tsallis formulas as particular cases for extreme large fluctuations in the limit c t 2 t 2 a new parameter free entropy probability relation is gained the corresponding canonical energy distribution is nearly boltzmannian for high probability but for low probability approaches the cumulative gompertz distribution the latter is met in several phenomena like earthquakes demography tumor growth models extreme value probability etc

structure properties of a doubly stochastic process on a network abstract in this paper we study how special patterns affect certain dynamic process on networks the process we analyze is an iteration to generate a doubly stochastic matrix consistent to the adjacent matrix of a network and the patterns can be described as h non interconnected vertices only connect other g vertices h g from the perspective of network structure we prove that the necessary and sufficient condition when the iteration converges is that these patterns do not exist in the network for ba networks there is a phase transition the diverge converge transition point is that the average degree is about 8 which is theoretically proved the existence of these patterns depends on two factors first higher moments of degree distribution of the network second the probability that vertices with degree 1 exist in the network simulation results also support our theory

group majorization and schur type inequalities abstract the problem of the reduction of a group majorization induced by an infinite group to a finite group majorization is discussed a necessary and sufficient condition for a possibility of the reduction is given the condition generalizes some known inequalities in the matrix theory involving eigenvalues or singular values of a matrix a corresponding result for a skew symmetric matrix is derived

multidimensional spectral order for selfadjoint operators abstract the notion of the spectral order is extended to finite families of pairwise commuting bounded and unbounded selfadjoint operators in hilbert space it is shown that the multidimensional spectral order is preserved by transformations represented by spectral integrals of separately increasing borel functions on r in particular the dimensional spectral order is the restriction of product of spectral orders for selfadjoint operators in the context of positive tuples of pairwise commuting selfadjoint operator the relation a b holds if and only if a b for every tuples of nonnegative integer numbers

quantum state determination estimates for information gain and some exact calculations it is known that mutually unbiased bases mubs whenever they exist are optimal in an information theoretic sense for the determination of the unknown state of a quantum ensemble such bases may not exist for most dimensions the present paper deals with information gain in some generalizations and approximations of mubs we give estimates of the information loss relative to mubs in these suboptimal choice of bases for some generalization of mubs we give exact calculations for the information gain it is calculated directly in terms of transition probabilities among the measurement bases we also give the formal solutions for the problem of quantum state tomography in these cases 1

eigenvalue equalities for ordinary and hadamard products of powers of positive semidefinite matrices abstract let a and b be n n positive semidefinite matrices and 0 let a b denote the hadamard product of a and b and a l denote the leading l l principal submatrix of a let 1 x n x denote the eigenvalues of an n n matrix x ordered when they are all real in this paper those matrices that satisfy any of the following equalities are determined i 1 a b i 1 a b i 1 n i 1 a l i 1 a l i 1 l i 1 a b i 1 a b i 1 n the results are extended to equalities involving more than one eigenvalue as an application for any 1 k n those a and b that satisfy i 1 k n i 1 a b i 1 k n i 1 a b where b b or b t the transpose of b are also determined

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

hadamard products and golden thompson type inequalities abstract by using hadamard products we give some reasonable upper and lower bounds of golden thompson type for e h 1 h m where h i i 1 2 m are arbitrary hermitian matrices and is an arbitrary unitarily invariant norm

a generalization of araki s log majorization abstract we generalize araki s log majorization to the log convexity theorem for the eigenvalues of a p 1 2 b p a p 1 2 as a function of p 0 where a b are positive semidefinite matrices and are positive linear maps between matrix algebras similar generalizations of the log majorization of ando hiai type for the weighted geometric mean a b are also given including a lemma on general operator means a e of a and a projection e

exact time dependent solutions of the renyi fokker planck equation and the fokker planck equations related to the entropies proposed by sharma and mittal abstract exact time dependent solutions representing generalized one dimensional ornstein uhlenbeck processes are derived from nonlinear fokker planck equations and the corresponding diffusion processes are studied the corresponding systems are related to the renyi entropy and entropies proposed by sharma and mittal in the former case normal diffusion is found with a diffusion rate depending on the order of the renyi entropy in the latter case we find anomalous diffusion processes

approximating the permanent via importance sampling with application to the dimer covering problem we estimate the asymptotic growth rate of the number of dimer covers of a cubic lattice our estimate 3 0 4466 0 0006 is consistent with the lower bounds obtained by hammersley and later schrijver and the more recent improved upper bound obtained by ciucu obtaining this estimate is an important step toward approximating the partition function of the cubic monomer dimer system from the partition function all of the standard thermodynamic quantities can be evaluated it is well known that computing 3 is equivalent to computing the permanent of a certain 0 1 matrix we describe an extremely efficient monte carlo algorithm for approximating the permanent previous work on monte carlo approaches includes the pioneering results of jerrum and sinclair who use a rapidly mixing random walk our method is inspired by results of soules on convergence of sinkhorn balancing to obtain a maximum entropy doubly stochastic matrix we use the sinkhorn balanced matrix to generate an importance function that allows us to do direct random sampling rather than a random walk that converges to a limiting distribution

monomer dimer problem on random planar honeycomb lattice we consider the monomer dimer md problem on a random planar honeycomb lattice model namely the random multiple chain this is a lattice system with non periodic boundary condition whose generating process is inspired by the growth of single walled zigzag carbon nanotubes by applying algebraic and combinatorial techniques we establish a calculating expression of the md partition function for bipartite graphs which corresponds to the permanent of a matrix further by using the transfer matrix argument we show that the computing problem of the permanent of high order matrix can be converted into some lower order matrices for this family of lattices based on which we derive an explicit recurrence formula for evaluating the md partition function of multiple chains and random multiple chains finally we analyze the expectation of the number of monomer dimer arrangements on a random multiple chain and the asymptotic behavior of the annealed md entropy when the multiple chain becomes infinite in width and length respectively

a simple probabilistic construction yielding generalized entropies and divergences escort distributions and q gaussians abstract we give a simple probabilistic description of a transition between two states which leads to a generalized escort distribution when the parameter of the distribution varies it defines a parametric curve that we call an escort path the renyi divergence appears as a natural by product of the setting we study the dynamics of the fisher information on this path and show in particular that the thermodynamic divergence is proportional to jeffreys divergence next we consider the problem of inferring a distribution on the escort path subject to generalized moment constraints we show that our setting naturally induces a rationale for the minimization of the renyi information divergence then we derive the optimum distribution as a generalized q gaussian distribution

convex hulls of unitary orbits of normal elements in c algebras with tracial rank zero abstract let a be a unital separable simple c algebra with tracial rank zero and let x y a be two normal elements we show that x is in the closure of the convex hull of the unitary orbit of y if and only if there exists a sequence of unital completely positive linear maps n from a to a such that the sequence n y converges to x in norm and also approximately preserves the trace values a purely measure theoretical description for normal elements in the closure of the convex hull of unitary orbit of y is also given in the case that a has a unique tracial state some classical results about the closure of the convex hull of the unitary orbits in von neumann algebras are proved to hold in the c algebraic setting

linear maps preserving group majorization abstract a necessary and sufficient condition for a linear map to preserve group majorizations is given the condition is applied to prove some preservation results

inequalities for monotonic arrangements of eigenvalues abstract this article corrects clarifies and extends results in 5 on inequalities for sequence rearrangements and for eigenvalues the prototypes for these results are the inequalities of hardy littlewood and polya about monotonic rearrangements we examine some analogous results for eigenvalues of matrices and of their products

free extreme values free probability analogues of the basics of extreme value theory are obtained based on ando s spectral order this includes clas sification of freely max stable laws and their domains of attraction using free extremal convolutions on the distributions these laws coincide with the limit laws in the classical peaks over threshold approach a free extremal projection valued process over a measure space is constructed which is related to the free poisson point process

possible generalization of boltzmann gibbs statistics with the use of a quantity normally scaled in multifractals a generalized form is postulated for entropy namelys q k 1 i 1 w p i q q 1 whereq characterizes the generalization andp i are the probabilities associated withw microscopic configurations w the main properties associated with this entropy are established particularly those corresponding to the microcanonical and canonical ensembles the boltzmann gibbs statistics is recovered as theq 1 limit

vigier s theorem for the spectral order and its applications abstract the paper mainly deals with suprema and infima of self adjoint operators in a von neumann algebra m with respect to the spectral order let m s a be the self adjoint part of m and let be the spectral order on m s a we show that a decreasing net in m s a with a lower bound has the infimum equal to the strong operator limit the similar statement is proved for an increasing net bounded above in m s a this version of vigier s theorem for the spectral order is used to describe suprema and infima of nonempty bounded sets of self adjoint operators in terms of the strong operator limit and operator means as an application of our results on suprema and infima we study the order topology on m s a with respect to the spectral order we show that it is finer than the restriction of the mackey topology

some eigenvalue inequalities for positive semidefinite matrix power products abstract we study the eigenvalues of positive semidefinite matrix power products and obtain some inequalities most of which are in terms of majorization in particular for a b 0 0 we prove log 1 a b log 1 a b the result is a generalization of some work of marcus lieb thirring le couteur bushell and trustrum

differentiable transformations preserving a cone preordering abstract a necessary and sufficient condition for a differentiable transformation to be isotone w r t cone preorderings on a convex set is given the result is applied to characterize monotone matrix functions a a p with integer p 2 and isotone differentiable transformations w r t g majorizations

an inequality for partition functions with disturbed hamiltonians we consider a thermodynamic system consisting ofn independent subsystems each subsystem is described by a hamiltonianh h 0 i h 1 i 1 2 n we answer the question of how the entirety 1 2 n must be varied in order to change the total partition function and the total free energy of the system monotonically

inequalities theory of majorization and its applications by albert w marshall and ingram olkin abstract in the years since the appearance of marshall and olkin s masterful survey the influence of this work has been so great that some comments however incomplete and idiosyncratic on subsequent developments will be of value figure 1 an appendix lists minor errata to the first printing of the book

weak majorization and doubly substochastic operators on lp i abstract we extend the notion of weak majorization on the banach space l p i where i is a non empty set and p 1 using doubly substochastic operators on l p i the close relationship between weak majorization and these operators are presented among others we show that f w g and g w f iff there is a partial permutation p such that g p f using the specific type of positive increasing convex functions a necessary and sufficient condition for f w g and g w f is given

on statistical properties of jizba arimitsu hybrid entropy abstract jizba arimitsu entropy also called hybrid entropy combines axiomatics of renyi and tsallis entropy it has many common properties with them on the other hand some aspects as e g maxent distributions are different in this paper we discuss statistical properties of hybrid entropy we define hybrid entropy for continuous distributions and its relation to discrete entropy additionally definition of hybrid divergence and its connection to fisher metric is also presented interestingly fisher metric connected to hybrid entropy differs from corresponding fisher metrics of renyi and tsallis entropy this motivates us to introduce average hybrid entropy which can be understood as an average between tsallis and renyi entropy

row and column majorization on mn m abstract let m n m be the set of all n m matrices with entries in r for a b m n m it is said that a is row majorized respectively column majorized by b if every row respectively column of a is majorized by the corresponding row respectively column of b i e for every i 1 i n there exists a doubly stochastic matrix d i such that a i b i d i where a i and b i are the i th rows of a and b respectively in this paper the relations row and column majorization on m n m are studied and also all linear operators t m n m m n m preserving or strongly preserving row or column majorization will be characterized

schur convexity and schur geometrically concavity of gini means the schur convexity and the schur geometric convexity with variables x y r 2 for fixed s t of gini means g r s x y are discussed some new inequalities are obtained

weak supermajorization and families as doubly superstochastic operators on lp i abstract we present necessary and sufficient conditions that a family a a i j i j i of real numbers may be considered as a bounded linear operator on banach spaces l 1 i and l i where i is an arbitrary non empty set moreover we get that these conditions are sufficient for a family to be a bounded linear operator on l p i for each p 1 within this class of operators the notion of doubly superstochastic operator is introduced as an extension of the doubly superstochastic matrix and some of its essentially properties are presented in the second part we extend the notion of weak supermajorization relation on the banach space l p i using doubly superstochastic operators and present close relationship between this relation and superstochastic operators as generalisation well known results in the theory of majorization among others for two functions f g l 1 i we show that relations f w s g and g w s f hold if and only if there exist a permutation p such that g p f

a transpose free quasi minimal residual variant of the cors method for solving non hermitian linear systems recently the conjugate a orthogonal residual squared cors method is often competitive and superior to other krylov subspace methods for solving complex non hermitian linear systems from many realistic problems however like the conjugate gradient squared cgs method the cors method often exhibits irregular convergence behavior with wild oscillations in the residual norm to try and remedy this problem we develop a transpose free quasi minimal residual tfqmr variant of the cors method which leads to smooth convergence curves with a convergence rate similar to the cors method we also give convergence results for this new method finally numerical examples are reported to illustrate the effectiveness of the proposed method

the complex bi conjugate gradient solver applied to large electromagnetic scattering problems computational costs and cost scalings the complex bi conjugate gradient iterative method is applied to an isoparametric boundary integral equation formulation for frequency domain electromagnetic scattering problems it is demonstrated to work well on large and geometrically complex examples including a 20 wavelength slender dipole the nasa almond and a resonant cavity on such problems with asymmetric curvilinear irregular meshes and nontrivial geometries the number of iterations required seems to increase rather more than linearly with body size indicating an overall spl sim sixth power cost scaling this scaling is essentially as for direct methods but with costs still a small fraction of the direct approach a method is proposed for the selection of a termination condition designed to avoid seeking the approximate answer too precisely it typically permits a further halving of costs