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

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]

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]

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]

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]

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]

Polynomial multiplication is the most computationally expensive part of the lattice-based cryptography algorithm. However, the existing acceleration schemes have problems, such as low performance and high hardware resource overhead. Based on the polynomial multiplication of number theoretic transformation (NTT), this paper proposed a simple element of Montgomery module reduction with pipeline structure to realize fast module multiplication. In order to improve the throughput of the NTT module, the block storage technology is used in the NTT hardware module to enable the computing unit to read and write data alternately. Based on the NTT hardware module, a precalculated parameter storage and real-time calculation method suitable for the hardware architecture of this paper is also proposed. Finally, the hardware of polynomial multiplier based on NTT module is implemented, and its function simulation and performance evaluation are carried out. The results show that the proposed hardware accelerator can have excellent computing performance while using fewer hardware resources, thus meeting the requirements of lattice cipher algorithms in security chips. Compared with the existing studies, the computing performance of the polynomial multiplier designed in this paper is improved by approximately 1 to 3 times, and the slice resources and storage resources used are reduced by approximately 60% and 17%, respectively. [ABSTRACT FROM AUTHOR]

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

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]

The present paper addresses linear parameter-varying systems with high-order time-varying parameter dependency known as polynomial LPV systems and their controller design. Throughout this work, a procedure ensuring a state-feedback controller from a parameterized linear matrix inequality (PLMI) solution is presented. As the main contribution of this paper, the controller is designed by considering the time-varying parameter rate as a tuning parameter with a continuous control gain in such a way that the closed-loop eigenvalues lie in a complex plane subset, with high-order time-varying parameters defining the system dynamics. Simulation results are presented, aiming to show the effectiveness of the proposed controller design. [ABSTRACT FROM AUTHOR]

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

This paper investigates blind identification methods for linear scramblers under non‐cooperative conditions, which are essential for the inverse analysis of communication protocols using scramblers. In this paper, a blind identification scheme for feedback polynomials of synchronous scramblers is proposed. A variable Υ¯$\overline{\Upsilon }$ is first proposed that measures the correctness of the test polynomial by using the soft information of the received sequence, then the mean and variance of the variable Υ¯$\overline{\Upsilon }$ in different cases are obtained, and finally the optimal threshold value to determine whether the test primitive polynomial is correct or not is obtained. That is, the blind identification problem is transformed into a hypothesis testing problem. The simulations verify that the proposed scheme requires a much smaller scrambled sequence length than existing blind identification schemes. Furthermore, the proposed scheme is more fault tolerant than existing schemes and has a signal‐to‐noise ratio (SNR) gain of at least 3 dB when the intercepted scrambled sequences are of the same length and high identification accuracy is achieved. [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]

Let R_k(t) := |P_k(e^{it})|^2 and S_k(t) := |Q_k(e^{it})|^2, where P_k and Q_k are the usual Rudin-Shapiro polynomials of degree n-1 with n=2^k. In a recent paper we combined close to sharp upper bounds for the modulus of the autocorrelation coefficients of the Rudin-Shapiro polynomials with a deep theorem of Littlewood to prove that there is an absolute constant A>0 such that the equations R_k(t) = (1+\eta)n and S_k(t) = (1+\eta)n have at least An^{0.5394282} distinct solutions in [0,2\pi) whenever \eta is real, |\eta | < 2^{-8}, and k is sufficiently large. In this paper we show that the equations R_k(t)=(1+\eta)n and S_k(t)=(1+\eta)n have at least (1/2-|\eta |-\varepsilon)n/2 distinct solutions in [0,2\pi) for every \eta \in (-1/2,1/2), \varepsilon > 0, and sufficiently large be k \geq k_{\eta,\varepsilon }. [ABSTRACT FROM AUTHOR]

This study aims to present the modified SVM polynomial method in order to evaluate insurance data. The research methodology discusses classical and modified SVM polynomial methods by R programming, and uses performance profiles to create the most preferable methods. It offers a new algorithm called an accurate evaluating algorithm as the way to construct the modified SVM polynomial method. The classical SVM polynomial method is also represented as the main idea in finding the modified polynomial SVM method. Model Performance Evaluation (MPE), Receiver Operating Characteristics (ROCs) Curve, Area Under Curve (AUC), partial AUC (pAUC), smoothing, confidence intervals, and thresholds are further named an accurate evaluating algorithm, employed to build the modified SVM polynomial method. The research paper also presents the best performance profiles based on the computing time and the number of iterations of both classical and modified SVM polynomial methods. Performance profiles show numerical comparisons based on both methods involving insurance data also displayed in this paper. It can be concluded that applying an accurate evaluating algorithm on the modified SVM polynomial method will improve the data accuracy up to 86% via computing time and iterations compared to the classical SVM polynomial method, which is only 79%. This accurate evaluating algorithm can be applied to various large-sized data by utilizing R programming with changing any suitable kernels for that data. This vital discovery will offer solutions for faster and more accurate data analysis that can benefit researchers, the private sector, or governments struggling with data. [ABSTRACT FROM AUTHOR]

China aims to reduce carbon dioxide emissions and achieve peak carbon and carbon neutrality goals. Natural gas, as a high-quality fossil fuel energy, is an important transition resource for China in the process of carbon reduction, so it is necessary to predict China's natural gas demand. In this paper, a novel natural gas demand combination forecasting model is constructed to accurately predict the future natural gas demand. The Lasso model and the polynomial model are used to build a combinatorial model, which overcomes the shortcomings of traditional models, which have low data dimensions and poor prediction abilities. In the modeling process, the cross-validation method is used to adjust the modeling parameters. By comparing the performance of the combinatorial forecasting model, the single forecasting model and other commonly used forecasting models, the results show that the error (2.99%) of the combinatorial forecasting model is the smallest, which verifies the high accuracy and good stability advantages of the combinatorial forecasting model. Finally, the paper analyzes the relevant data from 1999 to 2022 and predicts China's natural gas demand in the next 10 years. The results show that the annual growth rate of China's natural gas demand in the next 10 years will reach 13.33%, at 8.3 × 1011 m3 in 2033, which proves that China urgently needs to rapidly develop the gas supply capacity of gas supply enterprises. This study integrates the impact of multiple factors on the natural gas demand, predicts China's natural gas demand from 2023 to 2033, and provides decision-making support for China's energy structure adjustment and natural gas import trade. [ABSTRACT FROM AUTHOR]