arXiv subjects

Boris Shapiro

Publications and source records attributed to Boris Shapiro.

At least 109 records · Page 6Linked to original sources

arXiv2013

Around multivariate Schmidt-Spitzer theorem

Given an arbitrary complex-valued infinite matrix A and a positive integer n we introduce a naturally associated polynomial basis B_A of C[x0...xn]. We discuss some properties of the locus of common zeros of all polynomials in B_A having a given degree m; the latter locus can be interpreted as the spectrum of the m*(m+n)-submatrix of A formed by its m first rows and m+n first columns. We initiate the study of the asymptotics of these spectra when m goes to infinity in the case when A is a banded Toeplitz matrix. In particular, we present and partially prove a conjectural multivariate analog of the well-known Schmidt-Spitzer theorem which describes the spectral asymptotics for the sequence of principal minors of an arbitrary banded Toeplitz matrix. Finally, we discuss relations between polynomial bases B_A and multivariate orthogonal polynomials.

math.AG↗

arXiv2013

Vandermonde varieties and relations among Schur polynomials

Motivated by the famous Skolem-Mahler-Lech theorem we initiate in this paper the study of a natural class of determinantal varieties which we call {\em Vandermonde varieties}. They are closely related to the varieties consisting of all linear recurrence relations of a given order possessing a non-trivial solution vanishing at a given set of integers. In the regular case, i.e., when the dimension of a Vandermonde variety is the expected one, we present its free resolution, obtain its degree and the Hilbert series. Some interesting relations among Schur polynomials are derived. Many open problems and conjectures are posed.

math.AG↗

arXiv2012

Statistics of resonances in one-dimensional disordered systems

The paper is devoted to the problem of resonances in one-dimensional disordered systems. Some of the previous results are reviewed and a number of new ones is presented. These results pertain to different models (continuous as well as lattice) and various regimes of disorder and coupling strength. In particular, a close connection between resonances and the Wigner delay time is pointed out and used to obtain information on the resonance statistics.

cond-mat.dis-nn↗

arXiv2012

Spaces of locally convex curves in S^n and combinatorics of the group B_{n+1}

In the 1920's Marston Morse developed what is now known as Morse theory trying to study the topology of the space of closed curves on S^2. We propose to attack a very similar problem, which 80 years later remains open, about the topology of the space of closed curves on S^2 which are locally convex (i.e., without inflection points). One of the main difficulties is the absence of the covering homotopy principle for the map sending a non-closed locally convex curve to the Frenet frame at its endpoint. In the present paper we study the spaces of locally convex curves in S^n with a given initial and final Frenet frames. Using combinatorics of B^{+}_{n+1} = B_{n+1} \cap SO_{n+1}, where B_{n+1} \subset O_{n+1} is the usual Coxeter-Weyl group, we show that for any n \ge 2 these spaces fall in at most $\lceil\frac{n}{2}\rceil+1$ equivalence classes up to homeomorphism. We also study this classification in the double cover Spin(n+1). For $n = 2$ our results complete the classification of the corresponding spaces into two topologically distinct classes, or three classes in the spin case.

math.GT↗

arXiv2011

On the Waring problem for polynomial rings

In this note we discuss an analog of the classical Waring problem for C[x_0, x_1,...,x_n]. Namely, we show that a general homogeneous polynomial p \in C[x_0,x_1,...,x_n] of degree divisible by k\ge 2 can be represented as a sum of at most k^n k-th powers of homogeneous polynomials in C[x_0, x_1,...,x_n]. Noticeably, k^n coincides with the number obtained by naive dimension count.

math.AG↗

arXiv2011

Thermalization of Strongly Disordered Nonlinear Chains

Thermalization of systems described by the discrete non-linear Schrödinger equation, in the strong disorder limit, is investigated both theoretically and numerically. We show that introducing correlations in the disorder potential, while keeping the "effective" disorder fixed (as measured by the localization properties of wavepacket dynamics), strongly facilitate the thermalization process and lead to a standard grand canonical distribution of the probability norms associated to each site

cond-mat.dis-nn↗

arXiv2011

Hardy-Petrovitch-Hutchinson's problem and partial theta function

In 1907 M.Petrovitch initiated the study of a class of entire functions all whose finite sections are real-rooted polynomials. An explicit description of this class in terms of the coefficients of a series is impossible since it is determined by an infinite number of discriminantal inequalities one for each degree. However, interesting necessary or sufficient conditions can be formulated. In particular, J.I.Hutchinson has shown that an entire function p(x)=a_0+a_1x+...+a_nx^n+... with strictly positive coefficients has the property that any its finite segment a_ix^i+...+a_jx^j has all real roots if and only if for all i=1,2,... one has a_i^2/a_{i-1}a_{i+1} is greater than or equal to 4. In the present paper we give sharp lower bounds on the ratios a_i^2/a_{i-1}a_{i+1} for the class considered by M.Petrovitch. In particular, we show that the limit of these minima when i tends to infinity equals the inverse of the maximal positive value of the parameter for which the classical partial theta function belongs to the Laguerre-Polya class.

math.CA↗

arXiv2011

Discriminants, symmetrized graph monomials, and sums of squares

Motivated by the necessities of the invariant theory of binary forms J. J. Sylvester constructed in 1878 for each graph with possible multiple edges but without loops its symmetrized graph monomial which is a polynomial in the vertex labels of the original graph. In the 20-th century this construction was studied by several authors. We pose the question for which graphs this polynomial is a non-negative resp. a sum of squares. This problem is motivated by a recent conjecture of F. Sottile and E. Mukhin on discriminant of the derivative of a univariate polynomial, and an interesting example of P. and A. Lax of a graph with 4 edges whose symmetrized graph monomial is non-negative but not a sum of squares. We present detailed information about symmetrized graph monomials for graphs with four and six edges, obtained by computer calculations.

math.AG↗

arXiv2010

A Note on EVgrafov-Fedoryuk's theory and quadratic differentials

The purpose of this short paper is to recall the theory of the (homogenized) spectral problem for a Schroedinger equation with a polynomial potential developed in the 60's by M. Evgrafov with M. Fedoryuk, and, by Y. Sibuya and its relation with quadratic differentials. We derive from these results that the accumulation rays of the eigenvalues of this problem are in 1-1 -correspondence with the short geodesics of the singular planar metrics on CP^1 induced by the corresponding quadratic differential. Using this interpretation we show that for a polynomial potential of degree d the number of such accumulation rays can be any positive integer between (d-1) and d \choose 2.

math.SP↗

arXiv2010

Multiplier sequences and logarithmic mesh

In this note we prove a new result about (finite) multiplier sequences, i.e. linear operators acting diagonally in the standard monomial basis of R[x] and sending polynomials with all real roots to polynomials with all real roots. Namely, we show that any such operator does not decrease the logarithmic mesh when acting on an arbitrary polynomial having all roots real and of the same sign. The logarithmic mesh of such a polynomial is defined as the minimal quotient of its consecutive roots taken in the non-decreasing order.

math.CA↗

arXiv2010

Using homological duality in consecutive pattern avoidance

Using the approach suggested in [arXiv:1002.2761] we present below a sufficient condition guaranteeing that two collections of patterns of permutations have the same exponential generating functions for the number of permutations avoiding elements of these collections as consecutive patterns. In short, the coincidence of the latter generating functions is guaranteed by a length-preserving bijection of patterns in these collections which is identical on the overlappings of pairs of patterns where the overlappings are considered as unordered sets. Our proof is based on a direct algorithm for the computation of the inverse generating functions. As an application we present a large class of patterns where this algorithm is fast and, in particular, allows to obtain a linear ordinary differential equation with polynomial coefficients satisfied by the inverse generating function.

math.CO↗

arXiv2010

Homogenized spectral problems for exactly solvable operators: asymptotics of polynomial eigenfunctions

Consider a homogenized spectral pencil of exactly solvable linear differential operators $T_{\la}=\sum_{i=0}^k Q_{i}(z)\la^{k-i}\frac {d^i}{dz^i}$, where each $Q_{i}(z)$ is a polynomial of degree at most $i$ and $\la$ is the spectral parameter. We show that under mild nondegeneracy assumptions for all sufficiently large positive integers $n$ there exist exactly $k$ distinct values $\la_{n,j}$, $1\le j\le k$, of the spectral parameter $\la$ such that the operator $T_{\la}$ has a polynomial eigenfunction $p_{n,j}(z)$ of degree $n$. These eigenfunctions split into $k$ different families according to the asymptotic behavior of their eigenvalues. We conjecture and prove sequential versions of three fundamental properties: the limits $Ψ_{j}(z)=\lim_{n\to\infty} \frac{p_{n,j}'(z)}{\la_{n,j}p_{n,j}(z)}$ exist, are analytic and satisfy the algebraic equation $\sum_{i=0}^k Q_{i}(z) Ψ_{j}^i(z)=0$ almost everywhere in $\bCP$. As a consequence we obtain a class of algebraic functions possessing a branch near $\infty\in \bCP$ which is representable as the Cauchy transform of a compactly supported probability measure.

math.CA↗

arXiv2010

Thermal Fluctuations of the Electric Field in the Presence of Carrier Drift

We consider a semiconductor in a non-equilibrium steady state, with a dc current. On top of the stationary carrier motion there are fluctuations. It is shown that the stationary motion of the carriers (i.e., their drift) can have a profound effect on the electromagnetic field fluctuations in the bulk of the sample as well as outside it, close to the surface (evanescent waves in the near field). The effect is particularly pronounced near the plasma frequency. This is because drift leads to a significant modification of the dispersion relation for the bulk and surface plasmons.

cond-mat.other↗

arXiv2010

One Parameter Scaling Theory for Stationary States of Disordered Nonlinear Systems

We show, using detailed numerical analysis and theoretical arguments, that the normalized participation number of the stationary solutions of disordered nonlinear lattices obeys a one-parameter scaling law. Our approach opens a new way to investigate the interplay of Anderson localization and nonlinearity based on the powerful ideas of scaling theory.

cond-mat.dis-nn↗

arXiv2010

New Multiplier Sequences via Discriminant Amoebae

In their classic 1914 paper, Polya and Schur introduced and characterized two types of linear operators acting diagonally on the monomial basis of R[x], sending real-rooted polynomials (resp. polynomials with all nonzero roots of the same sign) to real-rooted polynomials. Motivated by fundamental properties of amoebae and discriminants discovered by Gelfand, Kapranov, and Zelevinsky, we introduce two new natural classes of polynomials and describe diagonal operators preserving these new classes. A pleasant circumstance in our description is that these classes have a simple explicit description, one of them coinciding with the class of log-concave sequences.

math.AG↗

arXiv2009

Exponentially Fragile PT-Symmetry in Lattices with Localized Eigenmodes

We study the effect of localized modes in lattices of size N with parity-time (PT) symmetry. Such modes are arranged in pairs of quasi-degenerate levels with splitting delta exp{-N/xi}, where ξis their localization length. The level "evolution" with respect to the PT breaking parameter gamma shows a cascade of bifurcations during which a pair of real levels becomes complex. The spontaneous PT symmetry breaking occurs at gamma min(delta), thus resulting in an exponentially narrow exact PT phase. As N/xi decreases, it becomes more robust with gamma (1/N)^2 and the distribution P(gamma) changes from log-normal to semi-Gaussian. Our theory can be tested in the frame of optical lattices.

cond-mat.other↗

arXiv2009

On spectral polynomials of the Heun equation. II

The well-known Heun equation has the form: Q(z)S''(z)+P(z)S'(z)+V(z)S(z)=0 where Q(z) is a cubic complex polynomial, P(z) and V(z) are polynomials of degrees at most 2 and 1 resp. One of the classical problems about the Heun equation is for a given positive integer N to find all possible linear polynomials V(z) such that the latter equation has a polynomial solution S(z) of degree N. Below we prove a conjecture of the 2nd author claiming that the union of roots of such V(z)'s for a given N tends when N->oo to a certain compact connecting the three roots of Q(z) and given by the condition that a certain natural abelian integral is real-valued.

math-ph↗

arXiv2009

On higher Heine-Stieltjes polynomials

Given a differential operator T=\sum_{i=1}^k Q_i(z)d^i/dz^i where each Q_i(z) is a polynomial define r=max_i deg(Q_i(z)-i). Assuming that r is nonnegative we consider the following multiparameter spectral problem: for each positive integer n find all polynomials V(z) of degree at most r such that the equation T(S(z))+V(z)S(z)=0 has a polynomial solution S(z) of degree n. We calculate for any converging sequence of normalized polynomials V_j(z) the root-counting measure of the corresponding sequence of polynomials S_j(z).

math.CA↗