arXiv subjects
Publications and source records attributed to Boris Shapiro.
We comment on some aspects of the paper entitled "Evanescent-wave Johnson noise in small devices" recently posted by V. N. Premakumar, M. G. Vavilov and R. Joynt.
The famous Descartes' rule of signs from 1637 giving an upper bound on the number of positive roots of a real univariate polynomials in terms of the number of sign changes of its coefficients, has been an indispensable source of inspiration for generations of mathematicians. Trying to extend and sharpen this rule, we consider below the set of all real univariate polynomials of a given degree, a given collection of signs of their coefficients, and a given number of positive and negative roots. In spite of the elementary definition of the main object of our study, it is a non-trivial question for which sign patterns and numbers of positive and negative roots the corresponding set is non-empty. The main result of the present paper is a discovery of a new infinite family of non-realizable combinations of sign patterns and the numbers of positive and negative roots.
In this paper, motivated by the classical notion of a Strebel qua- dratic differential on a compact Riemann surface without boundary, we in- troduce several classes of quadratic differentials (called non-chaotic, gradient, and positive gradient) which possess some properties of Strebel differentials and appear in applications. We discuss the relation between gradient differen- tials and special signed measures supported on their set of critical trajectories. We provide a characterisation of gradient differentials for which there exists a positive measure in the latter class.
Below we study theoretically and numerically the asymptotics of the algebraic part of the spectrum for the quasi-exactly solvable sextic potential, its level crossing points, and its monodromy in the complex plane of its parameter. We also discuss connection between the quasi-exactly solvable sextic and the classical quartic potential.
We study the critical dynamics of matter waves at the 3D Anderson mobility edge in cold-atom disorder quench experiments. General scaling arguments are supported by precision numerics for the spectral function, diffusion coefficient, and localization length in isotropic blue-detuned speckle potentials. We discuss signatures of critical slowdown in the time-dependent central column density of a spreading wave packet, and evaluate the prospects of observing anomalous diffusion right at criticality.
There exists an analogy between Maxwell equations in a rotating frame and Schrödinger equation for a charged particle in the presence of a magnetic field. We exploit this analogy to point out that electromagnetic phenomena in the rotating frame, under appropriate conditions, can exhibit periodicity with respect to the angular velocity of rotation. In particular, in disordered ring cavities one finds the optical analog of the Al'tshuler-Aronov-Spivak effect well known in mesoscopic physics of disordered metals.
For a point $p\in CP^2$ and a triple $(g,d,\ell)$ of non-negative integers we define a {\em Hurwitz--Severi number} ${\mathfrak H}_{g,d,\ell}$ as the number of generic irreducible plane curves of genus $g$ and degree $d+\ell$ having an $\ell$-fold node at $p$ and at most ordinary nodes as singularities at the other points, such that the projection of the curve from $p$ has a prescribed set of local and remote tangents and lines passing through nodes. In the cases $d+\ell\ge g+2$ and $d+2\ell \ge g+2 > d+\ell$ we express the Hurwitz--Severi numbers via appropriate ordinary Hurwitz numbers. The remaining case $d+2\ell<g+2$ is still widely open.
Two main topics of this paper are asymptotic distributions of zeros of Jacobi polynomials and topology of critical trajectories of related quadratic differentials. First, we will discuss recent developments and some new results concerning the limit of the root-counting measures of these polynomials. In particular, we will show that the support of the limit measure sits on the critical trajectories of a quadratic differential of the form Q(z)dz^2=(az^2+bz+c)dz^2/(z^2-1)^2. Then we will give a complete classification, in terms of complex parameters a, b, and c, of possible topological types of critical geodesics for the quadratic differential of this type.
We investigate sound propagation in a moving fluid confined in a randomly corrugated tube. For weak randomness and small fluid velocities $v^{(0)}$, the localization length $ξ$ shows extreme sensitivity to the variation of $v^{(0)}$. In the opposite limit of large fluid velocities, $ξ$ acquires a constant value which is independent of the frequency of the incident sound wave, the degree of randomness and $v^{(0)}$ itself. Finally, we find that the standard deviation $σ_{\ln T}$ of the logarithm of transmittance $\ln(T)$ is a universal function of the ensemble average $\langle \ln T\rangle $, which is not affected by the fluid velocity.
In this note we show that a linear ordinary differential equation with polynomial coefficients is globally non-oscillating in $\mathbb{C} P^1$ if and only if it is Fuchsian, and at every its singular point any two distinct characteristic exponents have distinct real parts. As a byproduct of our study, we obtain a new explicit upper bound for the number of zeros of exponential polynomials in a horizontal strip.
Below we discuss the partition of the space of real univariate polynomials according to the number of positive and negative roots and signs of the coefficients. We present several series of non-realizable combinations of signs together with the numbers of positive and negative roots. We provide a detailed information about possible non-realizable combinations as above up to degree 8 as well as a general conjecture about such combinations.
We analytically investigate the heat current $I$ and its thermal fluctuations $Δ$ in a branching network without loops (Cayley tree). The network consists of two type of harmonic masses: vertex masses $M$ placed at the branching points where phononic scattering occurs and masses $m$ at the bonds between branching points where phonon propagation take place. The network is coupled to thermal reservoirs consisting of one-dimensional harmonic chains of coupled masses $m$. Due to impedance missmatching phenomena, both ${\cal I}$ and $Δ$, are non-monotonic functions of the mass ratio $μ=M/m$. In particular, there are cases where they are strictly zero below some critical value $μ^*$.
In a recent experiment [McGehee et al., Phys. Rev. Lett. 111, 145303 (2013)], the expansion of non-interacting ultracold fermions was studied in a random speckle potential, and the observed density profiles were interpreted based on 3D Anderson localization. The purpose of this note is to demonstrate that slow diffusion of particles with a broad energy distribution and an energy-dependent diffusion coefficient leads to density profiles that agree with the measured data, but not with the behavior expected for 3D Anderson localization.
Below we discuss the existence of a motherbody measure for the exterior inverse problem in potential theory in the complex plane. More exactly, we study the question of representability almost everywhere (a.e.) in C of (a branch of) an irreducible algebraic function as the Cauchy transform of a signed measure supported on a finite number of compact semi-analytic curves and a finite number of isolated points. Firstly, we present a large class of algebraic functions for which there (conjecturally) always exists a positive measure with the above properties. This class was discovered in our earlier study %of the eigenpolynomials of exactly solvable linear differential operators. Secondly, we investigate in detail the representability problem in the case when the Cauchy transform satisfies a quadratic equation with polynomial coefficients a.e. in C. Several conjectures and open problems are posed.
We report direct observation of the development of long-range spatial intensity correlations and the growth of intensity fluctuations inside the random media. We fabricated quasi-two-dimensional disordered photonic structures and probed the light transport from a third dimension. Good agreements between experiment and theory are obtained. We were able to manipulate the long-range intensity correlations and intensity fluctuations inside the disordered waveguides by simply varying the waveguide geometry.
There are indications that some high temperature unconventional superconductors have a "complex" d-wave order parameter (with an admixture of s-wave) leading to nonzero energy gap. Since the coherence length is short and the Fermi energy is relatively small the quasiclassical approach is inapplicable and the more complicated Bogoliubov-deGennes equations should be used to investigate the excitation spectrum of such a material in a magneric field. It turns out that equations for the gapped chiral d-wave superconductor, simplify considerably and is the basis for any superconductor of that type with a sufficiently large gap. The spectrum of core excitations of the Abrikosov vortex in an anisotropic 3D sample exhibits several features. Unlike in conventional and gapless superconductors the core has a single excitation mode of order energy gap for each value of momentum along the field. This has a large impact on thermal transport and vortex dynamics.
A signed polygonal measure is the sum of finitely many real constant density measures supported on polygons. Given a finite set S in the plane, we study the existence of signed polygonal measures spanned by polygons with vertices in S, which have all harmonic moments vanishing. For S generic, we show that the dimension of the linear space of such measures is (|S|-3)(|S|-4)/2. We also investigate the situation where the resulting density is either 0, or 1, or -1, which corresponds to pairs of polygons of unit density having the same logarithmic potential at infinity. We show that such a signed measure does not exist if |S| is at most 5, but for each n at least 6 there exists an S, with |S|=n, giving rise to such a signed measure.
We consider an ensemble of fully connected networks of N oscillators coupled harmonically with random springs and show, using Random Matrix Theory considerations, that both the average phonon heat current and its variance are scale-invariant and take universal values in the large N-limit. These anomalous mesoscopic uctuations is the hallmark of strong correlations between normal modes.