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

majorization and a schur horn theorem for positive compact operators the nonzero kernel case abstract schur horn theorems focus on determining the diagonal sequences obtainable for an operator under all possible basis changes formally described as the range of the canonical conditional expectation of its unitary orbit following a brief background survey we prove an infinite dimensional schur horn theorem for positive compact operators with infinite dimensional kernel one of the two open cases posed recently by kaftal weiss there they characterized the diagonals of operators in the unitary orbits for finite rank or zero kernel positive compact operators here we show how the characterization problem depends on the dimension of the kernel when it is finite or infinite dimensional we obtain exact majorization characterizations of the range of the canonical conditional expectation of the unitary orbits of positive compact operators with infinite dimensional kernel unlike the approximate characterizations of arveson kadison but extending the exact characterizations of gohberg markus and kaftal weiss recent advances in this subject and related subjects like traces on ideals show the relevance of new kinds of sequence majorization as in the work of kaftal weiss e g strong majorization and another majorization similar to what here we call p majorization and of kalton sukochev e g uniform hardy littlewood majorization and of bownik jasper e g riemann and lebesgue majorization likewise key tools here are new kinds of majorization which we call p and approximate p majorization 0 p

spectral orders and isotone functionals it is shown that the spectral orders in lp are induced by some cones the result provides sufficient conditions of the schur ostrowski type for isotonicity of differentiable functionals w r t the orders 0 elsevier science inc 1997

counterexamples concerning the diagonal elements of normal matrices abstract an infinite class of conditions known to be satisfied by the diagonal elements of a normal matrix with prescribed spectrum is shown to be independent of other known conditions satisfied by the diagonal elements but nevertheless insufficient to characterize the diagonal

diagonals of normal operators with finite spectrum let x lambda1 lambdan be a finite set of complex numbers and let a be a normal operator with spectrum x that acts on a separable hilbert space h relative to a fixed orthonormal basis e1 e2 for h a gives rise to a matrix whose diagonal is a sequence d d1 d2 with the property that each of its terms dn belongs to the convex hull of x not all sequences with that property can arise as the diagonal of a normal operator with spectrum x the case where x is a set of real numbers has received a great deal of attention over the years and is reasonably well though incompletely understood in this work we take up the case in which x is the set of vertices of a convex polygon in the critical sequences d turn out to be those that accumulate rapidly in x in the sense that sigmainfinityn 1 dist dn x s d e gammax with the following property if s d not equal 0 then d is not the diagonal of any such operator a we also show that while this is the only obstruction when n 2 there are other as yet unknown obstructions when n 3

multivariable schur horn theorems fil massey pedro gustavo universidad nacional de la plata facultad de ciencias exactas departamento de matematicas argentina consejo nacional de investigaciones cientificas y tecnicas oficina de coordinacion administrativa saavedra 15 instituto argentino de matematica alberto calderon argentina

weak matrix majorization abstract given x y r n m we introduce the following notion of matrix majorization called weak matrix majorization x w y if there exists a row stochastic matrix a r n n such that ax y and consider the relations between this concept strong majorization s and directional majorization it is verified that s w but none of the reciprocal implications is true nevertheless we study the implications w s and s under additional hypotheses we give characterizations of strong directional and weak matrix majorization in terms of convexity we also introduce definitions for majorization between abelian families of selfadjoint matrices called joint majorizations they are induced by the previously mentioned matrix majorizations we obtain descriptions of these relations using convexity arguments

an extension of inequalities by ando abstract we give variations on ando s result comparing f b f a and f b a with respect to unitarily invariant norms on matrices

an infinite dimensional schur horn theorem and majorization theory the main result of this paper is the extension of the schur horn theorem to infinite sequences for two nonincreasing nonsummable sequences and that converge to 0 there exists a positive compact operator a with eigenvalue list and diagonal sequence if and only if j 1n j j 1n j for every n if and only if q for some orthostochastic matrix q when and are summable requiring additionally equality of their infinite series obtains the same conclusion extending a theorem by arveson and kadison our proof depends on the construction and analysis of an infinite product of t transform matrices

an infinite dimensional version of the schur horn convexity theorem the schur horn convexity theorem states that forain rnp u diag a u u u n conv sna wherepdenotes the projection on the diagonal in this paper we generalize this result to the setting of arbitrary separable hilbert spaces it turns out that the theorem still holds if we take thel closure on both sides we will also give a description of the left hand side for nondiagonalizable hermitian operators in the last section we use this result to get an extension theorem for invariant closed convex subsets of the diagonal operators

simulated annealing searching for an optimal temperature schedule a sizable part of the theoretical literature on simulated annealing deals with a property called convergence which asserts that the simulated annealing chain is in the set of global minimum states of the objective function with probability tending to 1 however in practice the convergent algorithms are considered too slow whereas a number of nonconvergent ones are usually preferred we attempt a detailed analysis of various temperature schedules examples will be given of when it is both practically and theoretically justified to use boiling fixed temperature or even fast cooling schedules which have a small probability of reaching global minima applications to traveling salesman problems of various sizes are also given

threshold accepting a general purpose optimization algorithm appearing superior to simulated annealing abstract a new general purpose algorithm for the solution of combinatorial optimization problems is presented the new threshold accepting method is even simpler structured than the wellknown simulated annealing approach the power of the new algorithm is demonstrated by computational results concerning the traveling salesman problem and the problem of the construction of error correcting codes moreover deterministic versions of the new heuristic turn out to perform nearly equally well consuming only a fraction of the computing time of the stochastic versions as an example the deterministic threshold accepting method yields very near to optimum tours for the famous 442 cities traveling salesman problem of grotschel within 1 to 2 s of cpu time

state convertibility in the von neumann algebra framework we establish a generalisation of the fundamental state convertibility theorem in quantum information to the context of bipartite quantum systems modelled by commuting semi finite von neumann algebras namely we establish a generalisation to this setting of nielsen s theorem on the convertibility of quantum states under local operations and classical communication locc schemes along the way we introduce an appropriate generalisation of locc operations and connect the resulting notion of approximate convertibility to the theory of singular numbers and majorisation in von neumann algebras as an application of our result in the setting of mathrm ii 1 factors we show that the entropy of the singular value distribution relative to the unique tracial state is an entanglement monotone in the sense of vidal thus yielding a new way to quantify entanglement in that context building on previous work in the infinite dimensional setting we show that trace vectors play the role of maximally entangled states for general mathrm ii 1 factors examples are drawn from infinite spin chains quasi free representations of the car and discretised versions of the ccr

the pythagorean theorem i the finite case the pythagorean theorem and variants of it are studied the variations evolve to a formulation in terms of noncommutative conditional expectations on von neumann algebras that displays the theorem as the basic result of noncommutative metric euclidean geometry the emphasis in the present article is finite dimensionality both discrete and continuous

the pythagorean theorem ii the infinite discrete case the study of the pythagorean theorem and variants of it as the basic result of noncommutative metric euclidean geometry is continued the emphasis in the present article is the case of infinite discrete dimensionality

optimization by simulated annealing an experimental evaluation part ii graph coloring and number partitioning this is the second in a series of three papers that empirically examine the competitiveness of simulated annealing in certain well studied domains of combinatorial optimization simulated annealing is a randomized technique proposed by s kirkpatrick c d gelatt and m p vecchi for improving local optimization algorithms here we report on experiments at adapting simulated annealing to graph coloring and number partitioning two problems for which local optimization had not previously been thought suitable for graph coloring we report on three simulated annealing schemes all of which can dominate traditional techniques for certain types of graphs at least when large amounts of computing time are available for number partitioning simulated annealing is not competitive with the differencing algorithm of n karmarkar and r m karp except on relatively small instances moreover if running time is taken into account natural annealing schemes cannot even outperform multiple random runs of the local optimization algorithms on which they are based in sharp contrast to the observed performance of annealing on other problems

new optimization heuristics in a former paper we introduced a very effective new general purpose optimization principle we compared this method which we called threshold accepting ta with the well known simulated annealing sa method for discrete optimization the empirical results demonstrated the superiority of the ta algorithm in further experiments with the ta principle we discovered two new powerful optimization heuristics the great deluge algorithm gda and the record to record travel rrt these algorithms resemble in their structure the formerly studied ta and the sa method the differences lie in their acceptance rules for worse intermediate solutions both the gda and the rrt are essentially one parameter algorithms i e for the achievement of best possible performance a good choice is necessary only for a single parameter this is in contrast for instance to the classical as algorithm where it is necessary to choose carefully a certain sequence of parameters the so called annealing schedule the quality of the computational results obtained so far by rrt and gda shows that the new algorithms behave equally well as ta and thus a fortiori better than sa

the cauchy problem for the homogeneous monge ampere equation i toeplitz quantization the cauchy problem for the homogeneous real complex monge ampere equation hrma hcma arises from the initial value problem for geodesics in the space of kahler metrics equipped with the mabuchi metric this cauchy problem is believed to be ill posed and a basic problem is to characterize initial data of weak solutions which exist up to time t in this article we use a quantization method to construct a subsolution of the hcma on a general projective variety and we conjecture that it solves the equation for as long as the unique solution exists the subsolution called the quantum analytic continuation potential is obtained by i toeplitz quantizing the hamiltonian flow determined by the cauchy data ii analytically continuing the quantization and iii taking a certain logarithmic classical limit we then prove that in the case of torus invariant metrics where the hcma reduces to the hrma the quantum analytic continuation potential coincides with the well known legendre transform potential and hence solves the equation as long as it is smooth in the sequel the cauchy problem for the homogeneous monge ampere equation ii legendre transform adv math 228 2011 2989 3025 it is proved that the legendre transform potential ceases to solve the hcma once it ceases to be smooth the results here and in the sequels in particular characterize the initial data of smooth geodesic rays

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

differential geometry of real monge ampere foliations on etudie les feuilletages qui proviennent de la condition de monge ampere recquerant le hessien d une fonction definie sur une variete affine d avoir un rang inferieur au maximal

the cauchy problem for the homogeneous monge ampere equation ii legendre transform abstract we continue our study of the cauchy problem for the homogeneous real and complex monge ampere equation hrma hcma in the prequel y a rubinstein and s zelditch 27 a quantum mechanical approach for solving the hcma was developed and was shown to coincide with the well known legendre transform approach in the case of the hrma in this article that uses tools of convex analysis and can be read independently we prove that the candidate solution produced by these methods ceases to solve the hrma even in a weak sense as soon as it ceases to be differentiable at the same time we show that it does solve the equation on its dense regular locus and we derive an explicit a priori upper bound on its monge ampere mass the technique involves studying regularity of legendre transforms of families of non convex functions

convexity properties of the moment mapping ii be its associated momen t mapping see w for definitions the set x is the union of co adjoint orbits the main result of this paper is a description of the orbit structure of this set to describe this result we first note that the coadjoint orbits in q can be parametr ized as follows let t be a caf tan subalgebra of g and let b be a positive definite g invariant bilinear form on by means of b we get a g equivariant identification q let t be the subspace of q corresponding to t then every co adjoint orbit intersects i in a weyl group orbit so there is a one one correspondence between co adjoint orbits in q and weyl group orbits in t to get an even sharper parametrization let t be a weyl chamber in then every weyl group orbit intersects t in a single point so t is a modul i space for the co adjoint orbits two orbits in x are said to be of the same type if they are isomorphic as homogeneous g spaces since x is compact there are at most a finite number of different types of orbits see 1113 let h be a closed subgroup of t of codimension r and let cit be the set of all orbits in x of type g h

geodesic convexity of small neighborhood in the space of kahler potentials abstract we show that given k 4 0 j min 1 4 k 4 4 any point in the space of non degenerate smooth kahler potentials has a small neighborhood with respect to c k norm s t any two points in this neighborhood can be connected by a geodesic of at least c k j regularity

noncommutative kothe duality using techniques drawn from the classical theory of rearrangement invariant banach function spaces we develop a duality theory in the sense of kothe for symmetric banach spaces of measurable operators affiliated with a semifinite von neumann algebra equipped with a distinguished trace a principal result of the paper is the identification of the kothe dual of a given banach space of measurable operators in terms of normality

extreme points of the set of elements majorised by an integrable function resolution of a problem by luxemburg and of its noncommutative counterpart abstract let f be an arbitrary integrable function on a finite measure space x we characterise the extreme points of the set f of all measurable functions on x majorised by f providing a complete answer to a problem raised by w a j luxemburg in 1967 moreover we obtain a noncommutative version of this result

the schur horn theorem for operators with three point spectrum abstract we characterize the set of diagonals of the unitary orbit of a self adjoint operator with three points in the spectrum our result gives a schur horn theorem for operators with three point spectrum analogous to kadison s pythagorean theorem and carpenter s theorem which characterize the diagonals of orthogonal projections

the late acceptance hill climbing heuristic abstract this paper introduces a new and very simple search methodology called late acceptance hill climbing lahc it is a local search algorithm which accepts non improving moves when a candidate cost function is better than it was a number of iterations before this number appears as a single algorithmic input parameter which determines the total processing time of the search procedure the properties of this method are experimentally studied in this paper with a range of travelling salesman and exam timetabling benchmark problems also we present a fair comparison of lahc with well known search techniques which employ a cooling schedule simulated annealing sa threshold accepting ta and the great deluge algorithm gda in addition we discuss the success of the method in winning an international competition to automatically solve the magic square problem our experiments have shown that the lahc approach is simple easy to implement and yet is an effective search procedure for most of the studied instances especially for the large sized ones its average performance is better than competitor methods moreover the lahc approach has an additional advantage in contrast to the above cooling schedule based methods in its scale independence we present an example where the rescaling of a cost function in the travelling salesman problem dramatically deteriorates the performance of three cooling schedule based techniques but has absolutely no influence upon the performance of lahc

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

frames of translates with prescribed fine structure in shift invariant spaces abstract for a given finitely generated shift invariant fsi subspace w l 2 r k we obtain a simple criterion for the existence of shift generated sg bessel sequences e f induced by finite sequences of vectors f w n that have a prescribed fine structure i e such that the norms of the vectors in f and the spectra of s e f are prescribed in each fiber of spec w t k we complement this result by developing an analogue of the so called sequences of eigensteps from finite frame theory in the context of sg bessel sequences that allows for a detailed description of all sequences with prescribed fine structure then given 1 n 0 we characterize the finite sequences f w n such that f i 2 i for 1 i n and such that the fine spectral structure of the shift generated bessel sequences e f has minimal spread i e we show the existence of optimal sg bessel sequences with prescribed norms in this context the spread of the spectra is measured in terms of the convex potential p w induced by w and an arbitrary convex function r r

an l p regularity theory for harmonic maps motivated by the harmonic map heat flow we consider maps between riemannian manifolds such that the tension field belongs to an lp space under an appropriate smallness condition a certain degree of regularity follows for suitable solutions of the harmonic map heat flow we have a partial regularity result as a consequence