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Boris Shapiro

Publications and source records attributed to Boris Shapiro.

At least 55 records · Page 3Linked to original sources
arXiv2023

Optical Kinetic Theory of Nonlinear Multi-mode Photonic Networks

Recent experimental developments in multimode nonlinear photonic circuits (MMNPC), have motivated the development of an optical thermodynamic theory that describes the equilibrium properties of an initial beam excitation. However, a non-equilibrium transport theory for these systems, when they are in contact with thermal reservoirs, is still {\it terra incognita}. Here, by combining Landauer and kinematics formalisms we develop a one-parameter scaling theory that describes the transport in one-dimensional MMNPCs from a ballistic to a diffusive regime. We also derive a photonic version of the Wiedemann -Franz law that connects the thermal and power conductivities. Our work paves the way toward a fundamental understanding of the transport properties of MMNPC and may be useful for the design of all-optical cooling protocols.

physics.optics↗
arXiv2023

Real polynomials with constrained real divisors. I. Fundamental groups

In the late 80s, V.~Arnold and V.~Vassiliev initiated the topological study of the space of real univariate polynomials of a given degree d and with no real roots of multiplicity exceeding a given positive integer. Expanding their studies, we consider the spaces of real monic univariate polynomials of degree d whose real divisors avoid sequences of root multiplicities taken from a given poset of compositions which is closed under certain natural combinatorial operations. In this paper, we concentrate on the fundamental group of such spaces. We find explicit presentations for the fundamental groups in terms of generators and relations and show that in a number of cases they are free with rank bounded from above by a quadratic function in d. We also show that the fundamental group stabilizes for d large. We further show that the fundamental groups admit an interpretation as special bordisms of immersions of 1-manifolds into the cylinder S^1 \times R, whose images avoid the tangency patterns from the poset with respect to the generators of the cylinder.

math.AT↗
arXiv2023

Rodrigues' descendants of a polynomial and Boutroux curves

Motivated by the classical Rodrigues' formula, we study the root asymptotic of the polynomial sequence $$R_{[αn],n,P}(z)=\frac{d^{[αn]}P^n(z)}{dz^{[αn]}}, n= 0,1,\dots$$ where ${P(z)}$ is a fixed univariate polynomial, $α$ is a fixed positive number smaller than deg $P$, and $[αn]$ stands for the integer part of $αn$. Our description of this asymptotic is expressed in terms of an explicit harmonic function uniquely determined by the plane rational curve emerging from the application of the saddle point method to the integral representation of the latter polynomials using Cauchy's formula for higher derivatives. As a consequence of our method, we conclude that this curve is birationally equivalent to the zero locus of the bivariate algebraic equation satisfied by the Cauchy transform of the asymptotic root-counting measure for the latter polynomial sequence. We show that this harmonic function is also associated with an abelian differential having only purely imaginary periods and the latter plane curve belongs to the class of Boutroux curves initially introduced by Bertola. As an additional relevant piece of information, we derive a linear ordinary differential equation satisfied by $\{R_{[αn],n,P}(z)\}$ as well as higher derivatives of powers of more general functions.

math.CA↗
arXiv2023

Stationary Charge Radiation in Anisotropic Photonic Time Crystals

Time metamaterials exhibit a great potential for wave manipulation, drawing increasing attention in recent years. Here, we explore the exotic wave dynamics in an anisotropic photonic time crystal (APTC), formed by an anisotropic medium whose optical properties are uniformly and periodically changed in time. Based on a temporal transfer matrix formalism, we show that a stationary charge embedded in an APTC can emit radiation, in contrast to the case of an isotropic photonic time crystal, and its distribution in momentum space is controlled by the APTC band structure. Our approach extends the functionalities of time metamaterials, offering new opportunities for simultaneous radiation generation and control, with implications for both classical and quantum applications.

physics.optics↗
arXiv2022

Non-self-adjoint Toeplitz matrices whose principal submatrices have real spectrum

We introduce and investigate a class of complex semi-infinite banded Toeplitz matrices satisfying the condition that the spectra of their principal submatrices accumulate onto a real interval when the size of the submatrix grows to $\infty$. We prove that a banded Toeplitz matrix belongs to this class if and only if its symbol has real values on a Jordan curve located in $\mathbb{C}\setminus\{0\}$. Surprisingly, it turns out that, if such a Jordan curve is present, the spectra of all the submatrices have to be real. The latter claim is also proven for matrices given by a more general symbol. Further, the limiting eigenvalue distribution of a real banded Toeplitz matrix is related to the solution of a determinate Hamburger moment problem. We use this to derive a formula for the limiting measure using a parametrization of the Jordan curve. We also describe a Jacobi operator, whose spectral measure coincides with the limiting measure. We show that this Jacobi operator is a compact perturbation of a tridiagonal Toeplitz matrix. Our main results are illustrated by several concrete examples; some of them allow an explicit analytic treatment, while some are only treated numerically. Update: The proof of Theorem 8 contains an error. An erratum is attached in the end

math.CA↗
arXiv2022

Deformed graphical zonotopal algebras

We study certain filtered deformations of the external zonotopal algebra of a given graph parametrized by univariate polynomials. We establish some general properties of these algebras, compute their Hilbert series for a number of graphs using Macaulay2, and formulate several conjectures.

math.CO↗
arXiv2022

Introducing isodynamic points for binary forms and their ratios

The isodynamic points of a plane triangle are known to be the only pair of its centers invariant under the action of the Mobius group on the set of triangles. Generalizing this classical result, we introduce below the isodynamic map associating to a univariate polynomial of degree d at least 3 with at most double roots a polynomial of degree (at most) 2d-4 such that this map commutes with the action of the Mobius group on the zero loci of the initial polynomial and its image. The roots of the image polynomial will be called the isodynamic points of the preimage polynomial. Our construction naturally extends from univariate polynomials to binary forms and further to their ratios.

math.CV↗
arXiv2022

Fluctuational electrodynamics in and out of equilibrium

Dispersion forces between neutral material bodies are due to fluctuations of the polarization of the bodies. For bodies in equilibrium these forces are often referred to as Casimir-Lifshitz forces. For bodies in relative motion, in addition to the Casimir-Lifshitz force, a lateral frictional force ("quantum friction", in the zero temperature limit) comes into play. The widely accepted theory of the fluctuation induced forces is based on the "fluctuational electrodynamics" , when the Maxwell equations are supplemented by random current sources responsible for the fluctuations of the medium polarization. The first part of our paper touches on some conceptual issues of the theory, such as the dissipation-less limit and the link between Rytov's approach and quantum electrodynamics. We point out the problems with the dissipation-less plasma model (with its unphysical double pole at zero frequency) which still appears in the literature. The second part of the paper is devoted to "quantum friction", in a broad sense, and it contains some novel material. In particular, it is pointed out that in weakly dissipative systems the friction force may not be a stationary process. It is shown, using an "exact" (nonpertubative) quantum treatment that under appropriate conditions, an instability can occur when the kinetic energy (due to the relative motion between the bodies) is transformed into coherent radiation, exponentially growing in intensity (the instability gets eventually limited by non-linear effects). We also discuss a setup when the two bodies are at rest but a constant electric current is flowing in one of the bodies. One may say that only the electron component of one body is dragged with respect to the other body, unlike the usual setup when the two bodies are in relative motion. Clearly there are differences in the frictional forces between the two setups.

quant-ph↗
arXiv2022

A critical discussion of different methods and models in Casimir effect

The Casimir-Lifhitz force acts between neutral material bodies and is due to the fluctuations (around zero) of the electrical polarizations of the bodies. This force is a macroscopic manifestation of the van der Waals forces between atoms and molecules. In addition to being of fundamental interest, the Casimir-Lifshitz force plays an important role in surface physics, nanotechnology and biophysics. There are two different approaches in the theory of this force. One is centered on the fluctuations inside the bodies, as the source of the fluctuational electromagnetic fields and forces. The second approach is based on finding the eigenmodes of the field, while the material bodies are assumed to be passive and non-fluctuating. In spite of the fact that both approaches have a long history, there are still some misconceptions in the literature. In particular, there are claims that (hypothetical) materials with a strictly real dielectric function $\varepsilon(ω)$ can give rise to fluctuational Casimir-Lifshitz forces. We review and compare the two approaches, using the simple example of the force in the absence of retardation. We point out that also in the second (the "field-oriented") approach one cannot avoid introducing an infinitesimal imaginary part into the dielectric function, i.e. introducing some dissipation. Furthermore, we emphasize that the requirement of analyticity of $ \varepsilon(ω)$ in the upper half of the complex $ω$ plane is not the only one for a viable dielectric function. There are other requirements as well. In particular, models that use a strictly real $\varepsilon(ω)$ (for all real positive $ω)$ are inadmissible and lead to various contradictions and inconsistencies. Specifically, we present a critical discussion of the "dissipation-less plasma model".

quant-ph↗
arXiv2021

Spaces of polynomials with constrained real divisors, II. (Co)homology & stabilization

In the late 80s, V.~Arnold and V.~Vassiliev initiated the topological study of the space of real univariate polynomials of a given degree which have no real roots of multiplicity exceeding a given positive integer. Expanding their studies, we consider the spaces P^{cΘ}_d of real monic univariate polynomials of degree d whose real divisors avoid given sequences of root multiplicities. These forbidden sequences are taken from an arbitrary poset Θof compositions that are closed under certain natural combinatorial operations. We reduce the computation of the homology H_*(P^{cΘ}_d) to the computation of the homology of a differential complex, defined purely combinatorially in terms of the given closed poset Θ. We also obtain the stabilization results about H^\ast(P^{c Θ}_d), as d goes to infinity. These results are deduced from our description of the homology of spaces B^{c Θ}_d whose points are binary real homogeneous forms, considered up to projective equivalence, with similarly Θ-constrained real divisors. In particular, we exhibit differential complexes that calculate the homology of these spaces and obtain some stabilization results for H^*(B^{c Θ}_d), as d goes to infinity. In particular, we compute the homology of the discriminants of projectivized binary real forms for which there is at least one line on which the form vanishes with multiplicity >= 2 and of their complements in \cB_d \cong RP^d.

math.AT↗
arXiv2021

Return of the plane evolute

Below we consider the evolutes of plane real-algebraic curves and discuss some of their complex and real-algebraic properties. In particular, for a given degree $d\ge 2$, we provide lower bounds for the following four numerical invariants: 1) the maximal number of times a real line can intersect the evolute of a real-algebraic curve of degree $d$; 2) the maximal number of real cusps which can occur on the evolute of a real-algebraic curve of degree $d$; 3) the maximal number of (cru)nodes which can occur on the dual curve to the evolute of a real-algebraic curve of degree $d$; 4) the maximal number of (cru)nodes which can occur on the evolute of a real-algebraic curve of degree $d$.

math.AG↗
arXiv2021

Finiteness of rank for Grassmann convexity

The Grassmann convexity conjecture gives a conjectural formula for the maximal total number of real zeros of the consecutive Wronskians of an arbitrary fundamental solution to a disconjugate linear ordinary differential equation with real time. The conjecture can be reformulated in terms of convex curves in the nilpotent lower triangular group. The formula has already been shown to be a correct lower bound and to give a correct upper bound in several small dimensional cases. In this paper we obtain a general explicit upper bound.

math.CA↗
arXiv2021

Controlling Optical Beam Thermalization via Band-Gap Engineering

We establish dispersion engineering rules that allow us to control the thermalization process and the thermal state of an initial beam propagating in a multimode nonlinear photonic circuit. To this end, we have implemented a kinetic equation (KE) approach in systems whose Bloch dispersion relation exhibits bands and gaps. When the ratio between the gap-width to the band-width is larger than a critical value, the KE has stationary solutions which differ from the standard Rayleigh-Jeans (RJ) distribution. The theory also predicts the relaxation times above which such non-conventional thermal states occur. We have tested the validity of our results for the prototype SSH model whose connectivity between the composite elements allows to control the band-gap structure. These spectral engineering rules can be extended to more complex photonic networks that lack periodicity but their spectra consist of groups of modes that are separated by spectral gaps.

physics.optics↗
arXiv2020

Grassmann convexity and multiplicative Sturm theory, revisited

In this paper we settle a special case of the Grassmann convexity conjecture formulated earlier by B.and M.Shapiro. We present a conjectural formula for the maximal total number of real zeros of the consecutive Wronskians of an arbitrary fundamental solution to a disconjugate linear ordinary differential equation with real time. We show that this formula gives the lower bound for the required total number of real zeros for equations of an arbitrary order and, using our results on the Grassmann convexity, we prove that the aforementioned formula is correct for equations of orders $4$ and $5$.

math.CA↗
arXiv2020

Moment Varieties of Measures on Polytopes

The uniform probability measure on a convex polytope induces piecewise polynomial densities on its projections. For a fixed combinatorial type of simplicial polytopes, the moments of these measures are rational functions in the vertex coordinates. We study projective varieties that are parametrized by finite collections of such rational functions. Our focus lies on determining the prime ideals of these moment varieties. Special cases include Hankel determinantal ideals for polytopal splines on line segments, and the relations among multisymmetric functions given by the cumulants of a simplex. In general, our moment varieties are more complicated than in these two special cases. They offer challenges for both numerical and symbolic computing in algebraic geometry.

math.AG↗
arXiv2020

Generalizing Tran's Conjecture

A conjecture of Khang Tran [6] claims that for an arbitrary pair of polynomials $A(z)$ and $B(z)$, every zero of every polynomial in the sequence $\{P_n(z)\}_{n=1}^\infty$ satisfying the three-term recurrence relation of length $k$ $$P_n(z)+B(z)P_{n-1}(z)+A(z)P_{n-k}(z)=0 $$ with the standard initial conditions $P_0(z)=1$, $P_{-1}(z)=\dots=P_{-k+1}(z)=0$ which is not a zero of $A(z)$ lies on the real (semi)-algebraic curve $\mathcal C \subset \mathbb {C}$ given by $$\Im \left(\frac{B^k(z)}{A(z)}\right)=0\quad {\rm and}\quad 0\le (-1)^k\Re \left(\frac{B^k(z)}{A(z)}\right)\le \frac{k^k}{(k-1)^{k-1}}.$$ In this short note, we show that for the recurrence relation (generalizing the latter recurrence of Tran) given by $$P_n(z)+B(z)P_{n-\ell}(z)+A(z)P_{n-k}(z)=0, $$ with coprime $k$ and $\ell$ and the same standard initial conditions as above, every root of $P_n(z)$ which is not a zero of $A(z)B(z)$ belongs to the real algebraic curve $\mathcal C_{\ell,k}$ given by $$\Im \left(\frac{B^k(z)}{A^\ell(z)}\right)=0.$$

math.CA↗
arXiv2020

On existence of quasi-Strebel structures for meromorphic k-differentials

In this paper, motivated by the classical notion of a Strebel quadratic differential on a compact Riemann surfaces without boundary we introduce the notion of a quasi-Strebel structure for a meromorphic differential of an arbitrary order. It turns out that every differential of even order k exceeding 2 satisfying certain natural conditions at its singular points admits such a structure. The case of differentials of odd order is quite different and our existence result involves some arithmetic conditions. We discuss the set of quasi-Stebel structures associated to a given differential and introduce the subclass of positive k-differentials. Finally, we provide a family of examples of positive rational differentials and explain their connection with the classical Heine-Stieltjes theory of linear differential equations with polynomial coefficients.

math.AG↗