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The enhanced power graph of a group G is a simple undirected graph with vertex set G and two vertices are adjacent if they belong to the same cyclic subgroup. In this paper, we obtain the Laplacian spectrum of the enhanced power graph of certain non-abelian groups, viz. semidihedral, dihedral and generalized quaternion. Also, we obtained the metric dimension and the resolving polynomial of the enhanced power graphs of these groups. At the final part of this paper, we study the distant properties and the detour distant properties, namely: closure, interior, distance degree sequence, eccentric subgraph of the enhanced power graph of semidihedral group, dihedral group and generalized quaternion group, respectively. [ABSTRACT FROM AUTHOR]

In this paper, the discovery of the Jones polynomial will be discussed, emphasizing the way in which it illustrated the remarkable unity between distinct parts of mathematics, each with its own language, but initially without a dictionary. [ABSTRACT FROM AUTHOR]

In this paper, we study algebraic properties of a family of certain polynomials arising from the functional equation for Dickson polynomials. We see that the roots and discriminants of those polynomials have very simple expressions, and each polynomial is cyclic. Further, we provide an irreducibility criterion analogous to the well-known criterion of Vahlen-Capelli. We finish the paper by showing that any cyclic extension of a certain field comes from a member of the family. [ABSTRACT FROM AUTHOR]

This paper deals with two new subclasses of holomorphic and bi-univalent functions in the open unit disk defined by generalized bivariate Fibonacci polynomials. In this paper the coefficient bounds are estimated for | a 2 | and | a 3 | which a 2 and a 3 are the Taylor–Maclaurin coefficients of the functions belonging to these new subclasses. Then, the Fekete–Szegö problem is handled for the functions in these subclasses. Also, several remarks are presented. The results of this paper generalize certain earlier results in the literature. [ABSTRACT FROM AUTHOR]

This paper surveys the theory of functional identities and its applications. No prior knowledge of the theory is required to follow the paper. [ABSTRACT FROM AUTHOR]

In this paper, we study orbits and fixed points of polynomials in a general skew polynomial ring D [ x , σ , δ ]. We extend results of the first author and Vishkautsan on polynomial dynamics in D [ x ]. In particular, we show that if a ∈ D and f ∈ D [ x , σ , δ ] satisfy f (a) = a , then f ∘ n (a) = a for every formal power of f. More generally, we give a sufficient condition for a point a to be r -periodic with respect to a polynomial f. Our proofs build upon foundational results on skew polynomial rings due to Lam and Leroy. [ABSTRACT FROM AUTHOR]

Let P (x) : = a d x d + ⋯ + a 0 ∈ ℚ [ x ] , a d > 0 , be a polynomial of degree d ≥ 2. Let (x n) be a sequence of integers satisfying x n + 1 = P (x n) for all  n = 0 , 1 , 2 , ... and x n → ∞ as  n → ∞. Set α : = lim n → ∞ x n d − n . Then, under the assumption a d 1 / (d − 1) ∈ ℚ , in a recent result by [A. Dubickas, Transcendency of some constants related to integer sequences of polynomial iterations, Ramanujan J. 57 (2022) 569–581], either α is transcendental or α can be an integer or a quadratic Pisot unit with α − 1 being its conjugate over ℚ. In this paper, we study the nature of such α without the assumption that a d 1 / (d − 1) is in ℚ , and we prove that either the number α is transcendental, or α h is a Pisot number with h being the order of the torsion subgroup of the Galois closure of the number field ℚ α , a d − 1 d − 1 . Other results presented in this paper investigate the solutions of the inequality | | q 1 α 1 n + ⋯ + q k α k n + β | | < n in (n , q 1 , ... , q k) ∈ ℕ × (K ×) k , considering whether β is rational or irrational. Here, K represents a number field, and ∈ (0 , 1). The notation | | x | | denotes the distance between x and its nearest integer in ℤ. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new method of applying the XOR and XNOR gates on exponentially large superpositions in Instantaneous Noise-Based Logic. These new gates are repeatable, and they can achieve an exponential speedup in computation with polynomial hardware complexity. [ABSTRACT FROM AUTHOR]

In this paper, we consider Tower of Hanoi graphs and study their matching properties. Explicit system of recurrences is derived for the matching polynomials of these graphs and their appropriate truncated variants. Consequently, we obtain exact formula for the numbers of maximum matchings in Hanoi graphs using matching polynomials, which is a new approach for old one problem. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the root distribution of the Ehrhart polynomials of lattice polytopes. When the lattice polytope is reflexive, the roots of the Ehrhart polynomial distribute symmetrically with respect to the line Re (z) = − 1 2 . A special case of this distribution is when all the roots lie on this line. Our main concern is to find out which reflexive polytopes satisfy this special condition. Such lattice polytopes are called CL-polytopes. Another special case opposite to this is when all the roots are real. Such lattice polytopes are called real polytopes. The first topic of this paper is the Ehrhart polynomials of equatorial spheres, which are related to graded posets. We discuss the CL-ness of the equatorial Ehrhart polynomials for complete graded posets and zig-zag posets. The second topic is to investigate the relation of the root distribution of the Ehrhart polynomials of a reflexive polytope Q and its dual Q ∨ . We discuss the CL-ness and realness of Q and Q ∨ in pair, of dimensions up to 4. Throughout this paper, we investigate these problems by computer calculation. [ABSTRACT FROM AUTHOR]

Let K be a field of characteristic zero and x a free variable. A K - ℰ -derivation of K [ x ] is a K -linear map of the form I , − , ϕ for some K -algebra endomorphism ϕ of K [ x ] , where I denotes the identity map of K [ x ]. In this paper, we study the image of an ideal of K [ x ] under some K -derivations and K - ℰ -derivations of K [ x ]. We show that the LFED conjecture proposed in [W. Zhao, Some open problems on locally finite or locally nilpotent derivations and ℰ -derivations, Commun. Contem. Math. 20(4) (2018) 1750056] holds for all K - ℰ -derivations and all locally finite K -derivations of K [ x ]. We also show that the LNED conjecture proposed in [W. Zhao, Some open problems on locally finite or locally nilpotent derivations and ℰ -derivations, Commun. Contem. Math. 20(4) (2018) 1750056] holds for all locally nilpotent K -derivations of K [ x ] , and also for all locally nilpotent K - ℰ -derivations of K [ x ] and the ideals u K [ x ] such that either u = 0 , or deg u ≤ 1 , or u has at least one repeated root in the algebraic closure of K. As a bi-product, the homogeneous Mathieu subspaces (Mathieu–Zhao spaces) of the univariate polynomial algebra over an arbitrary field have also been classified. [ABSTRACT FROM AUTHOR]

We obtain sharp Hardy inequalities on antisymmetric functions, where antisymmetry is understood for multi-dimensional particles. Partially it is an extension of the paper [Th. Hoffmann-Ostenhof and A. Laptev, Hardy inequalities with homogeneous weights, J. Funct. Anal. 268 (2015) 3278–3289], where Hardy's inequalities were considered for the antisymmetric functions in the case of the 1D particles. As a byproduct we obtain some Sobolev and Gagliardo–Nirenberg type inequalities that are applied to the study of spectral properties of Schrödinger operators. [ABSTRACT FROM AUTHOR]

In this paper, we study the relation between the images of polynomial derivations and their simplicity. We prove that the images of simple Shamsuddin derivations are not Mathieu–Zhao spaces. In addition, we show that the images of some simple derivations in dimension three are not Mathieu-Zhao spaces. Thus, we conjecture that the images of simple derivations in dimension greater than one are not Mathieu–Zhao spaces. We also prove that locally nilpotent derivations are not simple in dimension greater than one. [ABSTRACT FROM AUTHOR]

The bifurcation theory for homoclinic networks with singular and nonsingular equilibriums is a key to understand the global dynamics of nonlinear dynamical systems, which will help one determine the dynamical behaviors of physical and engineering nonlinear systems. In this paper, the appearing and switching bifurcations for homoclinic networks through equilibriums in planar polynomial dynamical systems are studied. The appearing and switching bifurcations are discussed for the homoclinic networks of nonsingular and singular sources, sinks, saddles with singular saddle-sources, saddle-sinks, and double-saddles in self-univariate polynomial systems. The first integral manifolds for nonsingular and singular equilibrium networks are determined. The illustrations of singular equilibriums to networks of nonsingular sources, sinks and saddles are given. The appearing and switching bifurcations are studied for homoclinic networks of singular and nonsingular saddles and centers with singular parabola-saddles and double-inflection saddles in crossing-univariate polynomial systems, and the first integral manifolds of such homoclinic networks are determined through polynomial functions. The illustrations of singular equilibriums to networks of nonsingular saddles and centers are given. This paper may help one understand higher-order bifurcation theory in nonlinear dynamical systems, which is completely different from the classic bifurcation theories. [ABSTRACT FROM AUTHOR]

Let R be a prime ring of characteristic different from 2 , Q r be its right Martindale quotient ring and C be its extended centroid, α be an automorphism of R , d be a skew derivation of R with associated automorphism α , F and G be two nonzero X -generalized skew derivation of R with associated term (b , α , d) and (b ′ , α , d) , respectively, S be the set of the evaluations of f (x 1 , ... , x n) on R , where f (x 1 , ... , x n) is a non-central multilinear polynomial over C in n non-commuting variables. Let 0 ≠ v ∈ R be such that F (x) x + G (x) x v = 0 for all x ∈ S. Then one of the following statements holds : (a) f (x 1 , ... , x n) 2 is central valued on R and there exist a , a ′ ∈ Q r such that F (x) = a x , G (x) = a ′ x for any x ∈ R with a + a ′ v = 0 ; (b) v ∈ C and F = − v G. In the last part of the paper, we present some applications on the basis of the foregoing proposed result. [ABSTRACT FROM AUTHOR]

Let G be a connected graph with n vertices and m edges. Let G ⃗ be the digraph obtained by orienting the edges of G arbitrarily. The digraph G ⃗ is called an orientation of G or oriented graph corresponding to G. The skew Laplacian matrix of the digraph G ⃗ is denoted by S L ̃ (G ⃗) and is defined as S L ̃ (G ⃗) = D ̃ (G ⃗) − S (G ⃗) , where S (G ⃗) is the skew matrix and D ̃ (G ⃗) is the diagonal matrix with i th diagonal entry d i + − d i − . In this paper, we obtain combinatorial representation for the first five coefficients of characteristic polynomial of skew Laplacian matrix of G ⃗. We provide examples of orientations of some well-known graphs to highlight the importance of our results. We conclude the paper with some observations about the skew Laplacian spectral determinations of the directed path and directed cycle. [ABSTRACT FROM AUTHOR]

We study the dynamics of the unicritical polynomial family f d , c (z) = z d + c ∈ ℂ [ z ]. The c -values for which f d , c has a strictly preperiodic postcritical orbit are called Misiurewicz parameters, and they are the roots of Misiurewicz polynomials. The arithmetic properties of these special parameters have found applications in both arithmetic and complex dynamics. In this paper, we investigate some new such properties. In particular, when d is a prime power and c is a Misiurewicz parameter, we prove certain arithmetic relations between the points in the postcritical orbit of f d , c . We also consider the algebraic integers obtained by evaluating a Misiurewicz polynomial at a different Misiurewicz parameter, and we ask when these algebraic integers are algebraic units. This question naturally arises from some results recently proven by Buff, Epstein, and Koch and by the second author. We propose a conjectural answer to this question, which we prove in many cases. [ABSTRACT FROM AUTHOR]

Recently, two partition functions p ω (n) and p ν (n) were introduced by Andrews et al. The partition function p ω (n) denotes the number of partitions of n in which each odd part is less than twice the smallest part, and p ν (n) counts the number of partitions of n into distinct non-negative parts such that all odd parts are less than twice the smallest part. Very recently, Silva et al. studied congruence properties of the restricted partition functions p ω (n , m) and p ν (n , m) which denote the number of partitions enumerated by p ω (n) and p ν (n) , respectively, into exactly m parts. In this paper, we give a polynomial method for discovering congruences for p ω (n , m) and p ν (n , m) by checking a finite number of initial values. Employing the polynomial method, we proved new and existing congruences for p ω (n , m) and p ν (n , m) based on a bounded number of calculations. For example, in order to prove that p ν (3 4 6 5 n + 1 5 6 , 5) ≡ 0 (mod 1 1) holds for all n ≥ 0 , it suffices to show that the congruence holds when 0 ≤ n ≤ 4. [ABSTRACT FROM AUTHOR]

For a simple graph G , the generalized adjacency matrix A α (G) is defined as A α (G) = α D (G) + (1 − α) A (G) , α ∈ [ 0 , 1 ] , where A (G) is the adjacency matrix and D (G) is the diagonal matrix of vertex degrees of G. This matrix generalizes the spectral theories of the adjacency matrix and the signless Laplacian matrix of G. In this paper, we find the A α -spectrum of the joined union of graphs in terms of the spectrum of the adjacency matrices of its components and the zeros of the characteristic polynomials of an auxiliary matrix determined by the joined union. We determine the A α -spectrum of join of two regular graphs, the join of a regular graph with the union of two regular graphs of distinct degrees. As applications, we investigate the A α -spectrum of certain power graphs of finite groups. [ABSTRACT FROM AUTHOR]

We study single-machine bicriteria scheduling problems with position-dependent weights. Due date costs and penalty costs can affect the decision-maker's profit. Therefore, how to optimize the benefits and balance of these costs for decision-makers is the focus of this research. This paper discusses three models: weighted-sum scheduling problem, constrained scheduling problem, Pareto scheduling problem. It analyzes the properties of bicriteria schedule, determines the optimal due dates (usually referred to as different due dates: DIF), and gives the corresponding polynomial solvable algorithms. [ABSTRACT FROM AUTHOR]

Noncatastrophic encoders are an important class of polynomial generator matrices of convolutional codes. When these polynomials have coefficients in a finite field, these encoders have been characterized as polynomial left prime matrices. In this paper, we study the notion of noncatastrophicity in the context of convolutional codes when the polynomial matrices have entries in the finite ring ℤ p r . In particular, we study the notion of zero left prime in order to fully characterize noncatastrophic encoders over the finite ring ℤ p r . The second part of the paper is devoted to investigate free and column distance of convolutional codes that are free finitely generated ℤ p r -modules. We introduce the notion of b -degree and provide new bounds on the free distances and column distance. We show that this class of convolutional codes is optimal with respect to the column distance and to the free distance if and only if its projection on ℤ p is. [ABSTRACT FROM AUTHOR]

This is a corrigendum to the authors paper [A reconstruction theorem for complex polynomials, Int. J. Math. 26(9) (2015) 1550073]. Here we present a completely rewritten and self-contained Sec. 4 of the aforementioned paper and provide the reader with the corrected proof of the main result that briefly states: "Given a generic polynomial P of degree at least 2 , there exist positive integers N , M such that Re (P k (z)) = Re (P k (w)) for all 0 ≤ k ≤ N implies P M (z) = P M (w) ". [ABSTRACT FROM AUTHOR]

Eisenstein polynomials, which were defined by the second author, are analogues to the concept of an Eisenstein series. The second author conjectured that there exist some analogous properties between Eisenstein series and Eisenstein polynomials. In the previous paper, the first author provided new analogous properties of Eisenstein polynomials and zeta polynomials for the Type II case. In this paper, the analogous properties of Eisenstein polynomials and zeta polynomials are shown to also hold for the Type I, Type III, and Type IV cases. These properties are finite analogies of certain properties of Eisenstein series. [ABSTRACT FROM AUTHOR]

In this paper, we introduced two classes of elements in the enveloping algebra U (g l (n)) : the double Young–Capelli bitableaux  [ S | T ] and the central Schur elements  S λ (n) , that act in a remarkable way on the highest weight vectors of irreducible Schur modules. Any element S λ (n) is the sum of all double Young–Capelli bitableaux [ S | S ] , S row (strictly) increasing Young tableaux of shape λ ̃. The Schur elements S λ (n) are proved to be the preimages — with respect to the Harish-Chandra isomorphism — of the shifted Schur polynomials  s λ | n ∗ ∈ Λ ∗ (n). Hence, the Schur elements are the same as the Okounkov quantum immanants, recently described by the present authors as linear combinations of Capelli immanants. This new presentation of Schur elements/quantum immanants does not involve the irreducible characters of symmetric groups. The Capelli elements H k (n) are column Schur elements and the Nazarov elements I k (n) are row Schur elements. The duality in ζ (n) follows from a combinatorial description of the eigenvalues of the H k (n) on irreducible modules that is dual (in the sense of shapes/partitions) to the combinatorial description of the eigenvalues of the I k (n). The passage n → ∞ for the algebras ζ (n) is obtained both as direct and inverse limit in the category of filtered algebras, via the Olshanski decomposition/projection. [ABSTRACT FROM AUTHOR]

In this paper, the improved localized method of approximated particular solutions (ILMAPS) using polyharmonic splines (PHS) together with a low-degree of polynomial basis is used to approximate solutions of various nonlinear elliptic Partial Differential Equations (PDEs). The method is completely meshfree, and it uses a radial basis function (RBF) that has no shape parameters. The discretization process is done through a simple collocation technique on a set of points in the local domain of influence. Resulted system of nonlinear algebraic equations is solved by the Picard method. The performance of the proposed method is tested on various nonlinear elliptical problems, including the Poisson-type problems in 2D and 3D with constant or variable coefficients on rectangular or irregular domains and the Poisson–Boltzmann equation with Dirichlet boundary conditions or mixed boundary conditions. The effect of domain shapes in 2D and 3D, types of boundary conditions, and degrees of PHS, and order of polynomial basis are examined. The performance of the method is compared with other bases such as multiquadrics (MQ) basis functions, and with results reported in the literature (method of particular solutions using polynomials). The numerical experiments suggest that ILMAPS with polyharmonic splines yields considerably superior accuracy than other RBFs as well as other approaches reported in the literature for solving nonlinear elliptic PDEs. [ABSTRACT FROM AUTHOR]

Although the connected sum of trivial knots is trivial in classical knot theory, there is a nontrivial connected sum of trivial knots in virtual knot theory. In this paper, we give connected sum formulae of the intersection polynomials of virtual knots introduced in the preceding paper [6]. As an application, we show that there are infinitely many connected sums of any pair of virtual knots. [ABSTRACT FROM AUTHOR]

For a partition λ of n ∈ ℕ , let I λ Sp be the ideal of R = K [ x 1 , ... , x n ] generated by all Specht polynomials of shape λ. We assume that char (K) = 0. Then R / I (n − 2 , 2) Sp is Gorenstein, and R / I (d , d , 1) Sp is a Cohen–Macaulay ring with a linear free resolution. In this paper, we construct minimal free resolutions of these rings. Zamaere et al. [Jack polynomials as fractional quantum Hall states and the Betti numbers of the (k + 1) -equals ideal, Commun. Math. Phys. 330 (2014) 415–434] already studied minimal free resolutions of R / I (n − d , d) Sp , which are also Cohen–Macaulay, using highly advanced technique of the representation theory. However, we only use the basic theory of Specht modules, and explicitly describe the differential maps. [ABSTRACT FROM AUTHOR]

In this paper, for a henselian valued field (K , v) of arbitrary rank and an extension w of v to K (X) , we use abstract key polynomials for w to obtain distinguished pairs and saturated distinguished chains. [ABSTRACT FROM AUTHOR]

Let (R ,) be a Noetherian local ring and M a finitely generated R -module. In this paper, we study the polynomial type and the sequential polynomial type of the idealization R ⋉ M of R and M introduced by Cuong [On the least degree of polynomials bounding above the differences between lengths and multiplicities of certain systems of parameters in local rings, Nagoya Math. J. 125 (1992) 105–114] and Nhan-Dung-Chau [A measure for non-sequential Cohen Macaulayness of finitely generated modules, J. Algebra 468 (2016) 275–295], respectively. In addition, we give a uniform bound for the reducibility index of parameter ideals of idealizations whose polynomial type and sequential polynomial type are at most one. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the uniqueness problems of meromorphic functions concerning differential polynomials sharing a small function. In particular, the resuts of the paper improve the results due to Waghamore and Anand [11]. [ABSTRACT FROM AUTHOR]

Entanglement is an important resource for quantum technologies that allow us to go beyond the classical world. Quantum systems can be entangled in various ways. Also, the degree of entanglement between the subsystems of a quantum system can be calculated in different ways. In our recent paper [Prog. Theor. Exp. Phys. (2022)], we directly measure the three-tangle of a pure three-qubit quantum state. In this paper, we extend this method to pure N-qubit quantum states (N is any odd number greater than 1) by considering polynomial invariant of degree 4 as a measure of entanglement. In this algorithm, we use quantum gates and output probabilities to calculate polynomial invariant of degree 4. The importance of this method lies in the possibility of experimental implementation of quantum calculations in the language of quantum gates. [ABSTRACT FROM AUTHOR]

The set of sampling and reconstruction in trigonometric polynomial spaces will play an important role in signal processing. However, in many applications, the frequencies in trigonometric polynomial spaces are not all integers. In this paper, we consider the problem of weighted random sampling and reconstruction of functions in general multivariate trigonometric polynomial spaces. The sampling set is randomly selected on a bounded cube with a probability distribution. We obtain that with overwhelming probability, the sampling inequality holds and the explicit reconstruction formula succeeds for all functions in the general multivariate trigonometric polynomial spaces when the sampling size is sufficiently large. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to present a new periodicity criterion. For that purpose, we study the HOMFLY-PT polynomial and the Kauffman polynomial of cables of periodic links. Furthermore, we exhibit a couple of examples for which our criterion is stronger than many previously known criteria, among which is the Khovanov homology criterion of Borodzik and the second author. [ABSTRACT FROM AUTHOR]

Let n ≥ 3 be an odd natural number. In 1738, Abraham de Moivre introduced a family of polynomials of degree n with rational coefficients, all of which are solvable. So far, the Galois groups of these polynomials have been investigated only for prime numbers n and under special assumptions. We describe the Galois groups for arbitrary odd numbers n ≥ 3 in the irreducible case, up to few exceptions. In addition, we express all zeros of such a polynomial as rational functions of three zeros, two of which are connected in a certain sense. These results are based on the reduction of an irrational of degree 2n to irrationals of degree ≤ n. Such a reduction was given in a previous paper of the author. Here, however, we present a much simpler approach that is based on properties of Chebyshev polynomials. We also give a simple proof of a result of Filaseta et al. [ABSTRACT FROM AUTHOR]

In this paper, using standard pairs, we present a method to construct symbols satisfying the sum rules of order p for any given p. It is shown that using matrix polynomial theory symbols, symmetric or non-symmetric, satisfying the sum rules of order p can be constructed efficiently. The construction is illustrated using various examples. [ABSTRACT FROM AUTHOR]

Given a finite subset A of a distributive lattice, its total orderization to(A) is a natural transformation of A into a totally ordered set. Recently, the author showed that multivariate maps on distributive lattices which remain invariant under total orderizations generalize various maps on vector lattices, including bounded orthosymmetric multilinear maps and finite sums of bounded orthogonally additive polynomials. Therefore, a study of total orderization invariant maps on distributive lattices provides new perspectives for maps widely researched in vector lattice theory. However, the unwieldy notation of total orderizations can make calculations extremely long and difficult. In this paper we resolve this complication by providing considerably simpler characterizations of total orderization maps. Utilizing these easier representations, we then prove that a lattice multi-homomorphism on a distributive lattice is total orderization invariant if and only if it is symmetric, and we show that the diagonal of a symmetric lattice multi-homomorphism is a lattice homomorphism, extending known results for orthosymmetric vector lattice homomorphisms. [ABSTRACT FROM AUTHOR]

Principal symmetric ideals were recently introduced by Harada et al. in [The minimal free resolution of a general principal symmetric ideal, preprint (2023), arXiv:2308.03141], where their homological properties are elucidated. They are ideals generated by the orbit of a single polynomial under permutations of variables in a polynomial ring. In this paper, we determine when a product of two principal symmetric ideals is principal symmetric and when the powers of a principal symmetric ideal are again principal symmetric ideals. We characterize the ideals that have the latter property as being generated by polynomials invariant up to a scalar multiple under permutation of variables. Recognizing principal symmetric ideals is an open question for the purpose of which we produce certain obstructions. We also demonstrate that the Hilbert functions of symmetric monomial ideals are not all given by symmetric monomial ideals, in contrast to the non-symmetric case. [ABSTRACT FROM AUTHOR]

In this paper, we present a simple proof of a famous result of Newton on unimodal sequences associated to polynomials with real roots. We then give a brief exposition of the recent work of Huh on such sequences arising from combinatorial geometries. [ABSTRACT FROM AUTHOR]

In this paper, we propose quantum mechanical operator formalism for Touchard polynomials whose generating function is T n (x) = ∂ n ∂ λ n e (e λ − 1) x | λ = 0. That is replacing e λ x by e λ a † a , where a † a is the number operator, and using the method of integration within ordered product we find that (a † a) n is just the normal ordering form : T n (a † a) :. Then by virtue of the Weyl ordering form of quantum mechanical operator, we also introduce a new special polynomial whose generating function is G n (x) = ∂ n ∂ λ n 2 e λ + 1 exp [ 2 x e λ − 1 e λ + 1 ] | λ = 0. With use of the Weyl ordering form of e λ a † a , we prove (a † a) n = : T n (a † a) : = : : G n (a † a) : : , where : : : : denotes Weyl ordering. It seems that quantum mechancal operator formalism presents a new and simpler approach for generalizing Touchard polynomial theory. [ABSTRACT FROM AUTHOR]

For a prime p, positive integers r,n, and a polynomial f with coefficients in 픽pr, let Wp,r,n(f) = fn 픽 pr ∖ fn+1 픽 pr. As n varies, the Wp,r,n(f) partition the set of strictly preperiodic points of the dynamical system induced by the action of f on 픽pr. In this paper, we compute statistics of strictly preperiodic points of dynamical systems induced by unicritical polynomials over finite fields by obtaining effective upper bounds for the proportion of 픽pr lying in a given Wp,r,n(f). Moreover, when we generalize our definition of Wp,r,n(f), we obtain both upper and lower bounds for the resulting averages. [ABSTRACT FROM AUTHOR]

In this paper, we present a generic construction of r-ary sequences with period pq2 based on the Euler quotient modulo pq, where p and q are odd primes satisfying that p divides q − 1 and r is any prime less than q. The minimal polynomial and the linear complexity of the proposed sequences are determined in most cases under the assumption that rq−1≢1 (mod q2). The result shows that each of the sequences has large linear complexity. [ABSTRACT FROM AUTHOR]

In this paper, we study the relation between the linear subspace of the pseudo-noise (PN)-sequences generated by a primitive polynomial and the simplex code. This family of sequences can be also seen as an Maximum Distance Separable (MDS) 2 -linear code over 2 r . Furthermore, we see how to compute the family of generalized sequences produced by a primitive polynomial by means of a first-order Reed–Muller code. [ABSTRACT FROM AUTHOR]

The authors of this research paper have introduced a new method for solving Hamilton–Jacobi (HJ) equations called third-order weighted essentially nonoscillatory (WENO) method. This method uses exponential polynomials to construct numerical fluxes and smoothness indicators, which helps distinguish between singular and smooth regions more efficiently. The smoothness indicators are created using a finite difference operator that eliminates exponential polynomials. The numerical flux is constructed using an interpolation method based on exponential polynomials, which results in better outcomes around steep gradients. The new method maintains a high level of accuracy (i.e. three) in smooth regions, even near critical points. The authors have presented some numerical results to demonstrate the effectiveness of the new method and compared it with other WENO methods. [ABSTRACT FROM AUTHOR]

Cilleruelo conjectured that if f ∈ ℤ [ x ] is an irreducible polynomial of degree d ≥ 2 then, log lcm { f (n) | n < x } ∼ (d − 1) x log x. In this paper, we investigate the analog of prime arguments, namely, lcm { f (p) | p < x } , where p denotes a prime and obtain nontrivial lower bounds on it. Further, we also show some results regarding the greatest prime divisor of f (p). [ABSTRACT FROM AUTHOR]

Let F be an infinite field and R be an algebraic F -algebra. It is known that if (R) , the unit group of R , satisfies a group identity, then R satisfies a polynomial identity. In this paper, using additive mappings, we generalize this result to some kinds of identities. Particularly, assume that ∗ is an involution of R. We show that if (R) satisfies a ∗-group identity, then R satisfies a polynomial identity. [ABSTRACT FROM AUTHOR]

This paper studies cobordism and concordance for virtual knots. We define the affine index polynomial, prove that it is a concordance invariant for knots and links (explaining when it is defined for links), show that it is also invariant under certain forms of labeled cobordism and study a number of examples in relation to these phenomena. Information on determinations of the four-ball genus of some virtual knots is obtained by via the affine index polynomial in conjunction with results on the genus of positive virtual knots using joint work with Dye and Kaestner. [ABSTRACT FROM AUTHOR]

This paper defines the Euler–Poincaré characteristic of joint reductions of ideals which concerns the maximal terms in the Hilbert polynomial; characterizes the positivity of mixed multiplicities in terms of minimal joint reductions; proves the additivity and other elementary properties for mixed multiplicities. The results of the paper together with the results of [Thanh and Viet, Mixed multiplicities of maximal degrees, J. Korean Math. Soc.55(3) (2018) 605–622] seem to show a natural and nice picture of mixed multiplicities of maximal degrees. [ABSTRACT FROM AUTHOR]

In this paper, we obtain the crosscap number of any alternating knots by using our recently-introduced diagrammatic knot invariant (Theorem 1). The proof is given by properties of chord diagrams (Kindred proved Theorem 1 independently via other techniques). For non-alternating knots, we give Theorem 2 that generalizes Theorem 1. We also improve known formulas to obtain upper bounds of the crosscap number of knots (alternating or non-alternating) (Theorem 3). As a corollary, this paper connects crosscap numbers and our invariant with other knot invariants such as the Jones polynomial, twist number, crossing number, and hyperbolic volume (Corollaries 1–7). In Appendix A, using Theorem 1, we complete giving the crosscap numbers of the alternating knots with up to 11 crossings including those of the previously unknown values for 1 9 3 knots (Tables A.1). [ABSTRACT FROM AUTHOR]

In this paper, we use a novel approach to describe generator polynomials of quasi-cyclic (QC) and generalized QC (GQC) codes over finite fields. Our study of QC- and GQC-codes will be general and not only restricted to one-generator codes. We prove that generator polynomials of QC-codes and GQC-codes are unique. Further, we use our results to obtain an expression for the dimensions of QC-codes and GQC-codes. As an application of our construction of these codes, we obtain many optimal linear codes over finite fields GF (2) , GF (3) , GF (4) and GF (5). [ABSTRACT FROM AUTHOR]

Given an integral domain D with quotient field K , the study of the ring of integer-valued polynomials Int (D) = { f ∈ K [ X ] | f (a) ∈ D for all a ∈ D } has attracted a lot of attention over the past decades. Recently, Werner has extended this study to the situation of skew polynomials. To be more precise, if σ is an automorphism of K , one may consider the set Int (D , σ) = { f ∈ K [ X , σ ] | f (a) ∈ D for all a ∈ D } , where K [ X , σ ] is the skew polynomial ring and f (a) is a "suitable" evaluation of f at a. For example, he gave sufficient conditions for Int (D , σ) to be a ring and study some of its properties. In this paper, we extend the study to the situation of the skew polynomial ring K [ X , σ , δ ] with a suitable evaluation, where δ is a σ -derivation. Moreover we prove, for example, that if σ is of finite order and D is a Dedekind domain with finite residue fields such that Int (D , σ) is a ring, then Int (D , σ) is non-Noetherian. [ABSTRACT FROM AUTHOR]