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A color correction method for balancing the color appearances, among a group of images, about a specified object or scene, such as panoramic images and object movies, is developed and tested in this paper. In order to increase the running speed of color correction and reduce the out-of-gamut color pixels, we introduce the selection of principal regions. The average color values of principal regions are used to construct the low-degree (up to degree two) polynomial mapping functions from the source images to the corrected images. The functions are run in the decorrelated color spaces. Our method is tested using real and synthetic images. The results of these tests show the proposed method can get a better performance than other existing methods. [Copyright &y& Elsevier]

In this paper, we construct a class of modules T over the Virasoro algebra by taking tensor products of the modules N (Ω (λ , b) , V) defined in [24,19] and the irreducible modules defined in [31]. This provides a unified description of many known examples of Virasoro modules, for example, in [28,34,35,12,20,5]. We obtain the necessary and sufficient conditions for T to be irreducible and study their submodule structure when they are reducible. Then we also determine the conditions for two such modules to be isomorphic. In the last part of the paper, we compare the tensor product modules with other known Virasoro modules, concluding that they provide new simple modules in general, while admit some interesting isomorphisms with modules defined in [40] and [33]. [ABSTRACT FROM AUTHOR]

Let F n be the free Lie algebra over R of rank n generated by y 1 , ... , y n , and let f ∈ F n ′ be a multilinear Lie polynomial contained in the commutator ideal F n ′ of F n. In this paper, we determine the image Im f = { f (w 1 , ... , w n) | w i ∈ L , i = 1 , ... , n } ⊂ L , for Lie algebras L of dimension ≤3, and of the Lie algebra of dimension 4 stated in a paper of Baker dating back to 1901. In all the cases studied, the L'vov-Kaplansky Conjecture has a positive answer. [ABSTRACT FROM AUTHOR]

This paper devotes to the study of the classical Abel equation d x d t = g (t) x 3 + f (t) x 2 , where g (t) and f (t) are trigonometric polynomials of degree m ≥ 1. We are interested in the problem that whether there is a uniform upper bound for the number of limit cycles of the equation with respect to m , which is known as the famous Smale-Pugh problem. In this work we generalize an idea from the recent paper (Yu, Chen and Liu, Disc. Cont. Dyn. Syst. Ser. B, 2023) and give a new criterion to estimate the maximum multiplicity of limit cycles of the above Abel equations. By virtue of this criterion and the previous results given by Álvarez et al. and Bravo et al., we completely solve the simplest case of the Smale-Pugh problem, i.e., the case when g (t) and f (t) are linear trigonometric, and obtain that the maximum number of limit cycles, is three, which gives a positive answer to the sixth of 33 open problem proposed by Gasull in paper (Gasull, SeMA J., 2021). [ABSTRACT FROM AUTHOR]

The study of P -polynomial association schemes (distance-regular graphs) and Q -polynomial association schemes, and in particular P - and Q -polynomial association schemes, has been a central theme not only in the theory of association schemes but also in the whole study of algebraic combinatorics in general. Leonard's theorem (1982) says that the spherical functions (or the character tables) of P - and Q -polynomial association schemes are described by Askey-Wilson orthogonal polynomials or their relatives. These polynomials are one-variable orthogonal polynomials. It seems that the new attempt to define and study higher rank P - and Q -polynomial association schemes had been hoped for, but had gotten only limited success. The first very successful attempt was initiated recently by Bernard-Crampé-d'Andecy-Vinet-Zaimi, and then followed by Bannai-Kurihara-Zhao-Zhu. The general theory and some explicit examples of families of higher rank (multivariate) P - and/or Q -polynomial association schemes have been obtained there. The main purpose of the present paper is to prove that some important families of association schemes are shown to be bivariate Q -polynomial. Namely, we show that all the nonbinary Johnson association schemes and all the association schemes obtained from attenuated spaces are bivariate Q -polynomial. It should be noted that the parameter restrictions needed in the previous papers are completely lifted in this paper. Our proofs are done by explicitly calculating the Krein parameters of these association schemes. At the end, we mention some speculations and indications of what we can expect in the future study. [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]

The Gyárfás-Sumner conjecture says that for every forest H and every integer k , if G is H -free and does not contain a clique on k vertices then it has bounded chromatic number. (A graph is H-free if it does not contain an induced copy of H.) Kierstead and Penrice proved it for trees of radius at most two, but otherwise the conjecture is known only for a few simple types of forest. More is known if we exclude a complete bipartite subgraph instead of a clique: Rödl showed that, for every forest H , if G is H -free and does not contain K t , t as a subgraph then it has bounded chromatic number. In an earlier paper with Sophie Spirkl, we strengthened Rödl's result, showing that for every forest H , the bound on chromatic number can be taken to be polynomial in t. In this paper, we prove a related strengthening of the Kierstead-Penrice theorem, showing that for every tree H of radius two and integer d ≥ 2 , if G is H -free and does not contain as a subgraph the complete d -partite graph with parts of cardinality t , then its chromatic number is at most polynomial in t. [ABSTRACT FROM AUTHOR]

Suppose that p is an odd prime, and R = G R (p 2 , p 2 r) is a Galois ring of characteristic p 2 , whose cardinality is p 2 r. For any subset D ⊆ R , let N (D , n , b) and N f (D , n , b) , respectively, be the number of n -subsets, denoted by S , in D such that ∑ x ∈ S x = b and ∑ x ∈ S f (x) = b , where f (x) ∈ R [ x ] is a polynomial with degree d. In this paper, we give the asymptotic formulae of N (D , n , b) and N f (D , n , b) under some certain constraints of D and f (x). The special cases D = { a k : a ∈ R ⁎ } and f (x) = x k were studied by the authors in a previous paper, where R ⁎ is the subset of all invertible elements of R and k is a given positive integer, which extends the former one in a more general setting. These extensions benefit from a refined form of the Li–Wan new sieve as well as a variant of the Weil bound on Galois rings. [ABSTRACT FROM AUTHOR]

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]

In this paper we give Monk rules for Macdonald polynomials which are analogous to the Monk rules for Schubert polynomials. These formulas are similar to the formulas given by Baratta [6] , but our method of derivation is to use Cherednik's intertwiners. Deriving Monk rules by this technique addresses the relationship between the work of Baratta and the product formulas of Yip [19]. Specializations of the Monk formula's at q = 0 and/or t = 0 provide Monk rules for Iwahori-spherical polynomials and for finite and affine key polynomials. [ABSTRACT FROM AUTHOR]

This is a paper about c -functions and Macdonald polynomials. There are c -function formulas for E -expansions of P λ and A λ + ρ , principal specializations of P λ and E μ , for Macdonald's constant term formulas, and for the norms of Macdonald polynomials. Most of these follow from the creation formulas for Macdonald polynomials, providing alternative proofs to several results from [19]. In addition, we prove the Boson-Fermion correspondence in the Macdonald polynomial setting and the Weyl character formula for Macdonald polynomials. [ABSTRACT FROM AUTHOR]

In this paper, we study the category of modules over the Smith algebra which are free of finite rank over the unital polynomial subalgebra generated by the Cartan element h and obtain families of such simple modules of arbitrary rank. In the case of rank one we obtain a full description of the isomorphism classes, a simplicity criterion, and an algorithm to produce all composition series. We show that all such modules have finite length and describe the composition factors and their multiplicity. [ABSTRACT FROM AUTHOR]

In this paper, we give characterization of quadratic ε -canonical number system (ε −CNS) polynomials for all values ε ∈ [ 0 , 1). Our characterization provides a unified view of the well-known characterizations of the classical quadratic CNS polynomials (ε = 0) and quadratic SCNS polynomials (ε = 1 / 2). This result is a consequence of our new characterization results of ε -shift radix systems (ε −SRS) in the two-dimensional case and their relation to quadratic ε −CNS polynomials. [ABSTRACT FROM AUTHOR]

Let T m (N , 2 k) be the m th Hecke operator on the space S (N , 2 k) of cuspforms of weight 2 k and level N. This paper shows that in all but finitely many cases, which we list, the second coefficient of the characteristic polynomial of T 2 (N , 2 k) does not vanish when 2 and N are coprime. [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]

In this paper we prove that polynomials F (x 1 , ⋯ , x n) ∈ Z [ x 1 , ⋯ , x n ] of degree d ≥ 3 , satisfying certain hypotheses, take on the expected density of (d − 1) -free values. This extends the authors' earlier result in [14] where a different method implied the similar statement for polynomials of degree d ≥ 5. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new algebraic winding number and prove that it computes the number of complex roots of a polynomial in a rectangle, including roots on edges or vertices with appropriate counting. The definition makes sense for the algebraic closure C = R [ i ] of a real closed field R , and the root counting result also holds in this case. We study in detail the properties of the algebraic winding number defined in [3] with respect to complex root counting in rectangles. We extend both winding numbers to rational functions, obtaining then algebraic versions of the argument principle for rectangles. [ABSTRACT FROM AUTHOR]

Let A be a finite-dimensional superalgebra with superautomorphism φ over a field of characteristic zero. In [11] the authors gave a positive answer to the Amitsur's conjecture in this setting showing that the φ -exponent of A exists and it is an integer. In the present paper we extend the notion of minimal variety to the context of superalgebras with superautomorphism and prove that a variety is minimal of fixed φ -exponent if, and only if, it is generated by a subalgebra of an upper block triangular matrix algebra equipped with a suitable elementary Z 2 -grading and superautomorphism. Along the way, we give a contribution on the isomorphism question within the theory of polynomial identities. [ABSTRACT FROM AUTHOR]

In 1964, J. Ax gave a ten line proof of the Chevalley–Warning Theorem. The novelty was an innocuous result about summing a polynomial f ∈ F q [ t 1 , ... , t N ] over all x ∈ F q N. Recent work of Aichinger–Moosbauer provides a context in which we can seek to generalize Ax's Lemma to maps f : A → B between any two finite commutative groups, and indeed Aichinger–Moosbauer proved such a result when the target group B has prime exponent. In this paper we define a summation invariant σ (A , B) that fits naturally into the Aichinger–Moosbauer calculus. We give upper and lower bounds on σ (A , B) and its exact value in many (but not all) cases. We also give Diophantine applications, including a qualitative Ax–Katz Theorem for polynomials over any finite rng. The bounds that we get turn out to be closely related to Ax's part of the Ax–Katz Theorem. [ABSTRACT FROM AUTHOR]

The present paper is devoted to investigating the two-dimensional real Jacobian conjecture. This conjecture claims that if F = (f , g) : R 2 → R 2 is a polynomial map with det ⁡ D F (x , y) ≠ 0 for all (x , y) ∈ R 2 , then F is globally injective. With the help of the Newton diagram, we provide a new sufficient condition such that the two-dimensional real Jacobian conjecture holds. Moreover, this sufficient condition generalizes the main result of Braun et al. (2016) [7]. Furthermore, two new classes of polynomial maps satisfying the two-dimensional real Jacobian conjecture are given. [ABSTRACT FROM AUTHOR]

In this paper, we classify the simple Harish-Chandra modules over the superconformal current algebra g ˆ , which is the semi-direct sum of the N = 1 superconformal algebra with the affine Lie superalgebra g ˙ ⊗ A ⊕ C C 1 , where g ˙ is a finite-dimensional simple Lie algebra, and A is the tensor product of the Laurent polynomial algebra and the Grassmann algebra. As an application, we can directly get the classification of the simple Harish-Chandra modules over the N = 1 Heisenberg-Virasoro algebra. [ABSTRACT FROM AUTHOR]

In this paper, we consider the bifurcation of small-amplitude limit cycles near the origin in perturbed pendulum systems of the form x ˙ = y , y ˙ = − sin ⁡ (x) + ε Q (x , y) , where Q (x , y) is a smooth or piecewise smooth polynomial in the triple (sin ⁡ (x) , cos ⁡ (x) , y) with free coefficients. We obtain sharp upper bounds on the number of positive zeros of its associated first order Melnikov function near h = 0 for Q (x , y) being smooth and piecewise smooth with the discontinuity at y = 0 or x = 0 , respectively. [ABSTRACT FROM AUTHOR]

Let D be a noncommutative division ring. In a recent paper, Lee and Lin proved that if char D ≠ 2 , the only solution of additive maps f , g on D satisfying the identity f (x) = x n g (x − 1) on D ∖ { 0 } with n ≠ 2 a positive integer is the trivial case, that is, f = 0 and g = 0. Applying Hua's identity and the theory of functional and generalized polynomial identities, we give a complete solution of the same identity for any nonnegative integer n if char D = 2. [ABSTRACT FROM AUTHOR]

Given a polynomial with integral coefficients, one can inquire about the possible residues it can take in its image modulo a prime p. The sum over the distinct residues can sometimes be computed independent of the prime p ; for example, Gauss showed that the sum over quadratic residues vanishes modulo a prime. In this paper we provide a closed form for the sum over distinct residues in the image of Dickson polynomials of arbitrary degree over finite fields of odd characteristic, and prove a complete characterization of the size of the value set. Our result provides the first non-trivial classification of such a sum for a family of polynomials of unbounded degree. [ABSTRACT FROM AUTHOR]

For a ∈ C ⁎ there are two natural evaluation maps ev a and ev a from the affine Hecke algebra H ▵ (r) C to the Hecke algebra H (r) C. The maps ev a and ev a induce evaluation maps ev ˜ a and ev ˜ a from the affine quantum Schur algebra S ▵ (n , r) C to the quantum Schur algebra S (n , r) C , respectively. In this paper we prove that the evaluation map ev ˜ a (resp. ev ˜ a) is compatible with the evaluation map Ev a (resp. Ev (− 1) n a q n ) for quantum affine sl n. Furthermore we compute the Drinfeld polynomials associated with the simple S ▵ (n , r) C -modules which come from the simple S (n , r) C -modules via the evaluation maps ev ˜ a. Then we characterize finite-dimensional irreducible S ▵ (n , r) C -modules which are irreducible as S (n , r) C -modules for n > r. As an application, we characterize finite-dimensional irreducible modules for the affine Hecke algebra H ▵ (r) C which are irreducible as modules for the Hecke algebra H (r) C. [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]

Let U q (g) denote the rational form of the quantized enveloping algebra associated to a complex simple Lie algebra g. Let λ be a nonzero dominant integral weight of g , and let V be the corresponding type 1 finite-dimensional irreducible representation of U q (g). Starting from this data, the R -matrix formalism for quantum groups outputs a Hopf algebra U R λ (g) defined in terms of a pair of generating matrices satisfying well-known quadratic matrix relations. In this paper, we prove that this Hopf algebra admits a Chevalley–Serre type presentation which can be recovered from that of U q (g) by adding a single invertible quantum Cartan element. We simultaneously establish that U R λ (g) can be realized as a Hopf subalgebra of the tensor product of the space of Laurent polynomials in a single variable with the quantized enveloping algebra associated to the lattice generated by the weights of V. The proofs of these results are based on a detailed analysis of the homogeneous components of the matrix equations and generating matrices defining U R λ (g) , with respect to a natural grading by the root lattice of g compatible with the weight space decomposition of End (V). [ABSTRACT FROM AUTHOR]

In this paper we classify all monic, quartic, polynomials d (x) ∈ Z [ x ] for which the Pell equation f (x) 2 − d (x) g (x) 2 = 1 has a non-trivial solution with f (x) , g (x) ∈ Z [ x ]. [ABSTRACT FROM AUTHOR]

In this paper, we present results on the uniqueness of the real zeros of the Hurwitz zeta function in given intervals. The uniqueness in question, if the zeros exist, has already been proved for the intervals (0 , 1) and (− N , − N + 1) for N ≥ 5 by Endo-Suzuki and Matsusaka, respectively. We prove the uniqueness of the real zeros in the remaining intervals by examining the behavior of certain associated polynomials. [ABSTRACT FROM AUTHOR]

The graph removal lemma is a fundamental result in extremal graph theory which says that for every fixed graph H and ε > 0 , if an n -vertex graph G contains ε n 2 edge-disjoint copies of H then G contains δ n v (H) copies of H for some δ = δ (ε , H) > 0. The current proofs of the removal lemma give only very weak bounds on δ (ε , H) , and it is also known that δ (ε , H) is not polynomial in ε unless H is bipartite. Recently, Fox and Wigderson initiated the study of minimum degree conditions guaranteeing that δ (ε , H) depends polynomially or linearly on ε. In this paper we answer several questions of Fox and Wigderson on this topic. [ABSTRACT FROM AUTHOR]

In 1971, Wilson (1971) proved that "almost all" finite games have an odd number of mixed Nash equilibria. Since then, several other proofs have been given, but always for mixed extensions of finite games. In this paper, we present a new oddness theorem for large classes of polynomial payoff functions and semi-algebraic sets of strategies. Additionally, we provide some applications to recent models of games on networks such that Patacchini-Zenou's model about juvenile delinquency and conformism (Patacchini and Zenou, 2012), Calvó-Armengol-Patacchini-Zenou's model about social networks in education (Calvó-Armengol et al., 2009), Konig-Liu-Zenou's model about R&D networks (König et al., 2019), Helsley-Zenou's model about social networks and interactions in cities (Helsley and Zenou, 2014). [ABSTRACT FROM AUTHOR]

Let f (x 1 , x 2) be a two-variable polynomial over a finite field of characteristic p , and assume that its convex hull is a triangle Δ. The goal of this paper is to study the Newton polygon NP (f , χ) of the L -function associated to f and a finite character χ of Z p. It is a partial generalization of [DWX] , in which the authors studied polynomials of one variable. In this paper, we prove an improved lower bound IHP (Δ) for NP (f , χ) , which, when f is non-ordinary, is strictly higher than the classical Hodge polygon. Moreover, we prove that if NP (f , χ) and IHP (Δ) coincide at one certain point, then they coincide at infinitely many points. This recovers the key property of the proof from [DWX]. [ABSTRACT FROM AUTHOR]

In this paper we introduce methods and algorithms that will help us solve connectivity queries of parameterized semi-algebraic sets. Answering these connectivity queries is applied in the design of robotic structures having similar kinematic properties (e.g. topology of the kinematic-singularity-free space). From these algorithms one also obtain solutions to connectivity queries of a specific parameter which is in turn related to kinematic-singularity free path-planning of a specific manipulator belonging to the family of robots with these properties; i.e. we obtain paths joining two given singularity free configurations lying in the same connected component of the singularity-free space. We prove in the paper how one reduces the problems related to connectivity queries of parameterized semi-algebraic sets to closed and bounded semi-algebraic sets. We then design an algorithm using computer-algebra methods for "solving" positive dimensional polynomial system depending on parameters. The meaning of solving here means partitioning the parameter space into semi-algebraic components over which the number of connected components of the semi-algebraic set defined by the input system is invariant. The complexity of this algorithm is singly exponential in the dimension of the ambient space. The algorithm scales enough to analyze automatically the family of UR-series robots. Finally we provide manual analysis of the family of UR-series robots, proving that the number of connected components of the complementary of kinematic singularity set of a generic UR-robot is eight. [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]

We study the category of representations of sl m + 2 n over a field of characteristic p with p ≫ 0 , whose p -character is a nilpotent whose Jordan type is the two-row partition (m + n , n). In a previous paper with Anno, we used Bezrukavnikov-Mirkovic-Rumynin's theory of positive characteristic localization and exotic t-structures to give a geometric parametrization of the simples using annular crossingless matchings. Building on this, here we give combinatorial dimension formulae for the simple objects, and compute the Jordan-Hölder multiplicities of the simples inside the baby Vermas. We use Cautis-Kamnitzer's geometric categorification of the tangle calculus to study the images of the simple objects under the BMR equivalence. Our results generalize Jantzen's formulae in the subregular nilpotent case (i.e. when n = 1), and may be viewed as a positive characteristic analogue of the combinatorial description for Kazhdan-Lusztig polynomials of Grassmannian permutations due to Lascoux and Schutzenberger. [ABSTRACT FROM AUTHOR]

In recent years, the notion of characteristic polynomials of representations of Lie algebras has been widely studied. This paper provides more properties of these characteristic polynomials. For simple Lie algebras, we characterize the linearization of characteristic polynomials. Additionally, we characterize nilpotent Lie algebras via characteristic polynomials of the adjoint representation. [ABSTRACT FROM AUTHOR]

Let F be an infinite field and M 1 , 1 (F) be the algebra of 2 × 2 matrices over F endowed with non-trivial Z 2 -grading. We consider the involutions ⁎ defined on M 1 , 1 (F) which preserve the homogeneous components of the grading. In this paper, we deal with the ⁎-superalgebra (M 1 , 1 (F) , ⁎) and determine the generators of its ideal of (Z 2 , ⁎) -identities, considering that F has characteristic zero and also, we explicitly construct the generators of its space of central (Z 2 , ⁎) -polynomials, when the characteristic of F is different from 2. [ABSTRACT FROM AUTHOR]

Let K be a finite field of characteristic 2, and U T 2 : = U T 2 (K) be the Lie algebra of 2 × 2 upper triangular matrices over K with the multiplication x ∘ y = x y + y x = x y − y x. In this paper, we exhibit a finite basis of graded identities for the variety of Lie algebras generated by U T 2 for any grading and show that it has the Specht property. It is important to highlight that the technique used in order to solve the Specht problem is independent of the characteristic of the field and also of its cardinality. [ABSTRACT FROM AUTHOR]

In this paper, we extend the slow divergence-integral from slow-fast systems, due to De Maesschalck, Dumortier and Roussarie, to smooth systems that limit onto piecewise smooth ones as ϵ → 0. In slow-fast systems, the slow divergence-integral is an integral of the divergence along a canard cycle with respect to the slow time and it has proven very useful in obtaining good lower and upper bounds of limit cycles in planar polynomial systems. In this paper, our slow divergence-integral is based upon integration along a generalized canard cycle for a piecewise smooth two-fold bifurcation (of type visible-invisible called V I 3). We use this framework to show that the number of limit cycles in regularized piecewise smooth polynomial systems is unbounded. [ABSTRACT FROM AUTHOR]

Interpolated multiple zeta values can be regarded as interpolation polynomials of multiple zeta values and multiple zeta-star values. In this paper, we give some algebraic relations of interpolated multiple zeta values, such as the shuffle regularized sum formula, a weighted sum formula and some evaluation formulas with even arguments. All the algebraic relations provided in this paper are deduced from the extended double shuffle relations. [ABSTRACT FROM AUTHOR]

This paper deals with the fundamental period π ˜ of the Carlitz module. The main theorem states that the Carlitz period and all its hyperderivatives are algebraically independent over the base field F q (θ). Our approach also reveals a connection of these hyperderivatives with the coordinates of a period lattice generator of the tensor powers of the Carlitz module which was already observed by M. Papanikolas in a yet unpublished paper. Namely, these coordinates can be obtained by explicit polynomial expressions in π ˜ and its hyperderivatives. Papanikolas also gave various presentations of these expressions which we also prove here. [ABSTRACT FROM AUTHOR]

Let q = 2 m. In this paper, we investigate permutation pentanomials over F q 2 of the form f (x) = x t + x r 1 (q − 1) + t + x r 2 (q − 1) + t + x r 3 (q − 1) + t + x r 4 (q − 1) + t with gcd (x r 4 + x r 3 + x r 2 + x r 1 + 1 , x t + x t − r 1 + x t − r 2 + x t − r 3 + x t − r 4 ) = 1. We transform the problem concerning permutation property of f (x) into demonstrating that the corresponding fractional polynomial permutes the unit circle U of F q 2 with order q + 1 via a well-known lemma, and then into showing that there are no certain solution in F q for some high-degree equations over F q associated with the fractional polynomial. According to numerical data, we have found all such permutations with 4 ≤ t < 100 , 1 ≤ r i ≤ t , i ∈ [ 1 , 4 ]. Several permutation polynomials are also investigated from the fractional polynomials permuting the unit circle U found in this paper. [ABSTRACT FROM AUTHOR]

An out-branching B u + (in-branching B u −) in a digraph D is a connected spanning subdigraph of D in which every vertex except the vertex u , called the root, has in-degree (out-degree) one. A good (u,v) - pair in D is a pair of branchings B u + , B v − which have no arc in common. Thomassen proved that it is NP-complete to decide if a digraph has any good pair. A digraph is semicomplete if it has no pair of non-adjacent vertices. A semicomplete composition is any digraph D which is obtained from a semicomplete digraph S by substituting an arbitrary digraph H x for each vertex x of S. Recently the authors of this paper gave a complete classification of semicomplete digraphs which have a good (u , v) -pair, where u , v are prescribed vertices. They also gave a polynomial algorithm which for a given semicomplete digraph D and vertices u , v of D , either produces a good (u , v) -pair in D or a certificate that D has no such pair. In this paper we show how to use the result for semicomplete digraphs to completely solve the problem of characterizing semicomplete compositions which have a good (u , v) -pair for given vertices u , v. Our solution implies that the problem of deciding the existence of a good (u , v) -pair and finding such a pair when it exists is polynomially solvable for all semicomplete compositions. We also completely solve the problem of deciding the existence of a good (u , v) -pair and finding one when it exists for digraphs that are compositions of transitive digraphs. Combining these two results we obtain a polynomial algorithm for deciding whether a given quasi-transitive digraph D has a good (u , v) -pair for given vertices u , v of D. This proves a conjecture of Bang-Jensen and Gutin from 1998. [ABSTRACT FROM AUTHOR]

Let A be an associative algebra over a fixed field F of characteristic zero. In this paper we focus our attention on those algebras A graded by Z 2 , the cyclic group of order 2. In this case A is said to be a superalgebra and it can be decomposed in the direct sum of homogeneous subspaces: A = A 0 ⊕ A 1. The main goal of this paper is to prove tight relations between some graded linear maps that can be defined on superalgebras, namely involutions, superinvolutions and pseudoinvolutions. Along the way, we shall present a classification of the pseudoinvolutions that one can define on the algebra U T n (F) of n × n upper-triangular matrices. In the final part of the paper we shall also give some consequences of these results in the context of the theory of polynomial identities. [ABSTRACT FROM AUTHOR]

Kapranov's theorem is a foundational result in tropical geometry. It states that the set of tropicalisations of points on a hypersurface coincides precisely with the tropical variety of the tropicalisation of the defining polynomial. The aim of this paper is to generalise Kapranov's theorem, replacing the role of a valuation, ν : K → R ∪ { − ∞ } , with a more general class of hyperfield homomorphisms, H → T , which satisfy a relative algebraic closure condition. The map η : T C → T , η (x) : = log (| x |) , where T C is the tropical complex hyperfield, provides an example of such a hyperfield homomorphism. [ABSTRACT FROM AUTHOR]

Let K be a field and let J n , k be the Jordan algebra of a degenerate symmetric bilinear form b of rank n − k over K. Then one can consider the decomposition J n , k = B n − k ⊕ D k , where B n − k represents the corresponding Jordan algebra, denoted as B n − k = K ⊕ V. In this algebra, the restriction of b on the (n − k) -dimensional subspace V is non-degenerate, while D k accounts for the degenerate part of J n , k. This paper aims to provide necessary and sufficient conditions to check if a given multilinear polynomial is an identity for J n , k. As a consequence of this result and under certain hypothesis on the base field, we exhibit a finite basis for the T -ideal of polynomial identities of J n , k. Over a field of characteristic zero, we also prove that the ideal of identities of J n , k satisfies the Specht property. Moreover, similar results are obtained for weak identities, trace identities and graded identities with a suitable Z 2 -grading as well. In all of these cases, we employ methods and results from Invariant Theory. Finally, as a consequence from the trace case, we provide a counterexample to the embedding problem given in [8] in case of infinite dimensional Jordan algebras with trace. [ABSTRACT FROM AUTHOR]

In this paper we derive a closed formula for the (Z × Z 2) -graded dimension of the cyclotomic quiver Hecke superalgebra R Λ (β) associated to an arbitrary Cartan superdatum (A , P , Π , Π ∨) , polynomials (Q i , j (x 1 , x 2)) i , j ∈ I , β ∈ Q n + and Λ ∈ P +. As applications, we obtain a necessary and sufficient condition for which e (ν) ≠ 0 in R Λ (β). We construct an explicit monomial basis for the bi-weight space e (ν ˜) R Λ (β) e (ν ˜) , where ν ˜ is a certain specific n -tuple defined in (1.4). In particular, this gives rise to a monomial basis for the cyclotomic odd nilHecke algebra. Finally, we consider the case when β = α 1 + α 2 + ⋯ + α n with α 1 , ⋯ , α n distinct. We construct an explicit monomial basis of R Λ (β) and show that it is indecomposable in this case. [ABSTRACT FROM AUTHOR]

A constant term sequence is a sequence of rational numbers whose n -th term is the constant term of P n (x) Q (x) , where P (x) and Q (x) are multivariate Laurent polynomials. While the generating functions of such sequences are invariably diagonals of multivariate rational functions, and hence special period functions, it is a famous open question, raised by Don Zagier, to classify diagonals that are constant terms. In this paper, we provide such a classification in the case of sequences satisfying linear recurrences with constant coefficients. We also consider the case of hypergeometric sequences and, for a simple illustrative family of hypergeometric sequences, classify those that are constant terms. [ABSTRACT FROM AUTHOR]

Spherical needlets were introduced by Narcowich, Petrushev, and Ward to provide a multiresolution sequence of polynomial approximations to functions on the sphere. The needlet construction makes use of integration rules that are exact for polynomials up to a given degree. The aim of the present paper is to relax the exactness of the integration rules by replacing them with QMC designs as introduced by Brauchart, Saff, Sloan, and Womersley (2014). Such integration rules (generalised here by allowing non-equal cubature weights) provide the same asymptotic order of convergence as exact rules for Sobolev spaces H s , but are easier to obtain numerically. With such rules we construct "generalised needlets". The paper provides an error analysis that allows the replacement of the original needlets by generalised needlets, and more generally, analyses a hybrid scheme in which the needlets for the lower levels are of the traditional kind, whereas the new generalised needlets are used for some number of higher levels. Numerical experiments complete the paper. [ABSTRACT FROM AUTHOR]