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This paper is devoted to studying the multiple recurrent property of topologically mildly mixing systems along generalized polynomials. We show that if a minimal system is topologically mildly mixing, then it is mild mixing of higher orders along generalized polynomials. Precisely, suppose that (X , T) is a topologically mildly mixing minimal system, d ∈ N , p 1 , ... , p d are integer-valued generalized polynomials with (p 1 , ... , p d) non-degenerate. Then for all non-empty open subsets U , V 1 , ... , V d of X , { n ∈ Z : U ∩ T − p 1 (n) V 1 ∩ ... ∩ T − p d (n) V d ≠ ∅ } is an IP⁎-set. [ABSTRACT FROM AUTHOR]

This paper devotes to the study of planar polynomial differential systems with homogeneous nonlinearities of degree n > 1. We are concerned with the maximum number of limit cycles surrounding the origin of such systems, denoted by H o (n). By means of the second order analysis using the theories of Melnikov functions, we provide new estimates for H o (n) restricted to the cases where the origin is a focus, a node, a saddle or a nilpotent singularity. In particular, H o (n) ≥ n for each n in the case of focus. To the best of our knowledge, this improves the previous works in the literature. [ABSTRACT FROM AUTHOR]

The maximum number of centers is an open problem proposed by Gasull for planar polynomial differential systems of degree n with n ≥ 4. In this paper we study the problem for planar polynomial Kolmogorov differential systems of degree n , prove that the maximum number of centers is exactly seven for planar quartic polynomial Kolmogorov differential systems, and give the upper and lower bound for the maximum number of centers that the Kolmogorov differential systems of degree n can have. [ABSTRACT FROM AUTHOR]

This paper investigates the expansions of Melnikov functions near a homoclinic loop with a nilpotent cusp of order m. It presents a methodology for calculating all coefficients in these expansions, which can be employed to study the problem of limit cycle bifurcation. As an application, by utilizing the obtained results, the paper rigorously establishes that a polynomial Liénard system of degree n + 1 has at least n + [ n 4 ] limit cycles near the homoclinic loop with a nilpotent cusp of order one. This work not only updates and generalizes existing results, but also provides a rigorous application of the obtained findings in the context of limit cycle bifurcation. [ABSTRACT FROM AUTHOR]

The aim of this paper is to investigate two important problems related to nilpotent center conditions and bifurcation of limit cycles in switching polynomial systems. Due to the difficulty in calculating the Lyapunov constants of switching polynomial systems at non-elementary singular points, it is extremely difficult to use the existing Poincaré-Lyapunov method to study these two problems. In this paper, we develop a higher-order Poincaré-Lyapunov method to consider the nilpotent center problem in switching polynomial systems, with particular attention focused on cubic switching Liénard systems. With proper perturbations, explicit center conditions are derived for switching Liénard systems at a nilpotent center. Moreover, with Bogdanov-Takens bifurcation theory, the existence of five limit cycles around the nilpotent center is proved for a class of switching Liénard systems, which is a new lower bound of cyclicity for such polynomial systems around a nilpotent center. [ABSTRACT FROM AUTHOR]

In this paper, we reduce pointwise convergence of polynomial ergodic averages of general measure-preserving systems acted by Z d to the case of measure-preserving systems acted by Z d with zero entropy. As an application, we can build pointwise convergence of polynomial ergodic averages for K -systems acted by Z d. [ABSTRACT FROM AUTHOR]

This paper focuses on the approximation of continuous functions on the unit sphere by spherical polynomials of degree n via hyperinterpolation. Hyperinterpolation of degree n is a discrete approximation of the L 2 -orthogonal projection of the same degree with its Fourier coefficients evaluated by a positive-weight quadrature rule that exactly integrates all spherical polynomials of degree at most 2 n. This paper aims to bypass this quadrature exactness assumption by replacing it with the Marcinkiewicz–Zygmund property proposed in a previous paper. Consequently, hyperinterpolation can be constructed by a positive-weight quadrature rule (not necessarily with quadrature exactness). This scheme is referred to as unfettered hyperinterpolation. This paper provides a reasonable error estimate for unfettered hyperinterpolation. The error estimate generally consists of two terms: a term representing the error estimate of the original hyperinterpolation of full quadrature exactness and another introduced as compensation for the loss of exactness degrees. A guide to controlling the newly introduced term in practice is provided. In particular, if the quadrature points form a quasi-Monte Carlo (QMC) design, then there is a refined error estimate. Numerical experiments verify the error estimates and the practical guide. [ABSTRACT FROM AUTHOR]

For any zero-dimensional polynomial ideal I and any nonzero polynomial F , this paper shows that the union of the multi-set of zeros of the ideal sum I + 〈 F 〉 and that of the ideal quotient I : 〈 F 〉 is equal to the multi-set of zeros of I , where zeros are counted with multiplicities. Based on this zero relation and the computation of Gröbner bases, a complete multiplicity-preserved algorithm is proposed to decompose any zero-dimensional polynomial set into finitely many squarefree normal triangular sets, resulting in a squarefree normal representation for the zeros of the polynomial set. In the representation the multiplicities of the zeros of the triangular sets can be read out directly. Examples and experiments are presented to illustrate the algorithm and its performance. [ABSTRACT FROM AUTHOR]

We develop the first algorithm for computing rational solutions of scalar integro-differential equations with polynomial coefficients. It starts by finding the possible poles of a rational solution. Then, bounding the order of each pole and solving an algebraic linear system, we compute the singular part of rational solutions at each possible pole. Finally, using partial fraction decomposition, the polynomial part of rational solutions is obtained by computing polynomial solutions of a non-homogeneous scalar integro-differential equation with a polynomial right-hand side. The paper is illustrated by examples where the computations are done with our Maple implementation. [ABSTRACT FROM AUTHOR]

Mixed trigonometric-polynomials (MTPs) are functions of the form f (x , sin ⁡ x , cos ⁡ x) where f is a trivariate polynomial with rational coefficients, and the argument x ranges over the reals. In this paper, an algorithm "isolating" all the real roots of an MTP is provided and implemented. It automatically divides the real roots into two parts: one consists of finitely many roots in an interval [ μ − , μ + ] while the other consists of countably many roots in R ﹨ [ μ − , μ + ]. For the roots in [ μ − , μ + ] , the algorithm returns isolating intervals and corresponding multiplicities while for those greater than μ + , it returns finitely many mutually disjoint small intervals I i ⊂ [ − π , π ] , integers c i > 0 and multisets of root multiplicity { m j , i } j = 1 c i such that any root > μ + is in the set (∪ i ∪ k ∈ N (I i + 2 k π)) and any interval I i + 2 k π ⊂ (μ + , ∞) contains exactly c i distinct roots with multiplicities m 1 , i ,... , m c i , i , respectively. The efficiency of the algorithm is shown by experiments. The method used to isolate the roots in [ μ − , μ + ] is applicable to any other bounded interval as well. The algorithm takes advantages of the weak Fourier sequence technique and deals with the intervals period-by-period without scaling the coordinate so to keep the length of the sequence short. The new approaches can easily be modified to decide whether there is any root, or whether there are infinitely many roots in unbounded intervals of the form (− ∞ , a) or (a , ∞) with a ∈ Q. [ABSTRACT FROM AUTHOR]

Gross, Mansour and Tucker introduced the partial-twuality polynomial of a ribbon graph. Chumutov and Vignes-Tourneret posed a problem: it would be interesting to know whether the partial duality polynomial and the related conjectures would make sense for general delta-matroids. In this paper we consider analogues of partial-twuality polynomials for delta-matroids. Various possible properties of partial-twuality polynomials of set systems are studied. We discuss the numerical implications of partial-twualities on a single element and prove that the intersection graphs can determine the partial-twuality polynomials of bouquets and normal binary delta-matroids, respectively. Finally, we give a characterization of vf-safe delta-matroids whose partial-twuality polynomials have only one term. [ABSTRACT FROM AUTHOR]

The reduction of a multidimensional system is closely related to the reduction of a multivariate polynomial matrix, for which the Smith normal form of the matrix plays a key role. In this paper, we investigate the reduction of weakly linear multivariate polynomial matrices to their Smith normal forms. Using hierarchical-recursive method and Quillen-Suslin Theorem, we derive some necessary and sufficient conditions ensuring that such matrices can be reduced to their Smith normal forms, and these conditions are easily checked by computing the reduced Gröbner bases of the relevant polynomial ideals. Based on the new results, we propose an algorithm for reducing weakly linear multivariate polynomial matrices to their Smith normal forms. [ABSTRACT FROM AUTHOR]

In this paper, we propose a symbolic-numeric algorithm to count the number of solutions of a zero-dimensional square polynomial system within a local region. We show that the algorithm succeeds under the condition that the region is sufficiently small and well-isolating for a k -fold solution z of the system. In our analysis, we derive a bound on the size of the region that guarantees success. We further argue that this size depends on local parameters such as the norm and multiplicity of z as well as the distances between z and all other solutions. Efficiency of our method stems from the fact that we reduce the problem of counting the roots of the original system to the problem of solving a truncated system of degree k. In particular, if the multiplicity k of z is small compared to the total degrees of the original polynomials, our method considerably improves upon known complete and certified methods. We see a series of applications of our approach. When combined with a numerical solver in the fashion of an a posteriori certification step, we obtain a certified and reliable method for solving polynomial systems while profiting both from the efficiency of the numerical algorithm and the reliability of the symbolic approach. An alternative application results from incorporating our algorithm as inclusion predicate into an elimination method. For the special case of bivariate systems, we experimentally show that this approach leads to a significant improvement over an existing state-of-the-art elimination method. [ABSTRACT FROM AUTHOR]

A paper by Boros, Little, Moll, Mosteig, and Stanley relates properties of a map defined on the space of rational functions to Eulerian polynomials. We link their work to the carries Markov chain, giving a new proof and slight generalization of one of their results. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]