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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, we describe the so-called homogeneous involution on finite-dimensional graded-division algebra over an algebraically closed field. We also compute their graded polynomial identities with involution. As pointed out by Fonseca and Mello, a homogeneous involution naturally appears when dealing with graded polynomial identities and a compatible involution. [ABSTRACT FROM AUTHOR]

Assume that we are given a sequence F of k homogeneous polynomials in n variables of degree at most d and the ideal ℐ generated by this sequence. The aim of this paper is to present a new and effective method to determine, within the arithmetic complexity d O (n) , whether F is regular. This algorithm has been implemented in Maple and its efficiency (compared to the classical approaches for regular sequence test) is evaluated via a set of benchmark polynomials. Furthermore, we show that if F is regular then we can transform ℐ into Nœther position and at the same time compute a reduced Gröbner basis for the transformed ideal within the arithmetic complexity d O (n 2) . Finally, under the same assumption, we establish the new upper bound 2 (d k / 2) 2 n − k − 1 for the maximum degree of the elements of any reduced Gröbner basis of ℐ in the case that n > k. [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]

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]

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]

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]

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 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 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]

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]

One of the interesting problems in arithmetic dynamics is to study the stability of polynomials over a field. A polynomial f (z) ∈ ℚ [ z ] is stable over ℚ if irreducibility of f (z) implies that all its iterates are also irreducible over ℚ , that is, f n (z) is irreducible over ℚ for all n ≥ 1 , where f n (z) denotes the n -fold composition of f (z). In this paper, we study the stability of f (z) = z d + 1 c for d ≥ 2 , c ∈ ℤ ∖ { 0 }. We show that for infinite families of d ≥ 3 , whenever f (z) is irreducible, all its iterates are irreducible, that is, f (z) is stable. Under the assumption of explicit a b c -conjecture, we further prove the stability of f (z) = z d + 1 c for the remaining values of d. Also for d = 3 , if f (z) is reducible, then the number of irreducible factors of each iterate of f (z) is exactly 2 for | c | ≤ 1 0 1 2 . [ABSTRACT FROM AUTHOR]

In this paper, we study the Newton polygon of the L -function of a generalized Kloosterman polynomial with two variables over finite fields. We give the explicit form of the monomial basis of the top dimensional cohomology space of the p -adic complex associated to the L -function. One consequence of this result is a concrete method for computing the Hodge polygon. Using decomposition theorems of Wan, we determine when a generalized Kloosterman polynomial is ordinary and when its Newton polytope is generically ordinary. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the explicit formulae for the uniformizers of the false Tate curve extension of ℚ p . More precisely, we establish the formulae for the fields p m , 1 = ℚ p (ζ p m , p 1 / p) with m ≥ 1 and for general n ≥ 2 , we prove the existence of the recurrence polynomials ℛ p m , n for general field extensions p m , n of ℚ p , which shows the possibility to construct the uniformizers systematically. [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]

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]

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]

In this paper, the complete discrimination system for the polynomial method is applied to solve the Wu–Zhang system, and all the possible exact solutions are obtained, these exact solutions can be applied to the exploration of nonlinear physical phenomena, the method in this paper is different from the existing literature studies on the Wu–Zhang equation. By taking different parameters, interesting graphs are plotted for all the obtained solutions. The results confirm that the proposed method is effective and can be used to solve a variety of nonlinear consistency time fractional partial differential equations. [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]

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]

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]

The 3-loop invariant (or, the 3-loop polynomial) of a knot is a rational form (or, a polynomial) presenting the 3-loop part of the Kontsevich invariant of knots. In this paper, we calculate the 3-loop polynomial of knots obtained by plumbing the doubles of two knots; this class of knots includes untwisted Whitehead doubles. We construct the 3-loop invariant by calculating the rational version of the Aarhus integral of a surgery presentation. As a consequence, we obtain an explicit presentation of the 3-loop polynomial for the knots. [ABSTRACT FROM AUTHOR]

In this paper, for the Allen–Cahn equation, we obtain the error estimate of the temporal semi-discrete scheme, and the fully-discrete finite element numerical scheme, both of which are based on the invariant energy quadratization (IEQ) time-marching strategy. We establish the relationship between the L 2 -error bound and the L ∞ / H 2 -stabilities of the numerical solution. Then, by converting the numerical schemes to a form compatible with the original format of the Allen–Cahn equation, using mathematical induction, the superconvergence property of nonlinear terms, and the spectrum argument, the optimal error estimates that only depends on the low-order polynomial degree of − 1 instead of e T / 2 for both of the semi and fully-discrete schemes are derived. Numerical experiment also validates our theoretical convergence analysis. [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]

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]

In this paper, we review some new results we recently obtained about the infrared physics of 3d 풩=2 SQCD with a unitary gauge group, in particular in the presence of a nonzero Fayet–Iliopoulos parameter and with generic values of the Chern–Simons levels. We review the 3d GLSM (also known as 3d A-model) approach to the computation of the 3d 풩=2 twisted chiral ring of half-BPS lines. For particular values of the Chern–Simons levels, this twisted chiral ring has a neat interpretation in terms of the quantum K-theory (QK) of the Grassmannian manifold. We propose a new set of line defects of the 3d gauge theory, dubbed Grothendieck lines, which represent equivariant Schubert classes in the QK ring. In particular, we show that double Grothendieck polynomials, which represent the equivariant Chern characters of the Schubert classes, arise physically as Witten indices of certain quiver supersymmetric quantum mechanics. We also explain two distinct ways how to compute K-theoretic enumerative invariants using the 3d GLSM approach. [ABSTRACT FROM AUTHOR]

As a generalization of the classical knots, knotoids deal with the open ended knot diagram in a surface. In recent years, many polynomial invariants for knotoids appeared, such as the bracket polynomial, the index polynomial and the nth polynomial. In this paper, we introduce a new polynomial invariant F-polynomial for knotoids and discuss some properties of the F-polynomial. Finally, we construct a family of knotoid diagrams which can be distinguished from each other by the F-polynomial but cannot be distinguished by the index polynomial and the nth polynomial. [ABSTRACT FROM AUTHOR]

For a prime p, positive integers r , n , and a polynomial f with coefficients in p r , let W p , r , n (f) = f n p r \ f n + 1 p r . As n varies, the W p , r , n (f) partition the set of strictly preperiodic points of the dynamical system induced by the action of f on p r . 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 p r lying in a given W p , r , n (f). Moreover, when we generalize our definition of W p , r , n (f) , we obtain both upper and lower bounds for the resulting averages. [ABSTRACT FROM AUTHOR]

In this paper, we consider an adaptive hybridizable discontinuous Galerkin (HDG) method based on the discrete fracture model for approximation of a single-phase flow in fractured porous media. We are interested in the case that the flow rate in fracture is large enough to justify the use of Forchheimer’s law for modeling flow within the fracture, while Darcy’s law is applied to the surrounding matrix. The HDG method could be designed to simulate the flow in porous media with reduced fractures which consist of many straight lines or planes. More specifically, we use piecewise polynomials of degree k to approximate the velocity and pressure in fracture and surrounding porous media. The existence and uniqueness of discrete solutions are proved by the Brouwer fixed point theorem, and an efficient and reliable a posteriori error estimator is obtained with respect to an energy norm. Moreover, the HDG scheme, the existence and uniqueness of discrete solutions, and the a posteriori error estimates are also extended to the problem with non-planar, embedded, and intersecting fractures. Finally, several numerical examples are provided to validate the performance of the obtained a posteriori error estimator. [ABSTRACT FROM AUTHOR]

In this paper we introduce a new graph polynomial, say connectivity polynomial. Let G be a simple graph of order n. The connectivity polynomial of G, denoted by 풞(G,x), is 풞(G,x) =∑i=0nc ixi, where c0 = 1 and ci (for i ≥ 1) is the number of connected induced subgraphs of G with order i. We investigate some properties of the connectivity polynomial. We find the connectivity polynomial of disjoint union and join of graphs. Finally, we find the coefficients of this polynomial in terms of the vertex connectivity and the order of graphs. [ABSTRACT FROM AUTHOR]

Monomial ordering is the process of putting single polynomial terms, or monomials, in a certain order depending on predefined principles. It facilitates the systematic grouping and comparison of monomials, as well as the division and factorization of polynomials and other algebraic operations. Let R := 핂[xi,j|i ∈ [c],j ∈ ℕ], be a polynomial ring over arbitrary field 핂, where c is any fixed natural number. In this paper, we have defined different orderings like lexicographical, degree lexicographical and reverse degree lexicographical ordering on polynomial ring. We have discussed the infinitely many Inc-compatible term ordering on polynomial ring R with respect to two different weight vectors Wc and Wc,n′. [ABSTRACT FROM AUTHOR]

In this paper, we estimate the number of limit cycles bifurcating from the periodic orbits of the weight-homogeneous polynomial centers of weight-degree 3, when they are perturbed inside the class of piecewise smooth polynomial systems of degree n. By using the first-order averaging method, we give the upper and lower bounds on the maximal number of limit cycles which can bifurcate from the period annuluses. Our results indicate that the nonsmooth systems can have more limit cycles than the smooth ones. [ABSTRACT FROM AUTHOR]

Polynomials of the form ∑ a i j x p i + p j + L (x) are called quadratic polynomials over a finite field p n and the quadratic polynomials of the form ∑ a i j x p i + p j are called Dembowski–Ostrom polynomials. In this paper, we propose four classes of permutation trinomials of Dembowski–Ostrom type polynomials over finite fields 2 n and two classes of quadratic permutation polynomials of the form ∑ a i j x 2 i + 2 j + L (x) over 2 n . [ABSTRACT FROM AUTHOR]

In this paper, we study linear codes of length n = mℓ that are invariant under an endomorphism T = T1 ⊕ T1 ⊕⋯ ⊕ T1 (ℓ copies of T1), where T1 is a cyclic endomorphism on 픽qm. As each endomorphism can be represented by a matrix, we restrict our study on linear codes that are under a matrix D = diag(M,M,…,M), where M is an m×m cyclic matrix, called quasi-M-cyclic codes of index ℓ, and quasi-f-cyclic codes when M is the companion matrix of a polynomial f(x). We prove a one-to-one correspondence between quasi-M-cyclic codes of index ℓ and Rf-submodules of Rfℓ, where Rf = 픽q[x]/〈f(x)〉 and f(x) is the minimal polynomial of M. We prove the BCH-like and Hartmann–Tzeng-like bounds for 1-generator quasi-f-cyclic codes. In addition, we study the additive structure of quasi-f-cyclic codes by mapping them to 픽qℓ/〈f(x)〉 via an 픽qℓ-module morphism. Finally, we provide examples of new quantum codes derived from quasi-f-cyclic codes as an application of our results. [ABSTRACT FROM AUTHOR]

Parabolic Finslerian Allen–Cahn equation is a quasilinear parabolic equation on Finsler metric measure spaces. In this paper, we establish a Harnack inequality for solutions to this equation on compact Finsler metric measure spaces using geometric methods. Moreover, we give a degree 4 polynomial gradient estimate of the traveling wave solution on flat-curved spaces. [ABSTRACT FROM AUTHOR]

Let λ = λ0 + uλ1 + ⋯ + uk−1λ k−1 be a Type 1 unit in ℜk = 픽pm + u픽pm + ⋯ + uk−1픽 pm(uk = 0), where p is an odd prime, m is a positive integer and λ0,λ1,…,λk−1 ∈ 픽pm,λ0≠0,λ1≠0. In this paper, we give the complete structure of all Type 1 λ-constacyclic codes and their duals of length 8ps over the finite commutative chain ring ℜk in terms of their generator polynomials. Using this structure, we determine the Hamming distance and the Rosenbloom–Tsfasman (RT) distance of all Type 1 λ-constacyclic codes. For pm ≡ 1(mod 4) and a unit λ ∈ ℜ2, we determine the b -symbol distances of all λ-constacyclic codes of length 8ps over ℜ2, where b ≤ 8. As illustrations, we provide several λ-constacyclic codes with new parameters with respect to Hamming, RT and b-symbol metrics. MDS codes are widely recognized for their optimal error-correction capability, and MDS b-symbol codes are generalization of MDS codes. We found some MDS b-symbol constacyclic codes of length 8ps over ℜ2. Additionally, for pm ≡ 1(mod 4), we provide a decoding algorithm for Type 1 constacyclic codes of length 8ps over ℜk with respect to the Hamming, RT and b-symbol metrics. [ABSTRACT FROM AUTHOR]

Let ℋ = {Hv : v ∈ V (G)} be a family of nonempty graphs indexed by the vertex set of a graph G. The corona graph G ∘ℋ of G and ℋ is the disjoint union of G and Hv, v ∈ V (G), with additional edges joining each vertex v ∈ V (G) to all the vertices of Hv. In this paper, we are deeply concerned in investigating the (induced) matching number of the graph G ∘ℋ. It consequently gives an explicit formula to compute the Castelnuovo–Mumford regularity of edge ideal of this graph. Moreover, we propose a close relationship between independence polynomial of corona graph and h-polynomial of its independence simplicial complexes. Thereby, the formula of h-polynomial is established and some results related to the unimodality and real-rootedness are presented. [ABSTRACT FROM AUTHOR]

In this paper, we describe a method of making an invariant of virtual links, generalizing Jones–Kauffman polynomial of virtual links. [ABSTRACT FROM AUTHOR]

Polynomial completeness results aim at characterizing those functions that are induced by polynomials. Each polynomial function is congruence preserving, but the opposite need not be true. A finite algebraic structure A is called strictly 1-affine complete if every unary partial function from a subset of A to A that preserves the congruences of A can be interpolated by a polynomial function of A. The problem of characterizing strictly 1-affine complete finite Mal'cev algebras is still open. In this paper, we extend the characterization by Aichinger and Idziak of strictly 1-affine complete expanded groups to finite congruence regular Mal'cev algebras. [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]