Stable Polynomials Research Articles

Categorical actions and multiplicities in the Deligne category [formula omitted

We study the categorical type A action on the Deligne category Dt=Rep_(GLt) (t∈C) and its “abelian envelope” Vt constructed in [13].For t∈Z, this action categorifies an action of the Lie algebra slZ on the tensor product of the Fock space F with Ft∨, its restricted dual “shifted” by t, as was suggested by I. Losev. In fact, this action makes the category Vt the tensor product (in the sense of Losev and Webster, [20]) of categorical slZ-modules Pol and Polt∨. The latter categorify F and Ft∨ respectively, the underlying category in both cases being the category of stable polynomial representations (also known as the category of Schur functors), as described in [16,18].When t∉Z, the Deligne category Dt is abelian semisimple, and the type A action induces a categorical action of slZ×slZ. This action categorifies the slZ×slZ-module F⊠F∨, making Dt the exterior tensor product of the categorical slZ-modules Pol, Pol∨.Along the way we establish a new relation between the Kazhdan–Lusztig coefficients and the multiplicities in the standard filtrations of tilting objects in Vt.

On the Stability of Independence Polynomials

The independence polynomial of a graph is the generating polynomial for the number of independent sets of each size and its roots are called independence roots. We investigate the stability of such polynomials, that is, conditions under which the independence roots lie in the left half-plane. We use results from complex analysis to determine graph operations that result in a stable or nonstable independence polynomial. In particular, we prove that every graph is an induced subgraph of a graph with stable independence polynomial. We also show that the independence polynomials of graphs with independence number at most three are necessarily stable, but for larger independence number, we show that the independence polynomials can have roots arbitrarily far to the right.

DAHA and plane curve singularities

We suggest a relatively simple and totally geometric conjectural description of uncolored DAHA superpolynomials of arbitrary algebraic knots (conjecturally coinciding with the reduced stable Khovanov-Rozansky polynomials) via the flagged Jacobian factors (new objects) of the corresponding unibranch plane curve singularities. This generalizes the Cherednik-Danilenko conjecture on the Betti numbers of Jacobian factors, the Gorsky combinatorial conjectural interpretation of superpolynomials of torus knots and that by Gorsky-Mazin for their constant term. The paper mainly focuses on non-torus algebraic knots. A connection with the conjecture due to Oblomkov-Rasmussen-Shende is possible, but our approach is different. A motivic version of our conjecture is related to p-adic orbital A-type integrals for anisotropic centralizers.

Coincidences among skew stable and dual stable Grothendieck polynomials

The question of when two skew Young diagrams produce the same skew Schur function has been well-studied. We investigate the same question in the case of stable Grothendieck polynomials, which are the K-theoretic analogues of the Schur functions. We prove a necessary condition for two skew shapes to give rise to the same dual stable Grothendieck polynomial. We also provide a necessary and sufficient condition in the case where the two skew shapes are ribbons.

Riemann Hypothesis for DAHA superpolynomials and plane curve singularities

Stable Khovanov-Rozansky polynomials of algebraic knots are expected to coincide with certain generating functions, superpolynomials, of nested Hilbert schemes and flagged Jacobian factors of the corresponding plane curve singularities. Also, these 3 families conjecturally match the DAHA superpolynomials. These superpolynomials can be considered as singular counterparts and generalizations of the Hasse-Weil zeta-functions. We conjecture that all $a$-coefficients of the DAHA superpolynomials upon the substitution $q\mapsto qt$ satisfy the Riemann Hypothesis for sufficiently small $q$ for uncolored algebraic knots, presumably for $q\le 1/2$ as $a=0$. This can be partially extended to algebraic links at least for $a=0$. Colored links are also considered, though mostly for rectangle Young diagrams. Connections with Kapranov's motivic zeta and the Galkin-Stohr zeta-functions are discussed.

The Inverse Kullback–Leibler Method for Fitting Vector Moving Averages

A new method for the estimation of a vector moving average (VMA) process is presented. The technique uses Kullback–Leibler discrepancy with inverse spectra, and yields a Yule–Walker system of equations in inverse autocovariances for the VMA coefficients. This provides a direct formula for the coefficients, which always results in a stable matrix polynomial. The paper provides asymptotic results, as well as an analysis of the method's performance, in terms of speed, bias, and precision. Applications to preliminary estimation of VMA models are discussed, and the method is illustrated on retail data.

Comments on \u2018On Hadamard powers of polynomials\u2019

Let f(x)=a_n x^n + a_{n-1} x^{n-1} + cdots +a_1 x +a_0 be a polynomial with real positive coefficients and pin mathbb {R}. The pth Hadamard power of f is the polynomial f^{[p]}(x):=a_n^p x^n + a_{n-1}^p x^{n-1}+ cdots + a_1^p x +a_0^p. We give sufficient conditions for f^{[p]} to be a Hurwitz polynomial (i.e., to be a stable polynomial) for all p>p_0 or p<p_1 with some positive p_0 and negative p_1 (without any assumption about stability of f). Theorem 5 by Gregor and Tišer (Math Control Signals Syst 11:372–378, 1998) asserts that if f is a stable polynomial with positive coefficients then f^{[p]} is stable for every pge 1. We construct a counterexample to this statement.

Self-interlacing polynomials

We describe a new subclass of the class of real polynomials with real simple roots called self-interlacing polynomials. This subclass is isomorphic to the class of real Hurwitz stable polynomials (all roots in the open left half-plane). In the work, we present basic properties of self-interlacing polynomials and their relations with Hurwitz and Hankel matrices as well as with Stiltjes type of continued fractions. We also establish “self-interlacing” analogues of the well-known Hurwitz and Liénard–Chipart criterions for stable polynomials. A criterion of Hurwitz stability of polynomials in terms of minors of certain Hankel matrices is established.

Computing effectively stabilizing controllers for a class of nD systems

In this paper, we study the internal stabilizability and internal stabilization problems for multidimensional (nD) systems. Within the fractional representation approach, a multidimensional system can be studied by means of matrices with entries in the integral domain of structurally stable rational fractions, namely the ring of rational functions which have no poles in the closed unit polydisc Ūn = {z = (z1,..., zn) ϵ Cn | |z1| ≤ 1,..., |zn| ≤ 1}. It is known that the internal stabilizability of a multidimensional system can be investigated by studying a certain polynomial ideal I = (p1,...,pr) that can be explicitly described in terms of the transfer matrix of the plant. More precisely the system is stabilizable if and only if V(I) = {z ϵ Cn | p1(z) =...= pr(z) = 0} Ո Ūn = ø. In the present article, we consider the specific class of linear nD systems (which includes the class of 2D systems) for which the ideal I is zero-dimensional, i.e., the pi’s have only a finite number of common complex zeros. We propose effective symbolic-numeric algorithms for testing if V(I) Ո Ūn = ø, as well as for computing, if it exists, a stable polynomial p ϵ I which allows the effective computation of a stabilizing controller. We finally illustrate our algorithms on an example.

Polynomial Chaos-Based Macromodeling of General Linear Multiport Systems for Time-Domain Analysis

In this paper, we present a new effective methodology to build stochastic macromodels for the time-domain analysis of generic linear multiport systems. The proposed technique allows one to calculate a stable and passive polynomial chaos (PC)-based macromodel of a system under stochastic variations. The Galerkin projections method and a PC-based model of the system scattering parameters are used along with the Vector Fitting algorithm, leading to an accurate and efficient description of the system variability features. The proposed technique results to be very versatile and, thus, is well suited to be applied to many different and complex modern electrical systems (e.g., interconnections and filters). The accuracy and computational efficiency of the proposed technique are verified through comparison with the standard Monte Carlo analysis for two pertinent numerical examples, showing a maximum simulation speedup of 765 times.

A Littlewood–Richardson rule for dual stable Grothendieck polynomials

For a given skew shape, we build a crystal graph on the set of all reverse plane partitions that have this shape. As a consequence, we get a simple extension of the Littlewood–Richardson rule for the expansion of the corresponding dual stable Grothendieck polynomial in terms of Schur polynomials.

BIBO-Stable Recursive Functional Link Polynomial Filters

The preeminent merit of linear or nonlinear recursive filters is their ability to represent unknown systems with far fewer coefficients than their nonrecursive counterparts. Unfortunately, nonlinear recursive filters become, in general, unstable for large amplitudes of the input signal. In this paper, we introduce a novel subclass of the linear-in-the-parameters nonlinear recursive filters whose members are stable according to the bounded-input bounded-output criterion. It is shown that these filters, called stable recursive functional link polynomial filters, are universal approximators for causal, time-invariant, infinite-memory, continuous, nonlinear systems according to the Stone-Weierstrass theorem. To further reduce the filter complexity, simplified structures are also investigated. Even though simplified filters are no more universal approximators, good performance is demonstrated exploiting either data measured on real systems or available in the literature as benchmarks for nonlinear system identification.

Geoacoustic inversion for the seabed transition layer using a Bernstein polynomial model.

This paper develops an inversion method for the seabed transition layer at the water-sediment interface, often found in muddy sediments, which provides density and sound-speed profiles that were previously not resolvable. The resolution improvements are achieved by introducing a parametrization that captures general depth-dependent gradients in geoacoustic parameters with a small number of parameters. In particular, the gradients are represented by a sum of Bernstein basis functions, weighted by unknown coefficients. Compared to previous forms found in the literature, the Bernstein-based parametrization can represent a wider range of depth-dependent geoacoustic profiles using fewer parameters which leads to reduced uncertainty and improved resolution. In addition, the Bernstein basis is the most stable polynomial representation in that small perturbations to the unknown coefficients result in small, localized perturbations to the geoacoustic profile, thereby providing an efficient exploration of the parameter space using Markov-chain methods in nonlinear inversion. Geoacoustic profiles at four mud sites on the Malta Plateau are studied with the proposed approach. Results show exceptional resolution of density profiles, estimated with low uncertainty and clear sensitivity to sediment features of centimeter scale.

Duality and deformations of stable Grothendieck polynomials

Stable Grothendieck polynomials can be viewed as a K-theory analog of Schur polynomials. We extend stable Grothendieck polynomials to a two-parameter version, which we call canonical stable Grothendieck functions. These functions have the same structure constants (with scaling) as stable Grothendieck polynomials and (composing with parameter switching) are self-dual under the standard involutive ring automorphism. We study various properties of these functions, including combinatorial formulas, Schur expansions, Jacobi–Trudi-type identities, and associated Fomin–Greene operators.

Refined Dual Stable Grothendieck Polynomials and Generalized Bender-Knuth Involutions

The dual stable Grothendieck polynomials are a deformation of the Schur functions, originating in the study of the $K$-theory of the Grassmannian. We generalize these polynomials by introducing a countable family of additional parameters, and we prove that this generalization still defines symmetric functions. For this fact, we give two self-contained proofs, one of which constructs a family of involutions on the set of reverse plane partitions generalizing the Bender-Knuth involutions on semistandard tableaux, whereas the other classifies the structure of reverse plane partitions with entries $1$ and $2$.

Structure constants for K-theory of Grassmannians, revisited

The problem of computing products of Schubert classes in the cohomology ring can be formulated as the problem of expanding skew Schur polynomials into the basis of ordinary Schur polynomials. In contrast, the problem of computing the structure constants of the Grothendieck ring of a Grassmannian variety with respect to its basis of Schubert structure sheaves is not equivalent to expanding skew stable Grothendieck polynomials into the basis of ordinary stable Grothendiecks. Instead, we show that the appropriate K-theoretic analogy is through the expansion of skew reverse plane partitions into the basis of polynomials which are Hopf-dual to stable Grothendieck polynomials. We combinatorially prove this expansion is determined by Yamanouchi set-valued tableaux. A by-product of our results is a dual approach proof for Buch's K-theoretic Littlewood–Richardson rule for the product of stable Grothendieck polynomials.

Optimal least squares model approximation for large-scale linear discrete-time systems

A new computationally simple and precise model approximation method is described for large-scale linear discrete-time systems. By least squares matching of a suitable number of time moment proportionals and Markov parameters about [Formula: see text] of the original higher order system within the approximate model, stable denominator polynomial coefficients of the approximate model are determined. To improvise the accuracy of the approximate model, numerator polynomial coefficients are determined by minimizing the integral squared error (ISE) between the unit impulse responses of the original system and its approximate model. A matrix formula is formulated for evaluating numerator coefficients of the approximate model that leads to minimum ISE, and also for evaluating ISE. The efficacy of the proposed method is shown by illustrating three typical numerical examples employed from the literature, and the results are compared with many familiar reduction methods in terms of the ISE and relative ISE values pertaining to impulse input. Furthermore, time and frequency responses of the original system and the respective approximate model are plotted.

DAHA and iterated torus knots

The theory of DAHA-Jones polynomials is extended from torus knots to their arbitrary iterations (for any reduced root systems and weights), which incudes the polynomiality, duality and other properties of the DAHA superpolynomials. Presumably they coincide with the reduced stable Khovanov-Rozansky polynomials in the case of non-negative coefficients. The new theory matches well the classical theory of algebraic knots and (unibranch) plane curve singularities; the Puiseux expansion naturally emerges. The corresponding DAHA superpolynomials are expected to coincide with the reduced ones in the Oblomkov-Shende-Rasmussen Conjecture upon its generalization to arbitrary dominant weights. For instance, the DAHA uncolored superpolynomials at a=0, q=1 are conjectured to provide the Betti numbers of the Jacobian factors of the corresponding singularities.

Notes on Schubert, Grothendieck and Key Polynomials

We introduce common generalization of (double) Schubert, Grothendieck, Demazure, dual and stable Grothendieck polynomials, and Di Francesco-Zinn-Justin polynomials. Our approach is based on the study of algebraic and combinatorial properties of the reduced rectangular plactic algebra and associated Cauchy kernels.

Robust Stability by Path-Connectivity of Fractional Order Polynomials

Given any pair of stable fractional order polynomials with fixed commensurate order ź$\alpha $, for 0<ź<2$0<\alpha <2$, a stable path joining them is provided by mean of convex combinations in the arguments and magnitudes of their roots. It is shown that any pair of paths obtained with this technique are homotopically equivalent. Consequently, a dense trajectory in the stable fractional order polynomials set has been exhibited. Finally, a stabilizing feedback control in terms of the continuous coefficients of the path has been designed and an illustrative example is given.