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The paper revisits the classical problem of evaluating f ( A ) for a real function f and a matrix A with real spectrum. The evaluation is based on expanding f in Chebyshev polynomials, and the focus of the paper is to study the convergence rates of these expansions. In particular, we derive bounds on the convergence rates which reveal the relation between the smoothness of f and the diagonalizability of the matrix A . We present several numerical examples to illustrate our analysis. [ABSTRACT FROM AUTHOR]

In recent years several papers have appeared that investigate the classical discrete logarithm problem for elliptic curves by means of the multivariate polynomial approach based on the celebrated summation polynomials, introduced by Semaev in 2004. With a notable exception by Petit et al. in 2016, all numerous papers on the subject have investigated only the composite-field case, leaving apart the laborious prime-field case. In this paper we propose a variation of Semaev's original approach that reduces to only one the relations to be found among points of the factor base, thus decreasing drastically the necessary Groebner basis computations. Our proposal holds for any finite field but it is particularly suitable for the prime-field case, where it outperforms both the original Semaev's method and the specialised algorithm by Petit et al.. [ABSTRACT FROM AUTHOR]

In this paper, we describe three systems of orthogonal polynomials belonging to the class of Meixner–Pollaczek polynomials, and establish some useful connections between them in terms of three basic operators that are related to them. Furthermore, we investigate boundedness properties of two other operators, both as convolution operators in the translation invariant case where we use Fourier transforms and for the weights related to the relevant orthogonal polynomials. We consider only the most important but also simplest case of L 2 -spaces. However, in subsequent papers, we intend to extend the study to L p -spaces ( 1 < p < ∞ ) . [ABSTRACT FROM AUTHOR]

This is the third paper on the study of gradient recovery for elliptic interface problems. In our previous works Guo and Yang (2017) [23] , we developed gradient recovery methods for elliptic interface problems based on body-fitted meshes and immersed finite element methods. Despite the efficiency and accuracy that these methods bring to recover the gradient, there are still some cases in unfitted meshes where skinny triangles appear in the generated local body-fitted triangulations that destroy the accuracy of recovered gradient near the interface. In this paper, we propose a gradient recovery technique based on Nitsche's method for elliptic interface problems, which avoids the loss of accuracy of gradient near the interface caused by skinny triangles. We analyze the supercloseness between the gradient of the numerical solution by the Nitsche's method and the gradient of the interpolation of the exact solution, which leads to the superconvergence of the proposed gradient recovery method. We also present several numerical examples to validate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper we introduce a new concept of key polynomials for a given valuation ν on K [ x ] . We prove that such polynomials have many of the expected properties of key polynomials as those defined by MacLane and Vaquié, for instance, that they are irreducible and that the truncation of ν associated to each key polynomial is a valuation. Moreover, we prove that every valuation ν on K [ x ] admits a sequence of key polynomials that completely determines ν (in the sense which we make precise in the paper). We also establish the relation between these key polynomials and pseudo-convergent sequences defined by Kaplansky. [ABSTRACT FROM AUTHOR]

This study investigates the influence on the wheel-rail contact forces of the running speed and the shape and position of weld defects along the track. For this purpose, a vertical dynamic model in the space domain is used. The model is obtained from the transformation between the domains of frequency and space using a Rational Fraction Polynomials (RFP) method, which is modified with multiobjective genetic algorithms in order to improve the fitting of track receptance and to assist integration during simulations. This produces a precise model with short calculation times, which is essential to this study. The wheel-rail contact is modelled using a non-linear Hertz spring. The contact forces are studied for several types of characteristic welds. The way in which forces vary as a function of weld position and running speed is studied for each type of weld. This paper studies some of the factors that affect the maximum forces when the vehicle moves over a rail weld, such as weld geometry, parametric excitation and contact stiffness. It is found that the maximum force in the wheel-rail contact when the vehicle moves over a weld is not always proportional to the running speed. The paper explains why it is not proportional in specific welds. [ABSTRACT FROM AUTHOR]

In this paper, we used the free fields of Wakimoto to construct a class of irreducible representations for the general linear Lie superalgebra gl m | n ( C ) . The structures of the representations over the general linear Lie superalgebra and the special linear Lie superalgebra are studied in this paper. Then we extend the construction to the affine Kac–Moody Lie superalgebra gl m | n ˆ ( C ) on the tensor product of a polynomial algebra and an exterior algebra with infinitely many variables involving one parameter μ , and we also obtain the necessary and sufficient condition for the representations to be irreducible. In fact, the representation is irreducible if and only if the parameter μ is nonzero. [ABSTRACT FROM AUTHOR]

In this paper we construct new invariants of knotoids including the odd writhe, the parity bracket polynomial, the affine index polynomial and the arrow polynomial, and give an introduction to the theory of virtual knotoids. The invariants in this paper are defined for both classical and virtual knotoids in analogy to corresponding invariants of virtual knots. We show that knotoids in S 2 have symmetric affine index polynomials. The affine index polynomial and the arrow polynomial provide bounds on the height (minimum crossing distance between endpoints) of a knotoid in S 2 . [ABSTRACT FROM AUTHOR]

In a surprising recent result, Gupta–Kamath–Kayal–Saptharishi have proved that over Q any n O ( 1 ) -variate and n -degree polynomial in VP can also be computed by a depth three ΣΠΣ circuit of size 2 O ( n log 3 / 2 ⁡ n ) . 2 Over fixed-size finite fields, Grigoriev and Karpinski proved that any ΣΠΣ circuit that computes the determinant (or the permanent) polynomial of a n × n matrix must be of size 2 Ω ( n ) . In this paper, for an explicit polynomial in VP (over fixed-size finite fields), we prove that any ΣΠΣ circuit computing it must be of size 2 Ω ( n log ⁡ n ) . The explicit polynomial that we consider is the iterated matrix multiplication polynomial of n generic matrices of size n × n . The importance of this result is that over fixed-size fields there is no depth reduction technique that can be used to compute all the n O ( 1 ) -variate and n -degree polynomials in VP by depth 3 circuits of size 2 o ( n log ⁡ n ) . The result of Grigoriev and Karpinski can only rule out such a possibility for ΣΠΣ circuits of size 2 o ( n ) . We also give an example of an explicit polynomial ( NW n , ϵ ( X ) ) in VNP (which is not known to be in VP ), for which any ΣΠΣ circuit computing it (over fixed-size fields) must be of size 2 Ω ( n log ⁡ n ) . The polynomial we consider is constructed from the combinatorial design of Nisan and Wigderson, and is closely related to the polynomials considered in many recent papers (by Kayal–Saha–Saptharishi, Kayal–Limaye–Saha–Srinivasan, and Kumar–Saraf), where strong depth 4 circuit size lower bounds are shown. [ABSTRACT FROM AUTHOR]

As a generalization of cyclic codes, constacyclic codes is an important and interesting class of codes due to their nice algebraic structures and various applications in engineering. This paper is devoted to the study of the q -polynomial approach to constacyclic codes. Fundamental theory of this approach will be developed, and will be employed to construct some families of optimal and almost optimal codes in this paper. [ABSTRACT FROM AUTHOR]

Let E be the infinite dimensional Grassmann algebra over an infinite field of characteristic p different from 2. Given an involution φ on E , denote by I d (E , φ) and C (E , φ) the set of all ⁎-polynomial identities and ⁎-central polynomials of (E , φ) respectively. In this paper we describe I d (E , φ) and C (E , φ). Moreover, we prove that C (E , φ) is not finitely generated as a T (⁎) -space if p > 2. [ABSTRACT FROM AUTHOR]

We investigate in this paper a generalization of Solomon-Terao formula for central equidimensional subspace arrangements. We introduce generalized Solomon-Terao functions based on the Hilbert-Poincaré series of the modules of multi-logarithmic forms and logarithmic multi-residues. We show that as in the case of hyperplane arrangements, these Solomon-Terao functions are polynomial. We then prove that if the Solomon-Terao polynomial of the modules of multi-residues satisfies a certain property, then this polynomial is related to the characteristic polynomial of the subspace arrangement. In particular, we prove that this generalized Solomon-Terao formula holds for any line arrangement of any codimension. [ABSTRACT FROM AUTHOR]

Let K be a finite field and let U T n (K) be the algebra of n × n upper triangular matrices over K. In this paper we describe the set of all G -graded polynomial identities of U T n (K) , where G is any group. Moreover, we describe a linear basis for the corresponding relatively free graded algebra. [ABSTRACT FROM AUTHOR]

Let (f n) n = 1 ∞ be a sequence of polynomials and α > 1. In this paper we study the distribution of the sequence (f n (α)) n = 1 ∞ modulo one. We give sufficient conditions for a sequence (f n) n = 1 ∞ to ensure that for Lebesgue almost every α > 1 the sequence (f n (α)) n = 1 ∞ has Poissonian pair correlations. In particular, this result implies that for Lebesgue almost every α > 1 , for any k ≥ 2 the sequence (α n k ) n = 1 ∞ has Poissonian pair correlations. [ABSTRACT FROM AUTHOR]

In this paper, we consider the stability of the Rao-Nakra sandwich beam equation with various boundary conditions, which consists of two wave equations for the longitudinal displacements of the top and bottom layers, and one Euler-Bernoulli beam equation for the transversal displacement. Polynomial stability of certain orders are obtained when there is only one viscous damping acting either on the beam equation or one of the wave equations. For a few special cases, optimal orders are confirmed. We also study the synchronization of the model with viscous damping on the transversal displacement. Our results reveal that the order of the polynomial decay rate is sensitive to various boundary conditions and to the damping locations. [ABSTRACT FROM AUTHOR]

In this paper we discuss conversion methods from the conjunctive normal form (CNF) to the algebraic normal form (ANF) of a Boolean function. Whereas the reverse conversion has been studied before, the CNF to ANF conversion has been achieved predominantly via a standard method which tends to produce many polynomials of high degree. Based on a block-building mechanism, we design a new blockwise algorithm for the CNF to ANF conversion which is geared towards producing fewer and lower degree polynomials. In particular, we look for as many linear polynomials as possible in the converted system and check that our algorithm finds them. As an application, we suggest a new algebraic extension of the resolution calculus which is tailored to the output of the standard CNF to ANF conversion algorithm. Experiments show that the ANF produced by our algorithm outperforms the standard conversion in "real life" examples originating from cryptographic attacks, and that our algebraic resolution algorithm solves some SAT instances which are notoriously hard for SAT solvers. [ABSTRACT FROM AUTHOR]

Cylindrical Algebraic Decomposition (CAD) has long been one of the most important algorithms within Symbolic Computation, as a tool to perform quantifier elimination in first order logic over the reals. More recently it is finding prominence in the Satisfiability Checking community, as a tool to identify satisfying solutions of problems in nonlinear real arithmetic. The original algorithm produces decompositions according to the signs of polynomials, when what is usually required is a decomposition according to the truth of a formula containing those polynomials. One approach to achieve that coarser (but hopefully cheaper) decomposition is to reduce the polynomials identified in the CAD to reflect a logical structure which reduces the solution space dimension: the presence of Equational Constraints (ECs). This paper may act as a tutorial for the use of CAD with ECs: we describe all necessary background and the current state of the art. In particular, we present recent work on how McCallum's theory of reduced projection may be leveraged to make further savings in the lifting phase: both to the polynomials we lift with and the cells lifted over. We give a new complexity analysis to demonstrate that the double exponent in the worst case complexity bound for CAD reduces in line with the number of ECs. We show that the reduction can apply to both the number of polynomials produced and their degree. [ABSTRACT FROM AUTHOR]

Let k (x) = k (x 1 , ... , x n) be the rational function field, and k ⊂ L ⊂ k (x) an intermediate field. Hilbert's fourteenth problem asks whether the k -algebra L ∩ k x 1 , ... , x n is finitely generated. Various counterexamples to this problem were already given, but there are still many open cases. In this paper, we study the problem in terms of the field-theoretic properties of L. Our result implies that, for any S ⊂ k x 1 , ... , x n − 1 with k (S) ≠ k (x 1 , ... , x n − 1) and tr. deg k k (S) ≥ 2 , there exists σ ∈ Aut k k (x) such that σ (k (S ∪ { x n })) is a counterexample to Hilbert's fourteenth problem. This implies the existence of several interesting new counterexamples, such as one with k (x) : L = 2. [ABSTRACT FROM AUTHOR]

A fundamental and challenging problem in spectral graph theory is to characterize which graphs are uniquely determined by their spectra. In Wang (2017), the author proved that an n -vertex graph G is uniquely determined by its generalized spectrum (DGS) whenever 2 − ⌊ n 2 ⌋ det W is odd and square-free. Here, W is the walk matrix of G , namely, W = [ e , A e , ... , A n − 1 e ] with e the all-one vector and A the adjacency matrix of G. In this paper, we focus on a larger family of graphs with d n square-free, where d n refers to the last invariant factor of W. We introduce a new kind of polynomial for a graph G associated with a prime p. Such a polynomial is invariant under generalized cospectrality. Using the newly defined polynomial, we obtain a sufficient condition for a graph in the larger family to be DGS. The main result of this paper improves upon the aforementioned result of Wang while the proof for the main result gives a new way to attack the problem of generalized spectral characterization of graphs. [ABSTRACT FROM AUTHOR]

In this paper, we consider the problem of identifying the type (local minimizer, maximizer or saddle point) of a given isolated real critical point c , which is degenerate, of a multivariate polynomial function f. To this end, we introduce the definition of faithful radius of c by means of the curve of tangency of f. We show that the type of c can be determined by the global extrema of f over the Euclidean ball centered at c with a faithful radius. We propose algorithms to compute faithful radius of c and determine its type. [ABSTRACT FROM AUTHOR]

In this paper, a symbolic, algorithmic procedure to compute an immersion that recasts a polynomial system into a linear one up to an output injection is proposed. Such a technique is based on computing, through algebraic geometry methods, the set of all the embeddings of the system and on matching the coefficients of these polynomials with the ones of the embeddings of a linear system up to an output injection. The given algorithm is then relaxed to compute an immersion that recasts a polynomial system into a form that is linear up to a finite order and an output injection and to compute an approximation of the immersion. [ABSTRACT FROM AUTHOR]

By a φ -variety V , we mean a supervariety or a ⁎-variety generated by an associative algebra over a field F of characteristic zero. In this case, we consider its sequence of φ -codimensions c n φ (V) and say that V is minimal of polynomial growth n k if c n φ (V) grows like n k , but any proper φ -subvariety grows like n t with t < k. In this paper, we deal with minimal φ -varieties generated by unitary algebras and prove that for k ≤ 2 there is only a finite number of them. We also explicit a list of finite dimensional algebras generating such minimal φ -varieties. For k ≥ 3 , we show that the number of minimal φ -varieties can be infinity and we classify all minimal φ -varieties of polynomial growth n k by giving a recipe for the construction of their T φ -ideals. [ABSTRACT FROM AUTHOR]

In this paper, we study the mean value of L (1 2 , χ D) where D runs over monic polynomials of degree 2 g + 2 as g tends to infinity. By extending Florea's method [5] , we show that there is an extra term of size | D | 1 3 log q ⁡ | D | and bound the error term by | D | 1 / 4 + ε in the asymptotic formula. [ABSTRACT FROM AUTHOR]

Let v = v 0 ¯ + v 1 ¯ be a Z 2 -graded (super) vector space with an even C × -action and χ ∈ v 0 ¯ ⁎ be a fixed point of the induced action. In this paper we prove an equivariant Darboux-Weinstein theorem for the formal polynomial algebras A ˆ = S [ v 0 ¯ ] ∧ χ ⊗ ⋀ (v 1 ¯ ). We also give a quantum version of it. Let g = g 0 ¯ + g 1 ¯ be a basic Lie superalgebra and e ∈ g 0 ¯ be a nilpotent element. We use the equivariant quantum Darboux-Weinstein theorem to give a Poisson geometric realization of the finite W -superalgebra U (g , e) in the sense of Losev. An indirect relation between finite W -(super)algebras U (g , e) and U (g 0 ¯ , e) is presented. Finally we use such a realization to study the finite-dimensional irreducible modules over U (g , e). [ABSTRACT FROM AUTHOR]

Let A be an algebra with involution ⁎ over a field of characteristic zero. In this paper we characterize in two different ways when the multiplicities of the ⁎-cocharacter of A are bounded by a constant. As a consequence, we characterize the algebras with involution of bounded colength. [ABSTRACT FROM AUTHOR]

A post-critically finite rational map ϕ of prime degree p and a base point β yield a tower of finitely ramified iterated extensions of number fields, and sometimes provide an arboreal Galois representation with a p -adic Lie image. In this paper, we take ϕ to be the monic Chebyshev polynomial x 2 − 2 , and we examine the size of the 2-part of the ideal class group of extensions in the resulting tower. In some cases, we prove an analogue of Greenberg's conjecture from Iwasawa theory. A key tool is a general theorem on p -indivisibility of class numbers of relative cyclic extensions of degree p 2. [ABSTRACT FROM AUTHOR]

The Jordan totient J k (n) can be defined by J k (n) = n k ∏ p | n (1 − p − k). In this paper, we study the average behavior of fractions P / Q of two products P and Q of Jordan totients, which we call Jordan totient quotients. To this end, we describe two general and ready-to-use methods that allow one to deal with a larger class of totient functions. The first one is elementary and the second one uses an advanced method due to Balakrishnan and Pétermann. As an application, we determine the average behavior of the Jordan totient quotient, the k t h normalized derivative of the n t h cyclotomic polynomial Φ n (z) at z = 1 , the second normalized derivative of the n t h cyclotomic polynomial Φ n (z) at z = − 1 , and the average order of the Schwarzian derivative of Φ n (z) at z = 1. [ABSTRACT FROM AUTHOR]

In general, a tensor product of Demazure crystals does not decompose into a disjoint union of Demazure crystals. However, under a certain condition, a tensor product decomposes into a disjoint union of Demazure crystals. In this paper, we introduce a necessary and sufficient condition for every connected component of a tensor product of two Demazure crystals to be isomorphic to some Demazure crystal. Moreover, we consider a recursive formula describing connected components of tensor products of arbitrary Demazure crystals. As an application, we discuss the key positivity problem, which is the problem whether a product of key polynomials is a linear combination of key polynomials with nonnegative integer coefficients or not. Also, we obtain a crystal-theoretic analog of the Leibniz rule for Demazure operators. [ABSTRACT FROM AUTHOR]

Counting problems whose solution is PORC were introduced in a famous paper by Higman (1960). We consider two specific counting problems with PORC solutions: the number of isomorphism types of d -generator class-2 Lie algebras over F q (as a function in q) and the number of isomorphism types of d -generator p -class 2 p -groups (as a function in p). We prove lower bounds for the degrees of their PORC formulae for all d ∈ N and we determine explicit PORC formulae for d ≤ 7. [ABSTRACT FROM AUTHOR]

We consider the class of Rudin–Shapiro-like polynomials, whose L 4 norms on the complex unit circle were studied by Borwein and Mossinghoff. The polynomial f (z) = f 0 + f 1 z + ⋯ + f d z d is identified with the sequence (f 0 , f 1 , ... , f d) of its coefficients. From the L 4 norm of a polynomial, one can easily calculate the autocorrelation merit factor of its associated sequence, and conversely. In this paper, we study the crosscorrelation properties of pairs of sequences associated to Rudin–Shapiro-like polynomials. We find an explicit formula for the crosscorrelation merit factor. A computer search is then used to find pairs of Rudin–Shapiro-like polynomials whose autocorrelation and crosscorrelation merit factors are simultaneously high. Pursley and Sarwate proved a bound that limits how good this combined autocorrelation and crosscorrelation performance can be. We find infinite families of polynomials whose performance approaches quite close to this fundamental limit. [ABSTRACT FROM AUTHOR]

In this paper, we consider the system x ˙ = y (1 + x) 2 − ϵ P (x , y) , y ˙ = − x (1 + x) 2 + ϵ Q (x , y) where P (x , y) and Q (x , y) are arbitrary quadratic polynomials. We study the maximum number of limit cycles bifurcating from the periodic orbits by using the Melnikov function of any order. We prove that the upper bound for the number of limit cycles is 3 and reached. [ABSTRACT FROM AUTHOR]

In this paper it is shown that, for a module M over a ring R with S = E n d R (M) , the endomorphism ring of the R [ x ] -module M [ x ] is isomorphic to a subring of S [ [ x ] ]. Also the endomorphism ring of the R [ [ x ] ] -module M [ [ x ] ] is isomorphic to S [ [ x ] ]. As a consequence, we show that for a module M R and an arbitrary nonempty set of not necessarily commuting indeterminates X , M R is quasi-Baer if and only if M [ X ] R [ X ] is quasi-Baer if and only if M [ [ X ] ] R [ [ X ] ] is quasi-Baer if and only if M [ x ] R [ x ] is quasi-Baer if and only if M [ [ x ] ] R [ [ x ] ] is quasi-Baer. Moreover, a module M R with IFP , is Baer if and only if M [ x ] R [ x ] is Baer if and only if M [ [ x ] ] R [ [ x ] ] is Baer. It is also shown that, when M R is a finitely generated module, and every semicentral idempotent in S is central, then M [ [ X ] ] R [ [ X ] ] is endo-p.q.-Baer if and only if M [ [ x ] ] R [ [ x ] ] is endo-p.q.-Baer if and only if M R is endo-p.q.-Baer and every countable family of fully invariant direct summand of M has a generalized countable join. Our results extend several existing results. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Littlewood's problem for the second order differential equation x ¨ + x 2 n + 1 + q (t) x l = p (t) , l < n , where the functions p (t) and q (t) do not need to be periodic. Using the theory of non-periodic twist mappings which is developed by Kunze and Ortega in [3–7] , we obtained longtime closeness results on the bounded and unbounded solutions of the equation. [ABSTRACT FROM AUTHOR]

Sylvester doubles sums, introduced first by Sylvester (see Sylvester, 1840, 1853), are symmetric expressions of the roots of two polynomials P and Q. Sylvester's definition of double sums makes no sense if P and Q have multiple roots, since the definition involves denominators that vanish when there are multiple roots. The aims of this paper are to give a new definition for Sylvester double sums making sense if P and Q have multiple roots, which coincides with the definition by Sylvester in the case of simple roots, to prove the fundamental property of Sylvester double sums, i.e. that Sylvester double sums indexed by (k , ℓ) are equal up to a constant if they share the same value for k + ℓ , and to prove the relationship between double sums and subresultants, i.e. that they are equal up to a constant. In the simple root case, proofs of these properties are already known (see Lascoux and Pragacz, 2002; d'Andrea et al., 2007; Roy and Szpirglas, 2011). The more general proofs given here are using generalized Vandermonde determinants and a new symmetric multivariate Hermite interpolation as well as an induction on the length of the remainder sequence of P and Q. [ABSTRACT FROM AUTHOR]

In this paper, an injection from the partition lattice Π n to a zero-dimensional affine algebra over the finite field F 2 is constructed. We study some interesting properties of the injection and present two direct sum decompositions of the zero-dimensional affine algebra. Based upon these, some properties of Π n are revealed in an algebraic manner, for instance, the rank-generating function and characteristic polynomial of Π n can be derived using the symbolic method. [ABSTRACT FROM AUTHOR]

In this paper, we study how the degrees of the elements in a minimal μ -basis of a parametrized surface behave. For an arbitrary rational surface parametrization P (s , t) = (a 1 (s , t) , a 2 (s , t) , a 3 (s , t) , a 4 (s , t)) ∈ F [ s , t ] 4 over an infinite field F , we show the existence of a μ -basis with polynomials bounded in degree by O (d 33) , where d = max ⁡ (deg ⁡ (a 1) , deg ⁡ (a 2) , deg ⁡ (a 3) , deg ⁡ (a 4)). Under additional assumptions we can obtain tighter bounds. [ABSTRACT FROM AUTHOR]

The paper discusses the construction of high dimensional spatial discretizations for arbitrary multivariate trigonometric polynomials, where the frequency support of the trigonometric polynomial is known. We suggest a construction based on the union of several rank-1 lattices as sampling scheme. We call such schemes multiple rank-1 lattices. This approach automatically makes available a fast discrete Fourier transform (FFT) on the data. The key objective of the construction of spatial discretizations is the unique reconstruction of the trigonometric polynomial using the sampling values at the sampling nodes. We develop different construction methods for multiple rank-1 lattices that allow for this unique reconstruction. The symbol M denotes the total number of sampling nodes within the multiple rank-1 lattice. In addition, we assume that the multivariate trigonometric polynomial is a linear combination of T trigonometric monomials. The ratio of the number M of sampling points that are sufficient for the unique reconstruction to the number T of distinct monomials is called oversampling factor in this context. The presented construction methods for multiple rank-1 lattices allow for estimates of this number M. Roughly speaking, the oversampling factor M / T is independent of the spatial dimension and, with high probability, only logarithmic in T , which is much better than the oversampling factor that is expected for a sampling method that uses one single rank-1 lattice. The newly developed approaches for the construction of spatial discretizations are probabilistic methods. The arithmetic complexity of these algorithms depend only linearly on the spatial dimension and, with high probability, only linearly on T up to some logarithmic factors. Furthermore, we analyze the computational complexities of the resulting FFT algorithms, that exploits the structure of the suggested multiple rank-1 lattice spatial discretizations, in detail and obtain upper bounds in O (M log ⁡ M) , where the constants depend only linearly on the spatial dimension. With high probability, we construct spatial discretizations where M / T ≤ C log ⁡ T holds, which implies that the complexity of the corresponding FFT converts to O (T log 2 ⁡ T). [ABSTRACT FROM AUTHOR]

The set of prime numbers p such that the supersingular j -invariants in characteristic p are all contained in the prime field is finite. And it is well known that this set of primes coincides with the set of prime divisors of the order of the Monster simple group. In this paper, we will present analogous coincidences of supersingular invariants in level 2 and 3 and the orders of the Baby monster group and the Fischer's group. The proof uses a connection between the number of supersingular invariants and class numbers of imaginary quadratic fields. [ABSTRACT FROM AUTHOR]

The Laguerre–Sobolev polynomials form an orthogonal polynomial system with respect to a Sobolev-type inner product associated with the Laguerre measure on the positive half-axis and two point masses M , N > 0 at the origin involving functions and derivatives. These polynomials have attracted much interest over the last two decades, since they became known to satisfy, for any value of the Laguerre parameter α ∈ N 0 , a spectral differential equation of finite order 4 α + 10. In this paper we establish a new explicit representation of the corresponding differential operator which consists of a number of elementary components depending on α , M , N. Their interaction reveals a rich structure both being useful for applications and as a model for further investigations in the field. In particular, the Laguerre–Sobolev differential operator is shown to be symmetric with respect to the Sobolev-type inner product. [ABSTRACT FROM AUTHOR]

In 2016, Ellenberg and Gijswijt employed a method of Croot, Lev, and Pach to show that a maximal cap in A G (n , q) has size O (q c n) for some c < 1. In this paper, we show more generally that if S is a subset of A G (n , q) containing no m points on any (m − 2) -flat, then | S | < q c m n for some c m < 1 , as long as q is odd or m is even. [ABSTRACT FROM AUTHOR]

Originally studied by Conway and Coxeter, friezes appeared in various recreational mathematics publications in the 1970s. More recently, in 2015, Baur, Parsons, and Tschabold constructed periodic infinite friezes and related them to matching numbers in the once-punctured disk and annulus. In this paper, we study such infinite friezes with an eye towards cluster algebras of type D and affine A, respectively. By examining infinite friezes with Laurent polynomial entries, we discover new symmetries and formulas relating the entries of this frieze to one another. Lastly, we also present a correspondence between Broline, Crowe and Isaacs's classical matching tuples and combinatorial interpretations of elements of cluster algebras from surfaces. [ABSTRACT FROM AUTHOR]

In this paper we consider weighted polynomial approximation on unbounded multidimensional domains in the spirit of the weighted version of the Weierstrass trigonometric theorem according to which every continuous function on the real line with equal finite limits at ± ∞ is a uniform limit on R of weighted algebraic polynomials of degree 2 n with varying weights (1 + t 2) − n. We will verify a similar statement in the multivariate setting for a general class of convex weights. We also consider the similar problem of multivariate polynomial approximation with varying weights for some typical non convex weights. In case of non convex weights of the form w α (x) ≔ (1 + | x 1 | α +... + | x d | α) 1 α , 0 < α < 1 in order for weighted polynomial approximation to hold for a given continuous function it is necessary that the function vanishes on a certain exceptional set consisting of all coordinate hyperplanes and ∞. Moreover, in case of rational α this condition is also sufficient for weighted polynomial approximation to hold. [ABSTRACT FROM AUTHOR]

In this paper, we study γ -positivity of descent-type polynomials by introducing the change of context-free grammars method. We first present a unified grammatical proof of the γ -positivity of Eulerian polynomials, type B Eulerian polynomials, derangement polynomials, Narayana polynomials and type B Narayana polynomials. We then provide partial γ -positive expansions for several multivariate polynomials associated to Stirling permutations, Legendre-Stirling permutations, Jacobi-Stirling permutations and type B derangements. The recurrence relations for the partial γ -coefficients of these expansions are also obtained. By using some variants of the Foata-Strehl group action, we provide combinatorial interpretations for the coefficients of most of these partial γ -positive expansions. [ABSTRACT FROM AUTHOR]

Matroid theory is often thought of as a generalisation of graph theory. In this paper we propose an analogous correspondence between embedded graphs and delta-matroids. We show that delta-matroids arise as the natural extension of graphic matroids to the setting of embedded graphs. We show that various basic ribbon graph operations and concepts have delta-matroid analogues, and illustrate how the connections between embedded graphs and delta-matroids can be exploited. Also, in direct analogy with the fact that the Tutte polynomial is matroidal, we show that several polynomials of embedded graphs from the literature, including the Las Vergnas, Bollabás-Riordan and Krushkal polynomials, are in fact delta-matroidal. [ABSTRACT FROM AUTHOR]

In this paper, we give some results on closed polynomials and factorially closed polynomials in n variables which are generalizations of results in [7] , [12] and [13]. In particular, we give a characterization of factorially closed polynomials in n variables over an algebraically closed field of any characteristic. Furthermore, as an application of results on closed polynomials, we determine kernels of non-zero monomial derivations on the polynomial ring in two variables over a UFD. Finally, by using this result and the argument in [16, §5] , for a field k , we determine the non-zero monomial derivations D on k [ x , y ] such that the quotient field of the kernel of D is not equal to the kernel of D in k (x , y). [ABSTRACT FROM AUTHOR]

Let K be a non-algebraically closed field and f ∈ K [ x ] be an irreducible polynomial of degree d ≥ 3. In this paper we study the existence of a polynomial h ∈ K [ x ] of degree ≤ d − 1 such that the polynomial f (h (x)) is reducible in K [ x ]. We prove the existence of such polynomial in case of d ≤ 4. A similar (and quite general) result is true in the case when f represents an even function, i.e., f (x) = f (− x). In particular, if char (K) = 2 , then we prove that the existence of a required polynomial h. [ABSTRACT FROM AUTHOR]

A Cullen number is a number of the form m 2 m + 1 , where m is a positive integer. In 2004, Luca and Stănică proved, among other things, that the largest Fibonacci number in the Cullen sequence is F 4 = 3. Actually, they searched for generalized Cullen numbers among some binary recurrence sequences. In this paper, we will work on higher order recurrence sequences. For a given linear recurrence (G n) n , under weak assumptions, and a given polynomial T (x) ∈ Z [ x ] , we shall prove that if G n = m x m + T (x) , then m ≪ 1 and n ≪ log ⁡ | x | , where the implied constants depend only on (G n) n and T (x). [ABSTRACT FROM AUTHOR]

We prove a partial fraction decomposition of a quotient of two functions E α (i t x) and I α (i t) which are defined in terms of the Bessel functions J α and J α + 1 of the first kind. This expansion leads naturally to the introduction of an orthonormal system with respect to the measure | x | 2 α + 1 d x 2 α + 1 Γ (α + 1) in [ − 1 , 1 ] , which we call the Fourier–Dunkl system of the second kind. Euler–Dunkl polynomials E n , α (x) of degree n are defined by considering E α (t x) ∕ I α (t) as a generating function. It is shown that the sum ∑ m = 1 ∞ 1 ∕ j m , α 2 k , where j m , α are the positive zeros of J α , is equal (up to an explicit factor) to E 2 k − 1 , α (1). For α = 1 ∕ 2 this leads to classical results of Euler since the function E 1 ∕ 2 (x) is the exponential function and E n , 1 ∕ 2 (x) are (essentially) the Euler polynomials. In the second part of the paper a sampling theorem of Whittaker–Shannon–Kotel'nikov type is established which is strongly related to the above-mentioned partial decomposition and which holds for all functions in the Payley–Wiener space defined by the Dunkl transform in [ − 1 , 1 ]. [ABSTRACT FROM AUTHOR]

The Turán inequality and its higher order analog arise in the study of Maclaurin coefficients of an entire function in the Laguerre-Pólya class. It is well known that if a real entire function ψ (x) is in the LP class, the Maclaurin coefficients satisfy both the Turán inequality and the higher order Turán inequality. Chen, Jia and Wang proved that for n ≥ 95 , the higher order Turán inequality holds for the partition function p (n) and the 3-rd associated Jensen polynomials p (n) + 3 p (n + 1) x + 3 p (n + 2) x 2 + p (n + 3) x 3 have only real zeros. Recently, Griffin, Ono, Rolen and Zagier showed that Jensen polynomials for a large family of functions, including those associated to ξ (s) and the partition function, are hyperbolic for sufficiently large n. This result gave evidence for Riemann hypothesis. In this paper, we give a unified approach to investigate the higher order Turán inequality for the sequences { a n / n ! } n ≥ 0 , where a n satisfy a three-term recurrence relation. In particular, we prove higher order Turán inequality for the sequences { a n / n ! } n ≥ 0 , where a n are the Motzkin numbers, the Fine number, the Franel numbers of order 3 and the Domb numbers. As a consequence, for these combinatorial sequences, the 3-rd associated Jensen polynomials a n n ! + 3 a n + 1 (n + 1) ! x + 3 a n + 2 (n + 2) ! x 2 + a n + 3 (n + 3) ! x 3 have only real zeros. Furthermore, for these combinatorial sequences we conjecture that for any given integer m ≥ 4 , there exists an integer N (m) such that for n > N (m) , the m -th associated Jensen polynomials ∑ i = 0 m ( m i ) a n + i (n + i) ! x i have only real zeros. [ABSTRACT FROM AUTHOR]

Let the polynomial b n (t) be defined as the n -th coefficient in the power series expansion (in variable x) of the function B (t , x) = ∏ n = 0 ∞ 1 1 − t x 2 n = ∑ n = 0 ∞ b n (t) x n. The polynomial b n (t) = ∑ i = 0 n a (i , n) t i has the following combinatorial interpretation: the i -th coefficient a (i , n) counts the number of representations of n as sums of exactly i powers of 2. The b n (t) can be seen as a polynomial analogue of the number b n (1) , which counts the number of binary partitions of n. In the present paper we obtain several results concerning arithmetic properties of the polynomials b n (t) as well as its coefficients. Moreover, we show an interesting connection between coefficients of b n (t) and the number counting so called s -partitions, i.e., the representations of n as sums of numbers of the form 2 k − 1. [ABSTRACT FROM AUTHOR]