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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 an essential tool in this program as well as various contexts in complex analysis, probability theory, combinatorics, matrix theory. We characterize all linear operators on finite or(More)
In the first part of this series we characterized all linear operators on spaces of multivariate polynomials preserving the property of being non-vanishing in products of open circular domains. For such sets this completes the multivariate generalization of the classification program initiated by Pólya-Schur for univariate real polynomials. We build on(More)
A multivariate polynomial is stable if it is nonvanishing whenever all variables have positive imaginary parts. We characterize all finite order linear differential operators that preserve stability. An important technical tool that we develop in the process is the multivariate generalization of the classical notion of multiplier sequence. We give a(More)
A notion of weighted multivariate majorization is defined as a preorder on sequences of vectors in Euclidean space induced by the Choquet ordering for atomic probability measures. We characterize this preorder both in terms of stochastic matrices and convex functions and use it to describe the distribution of equilibrium points of logarithmic potentials(More)
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 Brändén, a very successful multivariate generalization of this method has been developed. The first part of this(More)
For n × n matrices A and B define η(A, B) = X S det(A[S]) det`BˆS ′ ˜´, where the summation is over all subsets of {1,. .. , n}, S ′ is the complement of S, and A[S] is the principal submatrix of A with rows and columns indexed by S. We prove that if A ≥ 0 and B is Hermitian then (1) the polynomial η(zA, −B) has all real roots (2) the latter polynomial has(More)