the lee yang and polya schur programs i linear operators preserving stability in 1952 lee and yang proposed the program of analyzing phase transitions in terms of zeros of partition functions linear operators preserving non vanishing properties are essential in this program and various contexts in complex analysis probability theory combinatorics and matrix theory we characterize all linear operators on finite or infinite dimensional spaces of multivariate polynomials preserving the property of being non vanishing whenever the variables are in prescribed open circular domains in particular this solves the higher dimensional counterpart of a long standing classification problem originating from classical works of hermite laguerre hurwitz and polya schur on univariate polynomials with such properties

linear matrix inequality representation of sets this article concerns the question which subsets of m can be represented with linear matrix inequalities lmis this gives some perspective on the scope and limitations of one of the most powerful techniques commonly used in control theory also before having much hope of representing engineering problems as lmis by automatic methods one needs a good idea of which problems can and cannot be represented by lmis little is currently known about such problems in this article we give a necessary condition that we call rigid convexity which must hold for a set m in order for to have an lmi representation rigid convexity is proved to be necessary and sufficient when m 2 this settles a question formally stated by pablo parrilo and berndt sturmfels in 15 as shown by lewis parillo and ramana 11 our main result also establishes in the case of three variables a 1958 conjecture by peter lax on hyperbolic polynomials 2006 wiley periodicals inc

on a diffusive logistic equation we study a reaction diffusion version on all of rnof the logistic equation of population growth in which the birth rate depends on the spatial variable and may assume both positive and negative values our results which are obtained by the construction of sub and supersolutions and the study of asymptotic properties of solutions show the interplay between the birth rate of the species and the extent of diffusion in determining the existence or nonexistence of nontrivial steady state distributions of population

approximation of the average of some random matrices abstract rudelson s theorem states that if for a set of unit vectors u i and positive weights c i we have that c i u i u i is the identity operator i on r d then the sum of a random sample of c d ln d of these diadic products is close to i the ln d term cannot be removed on the other hand the recent fundamental result of batson spielman and srivastava and its improvement by marcus spielman and srivastava show that the ln d term can be removed if one wants to show the existence of a good approximation of i as the average of a few diadic products it is known that essentially the same proof as rudelson s yields a more general statement about the average of positive semi definite matrices first we give an example of an average of positive semi definite matrices where there is no approximation of this average by cd elements thus the result of batson spielman and srivastava cannot be extended to this wider class of matrices next we present a stability version of rudelson s result on positive semi definite matrices and thus extend it to certain non symmetric matrices this yields applications to the study of the banach mazur distance of convex bodies finally we show that in some cases one needs to take a subset of the vectors of order d 2 to approximate the identity

a non diagonalizable pure state we construct a pure state on the c algebra formula see text of all bounded linear operators on formula see text which is not diagonalizable i e it is not of the form formula see text for any orthonormal basis formula see text of formula see text and an ultrafilter u on n this constitutes a counterexample to anderson s conjecture without additional hypothesis and improves results of c akemann n weaver i farah and i smythe who constructed such states making additional set theoretic assumptions it follows from results of j anderson and the positive solution to the kadison singer problem due to a marcus d spielman and n srivastava that the restriction of our pure state to any atomic masa formula see text of diagonal operators with respect to an orthonormal basis formula see text is not multiplicative on formula see text

dilations of operator valued measures with bounded p variations and framings on banach spaces abstract the dilations for operator valued measures ovms and bounded linear maps indicate that the dilation theory is in general heavily dependent on the banach space nature of the dilation spaces this naturally led to many questions concerning special type of dilations in particular it is not known whether ultraweakly continuous normal maps can be dilated to ultraweakly continuous homomorphisms we answer this question affirmatively for the case when the domain algebra is an abelian von neumann algebra it is well known that completely bounded hilbert space operator valued measures correspond to the existence of orthogonal projection valued dilations in the sense of naimark and stinespring and ovms with bounded total variations are completely bounded but not the vice versa with the aim of classifying ovms from the dilation point of view we introduce the concept of total p variations for ovms we prove that any completely bounded ovm has finite 2 variation and any ovm with finite p variation can be dilated to a but usually non hilbertian projection valued measure of the same type with the help of framing induced ovms we prove that conventional minimal dilation space of a non trivial framing contains c 0 then does not have bounded p variation

hyperbolic polynomials interlacers and sums of squares hyperbolic polynomials are real polynomials whose real hypersurfaces are maximally nested ovaloids the innermost of which is convex these polynomials appear in many areas of mathematics including optimization combinatorics and differential equations here we investigate the special connection between a hyperbolic polynomial and the set of polynomials that interlace it this set of interlacers is a convex cone which we write as a linear slice of the cone of nonnegative polynomials in particular this allows us to realize any hyperbolicity cone as a slice of the cone of nonnegative polynomials using a sums of squares relaxation we then approximate a hyperbolicity cone by the projection of a spectrahedron a multiaffine example coming from the vamos matroid shows that this relaxation is not always exact using this theory we characterize the real stable multiaffine polynomials that have a definite determinantal representation and construct one when it exists

the lee yang and polya schur programs ii theory of stable polynomials and applications in the first part of this series we characterized all linear operators on spaces of multivariate polynomials preserving the property of being nonvanishing in products of open circular domains for such sets this completes the multivariate generalization of the classification program initiated by polya and schur for univariate real polynomials we build on these classification theorems to develop here a theory of multivariate stable polynomials applications and examples show that this theory provides a natural framework for dealing in a uniform way with lee yang type problems in statistical mechanics combinatorics and geometric function theory in one or several variables in particular we answer a question of hinkkanen on multivariate apolarity

applications of stable polynomials to mixed determinants johnson s conjectures unimodality and symmetrized fischer products for n x n matrices a and b define eta a b sigma s det a s det b s where the summation is over all subsets of 1 n s is the complement of s and a s is the principal submatrix

iterative actions of normal operators abstract let a be a normal operator in a hilbert space h and let g h be a countable set of vectors we investigate the relations between a g and l that make the system of iterations a n g g g 0 n l g complete bessel a basis or a frame for h the problem is motivated by the dynamical sampling problem and is connected to several topics in functional analysis including frame theory and spectral theory it also has relations to topics in applied harmonic analysis including wavelet theory and time frequency analysis

multivariate polya schur classification problems in the weyl algebra a multivariate polynomial is stable if it is nonvanishing whenever all variables have positive imaginary parts we classify all linear partial differential operators in the weyl algebra a n that preserve stability an important tool that we develop in the process is the higher dimensional generalization of polya schur s notion of multiplier sequence we characterize all multivariate multiplier sequences as well as those of finite order next we establish a multivariate extension of the cauchy poincare interlacing theorem and prove a natural analog of the lax conjecture for real stable polynomials in two variables using the latter we describe all operators in a 1 that preserve univariate hyperbolic polynomials by means of determinants and homogenized symbols our methods also yield homotopical properties for symbols of linear stability preservers and a duality theorem showing that an operator in a n preserves stability if and only if its fischer fock adjoint does these are powerful multivariate extensions of the classical hermite poulain jensen theorem polya s curve theorem and schur malo szegocomposition theorems examples and applications to strict stability preservers are also discussed

extracting a basis with fixed block inside a matrix abstract given u an n m matrix of rank n whose columns are denoted by u j j m several authors have already considered the problem of finding a subset 1 m such that u i i span r n and tr i u i u i t 1 is minimized in this paper we generalize this problem by selecting arbitrary rank matrices instead of rank 1 matrices another generalization is considering the same problem while allowing a part of the matrix to be fixed the methods of selection employed develop into algorithms

interlacing families ii mixed characteristic polynomials and the kadison singer problem we use the method of interlacing polynomials introduced in our previous article to prove two theorems known to imply a positive solution to the kadison singer problem the rst is weaver s conjecture ks 2 which is known to imply kadison singer via a projection paving conjecture of akemann and anderson the second is a formulation due to casazza et al of anderson s original paving conjecture s for which we are able to compute explicit paving bounds the proof involves an analysis of the largest roots of a family of polynomials that we call the mixed characteristic polynomials of a collection of matrices

mixed discriminants of positive semidefinite matrices abstract if ak akij k 1 2 n are n by n matrices then their mixed discriminant d a1 an is given by d a 1 a n 1 n s n a j ij where sn is the symmetric group of degree n and where denotes determinant we give certain alternative ways of defining the mixed discriminant and state some basic properties it is pointed out that a ryser type formula for the mixed discriminant exists in the literature and a simpler proof is given for it it is shown that the mixed discriminant can be expressed as an inner product a generalization of konig s theorem on 0 1 matrices is proved the following set d n which includes the set of n by n doubly stochastic matrices is defined and studied

twice ramanujan sparsifiers we prove that every graph has a spectral sparsifier with a number of edges linear in its number of vertices as linear sized spectral sparsifiers of complete graphs are expanders our sparsifiers of arbitrary graphs can be viewed as generalizations of expander graphs in particular we prove that for every d 1 and every undirected weighted graph g v e w on n vertices there exists a weighted graph h v f tilde w with at most lceil d n 1 rceil edges such that for every x in mathbb r v x t l g x leq x t l h x leq bigl frac d 1 2 sqrt d d 1 2 sqrt d bigr cdot x t l g x where l g and l h are the laplacian matrices of g and h respectively thus h approximates g spectrally at least as well as a ramanujan expander with dn 2 edges approximates the complete graph we give an elementary deterministic polynomial time algorithm for constructing h

the discretization problem for continuous frames abstract we characterize when a coherent state or continuous frame for a hilbert space may be sampled to obtain a frame which solves the discretization problem for continuous frames in particular we prove that every bounded continuous frame for a hilbert space may be sampled to obtain a frame we give multiple applications to different classes of frames such as scalable frames and gabor frames

random vectors in the isotropic position abstract letybe a random vector in r n satisfying e y y id letmbe a natural number and lety1 ymbe independent copies ofy we study the question of approximation of the identity operator by finite sums of the tensorsyi yi we prove that for some absolute constantc e 1 m i 1 m y i y i id c log n m e y log m 1 log m provided that the last expression is smaller than 1 we apply this estimate to improve a result of bourgain concerning the number of random points needed to bring a convex body into a nearly isotropic position

uniformly superconvergent analysis of an efficient two grid method for nonlinear bi wave singular perturbation problem abstract the main aim of this paper is to present a two grid method for the fourth order nonlinear bi wave singular perturbation problem with low order nonconforming finite element based on the ciarlet raviart scheme the existence and uniqueness of the approximation solution are demonstrated through the brouwer fixed point theorem and the uniform superconvergent estimates in the broken h 1 norm and l 2 norm are obtained which are independent of the perturbation parameter some numerical results indicate that the proposed method is indeed an efficient algorithm

multivariate stable polynomials theory and applications univariate polynomials with only real roots while special do occur often enough that their properties can lead to interesting conclusions in diverse areas due mainly to the recent work of two young mathematicians julius borcea and petter branden a very successful multivariate generalization of this method has been developed the first part of this paper surveys some of the main results of this theory of multivariate stable polynomials the most central of these results is the characterization of linear transformations preserving stability of polynomials the second part presents various applications of this theory in complex analysis matrix theory probability and statistical mechanics and combinatorics

strong converse for identification via quantum channels we present a simple proof of the strong converse for identification via discrete memoryless quantum channels based on a novel covering lemma the new method is a generalization to quantum communication channels of ahlswede s 1979 1992 approach to classical channels it involves a development of explicit large deviation estimates to the case of random variables taking values in self adjoint operators on a hilbert space this theory is presented separately in an appendix and we illustrate it by showing its application to quantum generalizations of classical hypergraph covering problems

the van der waerden conjecture for mixed discriminants we prove that the mixed discriminant of doubly stochastic n tuples of semidefinite hermitian n n matrices is bounded below by n nn and that this bound is uniquely attained at the n tuple 1ni 1ni this result settles a conjecture posed by r bapat in 1989 we consider various generalizations and applications of this result