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This paper corrects an error in an earlier work of the author. [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]

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]

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]

This paper extends the characteristic ratio approach using novel inequalities to ensure zero/low overshoot for linear‐time‐invariant systems with zeros. The extension provided by this paper is based on the maximally‐flatness property of a transfer function, where the square‐magnitude of the transfer function is ensured to be a low‐pass filter. In order to be able to design low‐order/fixed structure controllers, a partial pole‐assignment approach is used instead of the full pole‐assignment used in the Characteristic Ratio Assignment (CRA) method. The developed inequalities and additional stability conditions are combined into an optimization problem using time domain restrictions when necessary. Although the method given in the paper is general, particular inequalities are developed for PI and PI‐PD controller cases, due to their frequent use in industrial applications. Similarly, First‐Order‐Plus‐Delay‐Time (FOPDT) and Second‐Order‐Plus‐Delay‐Time (SOPDT) systems are considered specifically, since most of the practical systems can be approximated by one of these types. The study is extended to plants with uncertainties where a theorem is developed to decrease computation time dramatically. The benefits of the proposed methods are demonstrated by several examples. [ABSTRACT FROM AUTHOR]

Let G = Cp × H be a finite group, where Cp is a cyclic group of prime order p and H is a p'-group. Let F be an algebraically closed field in characteristic p. Let V be a direct sum of m non-trivial indecomposable G-modules such that the norm polynomials of the simple H-modules are the power of the product of the basis elements of the dual. In previous work, we proved the periodicity property of the polynomial ring F[V] with actions of G. In this paper, by the periodicity property, we showed that F[V]G is generated by m norm polynomials together with homogeneous invariants of degree at most m|G| - dim(V) and transfer invariants, which yields the well-known degree bound dim(V)·(|G|-1). More precisely, we found that this bound gets less sharp as the dimensions of simple H-modules increase. [ABSTRACT FROM AUTHOR]

The present paper deals with the approximation properties of the linear positive operators, including Δh -Gould-Hopper Appell polynomials. We investigate some theorems for convergence of the operators and their approximation degrees with the help of the classical approach, Peetre's K-functional, Lipschitz class and Voronovskajatype theorem. In the last section of the paper, we introduce the Kantorovich form of the operators and examine the approximation degree. [ABSTRACT FROM AUTHOR]

This paper mainly discussed the power sums of bi-periodic Fibonacci and Lucas polynomials. In addition, we generalized these results to obtain several congruences involving the divisible properties of bi-periodic Fibonacci and Lucas polynomials. [ABSTRACT FROM AUTHOR]

In this paper, we study parameter deformations of matrix valued orthogonal polynomials. These deformations are built on the use of certain matrix valued operators which are symmetric with respect to the matrix valued inner product defined by the orthogonality weight. We show that the recurrence coefficients associated with these operators satisfy generalizations of the non-Abelian lattice equations. We provide a Lax pair formulation for these equations, and an example of deformed Hermite-type matrix valued polynomials is discussed in detail. [ABSTRACT FROM AUTHOR]

COPA, introduced by Andreeva et al., is the first online authenticated encryption (AE) mode with nonce-misuse resistance, and it is covered in COLM, which is one of the final CAESAR portfolios. However, COPA has been proven to be insecure in the releasing unverified plaintext (RUP) setting. This paper mainly focuses on the integrity under RUP (INT-RUP) defect of COPA. Firstly, this paper revisits the INT-RUP security model for adaptive adversaries, investigates the possible factors of INT-RUP insecurity for "Encryption-Mix-Encryption"-type checksum-based AE schemes, and finds that these AE schemes with INT-RUP security vulnerabilities utilize a common poor checksum technique. Then, this paper introduces an improved checksum technique named polynomial intermediate checksum (PIC) for INT-RUP security and emphasizes that PIC is a sufficient condition for guaranteeing INT-RUP security for "Encryption-Mix-Encryption"-type checksum-based AE schemes. PIC is generated by a polynomial sum with full terms of intermediate internal states, which guarantees no information leakage. Moreover, PIC ensures the same level between the plaintext and the ciphertext, which guarantees that the adversary cannot obtain any useful information from the unverified decryption queries. Again, based on PIC, this paper proposes a modified scheme COPA-PIC to fix the INT-RUP defect of COPA. COPA-PIC is proven to be INT-RUP up to the birthday-bound security if the underlying primitive is secure. Finally, this paper discusses the properties of COPA-PIC and makes a comparison for AE modes with distinct checksum techniques. The proposed work is of good practical significance. In an interactive system where two parties communicate, the receiver can effectively determine whether the information received from the sender is valid or not, and thus perform the subsequent operation more effectively. [ABSTRACT FROM AUTHOR]

The walk matrix of an n-vertex graph G with adjacency matrix A, denoted by $ W(G) $ W (G) , is $ [e,Ae,\ldots,A^{n-1}e] $ [ e , Ae , ... , A n − 1 e ] , where e is the all-ones vector. Let $ G\circ P_m $ G ∘ P m be the rooted product of G and a rooted path $ P_m $ P m (taking an endvertex as the root), i.e. $ G\circ P_m $ G ∘ P m is a graph obtained from G and n copies of $ P_m $ P m by identifying each vertex of G with an endvertex of a copy of $ P_m $ P m . Mao et al. [A new method for constructing graphs determined by their generalized spectrum. Linear Algebra Appl. 2015;477:112–127.] and Mao and Wang [Generalized spectral characterization of rooted product graphs. Linear Multilinear Algebra. 2022. DOI:10.1080/03081087.2022.2098226.] proved that, for m = 2 and $ m\in \{3,4\} $ m ∈ { 3 , 4 } , respectively \[ \det W(G\circ P_m)=\pm a_0^{\lfloor\frac{m}{2}\rfloor}(\det W(G))^m, \] det W (G ∘ P m) = ± a 0 ⌊ m 2 ⌋ (det W (G)) m , where $ a_0 $ a 0 is the constant term of the characteristic polynomial of G. Furthermore, in the same paper, Mao and Wang conjectured that the formula holds for any $ m\ge 2 $ m ≥ 2. In this paper, we verify this conjecture using the technique of Chebyshev polynomials. [ABSTRACT FROM AUTHOR]

In this paper, we focus on the strong product of the pentagonal networks. Let R n be a pentagonal network composed of 2 n pentagons and n quadrilaterals. Let P n 2 denote the graph formed by the strong product of R n and its copy R n ′ . By utilizing the decomposition theorem of the normalized Laplacian characteristics polynomial, we characterize the explicit formula of the multiplicative degree-Kirchhoff index completely. Moreover, the complexity of P n 2 is determined. In this paper, we focus on the strong product of the pentagonal networks. Let be a pentagonal network composed of pentagons and quadrilaterals. Let denote the graph formed by the strong product of and its copy . By utilizing the decomposition theorem of the normalized Laplacian characteristics polynomial, we characterize the explicit formula of the multiplicative degree-Kirchhoff index completely. Moreover, the complexity of is determined. [ABSTRACT FROM AUTHOR]

In this paper, we considered the generalized bi-periodic Fibonacci polynomials, and obtained some identities related to generalized bi-periodic Fibonacci polynomials using the matrix theory. In addition, the generalized bi-periodic Lucas polynomial was defined by L n (x) = b p (x) L n − 1 (x) + q (x) L n − 2 (x) (if n is even) or L n (x) = a p (x) L n − 1 (x) + q (x) L n − 2 (x) (if n is odd), with initial conditions L 0 (x) = 2 , L 1 (x) = a p (x) , where p (x) and q (x) were nonzero polynomials in Q [ x ] . We obtained a series of identities related to the generalized bi-periodic Fibonacci and Lucas polynomials. In this paper, we considered the generalized bi-periodic Fibonacci polynomials, and obtained some identities related to generalized bi-periodic Fibonacci polynomials using the matrix theory. In addition, the generalized bi-periodic Lucas polynomial was defined by (if is even) or (if is odd), with initial conditions , , where and were nonzero polynomials in . We obtained a series of identities related to the generalized bi-periodic Fibonacci and Lucas polynomials. [ABSTRACT FROM AUTHOR]

The goal of the paper is to analyze a Gaudin model for a polynomial representation of the Kohno–Drinfeld Lie algebra associated with the multinomial distribution. The main result is the construction of an explicit basis of the space of polynomials consisting of common eigenfunctions of Gaudin operators in terms of Aomoto–Gelfand hypergeometric series. The construction shows that the polynomials in this basis are also common eigenfunctions of the operators for a dual Gaudin model acting on the degree indices, and therefore, they provide a solution to a multivariate discrete bispectral problem. [ABSTRACT FROM AUTHOR]

This paper considers the interactive effects between the ego vehicle and other vehicles in a dynamic driving environment and proposes an autonomous vehicle lane-changing behavior decision-making and trajectory planning method based on graph convolutional networks (GCNs) and multi-segment polynomial curve optimization. Firstly, hierarchical modeling is applied to the dynamic driving environment, aggregating the dynamic interaction information of driving scenes in the form of graph-structured data. Graph convolutional neural networks are employed to process interaction information and generate ego vehicle's driving behavior decision commands. Subsequently, collision-free drivable areas are constructed based on the dynamic driving scene information. An optimization-based multi-segment polynomial curve trajectory planning method is employed to solve the optimization model, obtaining collision-free motion trajectories satisfying dynamic constraints and efficiently completing the lane-changing behavior of the vehicle. Finally, simulation and on-road vehicle experiments are conducted for the proposed method. The experimental results demonstrate that the proposed method outperforms traditional decision-making and planning methods, exhibiting good robustness, real-time performance, and strong scenario generalization capabilities. [ABSTRACT FROM AUTHOR]

In this paper, we consider the problem of computing the entire sequence of the maximum degree of minors of a block-structured symbolic matrix (a generic partitioned polynomial matrix) A = (A α β x α β t d α β ) , where A α β is a 2 × 2 matrix over a field F , x α β is an indeterminate, and d α β is an integer for α = 1 , 2 , ⋯ , μ and β = 1 , 2 , ⋯ , ν , and t is an additional indeterminate. This problem can be viewed as an algebraic generalization of the maximum weight bipartite matching problem. The main result of this paper is a combinatorial -time algorithm for computing the entire sequence of the maximum degree of minors of a (2 × 2) -type generic partitioned polynomial matrix of size 2 μ × 2 ν . We also present a minimax theorem, which can be used as a good characterization (NP ∩ co-NP characterization) for the computation of the maximum degree of minors of order k. Our results generalize the classical primal-dual algorithm (the Hungarian method) and minimax formula (Egerváry's theorem) for the maximum weight bipartite matching problem. [ABSTRACT FROM AUTHOR]

This paper investigates various aspects of the distribution of roots and critical points of a complex polynomial, including their variance and the relationships between their moduli using an inequality due to de Bruijn. Making use of two other inequalities-again due to de Bruijn-we derive two probabilistic results concerning upper bounds for the average moduli of the imaginary parts of zeros and those of critical points, assuming uniform distribution of the zeros over a unit disc and employing the Markov inequality. The paper also provides an explicit formula for the variance of the roots of a complex polynomial for the case when all the zeros are real. In addition, for polynomials with uniform distribution of roots over the unit disc, the expected variance of the zeros is computed. Furthermore, a bound on the variance of the critical points in terms of the variance of the zeros of a general polynomial is derived, whereby it is established that the variance of the critical points of a polynomial cannot exceed the variance of its roots. Finally, we conjecture a relation between the real parts of the zeros and the critical points of a polynomial. [ABSTRACT FROM AUTHOR]

Let p(z) be a polynomial of degree n with zero of multiplicity s at the origin and the remaining zeros be 0. In this paper, we investigate the relative growth of a polynomial p(z) with respect to two circles z=r and z = R and obtain inequalities about the dependence of p(rz) on p(Rz), where z = 1, for 0 while taking into account the placement of the zeros of the underlying polynomial. Our results improve as well as generalize certain well-known polynomial inequalities. Some numerical examples are also given in order to illustrate and compare graphically the obtained inequalities with some recent results. [ABSTRACT FROM AUTHOR]

In this paper, we develop a novel method to construct Goethals–Seidel (GS) sequences with special structures. In the existing methods, utilizing Turyn sequences is an effective and convenient approach; however, this method cannot cover all GS sequences. Motivated by this, we are devoted to designing some sequences that can potentially construct all GS sequences. Firstly, it is proven that a quad of ± 1 polynomials can be considered a linear combination of eight polynomials with coefficients uniquely belonging to { 0 , ± 1 } . Based on this fact, we change the construction of a quad of Goethals–Seidel sequences to find eight sequences consisting of 0 and ± 1 . One more motivation is to obtain these sequences more efficiently. To this end, we make use of the k-block, of which some properties of (anti) symmetry are discussed. After this, we can then look for the sequences with the help of computers since the symmetry properties facilitate reducing the search range. Moreover, we find that one of the eight blocks, which we utilize to construct GS sequences directly, can also be combined with Williamson sequences to generate GS sequences with more order. Several examples are provided to verify the theoretical results. The main contribution of this work is in building a bridge linking the GS sequences and eight polynomials, and the paper also provides a novel insight through which to consider the existence of GS sequences. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to discuss a generalization of the bubble transform to differential forms. The bubble transform was discussed in Falk and Winther (Found Comput Math 16(1):297–328, 2016) for scalar valued functions, or zero-forms, and represents a new tool for the understanding of finite element spaces of arbitrary polynomial degree. The present paper contains a similar study for differential forms. From a simplicial mesh T of the domain Ω , we build a map which decomposes piecewise smooth k-forms into a sum of local bubbles supported on appropriate macroelements. The key properties of the decomposition are that it commutes with the exterior derivative and preserves the piecewise polynomial structure of the standard finite element spaces of k-forms. Furthermore, the transform is bounded in L 2 and also on the appropriate subspace consisting of k-forms with exterior derivatives in L 2 . [ABSTRACT FROM AUTHOR]

It is well known that in C [ X ] , the product of two polynomials is equal to the product of their greatest common divisor and their least common multiple. In a recent paper, we proved a similar relation between the ranks of matrix polynomials. More precisely, the sum of the ranks of two matrix polynomials is equal to the sum of the rank of the greatest common divisor of the polynomials applied to the respective matrix and the rank of the least common multiple of the polynomials applied to the respective matrix. In this paper, we present three new proofs for this result. In addition to these, we present two more applications. [ABSTRACT FROM AUTHOR]

The aim of this paper is to extend the domain of the Grace's theorem, Schur-Szegö composition theorem and Cohn-Egerváry theorem from the set of complex numbers to the set of bicomplex numbers. [ABSTRACT FROM AUTHOR]

In conventional finite element simulations, foil windings with thin foils and with a large number of turns require many mesh elements. This renders models quickly computationally infeasible. This paper uses a homogenized foil winding model and approximates the voltage distribution in the foil winding domain by globally supported polynomials. This way, the small-scale structure in the foil winding domain does not have to be resolved by the finite element mesh. The method is validated successfully for a stand-alone foil winding example and for a pot inductor example. Moreover, a transformer equipped with a foil winding at its primary side is simulated using a field-circuit coupled model. [ABSTRACT FROM AUTHOR]

This is the second paper in a series of two. The goal of the series is to give a polynomial-time algorithm for the 4-coloring problem and the 4-precoloring extension problem restricted to the class of graphs with no induced six-vertex path, thus proving a conjecture of Huang. Combined with previously known results, this completes the classification of the complexity of the 4-coloring problem for graphs with a connected forbidden induced subgraph. In this paper we give a polynomial time-algorithm that starts with a 4-precoloring of a graph with no induced six-vertex path and outputs a polynomial-sized collection of so-called excellent precolorings. Excellent precolorings are easier to handle than general ones, and, in addition, in order to determine whether the initial precoloring can be extended to the whole graph, it is enough to answer the same question for each of the excellent precolorings in the collection. The first paper in the series deals with excellent precolorings, thus providing a complete solution to the problem. [ABSTRACT FROM AUTHOR]

This paper presents an investigation of the uniqueness problem of derivatives of a meromorphic function and its difference polynomial in view of a partially sharing. As a consequence of the main result, we improve the recent result of W. J. Chen and Z. G. Huang with the weaker hypotheses and also supplement several results in particular cases. [ABSTRACT FROM AUTHOR]

An R-moduleM is called δ-weakly Hopfian if any δ-small surjective endomorphism of M is an automorphism. In this paper we explore various properties of δ-weakly Hopfian modules, shedding light on their distinct characteristics. Additionally, we examine the δ-weakly Hopficity of modules over polynomial and truncated polynomial rings. [ABSTRACT FROM AUTHOR]

In this paper, we introduce some new generalizations of classical descent and inversion statistics on signed permutations that arise from the work of Sack and Úlfarsson [18], and called k-width descents and k-width inversions of type A ([8]). Using the aforementioned new statistics, we derive new generalizations of Eulerian polynomials of type A, B and D. We establish also the 7-positivity of the Eulerian "width-k" polynomials. Referring to Petersen's paper [16], we give a combinatorial interpretation of finite sequences associated with these new polynomials using quasi-symmetric functions and a partition P. [ABSTRACT FROM AUTHOR]

Erdös–Lax inequality relates the sup norm of the derivative of a polynomial along the unit circle to that of the polynomial itself (on the unit circle). This paper aims to extend the classical Erdös–Lax inequality to the polar derivative of a polynomial by using the extreme coefficients of the given polynomial. The obtained results not only enrich the realm of Erdös–Lax-type inequalities but also offer a promising avenue for diverse applications where these inequalities play a pivotal role. To illustrate the practical significance of our results, we present a numerical example. It vividly demonstrates that our bounds are considerably sharper than the existing ones in the extensive literature on this captivating subject. [ABSTRACT FROM AUTHOR]

A dynamic visualization of Julia and Mandelbrot fractals involves creating animated representations of these fractals that change over time or in response to user interaction which allows users to gain deeper insights into the intricate structures and properties of these fractals. This paper explored the dynamic visualization of fractals within Julia and Mandelbrot sets, focusing on a generalized rational type complex polynomial of the form Sc(z) = azn + b/zm + c, where a; b; c ∈ C with |a| > 1 and n;m ∈ N with n > 1. By applying viscosity approximation-type iteration processes extended with s-convexity, we unveiled the intricate dynamics inherent in these fractals. Novel escape criteria was derived to facilitate the generation of Julia and Mandelbrot sets via the proposed iteration process. We also presented graphical illustrations of Mandelbrot and Julia fractals, highlighting the change in the structure of the generated sets with respect to the variations in parameters. [ABSTRACT FROM AUTHOR]

This paper presents a new approach for the unified Chebyshev polynomials (UCPs). It is first necessary to introduce the three basic formulas of these polynomials, namely analytic form, moments, and inversion formulas, which will later be utilized to derive further formulas of the UCPs. We will prove the basic formula that shows that these polynomials can be expressed as a combination of three consecutive terms of Chebyshev polynomials (CPs) of the second kind. New derivatives and connection formulas between two different classes of the UCPs are established. Some other expressions of the derivatives of UCPs are given in terms of other orthogonal and non-orthogonal polynomials. The UCPs are also the basis for additional derivative expressions of well-known polynomials. A new linearization formula (LF) of the UCPs that generalizes some well-known formulas is given in a simplified form where no hypergeometric forms are present. Other product formulas of the UCPs with various polynomials are also given. As an application to some of the derived formulas, some definite and weighted definite integrals are computed in closed forms. [ABSTRACT FROM AUTHOR]

In this paper, the vector extension operation is proposed to replace the de Boor‐Cox formula for a fast algorithm to B‐spline basis functions. This B‐spline basis function based on vector extending operation is implemented in the class and shape transformation (CST) parameterization method in place of the traditional Bézier polynomials to enhance the local ability of control and accuracy to represent an airfoil shape. To calculate the k‐degree B‐spline function's nonzero values, the algorithm can improve the computing efficiency by 2k+1 times. [ABSTRACT FROM AUTHOR]

The Boolean map χ n : F 2 n → F 2 n , x ↦ y defined by y i = x i + (x i + 1 + 1) x i + 2 (where i ∈ Z / n Z ) is used in various permutations that are part of cryptographic schemes, e.g., Keccak-f (the SHA-3-permutation), ASCON (the winner of the NIST Lightweight competition), Xoodoo, Rasta and Subterranean (2.0). In this paper, we study various algebraic properties of this map. We consider χ n (through vectorial isomorphism) as a univariate polynomial. We show that it is a power function if and only if n = 1 , 3 . We furthermore compute bounds on the sparsity and degree of these univariate polynomials, and the number of different univariate representations. Secondly, we compute the number of monomials of given degree in the inverse of χ n (if it exists). This number coincides with binomial coefficients. Lastly, we consider χ n as a polynomial map, to study whether the same rule ( y i = x i + (x i + 1 + 1) x i + 2 ) gives a bijection on field extensions of F 2 . We show that this is not the case for extensions whose degree is divisible by two or three. Based on these results, we conjecture that this rule does not give a bijection on any extension field of F 2 . [ABSTRACT FROM AUTHOR]