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In this paper, we study the asymptotic behavior of transmission system for coupled Kirchhoff plates, where one equation is conserved and the other has dissipative property, and the dissipation mechanism is given by fractional damping (- Δ) 2 θ v t with θ ∈ [ 1 2 , 1 ] . By using the semigroup method and the multiplier technique, we obtain the exact polynomial decay rates, and find that the polynomial decay rate of the system is determined by the inertia/elasticity ratios and the fractional damping order. Specifically, when the inertia/elasticity ratios are not equal and θ ∈ [ 1 2 , 3 4 ] , the polynomial decay rate of the system is t - 1 / (10 - 4 θ) . When the inertia/elasticity ratios are not equal and θ ∈ [ 3 4 , 1 ] , the polynomial decay rate of the system is t - 1 / (4 + 4 θ) . When the inertia/elasticity ratios are equal, the polynomial decay rate of the system is t - 1 / (4 + 4 θ) . Furthermore it has been proven that the obtained decay rates are all optimal. The obtained results extend the results of Oquendo and Suárez (Z Angew Math Phys 70(3):88, 2019) for the case of fractional damping exponent 2 θ from [0, 1] to [1, 2]. [ABSTRACT FROM AUTHOR]

The higher-order guaranteed lower eigenvalue bounds of the Laplacian in the recent work by Carstensen et al. (Numer Math 149(2):273–304, 2021) require a parameter C st , 1 that is found not robust as the polynomial degree p increases. This is related to the H 1 stability bound of the L 2 projection onto polynomials of degree at most p and its growth C st, 1 ∝ (p + 1) 1 / 2 as p → ∞ . A similar estimate for the Galerkin projection holds with a p-robust constant C st , 2 and C st , 2 ≤ 2 for right-isosceles triangles. This paper utilizes the new inequality with the constant C st , 2 to design a modified hybrid high-order eigensolver that directly computes guaranteed lower eigenvalue bounds under the idealized hypothesis of exact solve of the generalized algebraic eigenvalue problem and a mild explicit condition on the maximal mesh-size in the simplicial mesh. A key advance is a p-robust parameter selection. The analysis of the new method with a different fine-tuned volume stabilization allows for a priori quasi-best approximation and improved L 2 error estimates as well as a stabilization-free reliable and efficient a posteriori error control. The associated adaptive mesh-refining algorithm performs superior in computer benchmarks with striking numerical evidence for optimal higher empirical convergence rates. [ABSTRACT FROM AUTHOR]

In this paper, we consider a constructive proof of the termination of the normal form (NF) algorithm for multivariate polynomials, as well as the related concept of admissible ordering < on monomials. In classical mathematics, the well-quasiorder property of relation < is derived from Dickson's lemma, and this is sufficient to justify the termination of the NF algorithm. In provable programming based on constructive type theory (Coq and Agda), a somewhat stronger condition (in constructive mathematics) of the well-foundedness of the ordering (in its constructive version) is required. We propose a constructive proof of this theorem (T) for < , which is based on a known method that we refer to here as the "pattern method." This theorem on the well-foundedness of an arbitrary admissible ordering is also important in itself, independently of the NF algorithm. We are not aware of any other works on constructive proof of this theorem. However, it turns out that it follows, not very difficultly, from the results achieved by other researchers in 2003. We program this proof in the Agda language in the form of our library AdmissiblePPO-wellFounded of provable computational algebra programs. This development also uses the theorem to prove termination of the NF algorithm for polynomials. Thus, the library also contains a set of provable programs for polynomial algebra, which is significantly larger than that needed to prove Theorem T. [ABSTRACT FROM AUTHOR]

This paper finds new quasi-polynomials over Z for the number p k (n) of partitions of n with parts at most k. Methods throughout are elementary. We derive a small number of polynomials (e.g., one for k = 3 , two for k = 4 or 5, six for k = 6 ) that, after addition of appropriate constant terms, take the value p k (n) . For example, for 0 ≤ r < 6 and for all q ≥ 0 , p 3 (6 q + r) = p 3 (r) + π 0 (q , r) , a polynomial of total degree 2 in q and r. In general there are M ⌊ k / 2 ⌋ = lcm { 1 , 2 , ... , ⌊ k / 2 ⌋ } such polynomials. In two variables q and s, they take the form ∑ a i , j q i s j with a i , j ∈ Z , which we call the proper form for an integer-valued polynomial. They constitute a quasi-polynomial of period M ⌊ k / 2 ⌋ for the sequence (p k (n) - p k (r)) with n ≡ r (mod M k) . For each k the terms of highest total degree are the same in all the polynomials and have coefficients dependent only on k. A second theorem, combining partial fractions and the above approach, finds hybrid polynomials over Q for p k (n) that are easier to determine than those above. We compare our results to those of Cayley, MacMahon, and Arkin, whose classical results, as recast here, stand up well. We also discuss recent results of Munagi and conclude that circulators in some form are inevitable. At k = 6 we find serious errors in Sylvester's calculation of his "waves." Sylvester JJ (Q J Pure Appl Math 1:141–152, 1855). The results are generalized to the (not very different) problem called "making change," where significant improvements to existing approaches are found. We find an infinitude of new congruences for p k (n) for k = 3 , 4 , and one new one for k = 5 . Reduced modulo m the periodic sequence (p k (n)) is investigated for periodicity and zeros: we find, from scratch, a simple proof of a known result in a special case. [ABSTRACT FROM AUTHOR]

We study the problem of approximating the cone of positive semidefinite (PSD) matrices with a cone that can be described by smaller-sized PSD constraints. Specifically, we ask the question: "how closely can we approximate the set of unit-trace n × n PSD matrices, denoted by D, using at most N number of k × k PSD constraints?" In this paper, we prove lower bounds on N to achieve a good approximation of D by considering two constructions of an approximating set. First, we consider the unit-trace n × n symmetric matrices that are PSD when restricted to a fixed set of k-dimensional subspaces in R n . We prove that if this set is a good approximation of D, then the number of subspaces must be at least exponentially large in n for any k = o (n) . Second, we show that any set S that approximates D within a constant approximation ratio must have superpolynomial S + k -extension complexity. To be more precise, if S is a constant factor approximation of D, then S must have S + k -extension complexity at least exp (C · min { n , n / k }) where C is some absolute constant. In addition, we show that any set S such that D ⊆ S and the Gaussian width of S is at most a constant times larger than the Gaussian width of D must have S + k -extension complexity at least exp (C · min { n 1 / 3 , n / k }) . These results imply that the cone of n × n PSD matrices cannot be approximated by a polynomial number of k × k PSD constraints for any k = o (n / log 2 n) . These results generalize the recent work of Fawzi (Math Oper Res 46(4):1479–1489, 2021) on the hardness of polyhedral approximations of S + n , which corresponds to the special case with k = 1 . [ABSTRACT FROM AUTHOR]

Theory of quasi-orthogonal polynomials is significantly related to constructing vector Padé approximations. The present paper introduces an efficient procedure to compute adjacent families of quasi-orthogonal polynomials as defined in Sadaka (Appl. Numer. Math. 21:57–70, 1996) and Sadaka (Appl. Numer. Math. 24:483–499, 1997). The derived algorithm uses short recursive relations whose coefficients are written in terms of lower triangular block determinants. The strategy of computing such determinants is given and analyzed. [ABSTRACT FROM AUTHOR]

In this paper we develop new tensor methods for unconstrained convex optimization, which solve at each iteration an auxiliary problem of minimizing convex multivariate polynomial. We analyze the simplest scheme, based on minimization of a regularized local model of the objective function, and its accelerated version obtained in the framework of estimating sequences. Their rates of convergence are compared with the worst-case lower complexity bounds for corresponding problem classes. Finally, for the third-order methods, we suggest an efficient technique for solving the auxiliary problem, which is based on the recently developed relative smoothness condition (Bauschke et al. in Math Oper Res 42:330–348, 2017; Lu et al. in SIOPT 28(1):333–354, 2018). With this elaboration, the third-order methods become implementable and very fast. The rate of convergence in terms of the function value for the accelerated third-order scheme reaches the level O 1 k 4 , where k is the number of iterations. This is very close to the lower bound of the order O 1 k 5 , which is also justified in this paper. At the same time, in many important cases the computational cost of one iteration of this method remains on the level typical for the second-order methods. [ABSTRACT FROM AUTHOR]

A graph is well-covered if all its maximal independent sets are of the same cardinality (Plummer in J Comb Theory 8:91-98, 1970). If G is a well-covered graph with at least two vertices, and $$G{-}v$$ is well-covered for every vertex v, then G is a 1- well-covered graph (Staples in On some subclasses of well-covered graphs. Ph.D. Thesis, Vanderbilt University, 1975). The graph G is $$\lambda $$ - quasi-regularizable if $$\lambda >0$$ and $$\lambda \cdot \vert S\vert \le \vert N( S) \vert $$ for every independent set S of G. It is known that every well-covered graph without isolated vertices is 1-quasi-regularizable (Berge in Ann Discret Math 12:31-44, 1982). The independence polynomial $$I(G;x)= {\sum _{k=0}^{\alpha }} s_{k}x^{k}$$ is the generating function of independent sets in a graph G (Gutman and Harary in Util Math 24:97-106, 1983), where $$\alpha $$ is the independence number of G. The Roller-Coaster Conjecture (Michael and Traves in Graphs Comb 19:403-411, 2003), saying that for every permutation $$\sigma $$ of the set $$\{\lceil \frac{\alpha }{2}\rceil ,\ldots ,\alpha \}$$ there exists a well-covered graph G with the independence number $$\alpha $$ such that the coefficients $$( s_{k}) $$ of I( G; x) satisfy has been validated in Cutler and Pebody (J Comb Theory A 145:25-35, 2017). In this paper we show that independence polynomials of $$\lambda $$ -quasi-regularizable graphs are partially unimodal. More precisely, the coefficients of an upper part of I( G; x) are in non-increasing order. Based on this finding, we prove that the unconstrained part of the independence sequence is: for well-covered graphs, and for 1-well-covered graphs, where $$\alpha $$ stands for the independence number, and n is the cardinality of the vertex set. [ABSTRACT FROM AUTHOR]

Let P(z) be a polynomial of degree n which does not vanish in | z | < 1 . Then it was proved by Hans and Lal (Anal Math 40:105–115, 2014) that | z s P (s) + β n s 2 s P (z) | ≤ n s 2 (| 1 + β 2 s | + | β 2 s |) max | z | = 1 | P (z) | , for every β ∈ C with | β | ≤ 1 , 1 ≤ s ≤ n and | z | = 1. The L γ analog of the above inequality was recently given by Gulzar (Anal Math 42:339–352, 2016) who under the same hypothesis proved { ∫ 0 2 π | e i s θ P (s) (e i θ) + β n s 2 s P (e i θ) | γ d θ } 1 γ ≤ n s { ∫ 0 2 π | (1 + β 2 s ) e i α + β 2 s | γ d α } 1 γ { ∫ 0 2 π | P (e i θ) | γ d θ } 1 γ { ∫ 0 2 π | 1 + e i α | γ d α } 1 γ , where n s = n (n - 1) ... (n - s + 1) and 0 ≤ γ < ∞ . In this paper, we generalize this and some other related results. [ABSTRACT FROM AUTHOR]

For rational functions , where is a polynomial of degree at the most and , with we use simple but elegant techniques to strengthen generalizations of certain results which extend some widely known polynomial inequalities of Erdős-Lax and Turán to rational functions . In return these reinforced results, in the limiting case, lead to the corresponding refinements of the said polynomial inequalities. As an illustration and as an application of our results, we obtain some new improvements of the Erdős-Lax and Turán type inequalities for polynomials. These improved results take into account the size of the constant term and the leading coefficient of the given polynomial. As a further factor of consideration, during the course of this paper we will demonstrate how some recently obtained results could have been proved without invoking the results of Dubinin [Distortion theorems for polynomials on the circle, Sb. Math. 191(12) (2000) 1797–1807]. [ABSTRACT FROM AUTHOR]

We propose a doubly nonnegative (DNN) relaxation for polynomial optimization problems (POPs) with binary and box constraints. This work is an extension of the work by Kim, Kojima and Toh in 2016 from quadratic optimization problems to POPs. The dense and sparse DNN relaxations are reduced to a simple conic optimization problem (COP) to which an accelerated bisection and projection (BP) algorithm is applied. The COP involves a single equality constraint in a matrix variable which is restricted to the intersection of the positive semidefinite cone and a polyhedral cone representing the algebraic properties of binary and box constraints as well as the sparsity structure of the objective polynomial. Our DNN relaxation serves as a variant of the Lasserre moment-SOS hierarchy of relaxations for binary and box constrained POPs when the size of the cones is gradually increased to compute tighter lower bounds for their optimal values. A significant part of the paper is devoted to deriving and analyzing a class of polyhedral cones to strengthen the DNN relaxation, and the design of efficient computation of the metric projections onto these cones in the accelerated BP algorithm. Numerical results on various POPs are available in Ito et al. (ACM Trans Math Softw 45, Article 34, 2019). [ABSTRACT FROM AUTHOR]

The aim of this paper is to fill in the gaps in the formulation and the proof of a theorem contained in the paper by K.Ozeki (Aequ Math 25:247–252, 1982) published in this journal. We also give a short proof of this theorem and use it to obtain certain information about the factorization of polynomials of the form f (x) - f (y) . [ABSTRACT FROM AUTHOR]

The aim of this paper is to prove that Markov's theorem on variation of zeros of orthogonal polynomials on the real line (Markoff in Math Ann 27:177–182, 1886) remains essentially valid in the case of paraorthogonal polynomials on the unit circle. [ABSTRACT FROM AUTHOR]

The classical result of L. Székelyhidi states that (under some assumptions) every solution of a general linear equation must be a polynomial function. It is known that Székelyhidi's result may be generalized to equations where some occurrences of the unknown functions are multiplied by a linear combination of the variables. In this paper we study the equations where two such combinations appear. The simplest nontrivial example of such a case is given by the equation F (x + y) - F (x) - F (y) = y f (x) + x f (y) considered by Fechner and Gselmann (Publ Math Debrecen 80(1–2):143–154, 2012). In the present paper we prove several results concerning the systematic approach to the generalizations of this equation. [ABSTRACT FROM AUTHOR]

In the theory of orthogonal polynomials, as well as in its intersection with harmonic analysis, it is an important problem to decide whether a given orthogonal polynomial sequence (P n (x)) n ∈ N 0 satisfies nonnegative linearization of products, i.e., the product of any two P m (x) , P n (x) is a conical combination of the polynomials P | m - n | (x) , ... , P m + n (x) . Since the coefficients in the arising expansions are often of cumbersome structure or not explicitly available, such considerations are generally very nontrivial. Gasper (Can J Math 22:582–593, 1970) was able to determine the set V of all pairs (α , β) ∈ (- 1 , ∞) 2 for which the corresponding Jacobi polynomials (R n (α , β) (x)) n ∈ N 0 , normalized by R n (α , β) (1) ≡ 1 , satisfy nonnegative linearization of products. Szwarc (Inzell Lectures on Orthogonal Polynomials, Adv. Theory Spec. Funct. Orthogonal Polynomials, vol 2, Nova Sci. Publ., Hauppauge, NY pp 103–139, 2005) asked to solve the analogous problem for the generalized Chebyshev polynomials (T n (α , β) (x)) n ∈ N 0 , which are the quadratic transformations of the Jacobi polynomials and orthogonal w.r.t. the measure (1 - x 2) α | x | 2 β + 1 χ (- 1 , 1) (x) d x . In this paper, we give the solution and show that (T n (α , β) (x)) n ∈ N 0 satisfies nonnegative linearization of products if and only if (α , β) ∈ V , so the generalized Chebyshev polynomials share this property with the Jacobi polynomials. Moreover, we reconsider the Jacobi polynomials themselves, simplify Gasper's original proof and characterize strict positivity of the linearization coefficients. Our results can also be regarded as sharpenings of Gasper's one. [ABSTRACT FROM AUTHOR]

In this paper, we mainly investigate upper semicontinuity and regularity of attractors for nonclassical diffusion equations with perturbed parameters ν and the nonlinear term f satisfying the polynomial growth of arbitrary order p − 1 (p ≥ 2 ). We extend the asymptotic a priori estimate method (see (Wang et al. in Appl. Math. Comput. 240:51–61, 2014)) to verify asymptotic compactness and upper semicontinuity of a family of semigroups for autonomous dynamical systems (see Theorems 2.2 and 2.3). By using the new operator decomposition method, we construct asymptotic contractive function and obtain the upper semicontinuity for our problem, which generalizes the results obtained in (Wang et al. in Appl. Math. Comput. 240:51–61, 2014). In particular, the regularity of global attractors is obtained, which extends and improves some results in (Xie et al. in J. Funct. Spaces 2016:5340489, 2016; Xie et al. in Nonlinear Anal. 31:23–37, 2016). [ABSTRACT FROM AUTHOR]

In this paper, we first present a family of iterative algorithms for simultaneous determination of all zeros of a polynomial. This family contains two well-known algorithms: Dochev-Byrnev's method and Ehrlich's method. Second, using Proinov's approach to studying convergence of iterative methods for polynomial zeros, we provide a semilocal convergence theorem that unifies the results of Proinov (Appl. Math. Comput. 284: 102-114, 2016) for Dochev-Byrnev's and Ehrlich's methods. [ABSTRACT FROM AUTHOR]

In this paper, a kind of univariate multiquadric quasi-interpolants with the derivatives of approximated function is proposed by combining a univariate multiquadric quasi-interpolant with Lidstone interpolation polynomials proposed in Lidstone (Proc Edinb Math Soc 2:16–19, 1929), Costabile and Dell' Accio (App Numer Math 52:339–361, 2005) and Catinas (J Appl Funct Anal 4:425–439, 2006). For practical purposes, another kind of approximation operators without any derivative of the approximated function is given using divided differences to approximate the derivatives. Some error bounds and the convergence rates of new operators are derived, which demonstrates that our operators could provide the desired precision by choosing a suitable shape-preserving parameter c and a non-negative integer n. Finally, we make extensive comparison with the other existing methods and give some numerical examples. Moreover, the associated algorithm is easily implemented. [ABSTRACT FROM AUTHOR]

The isospectral deformations of the Frobenius–Stickelberger–Thiele (FST) polynomials introduced in Spiridonov et al. (Commun Math Phys 272:139–165, 2007) are studied. For a specific choice of the deformation of the spectral measure, one is led to an integrable lattice (FST lattice), which is indeed an isospectral flow connected with a generalized eigenvalue problem. In the second part of the paper the spectral problem used previously in the study of the modified Camassa–Holm (mCH) peakon lattice is interpreted in terms of the FST polynomials together with the associated FST polynomials, resulting in a map from the mCH peakon lattice to a negative flow of the finite FST lattice. Furthermore, it is pointed out that the degenerate case of the finite FST lattice unexpectedly maps to the interlacing peakon ODE system associated with the two-component mCH equation studied in Chang et al. (Adv Math 299:1–35, 2016). [ABSTRACT FROM AUTHOR]

In this paper, we consider a system of two viscoelastic equations with Dirichlet boundary conditions. For certain class of relaxation functions and initial data, we establish general and optimal decay results. This result extends earlier one of Liu (Nonlinear Anal. TMA 71:2257–2267, 2010), in which only the usual exponential and polynomial decay rates are considered. The conditions of the relaxation functions g 1 (t) and g 2 (t) in our work appeared first in Messaoudi and Khulaifi (Appl. Math. Lett. 66:16–22, 2017). [ABSTRACT FROM AUTHOR]

An old conjecture of Erdős and Graham states that only finitely many integer squares could be obtained from product of disjoint blocks of consecutive integers of length greater than or equal to four. It is known by counterexamples that the conjecture is false for product of disjoint blocks of four and five consecutive integers. In this paper, we present new algorithms generating new polynomial parametrizations that extend the polynomial parametrization given by Bennett and Luijk (Indag Math (N.S.) 23(1–2):123–127, 2012). Moreover, we produce the first examples of integer squares obtained from product of disjoint blocks of consecutive integers such that each block has length six or seven. [ABSTRACT FROM AUTHOR]

The paper continues (Shcherbina and Shcherbina in Commun Math Phys 351:1009–1044, 2017); Shcherbina in Commun Math Phys 328:45–82, 2014) which study the behaviour of second correlation function of characteristic polynomials of the special case of n × n one-dimensional Gaussian Hermitian random band matrices, when the covariance of the elements is determined by the matrix J = (- W 2 ▵ + 1) - 1 . Applying the transfer matrix approach, we study the case when the bandwidth W is proportional to the threshold n . [ABSTRACT FROM AUTHOR]

This paper is devoted to the uniqueness of q-difference-differential polynomials of different types. Using the idea of common zeros and common poles (Chin. Ann. Math., Ser. A 35:675–684, 2014), we improve the conditions of the former theorems and obtain some new results on the uniqueness of q-difference-differential polynomials of meromorphic functions. [ABSTRACT FROM AUTHOR]

Let D be a closed disk in the complex plane centered at the origin and f, g be a complex-valued continuous function on D. Let P[f, g; D] (res. R[f, g; D])) be the uniform closure on D of polynomials (res. rational functions) in variables f and g. In O'Farrell et al. (Bull Lond Math Soc 34(4):490–494, 2002), using complex dynamical systems, O'Farrell and Sanabria-Garcia proved that (z 2 , z ¯ 1 + z ¯ ) : z ∈ D is not polynomially convex when D is sufficiently small and that P z 2 , z ¯ 1 + z ¯ ; D ≠ C (D) if D is sufficiently small. In this paper, we first give certain conditions for rational convexity for the union of two compact sets of C n and apply them to show that R z 2 , z ¯ 1 + z ¯ ; D = C (D) for all sufficiently small Ds. [ABSTRACT FROM AUTHOR]

The notion of coupled fixed point was introduced in 2006 by Bhaskar and Lakshmikantham. On the other hand, Filipovićet al. [M. Filipovićet al., 'Remarks on 'Cone metric spaces and fixed-point theorems of T-Kannan and T-Chatterjea contractive mappings',' Math. Comput. Modelling 54, 1467-1472 (2011)] proved several fixed and periodic point theorems for solid cones on cone metric spaces. In this paper we prove some coupled fixed-point theorems for certain T-contractions and study the existence of solutions of a system of nonlinear integral equations using the results of our work. The results of this paper extend and generalize well-known comparable results in the literature. [ABSTRACT FROM AUTHOR]

Preferential attachment networks with power law exponent τ>3 are known to exhibit a phase transition. There is a value ρc>0 such that, for small edge densities ρ≤ρc every component of the graph comprises an asymptotically vanishing proportion of vertices, while for large edge densities ρ>ρc there is a unique giant component comprising an asymptotically positive proportion of vertices. In this paper we study the decay in the size of the giant component as the critical edge density is approached from above. We show that the size decays very rapidly, like exp(-c/ρ-ρc) for an explicit constant c>0 depending on the model implementation. This result is in contrast to the behaviour of the class of rank-one models of scale-free networks, including the configuration model, where the decay is polynomial. Our proofs rely on the local neighbourhood approximations of Dereich and Mörters (Ann Probab 41(1):329-384, 2013) and recent progress in the theory of branching random walks (Gantert et al. in Ann Inst Henri Poincaré Probab Stat 47(1):111-129, 2011). [ABSTRACT FROM AUTHOR]

In this paper, we consider polytopes P that are contained in the unit hypercube. We provide conditions on the set of 0, 1 vectors not contained in P that guarantee that P has a small Chvátal rank. Our conditions are in terms of the subgraph induced by these infeasible 0, 1 vertices in the skeleton graph of the unit hypercube. In particular, we show that when this subgraph contains no 4-cycle, the Chvátal rank is at most 3; and when it has tree width 2, the Chvátal rank is at most 4. We also give polyhedral decomposition theorems when this graph has a vertex cutset of size one or two. [ABSTRACT FROM AUTHOR]

We examine the family of all (at most) three-point symmetric quadratures on [ - 1 , 1 ] which are exact on polynomials of order 3 to find all possible inequalities between them in the class of 3-convex functions. Next we optimise them by using convex combinations of the quadratures considered. We find the optimal quadrature and use it to construct the adaptive method of approximate integration. An effective method to estimate the error of this method is also given. It needs a considerably fewer number of subdivisions of the interval of integration than the classical adaptive methods as well as the method developed by the second-named author in his recent paper (Wa̧sowicz in Aequ Math 94(5):887–898, 2020). [ABSTRACT FROM AUTHOR]

The concept of the minimal polyanalytic polynomial was introduced by M. Huhtanen in connection with the generalized Lanczos process as applied to a normal matrix. The paper discusses the possibility of finding an equivalent of the characteristic polynomial in the set of polyanalytic polynomials. [ABSTRACT FROM AUTHOR]

Tropical algebra is an emerging field with a number of applications in various areas of mathematics. In many of these applications appeal to tropical polynomials allows studying properties of mathematical objects such as algebraic varieties from the computational point of view. This makes it important to study both mathematical and computational aspects of tropical polynomials. In this paper we prove a tropical Nullstellensatz, and moreover, we show an effective formulation of this theorem. Nullstellensatz is a natural step in building algebraic theory of tropical polynomials and its effective version is relevant for computational aspects of this field. On our way we establish a simple formulation of min-plus and tropical linear dualities. We also observe a close connection between tropical and min-plus polynomial systems. [ABSTRACT FROM AUTHOR]

For positive integers N and M, the general hypergeometric Cauchy polynomials c ( z) ( M, N ≥ 1; n ≥ 0) are defined by where $${{}_2 F_1(a,b;c;z)}$$ is the Gauss hypergeometric function. When M = N = 1, c = c are the classical Cauchy numbers. In 1875, Glaisher gave several interesting determinant expressions of numbers, including Bernoulli, Cauchy and Euler numbers. In the aspect of determinant expressions, hypergeometric Cauchy numbers are the natural extension of the classical Cauchy numbers, though many kinds of generalizations of the Cauchy numbers have been considered by many authors. In this paper, we show some interesting expressions of generalized hypergeometric Cauchy numbers. We also give a convolution identity for generalized hypergeometric Cauchy polynomials. [ABSTRACT FROM AUTHOR]

An $$m\times n\,\{0,1\}$$ -matrix $$A$$ is said to be totally convertible if there exists a matrix $$B$$ obtained from $$A$$ by changing some 1's in $$A$$ to $$-1$$ 's such that for any submatrix $$A^{\prime }$$ of $$A$$ of order $$m$$ , the corresponding submatrix $$B^{\prime }$$ of $$B$$ satisfies $$\mathrm{per}(xI-A^{\prime })=\det (xI-B^{\prime })$$ . In this paper, motivated by the well-known Pólya's problem, our object is to characterize those totally convertible matrices. Associate a matrix $$A$$ with a bipartite graph $$G^{*}_A$$ . We first prove that a square matrix $$A$$ is totally convertible if and only if $$G^{*}_A$$ is Pfaffian, and then we generalize this result to an $$m\times n$$ $$\{0,1\}$$ -matrix. Moreover, the characterization of a totally convertible matrix provides an equivalent condition to compute the permanental polynomial of a bipartite graph by the characteristic polynomial of the skew adjacency matrix of its orientation graph. As applications, we give some explicit expressions of the permanental polynomials of two totally convertible matrices by the technique of Pfaffian orientation. [ABSTRACT FROM AUTHOR]

In this paper, some converses of Jessen's inequality, in terms of the time scale calculus, are proved. The obtained results are used to derive new converse inequalities for generalized means, power means and Hölder's inequality. [ABSTRACT FROM AUTHOR]

It was a question of Taibleson, open for a long time that the almost everywhere convergence of Fejér (or ( C, 1)) means of Fourier series of integrable functions with respect the character system of the group of 2-adic integers. This question was answered by Gá t in 1997. The aim of this paper is to investigate the maximal operator of the sup |σ|. Among other things, we prove that this operator is bounded from the Hardy space H to the Lebesgue space L if and only if 1/2 < p < ∞. The two-dimensional maximal operator is also discussed. [ABSTRACT FROM AUTHOR]

Earlier, for odd n, the author described unitary n × n matrices U with the following property: if H is a Hankel matrix, then U* HU is a Hankel matrix as well. In the present paper, it is shown that this description also holds for even n starting from n = 4. [ABSTRACT FROM AUTHOR]

Formulas for the regularized trace of the one-dimensional non-self-adjoint Dirac operator with L-potential are obtained. The cases of periodic and antiperiodic boundary conditions as well as of the Dirichlet boundary conditions are considered. The formulas are obtained by using the method of similar operators on the basis of results from the papers [1] and [2]. [ABSTRACT FROM AUTHOR]

Let g n , n = 1 , 2 , ... , be the logarithmic derivative of a complex polynomial having all zeros on the unit circle, i.e., a function of the form g n (z) = (z - z 1) - 1 + ⋯ + (z - z n) - 1 , | z 1 | = ⋯ = | z n | = 1 . For any p > 0 , we establish the bound ∫ - 1 1 | g n (x) | p d x > C p n p - 1 , sharp in the order of the quantity n, where C p > 0 is a constant, depending only on p. The particular case p = 1 of this inequality can be considered as a stronger variant of the well-known estimate ∬ | z | < 1 | g n (z) | d x d y > c > 0 for the area integral of g n , obtained by Newman (Am Math Mon 79(9):1015–1016, 1972). The result also shows that the set { g n } is not dense in the spaces L p [ - 1 , 1 ] , p ≥ 1 . [ABSTRACT FROM AUTHOR]

We consider the following variational problem: minimize the (n + 1) st polynomial conserved quantity of KdV over H n (R) with the first n conserved quantities constrained. Maddocks and Sachs (Comm Pure Appl Math 46:867–901, 1993) used that n-solitons are local minimizers for this problem in order to prove that n-solitons are orbitally stable in H n (R) . Given n constraints that are attainable by an n-soliton, we show that there is a unique set of n amplitude parameters so that the corresponding multisolitons satisfy the constraints. Moreover, we prove that these multisolitons are the unique global constrained minimizers. We then use this variational characterization to provide a new proof of the orbital stability result from Maddocks and Sachs (Comm Pure Appl Math 46:867–901, 1993) via concentration compactness. In the case when the constraints can be attained by functions in H n (R) but not by an n-soliton, we discover new behavior for minimizing sequences. [ABSTRACT FROM AUTHOR]

We revisit the classical subject of equidistribution of the roots of Littlewood-type polynomials. More precisely, we show that the roots of the family of polynomials Ψk(z) = zk−zk−1−⋯−1, k ⩾ 1, are uniformly distributed around the unit circle in the strong quantitative form, confirming a conjecture from [C.-A. Gómez and F. Luca, Commentat. Math. Univ. Carol., On the distribution of roots of zk − zk−1−⋯−z − 1, 62(3):291–296, 2021]. [ABSTRACT FROM AUTHOR]

We recall the 'addition formula' for a commutative association scheme. From this formula we construct a linear programming for the size of harmonic index designs and define the notion of tight designs using Fisher type inequality. We examine the existence of tight harmonic index T-designs for some T in binary Hamming association schemes H( d, 2). [ABSTRACT FROM AUTHOR]

To improve the MRPSS preconditioner developed by Cao et al. (BIT Numer Math 56:423–439, 2016), a generalized variant of the MRPSS (GMRPSS) preconditioner is proposed. The invertibility of the GMRPSS preconditioner, the convergence of the GMRPSS iteration method and some properties of the corresponding preconditioned matrix are presented. The degree of minimal polynomials of the preconditioned matrix is obtained. Simultaneously, the algorithm implementation and the selection of the parameter of the GMRPSS preconditioner are discussed. Numerical experiments are performed to illustrate the feasibility of the new preconditioner. [ABSTRACT FROM AUTHOR]

In the Maximum Minimal Vertex Cover (MMVC) problem, we are given a graph G and a positive integer k, and the objective is to decide whether G contains a minimal vertex cover of size at least k. Motivated by the kernelization of MMVC with parameter k, our main contribution is to introduce a simple general framework to obtain kernelization lower bounds for a certain type of kernels for optimization problems, which we call lop-kernels. Informally, this type of kernel is required to preserve large optimal solutions in the reduced instance, and captures the vast majority of existing kernels in the literature. As a consequence of this framework, we show that the trivial quadratic kernel for MMVC is essentially optimal, answering a question of Boria et al. Discrete Appl Math 196:62–71, 2015. https://doi.org/10.1016/j.dam.2014.06.001), and that the known cubic kernel for Maximum Minimal Feedback Vertex Set is also essentially optimal. We present further applications for Tree Deletion Set and for Maximum Independent Set on K t -free graphs. Back to the MMVC problem, given the (plausible) non-existence of subquadratic kernels for MMVC on general graphs, we provide subquadratic kernels on H-free graphs for several graphs H, such as the bull, the paw, or the complete graphs, by making use of the Erdős–Hajnal property. Finally, we prove that MMVC does not admit polynomial kernels parameterized by the size of a minimum vertex cover of the input graph, even on bipartite graphs, unless NP ⊆ coNP / poly . [ABSTRACT FROM AUTHOR]

The present paper deals with the limit behavior of iterates of Lupaş q- and (p,q)-Bernstein operators. We obtain the convergence for Lupaş q- and (p,q)-Bernstein operators by using the contraction principle. [ABSTRACT FROM AUTHOR]

The finite-gap approach to constructing the discrete Schrödinger operator on a quad graph expressed as a two-dimensional integer sublattice in d-dimensional space is used. The Green function for this operator is explicitly expressed as an integral over special contours of the differential constructed from spectral data. The resulting function has a well-known asymptotics. [ABSTRACT FROM AUTHOR]

A ring K is called a unique addition ring (a UA-ring) if there exists a unique binary operation + on the multiplicative semigroup ( K, · ) of K such that ( K, ·, +) is a ring. We say that an abelian group is an End-UA-group if its endomorphism ring is a UA-ring. We find End-UA-groups in the class of completely decomposable quotient divisible abelian groups. [ABSTRACT FROM AUTHOR]

Given rational vector spaces V, W a mapping $${f \colon V \to W}$$ is called a generalized polynomial of degree at most n, if there are homogeneous generalized polynomials f of degree i such that $${f = \sum_{i = 0}^n f_i}$$ . Homogeneous generalized polynomials f of degree i are mappings of the form $${f_i (x) = f_i^*(x, x, \ldots , x)}$$ with $${f_i^* \colon V^i \to W i}$$ -linear. In the literature one may find quite a lot of functional equations such that their general solution is of the form f or $${f_n + f_{n - 1}}$$ where n is a small positive integer ( ≤ 6 or ≤ 4 respectively). In this paper, given an arbitrary positive integer n and an arbitrary subset $${L \subseteq \{0, 1, \ldots, n\}}$$ such that $${n \in L}$$ , a method is described to find (many) functional equations, such that their general solution is given by $${\sum_{i \in L} f_i}$$ . For the cases $${L = \{n\}}$$ and $${L = \{n - 1, n\}}$$ additional equations are given. [ABSTRACT FROM AUTHOR]

Generalizing Frassek et al. (Adv. Math. 401, 108283 (2022). https://doi.org/10.1016/j.aim.2022.108283), we construct a family of SO(2r), Sp(2r), S O (2 r + 1) rational Lax matrices T D (z) , polynomial in the spectral parameter z, parametrized by Λ + -valued divisors D on P 1 . To this end, we provide the RTT realization of the antidominantly shifted extended Drinfeld Yangians of g = so 2 r , sp 2 r , so 2 r + 1 , and of their coproduct homomorphisms. [ABSTRACT FROM AUTHOR]

A numerical semigroup S is cyclotomic if its semigroup polynomial P S is a product of cyclotomic polynomials. The number of irreducible factors of P S (with multiplicity) is the polynomial length ℓ (S) of S. We show that a cyclotomic numerical semigroup is complete intersection if ℓ (S) ≤ 2 . This establishes a particular case of a conjecture of Ciolan et al. (SIAM J Discrete Math 30(2):650–668, 2016) claiming that every cyclotomic numerical semigroup is complete intersection. In addition, we investigate the relation between ℓ (S) and the embedding dimension of S. [ABSTRACT FROM AUTHOR]

We construct "quantum theta bases," extending the set of quantum cluster monomials, for various versions of skew-symmetric quantum cluster algebras. These bases consist precisely of the indecomposable universally positive elements of the algebras they generate, and the structure constants for their multiplication are Laurent polynomials in the quantum parameter with non-negative integer coefficients, proving the quantum strong cluster positivity conjecture for these algebras. The classical limits recover the theta bases considered by Gross–Hacking–Keel–Kontsevich (J Am Math Soc 31(2):497–608, 2018). Our approach combines the scattering diagram techniques used in loc. cit. with the Donaldson–Thomas theory of quivers. [ABSTRACT FROM AUTHOR]

We study the problems of uniqueness of some types of differential-difference polynomials sharing a small function. We not only solve an open problem formulated in [A. Banerjee and S. Majumder, Anal. Math., 43, 415–444 (2017)] but also present our main results in a more generalized way. [ABSTRACT FROM AUTHOR]