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The paper presents a technique for determining the time index of growth of the slope of reverse voltage for double insulated cables, based on the body of the theory of series. It is proved that in the vicinity of the extremum point (maximum) the function of the reverse voltage is approximated by polynomials of the nth power. It is proposed to use second-degree polynomials for practical calculations. The method for calculating relevant indicators is illustrated using real data. Analysis of deviations made it possible to conclude that the calculation method proposed in the paper is far more accurate. In the final part of the study, it was concluded that there is a promising outlook for further development of methodological guidelines for determining complex indices of the remaining life of the cable, including but not limited to the use of various mathematical methods. [ABSTRACT FROM AUTHOR]

This article has two main parts. In the first part we show that some of the basic theory of generalized polynomials on commutative semi-groups can be extended to all semigroups. In the second part we show that if a sub-semigroup S of a group G generates G in the sense that G = S · S−1, then a generalized polynomial on S with values in an Abelian group H can be extended to a generalized polynomial on G into H. Finally there is a short discussion of the extendability of exponential functions and generalized exponential polynomials. [ABSTRACT FROM AUTHOR]

The paper presents a technique for determining the time index of growth of the slope of reverse voltage for double insulated cables, based on the body of the theory of series. It is proved that in the vicinity of the extremum point (maximum) the function of the reverse voltage is approximated by polynomials of the nth power. It is proposed to use second-degree polynomials for practical calculations. The method for calculating relevant indicators is illustrated using real data. Analysis of deviations made it possible to conclude that the calculation method proposed in the paper is far more accurate. In the final part of the study, it was concluded that there is a promising outlook for further development of methodological guidelines for determining complex indices of the remaining life of the cable, including but not limited to the use of various mathematical methods. [ABSTRACT FROM AUTHOR]

Three types of orthogonal polynomials (Chebyshev, Chelyshkov, and Legendre) are employed as basis functions in a collocation scheme to solve a nonlinear cubic initial value problem arising in population growth models. The method reduces the given problem to a set of algebraic equations consist of polynomial coefficients. Our main goal is to present a comparative study of these polynomials and to asses their performances and accuracies applied to the logistic population equation. Numerical applications are given to demonstrate the validity and applicability of the method. Comparisons are also made between the present method based on different basis functions and other existing approximation algorithms. [ABSTRACT FROM AUTHOR]

For the perturbed Hamiltonian of a multifrequency resonance harmonic oscillator, a new approach to calculating the coefficients in the procedure of quantum averaging is proposed. The procedure of quantum averaging is transferred to the space of the graded algebra of symbols by using twisted product introduced in the paper. As a result, the averaged Hamiltonian is represented as a function of generators of the quantum symmetry algebra of the harmonic part of the Hamiltonian. The proposed method is applied to the spectral problem for the Hamiltonian of the cylindrical Penning trap. [ABSTRACT FROM AUTHOR]

This paper considers MEP - Mixed Exponential Polynomials as one class of real exponential polynomials. We introduce a method for proving the positivity of MEP inequalities over positive intervals using the Maclaurin series to approximate the exponential function precisely. Additionally, we discuss the relation between MEPs and stratified families of functions from [1] through two applications, referring to inequalities from papers [2] and [3]. [ABSTRACT FROM AUTHOR]

In their celebrated paper (Furst et al., Math. Syst. Theory 17(1), 13-27 (12)), Furst, Saxe, and Sipser used random restrictions to reveal the weakness of Boolean circuits of bounded depth, establishing that constant-depth and polynomial-size circuits cannot compute the parity function. Such local restrictions have played important roles and have found many applications in complexity analysis and algorithm design over the past three decades. In this article, we give a brief overview of two intriguing applications of local restrictions: the first one is for the Isomorphism Conjecture and the second one is for moderately exponential time algorithms for the Boolean formula satisfiability problem. [ABSTRACT FROM AUTHOR]

Refereed conferences require every submission to be reviewed by members of a program committee (PC) in charge of selecting the conference program. There are many software packages available to manage the review process. Typically, in a bidding phase PC members express their personal preferences by ranking the submissions. This information is used by the system to compute an assignment of the papers to referees (PC members). We study the problem of assigning papers to referees. We propose to optimize a number of criteria that aim at achieving fairness among referees/papers. Some of these variants can be solved optimally in polynomial time, while others are NP-hard, in which case we design approximation algorithms. Experimental results strongly suggest that the assignments computed by our algorithms are considerably better than those computed by popular conference management software. [ABSTRACT FROM AUTHOR]

Self-interference (SI) represents a bottleneck in the performance of full-duplex (FD) communication systems, necessitating robust offsetting techniques to unlock the potential of FD systems. Currently, deep learning has been leveraged within the communication domain to address specific challenges and enhance efficiency. Inspired by this, this paper reviews the self-interference cancellation (SIC) process in the digital domain focusing on SIC capability. The paper introduces a model architecture that integrates CNN and gated recurrent unit (GRU), while also incorporating residual networks and self-attention mechanisms to enhance the identification and elimination of SI. This model is named CGRSA-Net. Firstly, CNN is employed to capture local signal features in the time–frequency domain. Subsequently, a ResNet module is introduced to mitigate the gradient vanishing problem. Concurrently, GRU is utilized to dynamically capture and retain both long- and short-term dependencies during the communication process. Lastly, by integrating the self-attention mechanism, attention weights are flexibly assigned when processing sequence data, thereby focusing on the most important parts of the input sequence. Experimental results demonstrate that the proposed CGRSA-Net model achieves a minimum of 28% improvement in nonlinear SIC capability compared to polynomial and existing neural network-based eliminator. Additionally, through ablation experiments, we demonstrate that the various modules utilized in this paper effectively learn signal features and further enhance SIC performance. [ABSTRACT FROM AUTHOR]

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]

For a simple, finite, and connected graph G, the spherical maximal operator is defined as ... where ... is the sphere with center at t and having radius r. In this paper, we consider the spherical maximal operator ... on ... spaces and calculate the ... for ... and estimate the ... for ..., when G is Km. Furthermore, We establish the maximum and minimum bounds for the spherical maximum operator on finite graphs and indicate the graphs that achieve these bounds. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to establish a numerical method for solving time-fractional diffusion equations. To obtain the numerical solution, a binary reproducing kernel space is defined, and the orthonormal basis is constructed based on Legendre polynomials in this space. In order to find the ε-approximation solution of time-fractional diffusion equations, which is defined in this paper, the algorithm is designed using the constructed orthonormal basis. Some numerical examples are analyzed to illustrate the procedure and confirm the performance of the proposed method. The results faithfully reveal that the presented method is considerably accurate and effective, as expected. [ABSTRACT FROM AUTHOR]

This paper introduces cutting planes that involve minimal structural assumptions, enabling the generation of strong polyhedral relaxations for a broad class of problems. We consider valid inequalities for the set S ∩ P , where S is a closed set, and P is a polyhedron. Given an oracle that provides the distance from a point to S, we construct a pure cutting plane algorithm which is shown to converge if the initial relaxation is a polyhedron. These cuts are generated from convex forbidden zones, or S-free sets, derived from the oracle. We also consider the special case of polynomial optimization. Accordingly we develop a theory of outer-product-free sets, where S is the set of real, symmetric matrices of the form x x T . All maximal outer-product-free sets of full dimension are shown to be convex cones and we identify several families of such sets. These families are used to generate strengthened intersection cuts that can separate any infeasible extreme point of a linear programming relaxation efficiently. Computational experiments demonstrate the promise of our approach. [ABSTRACT FROM AUTHOR]

Abstract: We study a family of orthogonal polynomials which generalizes a sequence of polynomials considered by L. Carlitz. We show that they are a special case of the Sheffer polynomials and point out some interesting connections with certain Sobolev orthogonal polynomials. [Copyright &y& Elsevier]

Let k be a field of characteristic 0, and let f : kn→kn be a polynomial map with components of the form fi=xi+hi, where the hi are monomials. If the Jacobian determinant of the map f is a nonzero constant, then f is a tame automorphism. If, in addition, each hi is either constant or of degree 2 or more, then f is linearly triangularizable. [Copyright &y& Elsevier]

Fuzzy logic is becoming one of the most-influential fields of modern mathematics with applications that impact not only other sciences, but society in general. This newly found interest in fuzzy logic is in part due to the crucial role it plays in the development of artificial intelligence. As a result, new tools and practices for the development of the above-mentioned field are in high demand. This is one of the issues this paper was composed to address. To be more specific, a sizable part of fuzzy logic is the study of fuzzy connectives. However, the current method used to generalize them is restricted to the use of basic automorphisms, which hinders the creation of new fuzzy connectives. For this reason, in this paper, a new method of generalization is conceived of that aims to generalize the fuzzy connectives using polynomial automorphism functions instead. The creation of these automorphisms is achieved through numerical analysis, an endeavor that is supported with programming applications that, using mathematical modeling, validate and visualize the research. Furthermore, the automorphisms satisfy all the necessary criteria that have been established for use in the generalization process and, consequently, are used to successfully generalize fuzzy connectives. The result of the new generalization method is the creation of new usable and flexible fuzzy connectives, which is very promising for the future development of the field. [ABSTRACT FROM AUTHOR]

Abstract: Ahn, Lee, Park and Yoo proved that the best constrained degree reduction of a polynomial f in -norm equals the best weighted Euclidean approximation of the Bernstein–Bézier coefficients of f in a paper, published in the journal, Computer Aided Geometric Design 21 (2) (2004) 181–191. In this note, we point out an error in their paper and give the correct result. [Copyright &y& Elsevier]

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

Monogenity is a classical area of algebraic number theory that continues to be actively researched. This paper collects the results obtained over the past few years in this area. Several of the listed results were presented at a series of online conferences titled "Monogenity and Power Integral Bases". We also give a collection of the most important methods used in several of these papers. A list of open problems for further research is also given. [ABSTRACT FROM AUTHOR]

This paper introduces the operational rule for 2-iterated 2D Appell polynomials and derives its generalized form using fractional operators. It also presents the generating relation and explicit forms that characterize the generalized 2-iterated 2D Appell polynomials. Additionally, it establishes the monomiality principle for these polynomials and obtains their recurrence relations. The paper also establishes corresponding results for the generalized 2-iterated 2D Bernoulli, 2-iterated 2D Euler, and 2-iterated 2D Genocchi polynomials. [ABSTRACT FROM AUTHOR]

The modified RSA provides high efficiency against attacks and, as a result, it is considered the ideal choice for many applications. In this paper, we introduce an alternative to the modified RSA key encryption system called TPRSA, based on Tri-Cartesian algebra and polynomials, by modifying the mathematical structure of text encryption and decryption keys to obtain a high level of security. [ABSTRACT FROM AUTHOR]

In this paper, a new family of iterative methods for the simultane-ous approximation of simple complex polynomial zeros is presented. The proposed family of simultaneous methods is constructed on the basis of the well-known third order Ehrlich iteration, combined with an iterative correction from the sixth order Neta's method for nonlinear equations. It is proved that the use of this iterative correction allows to increase the convergence order of the basic method from three to eight. Numerical examples are given to illustrate the convergence and effectiveness of the proposed combined method. [ABSTRACT FROM AUTHOR]

This paper presents a modified version of Leibman's group-theoretic generalizations of the difference calculus for polynomial maps from nonempty commutative semigroups to groups, and proves that it has many desirable formal properties when the target group is locally nilpotent and also when the semigroup is the set of nonnegative integers. We will apply it to solve Waring's problem for general discrete Heisenberg groups in a sequel to this paper. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the effect of dependent errors in the fixed design nonparametric regression models. Under some mild conditions, we obtain the complete consistency and asymptotic normality for the weighted estimator in the fixed design nonparametric regression models. In addition, a simulation study is undertaken to investigate finite sample behavior of the estimator. [ABSTRACT FROM AUTHOR]

The ordering betweenWigner-Yanase-Dyson function and logarithmic mean is known. Also bounds for logarithmic mean are known. In this paper, we give two reverse inequalities for Wigner-Yanase-Dyson function and logarithmic mean. We also compare the obtained results with the known bounds of the logarithmic mean. Finally, we give operator inequalities based on the obtained results. [ABSTRACT FROM AUTHOR]

In this paper, we study the permutation properties of the class of trinomials of the form f(x) = x4q+1 + λ1xq+4 + λ2x2q+3 ∈ Fq²[x] where λ1, λ2 ∈ Fq and they are not simultaneously zero. We find all necessary and sufficient conditions on λ1 and λ2 such that f(x) permutes Fq², where q is odd and q = 22k+1, k ∈ N. [ABSTRACT FROM AUTHOR]

This paper presents an overview of cosine and sine Apostol-type Frobenius–Euler–Fibonacci polynomials, as well as several identities that are associated with these polynomials. By applying a partial derivative operator to the generating functions, the authors obtain derivative formulae and finite combinatorial sums involving these polynomials and numbers. Additionally, the paper establishes connections between cosine and sine Apostol-type Frobenius–Euler–Fibonacci polynomials of order α and several other polynomial sequences, such as the Apostol-type Bernoulli–Fibonacci polynomials, the Apostol-type Euler–Fibonacci polynomials, the Apostol-type Genocchi–Fibonacci polynomials, and the Stirling–Fibonacci numbers of the second kind. The authors also provide computational formulae and graphical representations of these polynomials using the Mathematica program. [ABSTRACT FROM AUTHOR]

A polynomial f(x) over a field K is said to be stable if all its iterates are irreducible over K. L. Danielson and B. Fein have shown that over a large class of fields K, if f(x) is an irreducible monic binomial, then it is stable over K. In this paper it is proved that this result no longer holds over finite fields. Necessary and sufficient conditions are provided under which a given binomial is stable over Fq. These conditions are used to construct a table listing the stable binomials over Fq of the form f(x) = xd - a, a ∈ Fq \ {0, 1}, for q ≤ 27 and d ≤ 10. The paper ends with a brief link to Mersenne primes. [ABSTRACT FROM AUTHOR]

This paper shows a 2-step procedure for obtaining a variable-bandwidth recursive digital filter whose structure contains cascaded second-order (2nd-order) sections. Such a cascade-form structure is insensitive to the round-off noises that come from filter-coefficient quantizations in hardware implementations. This paper also shows how to utilize a 2-step procedure to get a variable-bandwidth recursive filter that is absolutely stable. The first step (Step-1) of the 2-step procedure designs a series of constant-bandwidth filters for approximating a series of evenly discretized variable specifications, and the second step (Step-2) fits the coefficient values obtained from Step-1 by employing individual polynomials. To ensure the stability of the resultant constant-bandwidth filters in Step-1, coefficient transformations are first executed on the 2nd-order transfer function's denominator-coefficients, and then each coefficient of both numerator and transformed denominator is found as an individual polynomial. Once all the polynomials are obtained, the polynomials corresponding to the transformed denominator are further converted to composite functions for ensuring the stability. Hence, the 2-step procedure not only produces a cascade-form variable-bandwidth filter that has low quantization errors, but also ensures the stability. A lowpass example is included for verifying the achieved stability and showing the high approximation accuracy. [ABSTRACT FROM AUTHOR]

Orthogonal polynomials and their moments serve as pivotal elements across various fields. Discrete Krawtchouk polynomials (DKraPs) are considered a versatile family of orthogonal polynomials and are widely used in different fields such as probability theory, signal processing, digital communications, and image processing. Various recurrence algorithms have been proposed so far to address the challenge of numerical instability for large values of orders and signal sizes. The computation of DKraP coefficients was typically computed using sequential algorithms, which are computationally extensive for large order values and polynomial sizes. To this end, this paper introduces a computationally efficient solution that utilizes the parallel processing capabilities of modern central processing units (CPUs), namely the availability of multiple cores and multithreading. The proposed multi-threaded implementations for computing DKraP coefficients divide the computations into multiple independent tasks, which are executed concurrently by different threads distributed among the independent cores. This multi-threaded approach has been evaluated across a range of DKraP sizes and various values of polynomial parameters. The results show that the proposed method achieves a significant reduction in computation time. In addition, the proposed method has the added benefit of applying to larger polynomial sizes and a wider range of Krawtchouk polynomial parameters. Furthermore, an accurate and appropriate selection scheme of the recurrence algorithm is introduced. The proposed approach introduced in this paper makes the DKraP coefficient computation an attractive solution for a variety of applications. [ABSTRACT FROM AUTHOR]

The first part of this work provides explicit solutions for two integral equations; both are solved by means of Fourier transform. In the second part of this paper, sufficient conditions for the existence and uniqueness of the solutions satisfying sandwich constraints for two types of full moment problems are provided. The only given data are the moments of all positive integer orders of the solution and two other linear, not necessarily positive, constraints on it. Under natural assumptions, all the linear solutions are continuous. With their value in the subspace of polynomials being given by the moment conditions, the uniqueness follows. When the involved linear solutions and constraints are positive, the sufficient conditions mentioned above are also necessary. This is achieved in the third part of the paper. All these conditions are written in terms of quadratic expressions. [ABSTRACT FROM AUTHOR]

In this paper, we give a simple sufficient condition for the eigenvalue-separation properties of real tridiagonal matrices T. This result is much more than the statement that the pertinent eigenvalues are distinct. Its derivation is based on recurrence formulae satisfied by the polynomials made up by the minors of the characteristic polynomial det(xE - T) that are proven to form a Sturm sequence. This is a new result, and it proves the simple spectrum property of a symmetric tridiagonal matrix studied in a Grünbaum paper. Two numerical examples underpin the theoretical findings. The style of the paper is expository in order to address a large readership. [ABSTRACT FROM AUTHOR]