Palindromic Polynomials Research Articles

An exactly solvable asymmetric K-exclusion process

We study an interacting particle process on a finite ring with L sites with at most K particles per site, in which particles hop to nearest neighbors with rates given in terms of t-deformed integers and asymmetry parameter q, where t > 0 and q⩾0 are parameters. This model, which we call the (q, t) asymmetric simple K-exclusion process (ASEP), reduces to the usual ASEP on the ring when K = 1 and to a model studied by Schütz and Sandow (Phys. Rev. E, 1994) when t=q=1 . This is a special case of the misanthrope process and as a consequence, the steady state does not depend on q and is of product form, generalizing the same phenomena for the ASEP. What is interesting here is the steady state weights are given by explicit formulas involving t-binomial coefficients, and are palindromic polynomials in t. Interestingly, although the (q, t) K-ASEP does not satisfy particle-hole symmetry, its steady state does. We analyze the density and calculate the most probable number of particles at a site in the steady state in various regimes of t. Lastly, we construct a two-dimensional exclusion process on a discrete cylinder with height K and circumference L which projects to the (q, t) K-ASEP and whose steady state distribution is also of product form. We believe this model will serve as an illustrative example in constructing two-dimensional analogues of misanthrope processes.Simulations are attached as ancillary files.

High frequency multi-field continualization scheme for layered magneto-electro-elastic materials

Magneto-electro-elastic (MEE) heterogeneous materials with periodic microstructure are investigated, with special focus on the particular case of a layered MEE material. By exploiting the transfer matrix method in combination with the Floquet–Bloch boundary conditions, the frequency dispersion spectrum can be derived by solving an eigenproblem, involving a 12 × 12 transfer matrix with palindromic characteristic polynomial. In particular, by assuming the cubic symmetry of layers, the problems involving longitudinal and transverse elastic waves are uncoupled. This circumstance has the advantage of simplifying the treatment and the size of the subproblems to be solved, corresponding to a 4 × 4 transfer matrix for the longitudinal case, and a 8 × 8 transfer matrix of the transverse problem, both characterized by palindromic characteristic polynomials. Afterward, a dynamic multi-field continualization approach is proposed to investigate wave propagation in MEE layered periodic material by matching the Z-transform of the vector collecting the nodal fields to the two-sided Laplace transform of the same vector at the macroscale. The continualization technique allows identifying multi-field integral-type non-local continua as well as multi-field generalized gradient-type (higher-order) non-local continua. Finally, to verify the accuracy of the proposed approach, the dispersion curves derived from the continualization technique are compared with the curves provided by the Floquet–Bloch theory.

Some results related to Hurwitz stability of combinatorial polynomials

Many important problems are closely related to the zeros of certain polynomials derived from combinatorial objects. The aim of this paper is to observe some results and applications for the Hurwitz stability of polynomials in combinatorics and study other related problems.We first present a criterion for the Hurwitz stability of the Turán expressions of recursive polynomials. In particular, it implies the q-log-convexity or q-log-concavity of the original polynomials. We also give a criterion for the Hurwitz stability of recursive polynomials and prove that the Hurwitz stability of any palindromic polynomial implies its semi-γ-positivity, which illustrates that the original polynomial with odd degree is unimodal. In particular, we get that the semi-γ-positivity of polynomials implies their parity-unimodality and the Hurwitz stability of polynomials implies their parity-log-concavity. Those results generalize the connections between real-rootedness, γ-positivity, log-concavity and unimodality to Hurwitz stability, semi-γ-positivity, parity-log-concavity and parity-unimodality (unimodality). As applications of these criteria, we derive some Hurwitz stability results occurred in the literature in a unified manner. In addition, we obtain the Hurwitz stability of Turán expressions for alternating run polynomials of types A and B and the Hurwitz stability for alternating run polynomials defined on a dual set of Stirling permutations.Finally, we study a class of recursive palindromic polynomials and derive many nice properties including Hurwitz stability, semi-γ-positivity, non-γ-positivity, unimodality, strong q-log-convexity, the Jacobi continued fraction expansion and the relation with derivative polynomials. In particular, these properties of the alternating descents polynomials of types A and B can be implied in a unified approach.

Comb-Based Filter Design Using Sharpening Technique and Certain Palindromic Polynomial

Comb-Based Filter Design Using Sharpening Technique and Certain Palindromic Polynomial

Dispersive waves in magneto-electro-elastic periodic waveguides

Magneto-electro-elastic (MEE) materials are attracting an increasing amount of scientific interest in physics, for their valuable potential in the functional management of mechanical-to-magnetoelectric energy conversions. The present paper focuses on heterogeneous MEE media – artfully realized by virtue of the periodic repetition of a multi-phase elementary cell – which represent promising solutions to passively control the propagation of multifield Bloch waves. Specifically, the linear continuum model governing the coupled free dynamics of a cellular non-dissipative MEE material is formulated for the unbounded three-dimensional domain. Then, the governing equations are specialized for a periodic layered waveguide with perfect inter-layer interfaces. The Floquet–Bloch decomposition is employed to analytically determine the uni-modular transfer matrix of the heterogeneous multi-layered cell. The related eigenproblem, governed by a palindromic characteristic polynomial in the Floquet multipliers, is solved in closed form to analyze the dispersion properties in the real-valued frequency domain and complex-valued wavenumber domain. Alternatively, a perturbation scheme is outlined to asymptotically approximate the exact eigensolutions. The dispersion spectra, composed of propagation and attenuation branches, are parametrically investigated for a two-layered periodic waveguide. The corresponding band structures are assessable a priori by leveraging the formal analogy with the stability analysis for non-autonomous dynamical systems. As main findings, slow-propagating quasi-shear-elastic waves and fast-propagating quasi-magneto-electric waves are found to dominate the low-frequency and high-frequency ranges, respectively, although some crossing points between the respective spectral branches can be detected. Quantitatively different but qualitatively similar behaviors of propagation and/or attenuation are observed for the quasi-shear-elastic waves and quasi-magneto-electric waves. Consequently, pass–pass, pass–stop and stop–stop bands of frequencies can be recognized and tuned by varying the key physical parameters. This rich spectral scenario opens the way for the functional customization of MEE-based metafilters or metapropagators, purposely designed to govern the transfer, localization, conversion, harvesting of energy across large frequency ranges.

Q-Deformations in the modular group and of the real quadratic irrational numbers

We develop further the theory of q-deformations of real numbers introduced in [10] and [9] and focus in particular on the class of real quadratic irrationals. Our key tool is a q-deformation of matrices of the modular group PSL(2,Z). The action of the modular group by Möbius transformations commutes with the q-deformations. We prove that the traces of the q-deformed matrices are palindromic polynomials with positive coefficients. These traces appear in the explicit expressions of the q-deformed quadratic irrationals.

Differential Identities for the Structure Function of Some Random Matrix Ensembles

The structure function of a random matrix ensemble can be specified as the covariance of the linear statistics $\sum_{j=1}^N e^{i k_1 \lambda_j}$, $\sum_{j=1}^N e^{-i k_2 \lambda_j}$ for Hermitian matrices, and the same with the eigenvalues $\lambda_j$ replaced by the eigenangles $\theta_j$ for unitary matrices. As such it can be written in terms of the Fourier transform of the density-density correlation $\rho_{(2)}$. For the circular $\beta$-ensemble of unitary matrices, and with $\beta$ even, we characterise the bulk scaling limit of $\rho_{(2)}$ as the solution of a linear differential equation of order $\beta + 1$ -- a duality relates $\rho_{(2)}$ with $\beta$ replaced by $4/\beta$ to the same equation. Asymptotics obtained in the case $\beta = 6$ from this characterisation are combined with previously established results to determine the explicit form of the degree 10 palindromic polynomial in $\beta/2$ which determines the coefficient of $|k|^{11}$ in the small $|k|$ expansion of the structure function for general $\beta > 0$. For the Gaussian unitary ensemble we give a reworking of a recent derivation and generalisation, due to Okuyama, of an identity relating the structure function to simpler quantities in the Laguerre unitary ensemble first derived in random matrix theory by Br\'ezin and Hikami. This is used to determine various scaling limits, many of which relate to the dip-ramp-plateau effect emphasised in recent studies of many body quantum chaos, and allows too for rates of convergence to be established.

The generalized Floquet-Bloch spectrum for periodic thermodiffusive layered materials

The dynamic behaviour of periodic thermodiffusive multi-layered media excited by harmonic oscillations is studied. In the framework of linear thermodiffusive elasticity, periodic laminates, whose elementary cell is composed by an arbitrary number of layers, are considered. The generalized Floquet-Bloch conditions are imposed, and the universal dispersion relation of the composite is obtained by means of an approach based on the formal solution for a single layer together with the transfer matrix method. The eigenvalue problem associated with the dispersion equation is solved by means of an analytical procedure based on the symplecticity properties of the transfer matrix to which corresponds a palindromic characteristic polynomial, and the frequency band structure associated to wave propagating inside the medium is finally derived. The proposed approach is tested through illustrative examples where thermodiffusive multilayered structures of interest for renewable energy devices fabrication are analyzed. The effects of thermodiffusion coupling on both the propagation and attenuation of Bloch waves in these systems are investigated in detail.

Palindromic Linearizations of Palindromic Matrix Polynomials of Odd Degree Obtained from Fiedler-Like Pencils

Palindromic matrix polynomials arise in many applications. Structure-preserving linearizations of palindromic matrix polynomials have been proposed in the literature so as to preserve the spectral symmetry in the eigenvalues. We present a new family of palindromic strong linearizations of a palindromic matrix polynomial of odd degree. A salient feature of the new family is that it allows the construction of banded palindromic linearizations of block-bandwidth k + 1 for any k = 0 : m − 2, where m is the degree of the palindromic matrix polynomial. Low bandwidth palindromic pencils may be useful for numerical computations. Our construction of the new family is based on Fiedler companion matrices associated with matrix polynomials and the construction is operation-free. Moreover, the new family of palindromic pencils allows operation-free recovery of eigenvectors and minimal bases, and an easy recovery of minimal indices of matrix polynomials from those of the palindromic linearizations. We also present an operation-free algorithm for construction of palindromic pencils belonging to the new family.

On the roots of the Poupard and Kreweras polynomials

The Poupard polynomials are polynomials in one variable with integer coefficients, with some close relationship to Bernoulli and tangent numbers. They also have a combinatorial interpretation. We prove that every Poupard polynomial has all its roots on the unit circle. We also obtain the same property for another sequence of polynomials introduced by Kreweras and related to Genocchi numbers. This is obtained through a general statement about some linear operators acting on palindromic polynomials.

A Class of Littlewood Polynomials that are Not L α -Flat

Abstract We exhibit a class of Littlewood polynomials that are not L α -flat for any α ≥ 0. Indeed, it is shown that the sequence of Littlewood polynomials is not L α -flat, α ≥ 0, when the frequency of −1 is not in the interval ] 1 4 {1 \over 4} , 3 4 {3 \over 4} [ We further obtain a generalization of Jensen-Jensen-Hoholdt’s result by establishing that the sequence of Littlewood polynomials is not L α -flat for any α> 2 if the frequency of −1 is not 1 2 {1 \over 2} . Finally, we prove that the sequence of palindromic Littlewood polynomials with even degrees are not L α -flat for any α ≥ 0, and we provide a lemma on the existence of c-flat polynomials.

Binomial Eulerian polynomials for colored permutations

Binomial Eulerian polynomials first appeared in work of Postnikov, Reiner and Williams on the face enumeration of generalized permutohedra. They are γ-positive (in particular, palindromic and unimodal) polynomials which can be interpreted as h-polynomials of certain flag simplicial polytopes and which admit interesting Schur γ-positive symmetric function generalizations. This paper introduces analogues of these polynomials for r-colored permutations with similar properties and uncovers some new instances of equivariant γ-positivity in geometric combinatorics.

On Inverse Eigenvalue Problems of Quadratic Palindromic Systems with Partially Prescribed Eigenstructure

The palindromic inverse eigenvalue problem (PIEP) of constructing matrices $A$ and $Q$ of size $n \times n$ for the quadratic palindromic polynomial $P(\lambda) = \lambda^2 A^{\star} + \lambda Q + A$ so that $P(\lambda)$ has $p$ prescribed eigenpairs is considered. This paper provides two different methods to solve PIEP, and it is shown via construction that PIEP is always solvable for any $p$ ($1 \leq p \leq (3n+1)/2$) prescribed eigenpairs. The eigenstructure of the resulting $P(\lambda)$ is completely analyzed.

Design of multiplierless minimum-phase FIR filters using palindromic polynomials and sharpening

Minimum phase (MP) filters have all zeros inside the unit circle and have a minimum group delay and a minimum energy delay properties, especially useful in communication systems. The special class of MP filters has all zeros inside and on the unit circle. This paper presents a method for design of this class of MP FIR (Finite Impulse Response) digital filters without multipliers The method is based on palindromic polynomials and sharpening technique. The coefficients of palindromic polynomials are presented as a sum of powers of two, and consequently can be implemented by using only adders and shifts, resulting in a multiplierless design. Two classes of palindromic polynomials, having all zeros on the unit circle, are considered. Next, the sharpening technique was applied to improve the magnitude characteristic and to move zeros inside of unit circle. The choice of the design parameters is also presented. All zeros, of the designed MP filter, are inside or on the unit circle. Method is illustrated with one example.

Counting the Ideals of Given Codimension of the Algebra of Laurent Polynomials in Two Variables

We establish an explicit formula for the number Cn(q) of ideals of codimension (colength) n of the algebra Fq[x,y,x−1,y−1] of Laurent polynomials in two variables over a finite field Fq of cardinality q. This number is a palindromic polynomial of degree 2n in q. Moreover, Cn(q)=(q−1)2Pn(q), where Pn(q) is another palindromic polynomial; the latter is a q-analogue of the sum of divisors of n, which happens to be the number of subgroups of Z2 of index n.

Polynomials with palindromic and unimodal coefficients

Let $f(q)=a_rq^r+\cdots+a_sq^s$, with $a_r\neq 0$ and $a_s\neq 0$, be a real polynomial. It is a palindromic polynomial of darga $n$ if $r+s=n$ and $a_{r+i}=a_{s-i}$ for all $i$. Polynomials of darga $n$ form a linear subspace $\mathcal{P}_n(q)$ of $\mathbb{R}(q)_{n+1}$ of dimension $\lfloor{n/2}\rfloor+1$. We give transition matrices between two bases $\left\{q^j(1+q+\cdots+q^{n-2j})\right\}, \left\{q^j(1+q)^{n-2j}\right\}$ and the standard basis $\left\{q^j(1+q^{n-2j})\right\}$ of $\mathcal{P}_n(q)$. We present some characterizations and sufficient conditions for palindromic polynomials that can be expressed in terms of these two bases with nonnegative coefficients. We also point out the link between such polynomials and rank-generating functions of posets.

Structured Eigenvalue Backward Errors of Matrix Pencils and Polynomials with Palindromic Structures

We derive formulas for the backward error of an approximate eigenvalue of a $*$-palindromic matrix polynomial with respect to $*$-palindromic perturbations. Such formulas are also obtained for complex $T$-palindromic pencils and quadratic polynomials. When the $T$-palindromic polynomial is real, then we derive the backward error of a real number considered as an approximate eigenvalue of the matrix polynomial with respect to real $T$-palindromic perturbations. In all cases the corresponding minimal structure preserving perturbations are obtained as well. The results are illustrated by numerical experiments. These show that there is a significant difference between the backward errors with respect to structure preserving and arbitrary perturbations in many cases.

Statistics on Lattice Walks and q-Lassalle Numbers

This paper contains two results. First, I propose a $q$-generalization of a certain sequence of positive integers, related to Catalan numbers, introduced by Zeilberger, see Lassalle (2010). These $q$-integers are palindromic polynomials in $q$ with positive integer coefficients. The positivity depends on the positivity of a certain difference of products of $q$-binomial coefficients.To this end, I introduce a new inversion/major statistics on lattice walks. The difference in $q$-binomial coefficients is then seen as a generating function of weighted walks that remain in the upper half-plan. Cet document contient deux résultats. Tout d’abord, je vous propose un $q$-generalization d’une certaine séquence de nombres entiers positifs, liés à nombres de Catalan, introduites par Zeilberger (Lassalle, 2010). Ces $q$-integers sont des polynômes palindromiques à $q$ à coefficients entiers positifs. La positivité dépend de la positivité d’une certaine différence de produits de $q$-coefficients binomial.Pour ce faire, je vous présente une nouvelle inversion/major index sur les chemins du réseau. La différence de $q$-binomial coefficients est alors considérée comme une fonction de génération de trajets pondérés qui restent dans le demi-plan supérieur.

Decomposability of symplectic matrices over principal ideal domain

The characteristic polynomial of any integral symplectic matrix is palindromic. First, we say that the inverse is also true, that is for any palindromic monic polynomial f(x) of even degree, there is an integral symplectic matrix whose characteristic polynomial is f(x). Furthermore, we give a sufficient condition on decomposability of integral symplectic matrices as a symplectic direct sum of two symplectic matrices which have smaller genera and coprime characteristic polynomials. Finally, we find the number of conjugacy classes of pq-torsion in the symplectic group of genus (p+q−2)/2 over rational integers, where p and q are distinct odd primes.

Noncommutative recursions and the Laurent phenomenon

We exhibit a family of sequences of noncommutative variables, recursively defined using monic palindromic polynomials in Q[x], and show that each possesses the Laurent phenomenon. This generalizes a conjecture by Kontsevich.