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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]

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]

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]

This paper studies the H∞ control problem for polynomial time‐varying systems. The H∞ control problem has been much less investigated for time‐varying systems in comparison to the time‐invariant systems. Approximate dynamic programming (ADP) is an optimal method to solve the control problems. Therefore, it is valuable to solve the polynomial time‐varying H∞ control problem with the ADP approach. Considering the time as an independent variable for sum‐of‐squares (SOS) optimization problems, an SOS‐based ADP method is proposed to solve this problem. A policy iteration algorithm is presented, where in its policy evaluation step it is sufficient to solve an optimization problem. Some constraints are added to this optimization problem to guarantee the closed‐loop exponential stability. The convergence and stability properties of the proposed algorithm are stated and proven. Moreover, in order to design an H∞ controller with a smaller disturbance attenuation coefficient, a two‐loop algorithm is suggested. Finally, the effectiveness of the proposed method is demonstrated by simulation examples. [ABSTRACT FROM AUTHOR]

According to Löwenstein's rule, Al–O–Al bridges are forbidden in the aluminosilicate framework of zeolites. A graph‐theoretical interpretation of the rule, based on the concept of independent sets, was proposed earlier. It was shown that one can apply the vector method to the associated periodic net and define a maximal Al/(Al+Si) ratio for any aluminosilicate framework following the rule; this ratio was called the independence ratio of the net. According to this method, the determination of the independence ratio of a periodic net requires finding a subgroup of the translation group of the net for which the quotient graph and a fundamental transversal have the same independence ratio. This article and a companion paper deal with practical issues regarding the calculation of the independence ratio of mainly 2‐periodic nets and the determination of site distributions realizing this ratio. The first paper describes a calculation technique based on propositional calculus and introduces a multivariate polynomial, called the independence polynomial. This polynomial can be calculated in an automatic way and provides the list of all maximal independent sets of the graph, hence also the value of its independence ratio. Some properties of this polynomial are discussed; the independence polynomials of some simple graphs, such as short paths or cycles, are determined as examples of calculation techniques. The method is also applied to the determination of the independence ratio of the 2‐periodic net dhc. [ABSTRACT FROM AUTHOR]

In this paper, using the Faà di Bruno formula and some properties of the Bell polynomials of the second kind, we obtain a new explicit formula for the generalized Humbert–Hermite polynomials. We provide determinantal representations for the ratio of two differentiable functions. We obtain a recursive relation for the generalized Humbert–Hermite polynomials. As a practice, we derive an alternative recursive relation for generalized Humbert–Hermite polynomials via the Hessenberg determinant. Finally, we derive several families of multilinear and multilateral generating functions for the generalized Humbert–Hermite‐type polynomials and other polynomials which are mentioned in this paper. Our results also include many well‐known polynomials in the literature. [ABSTRACT FROM AUTHOR]

When a convolutional neural network (CNN) model was built, the size and resolution of its input data were fixed. However, Raman spectra collected by different Raman spectrometers usually had different length, intensity range, and wavenumber interval between two adjacent data points, which made the existing CNN model difficult to be applied to a new Raman spectral data set. Therefore, this paper proposed a polynomial reconstruction algorithm as pretreatment method to obtain reconstructed spectra that would be imported into CNN model with consistent length, intensity range, and wavenumber interval. To test the effectiveness of this method, a big data set with 2563 Raman spectra of 831 minerals and synthetic organic pigments samples was constructed from the RRUFF and SOP database to pretrain a one‐dimensional CNN (1D‐CNN) model. The pretraining results showed that polynomial reconstruction algorithm used as pretreatment method was better than SG smoothing combined spline interpolation algorithm. Then two data sets were collected by different Raman spectrometers for evaluating the transfer learning performance of the trained 1D‐CNN model. Both data sets contained 390 Raman spectra from the same 39 samples of inorganic salts, organic compounds, and amino acids. One was used as calibration data to retrain the 1D‐CNN model, while the other was used as test. Based on data augmentation and 75% calibration data for retraining, the transfer learning performances of 1D‐CNN model were clearly shown in the excellent identification accuracies of 99.58%, 99.32%, and 97.69% for training, validation, and test sets, respectively, which were better than those of K‐nearest neighbor classifier. This paper provides a significant way for the wide application of CNN model in Raman spectroscopy with much more advantages in simplicity and rapidity. [ABSTRACT FROM AUTHOR]

In the design of antenna arrays which require fast and robust flat‐top beam synthesis, computationally efficient methods are preferred. This feature is usually met by analytical techniques or simple optimisation procedures. On the other hand, in the flat‐top beam synthesis, a common requirement is the ability to control beamwidth or sidelobe level. However, this can result in a high dynamic range ratio (DRR) of array's excitation coefficients. In this paper, a straightforward method for the design of symmetrical flat‐top arrays with controllable sidelobe level or DRR is proposed. The method is based on quadratic and cubic transforms of Gaussian excitations. In addition, the method utilises zero coefficients whose positions are used to control the DRR, including the ability to achieve its minimum. Compared to other flat‐top arrays with analytically shaped beams, the proposed arrays have lower DRRs for the same sidelobe level. [ABSTRACT FROM AUTHOR]

In this paper, we discuss a generalization of Vieta theorem (Vieta's formulas) to the case of Clifford geometric algebras. We compare the generalized Vieta formulas with the ordinary Vieta formulas for characteristic polynomial containing eigenvalues. We discuss Gelfand–Retakh noncommutative Vieta theorem and use it for the case of geometric algebras of small dimensions. We introduce the notion of a simple basis‐free formula for a determinant in geometric algebra and prove that a formula of this type exists in the case of arbitrary dimension. Using this notion, we present and prove generalized Vieta theorem in geometric algebra of arbitrary dimension. The results can be used in symbolic computation and various applications of geometric algebras in computer science, computer graphics, computer vision, physics, and engineering. [ABSTRACT FROM AUTHOR]

In this paper, we develop the Hermitian refinement of symplectic Clifford analysis, by introducing a complex structure 핁 on the canonical symplectic manifold (ℝ2n,ω0)$$ \left({\mathrm{\mathbb{R}}}^{2n},{\omega}_0\right) $$. This gives rise to two symplectic Dirac operators Ds$$ {D}_s $$ and Dt$$ {D}_t $$ (in the sense of Habermann), leading to a u(n)$$ \mathfrak{u}(n) $$‐invariant system of equations on ℝ2n$$ {\mathrm{\mathbb{R}}}^{2n} $$. We discuss the solution space for this system, culminating in a Fischer decomposition for the space of (harmonic) polynomials on ℝ2n$$ {\mathrm{\mathbb{R}}}^{2n} $$ with values in the symplectic spinors. To make this decomposition explicit, we will construct the associated embedding factors using a transvector algebra. [ABSTRACT FROM AUTHOR]

In this paper, we derive the equations that constitute the nonlinear mathematical model of an extensible thermoelastic Timoshenko system. The nonlinear governing equations are derived by applying the Hamilton principle to full von Kármán equations. The model takes account of the effects of extensibility, where the dissipations are entirely contributed by temperature. Based on the semigroups theory, we establish existence and uniqueness of weak and strong solutions to the derived problem. By using a resolvent criterion, developed by Borichev and Tomilov, we prove the optimality of the polynomial decay rate of the considered problem under the condition (65). Moreover, by an approach based on the Gearhart–Herbst–Prüss–Huang theorem, we show the non‐exponential stability of the same problem; but strongly stable by following a result due to Arendt–Batty. In the absence of additional mechanical dissipations, the system is often not highly stable. By adding a damping frictional function to the first equation of the nonlinear derived model with extensibility and using the multiplier method, we show that the solutions decay exponentially if Equation (85) holds. [ABSTRACT FROM AUTHOR]

For 0<λ<1$$ 0<\lambda <1 $$ and n→∞$$ n\to \infty $$ pick uniformly at random λn$$ \lambda n $$ vectors in {0,1} n$$ {\left\{0,1\right\}}^n $$ and let C$$ C $$ be the orthogonal complement of their span. Given 0<γ<12$$ 0<\gamma <\frac{1}{2} $$ with 0<λ

In this paper, we investigate to the existence and uniqueness of periodic solutions for the parabolic–elliptic Keller–Segel system on whole spaces detailized by Euclidean space Rn(wheren⩾4)$\mathbb {R}^n\,\,(\hbox{ where }n \geqslant 4)$ and real hyperbolic space Hn(wheren⩾2)$\mathbb {H}^n\,\, (\hbox{where }n \geqslant 2)$. We work in framework of critical spaces such as on weak‐Lorentz space Ln2,∞(Rn)$L^{\frac{n}{2},\infty }(\mathbb {R}^n)$ to obtain the results for the Keller–Segel system on Rn$\mathbb {R}^n$ and on Lp2(Hn)$L^{\frac{p}{2}}(\mathbb {H}^n)$ for n

The linear weakly singular fractional Volterra integro‐differential equations involving the Caputo derivative have solutions whose derivatives are unbounded at the left endpoint of the integration interval. In this paper, we use suitable transformations to prevail on this nonsmooth behavior. We used the product integration method based on the new fractional basis function to solve these equations, which led to the production of the new interpolation formula and weights for the method. We investigate the convergence of the presented method. The proposed scheme is employed to solve some numerical examples to test its efficiency and accuracy. [ABSTRACT FROM AUTHOR]

Summary: In this paper, we describe and analyze the spectral properties of several exact block preconditioners for a class of double saddle point problems. Among all these, we consider an inexact version of a block triangular preconditioner providing extremely fast convergence of the (F)GMRES method. We develop a spectral analysis of the preconditioned matrix showing that the complex eigenvalues lie in a circle of center (1,0)$$ \left(1,0\right) $$ and radius 1, while the real eigenvalues are described in terms of the roots of a third order polynomial with real coefficients. Numerical examples are reported to illustrate the efficiency of inexact versions of the proposed preconditioners, and to verify the theoretical bounds. [ABSTRACT FROM AUTHOR]

The problem of generalized tensor eigenvalue is the focus of this paper. To solve the problem, we suggest using the normalized Newton generalized eigenproblem approach (NNGEM). Since the rate of convergence of the spectral gradient projection method (SGP), the generalized eigenproblem adaptive power (GEAP), and other approaches is only linear, they are significantly improved by our proposed method, which is demonstrated to be locally and cubically convergent. Additionally, the modified normalized Newton method (MNNM), which converges to symmetric tensors Z‐eigenpairs under the same γ$$ \gamma $$‐Newton stability requirement, is extended by the NNGEM technique. Using a Gröbner basis, a polynomial system solver (NSolve) generates all of the real eigenvalues for us. To illustrate the efficacy of our methodology, we present a few numerical findings. [ABSTRACT FROM AUTHOR]

In this paper, we for the first time prove local solvability and stability of the inverse Sturm‐Liouville problem with complex‐valued singular potential and with polynomials of the spectral parameter in the boundary conditions. The proof method is constructive. It is based on the reduction of the inverse problem to a linear equation in the Banach space of bounded infinite sequences. We prove that, under a small perturbation of the spectral data, the main equation of the inverse problem remains uniquely solvable. Furthermore, we derive new reconstruction formulas for obtaining the problem coefficients from the solution of the main equation and get stability estimates for the recovered coefficients. [ABSTRACT FROM AUTHOR]

In this paper, we give some properties of the Dunkl–Gamma‐type operators including Appell polynomials, using into consideration the generalized power summability method. In the first section are given moments of the new defined operators. In second section are given some direct estimation related to the Dunkl–Gamma‐type operators, including Korovkin‐type theorem. In the third section, we give some results related to the weighted spaces of continuous functions, and in the last section, we give some properties in the sense of A$$ A $$‐statistically convergence, including Voronovskaya and Grüss–Voronovskaya‐type theorem. [ABSTRACT FROM AUTHOR]

The study of −1$-1$ orthogonal polynomials viewed as q→−1$q\rightarrow -1$ limits of the q$q$‐orthogonal polynomials is pursued. This paper presents the continuous polynomials part of the −1$-1$ analog of the q$q$‐Askey scheme. A compendium of the properties of all the continuous −1$-1$ hypergeometric polynomials and their connections is provided. [ABSTRACT FROM AUTHOR]

Numerical computation of capacitance matrices using conventional finite‐difference (FD) technique for arbitrarily shaped multi‐conductor problems which typically involve thin dielectric coating is challenging due to the fact that an extremely fine discretization of the computational domain is required to capture the nuances of the geometries involved, which, in turn, exacts a high computational resource cost—both in terms of memory and time. In this paper, we present a novel finite‐difference‐based technique which utilizes polynomial interpolation and extrapolation techniques in conjunction with the conventional finite‐difference technique to handle 2‐dimensional problems involving multiple conductors, typically with a thin dielectric coating. The proposed technique does not require fine discretization of the computational domain and provides accurate results in an efficient manner. [ABSTRACT FROM AUTHOR]

The bicomplex magnetic Laplacian (shortly bc‐magnetic Laplacian) is defined as a couple of magnetic Laplacians on two separate complex planes. In the present paper, we provide an explicit characterization of its L2$$ {L}&#x0005E;2 $$‐eigenspaces when acting on the so‐called bicomplex p‐Hilbert space. The common eigenfunction problem for the bc‐magnetic Laplacian and its †$$ \dagger $$‐conjugate is also tackled. The corresponding eigenspaces are described and the explicit expression of their reproducing kernels is given. [ABSTRACT FROM AUTHOR]

In this paper, we study a porous thermoelastic system with microtemperature and strong time delay acting on the volume fraction equation. The thermal effect of microtemperature is based on the Lord–Shulman theory (J Mech Phys Solids. 15(5) (1967), 299–309.), while the strong delay is motivated by Makheloufi's et al. recent work (Math Meth Appl Sci. 44 (2021), 6301–6317.). To prove the well‐posedness of the system, lack of exponential stability and the polynomial decay with optimal rate, we use the semigroup theory of linear operators. [ABSTRACT FROM AUTHOR]

An st $st$‐ordering of a graph G=(V,E) $G=(V,E)$ is an ordering v1,v2,...,vn ${v}_{1},{v}_{2},\ldots ,{v}_{n}$ of its vertex set such that s=v1,t=vn $s={v}_{1},t={v}_{n}$ and every vertex vi ${v}_{i}$ with i=2,3,...,n−1 $i=2,3,\ldots ,n-1$ has both a lower numbered and a higher numbered neighbor. Such orderings have played an important role in algorithms for planarity testing. It is well‐known that every 2‐connected graph has an st $st$‐ordering for every choice of distinct vertices s,t $s,t$. An st $st$‐ordering of a graph G $G$ corresponds directly to a so‐called bipolar orientation of G $G$, that is, an acyclic orientation D $D$ of G $G$ in which s $s$ is the unique source and t $t$ is the unique sink. Clearly every bipolar orientation of a graph has an out‐branching rooted at the source vertex and an in‐branching rooted at the sink vertex. In this paper, we study graphs which admit a bipolar orientation that contains an out‐branching and in‐branching which are arc‐disjoint (such an orientation is called good). A 2T‐graph is a graph whose edge set can be decomposed into two edge‐disjoint spanning trees. Clearly a graph has a good orientation if and only if it contains a spanning 2T‐graph with a good orientation, implying that 2T‐graphs play a central role. It is a well‐known result due to Tutte and Nash‐Williams, respectively, that every 4‐edge‐connected graph contains a spanning 2T‐graph. Vertex‐minimal 2T‐graphs with at least two vertices, also known as generic circuits, play an important role in rigidity theory for graphs. Recently with Bessy and Huang we proved that every generic circuit has a good orientation. In fact, we may specify the roots of the two branchings arbitrarily as long as they are distinct. Using this, several results on good orientations of 2T‐graphs were obtained. It is an open problem whether there exists a polynomial algorithm for deciding whether a given 2T‐graph has a good orientation. Complex constructions of 2T‐graphs with no good orientation were given in work by Bang‐Jensen, Bessy, Huang and Kriesell (2021) indicating that the problem might be very difficult. In this paper, we focus on so‐called quartics which are 2T‐graphs where every vertex has degree 3 or 4. We identify a sufficient condition for a quartic to have a good orientation, give a polynomial algorithm to recognize quartics satisfying the condition and a polynomial algorithm to produce a good orientation when this condition is met. As a consequence of these results we prove that every 4‐regular and 4‐connected graph has a good orientation, where, as for generic circuits, we may specify the roots of the two branchings arbitrarily as long as they are distinct. We also provide evidence that even for quartics it may be difficult to find a characterization of those instances which have a good orientation. We also show that every graph on n≥8 $n\ge 8$ vertices and of minimum degree at least ⌊n∕2⌋ $\lfloor n\unicode{x02215}2\rfloor $ has a good orientation. Finally we pose a number of open problems. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new model to study network reliability with node failures. This model, strongly connected node reliability, is the directed variant of node reliability and measures the probability that the operational vertices induce a subdigraph that is strongly connected. If we are restricted to directed graphs with n$$ n $$ vertices and n+1≤m≤2n−3$$ n+1\le m\le 2n-3 $$ or m=2n$$ m=2n $$ arcs, an optimally‐greatest digraph does not exist. Furthermore, we study optimally‐greatest directed circulant graphs when the vertices operate with probability p$$ p $$ near zero and near one. In particular, we show that the graph Γℤn,{1,−1}$$ \Gamma \left({\mathbb{Z}}_n,\left\{1,-1\right\}\right) $$ is optimally‐greatest for values of p$$ p $$ near zero. Then, we determine that the graph Γℤn,{1,n+22}$$ \Gamma \left({\mathbb{Z}}_n,\left\{1,\frac{n+2}{2}\right\}\right) $$ is optimally‐greatest for values of p$$ p $$ near one when n$$ n $$ is even. Next, we show that the graph Γℤn,{1,2(3−1)}$$ \Gamma \left({\mathbb{Z}}_n,\Big\{1,2\left({3}^{-1}\right)\Big\}\right) $$ is optimally‐greatest for values of p$$ p $$ near one when n$$ n $$ is odd and not divisible by three and that Γℤn,{1,3(2−1)}$$ \Gamma \left({\mathbb{Z}}_n,\Big\{1,3\left({2}^{-1}\right)\Big\}\right) $$ is optimally‐greatest for values of p$$ p $$ near one when n$$ n $$ is odd and divisible by three. We conclude with a discussion of open problems. [ABSTRACT FROM AUTHOR]

This paper introduces a fractional version of reaction‐diffusion equations with non‐local boundary conditions via a non‐singular fractional derivative defined by Atangana and Baleanu. The orthonormal discrete Legendre polynomials are introduced as suitable family of basis functions to find the solution of these equations. An operational matrix is derived for fractional derivative of these polynomials. A collocation method based on the expressed polynomials and their operational matrices is developed for solving such problems. The established method transforms solving the original problem under consideration into solving a system of algebraic equations. Some numerical examples are used to investigate the validity of the presented method. [ABSTRACT FROM AUTHOR]

The aim here is to consider the orthogonal polynomials Sn(ν0,ν1,s;x)$\mathcal {S}_{n}(\nu _0, \nu _1,s; x)$ with respect to an inner product of the type ⟨f,g⟩s=∫f(x)g(x)dν0(x)+s∫f′(x)g′(x)dν1(x)$ \langle f, g \rangle _{\mathfrak{s}} = \int f(x)g(x) d \nu _0(x) + s \int f^{\prime }(x) g^{\prime }(x) d \nu _1(x)$, where s>0$s > 0$ and {ν0,ν1}$\lbrace \nu _0, \nu _1\rbrace$ is a coherent pair of positive measures of the second kind on the real line (CPPM2K on the real line). Properties of Sn(ν0,ν1,s;x)$\mathcal {S}_{n}(\nu _0, \nu _1, s; x)$ and the connection formulas they satisfy with the orthogonal polynomials associated with the measure ν0 are analyzed. It is also shown that the zeros of Sn(ν0,ν1,s;x)$\mathcal {S}_{n}(\nu _0, \nu _1, s; x)$ are the eigenvalues of a matrix, which is a single line modification of the n×n$n \times n$ Jacobi matrix associated with the measure ν0. The paper also looks at a special example of a CPPM2K on the real line, where one of the measures is the Jacobi measure, and provides a much more detailed study of the properties of the orthogonal polynomials and the corresponding connection coefficients. In particular, the relation that these connection coefficients have with the Wilson polynomials is exposed. [ABSTRACT FROM AUTHOR]

In this paper we consider Diophantine equations of the form f(x)=g(y)$f(x)=g(y)$ where f$f$ has simple rational roots and g$g$ has rational coefficients. We give strict conditions for the cases where the equation has infinitely many solutions in rationals with a bounded denominator. We give examples illustrating that the given conditions are necessary. It turns out that such equations with infinitely many solutions are strongly related to Prouhet–Tarry–Escott tuples. In the special, but important case when g$g$ has only simple rational roots as well, we can give a simpler statement. Also we provide an application to equal products with terms belonging to blocks of consecutive integers of bounded length. The latter theorem is related to problems and results of Erdős and Turk, and of Erdős and Graham. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to give computational and implementational analysis of generating functions for some extensions of combinatorial numbers and polynomials attached to Dirichlet characters. By using generating function methods and p‐adic q‐integral techniques, it is also aimed to derive some computational formulas for these numbers and polynomials. By implementing both the derived computational formulas and the constructed generating functions in Mathematica, we present some tables and plots for these numbers and polynomials, and their generating functions to illustrate the effects of their parameters for some special cases. Moreover, by making an observation on some results of this paper, we derive some novel computational formulas for the finite sums that contain the Dirichlet characters and falling factorials. We also assess some cases of these special finite sums. In the sequel, we pose some open problems regarding these special finite sums. Apart from these findings, we also give not only Fourier series expansion but also asymptotic estimates for the mentioned combinatorial numbers. In addition, we give some Raabe‐type multiplication formulas for the mentioned combinatorial polynomials. Finally, we give an observation on decompositions of the generating functions attached to any group homomorphism. [ABSTRACT FROM AUTHOR]

The paper studies the probability for a Galois group of a random polynomial to be An$A_n$. We focus on the so‐called large box model, where we choose the coefficients of the polynomial independently and uniformly from {−L,...,L}$\lbrace -L,\ldots , L\rbrace$. The state‐of‐the‐art upper bound is O(L−1)$O(L^{-1})$, due to Bhargava. We conjecture a much stronger upper bound L−n/2+ε$L^{-n/2 +\epsilon }$, and that this bound is essentially sharp. We prove strong lower bounds both on this probability and on the related probability of the discriminant being a square. [ABSTRACT FROM AUTHOR]

The goal of this paper is to study the long time behavior of the extensible beams equation with the nonlocal weak damping utt + Δ²u - m(||∇u||²)Δu + ||ut||put + ƒ(u) = h. In the case of critical growth of the nonlinearity, the polynomial attractor which can quantitatively describe the attractive rate of attractors for the equation is studied. The work generalizes and improves previous results for the extensible beams equation. [ABSTRACT FROM AUTHOR]

An out‐branching Bu+ ${B}_{u}^{+}$ (in‐branching Bu− ${B}_{u}^{-}$) in a digraph D $D$ is a connected spanning subdigraph of D $D$ in which every vertex except the vertex u $u$, called the root, has in‐degree (out‐degree) one. It is well known that there exists a polynomial algorithm for deciding whether a given digraph has k $k$ arc‐disjoint out‐branchings with prescribed roots (k $k$ is part of the input). In sharp contrast to this, it is already NP‐complete to decide if a digraph has one out‐branching which is arc‐disjoint from some in‐branching. A digraph is semicomplete if it has no pair of nonadjacent vertices. A tournament is a semicomplete digraph without directed cycles of length 2. In this paper we give a complete classification of semicomplete digraphs that have an out‐branching Bu+ ${B}_{u}^{+}$ which is arc‐disjoint from some in‐branching Bv− ${B}_{v}^{-}$ where u,v $u,v$ are prescribed vertices of D $D$. Our characterization, which is surprisingly simple, generalizes a complicated characterization for tournaments from 1991 by the first author and our proof implies the existence of a polynomial algorithm for checking whether a given semicomplete digraph has such a pair of branchings for prescribed vertices u,v $u,v$ and construct a solution if one exists. This confirms a conjecture of Bang‐Jensen for the case of semicomplete digraphs. [ABSTRACT FROM AUTHOR]

In this paper, we consider a dissipative system of one-dimensional piezoelectric beam with magnetic effect and a tip load at the free end of the beam, which is modeled as a special form of double boundary dissipation. Our main aim is to study the well-posedness and asymptotic behavior of this system. By introducing two functions defined on the right boundary, we first transform the original problem into a new abstract form, so as to show the well-posedness of the system by using Lumer–Phillips theorem. We then divide the original system into a conservative system and an auxiliary system, and show that the auxiliary problem generates a compact operator. With the help of Weyl’s theorem, we obtain that the system is not exponentially stable. Moreover, we prove the polynomial stability of the system by using a result of Borichev and Tomilov (Math. Ann. 347 (2010), 455–478). [ABSTRACT FROM AUTHOR]

Many graph polynomials may be derived from the coefficients of the chromatic symmetric function XG ${X}_{G}$ of a graph G $G$ when expressed in different bases. For instance, the chromatic polynomial is obtained by mapping pn→x ${p}_{n}\to x$ for each n $n$ in this function, while a polynomial whose coefficients enumerate acyclic orientations is obtained by mapping en→x ${e}_{n}\to x$ for each n $n$. In this paper, we study a new polynomial we call the tree polynomial arising by mapping XPn→x ${X}_{{P}_{n}}\to x$, where XPn ${X}_{{P}_{n}}$ is the chromatic symmetric function of a path with n $n$ vertices. In particular, we show that this polynomial has a deletion‐contraction relation and has properties closely related to the chromatic polynomial while having coefficients that enumerate certain spanning trees and edge cutsets. [ABSTRACT FROM AUTHOR]

A graph G$$ G $$ is stronger than a graph H$$ H $$ if G$$ G $$ has at least as many connected spanning subgraphs of size k$$ k $$ as H$$ H $$ for any positive integer k$$ k $$. Counting the number of connected spanning subgraphs of fixed size allows us to compute the reliability of a graph. Formally, the reliability polynomial of a graph is the probability that the graph is connected when each edge is deleted independently with the same fixed probability. A graph G$$ G $$ is uniformly more reliable than H$$ H $$ if its reliability polynomial is greater than or equal to the reliability polynomial of H$$ H $$ for all probabilities. As a direct consequence of the definition, a sufficient condition for G$$ G $$ to be uniformly more reliable than H$$ H $$ is for G$$ G $$ to be stronger than H$$ H $$. In this paper, we show that the sufficient condition is not necessary by providing an example of two infinite families of graphs, Gk$$ {G}_k $$ and Hk$$ {H}_k $$, such that Gk$$ {G}_k $$ is uniformly more reliable than Hk$$ {H}_k $$ but is not stronger than Hk$$ {H}_k $$. [ABSTRACT FROM AUTHOR]

Many important special numbers appear in the expansions of some polynomials in terms of central factorials and vice versa, for example, central factorial numbers, degenerate central factorial numbers, and central Lah numbers which are recently introduced. Here we generalize this to any sequence of polynomials P=pn(x)n=0∞$$ \boldsymbol{P}={\left\{{p}_n(x)\right\}}_{n=0}^{\infty } $$ such that deg pn(x)=n,p0(x)=1$$ {p}_n(x)=n,{p}_0(x)=1 $$. The aim of this paper is to study the central factorial numbers of the second kind associated with any sequence of polynomials and of the first kind associated with any sequence of polynomials, in a unified and systematic way with the help of umbral calculus technique. The central factorial numbers associated with any sequence of polynomials enjoy orthogonality and inverse relations. We illustrate our results with many examples and obtain interesting orthogonality and inverse relations by applying such relations for the central factorial numbers associated with any sequence of polynomials to each of our examples. [ABSTRACT FROM AUTHOR]

Fascioliasis is a liver fluke disease in which food and water are the transmitting agents. The disease is caused by a genus of Fasciola, parasitic Trematoda. The genus Fasciola includes two species of Fasciola gigantica and Fasciola hepatica. The study of this disease is of particular importance as the infected cases are globally growing, especially in endemic areas. In this paper, a fractional‐order mathematical model is presented for fascioliasis disease. The optimization method is an effective choice in solving mathematical models of epidemic diseases. The study aims at presenting a new set of generalized Fibonacci polynomials as basic functions and some innovative operational matrices of derivatives in the Caputo sense to solve the fascioliasis disease model. The numerical experiments reveal better results and simulations show effectiveness of the presented method. [ABSTRACT FROM AUTHOR]

In this paper, we study the stabilization of a coupled wave system formed by one localized nonregular fractional viscoelastic damping of Kelvin–Voigt type and localized nonsmooth coefficients. Our main aim is to prove that the C0$$ {C}_0 $$‐semigroup associated with this model is strong stability and decays polynomially at a rate of t−1$$ {t}^{-1} $$. By introducing a new system to deal with fractional Kelvin–Voigt damping, we obtain a new equivalent augmented system, so as to show the well‐posedness of the system based on Lumer–Phillips theorem. We achieve the strong stability for the C0$$ {C}_0 $$‐semigroup associated with this new model by using a general criteria of Arendt–Batty and then turn out a polynomial energy decay rate of order t−1$$ {t}^{-1} $$ with the help of a frequency domain approach. [ABSTRACT FROM AUTHOR]

Lyapunov functions are a widely used tool to evaluate stability properties of nonlinear dynamical systems' equilibria. In this paper quantifier elimination is used to construct Lyapunov functions for polynomial systems from a parametric ansatz. Because the existing algorithms for quantifier elimination are inherently computationally expensive, a strategy is to simplify the quantifier elimination problem beforehand. A method is presented that may help simplifying the problem of finding suitable Lyapunov functions by deriving easier to evaluate necessary conditions. Finally, the introduced method is applied to an example system to show local asymptotic stability of an equilibrium point by constructing a Lyapunov function. [ABSTRACT FROM AUTHOR]

In this paper, we focus on the development and study of a numerical method based on the idea of kernel‐based approximation and finite difference discretization to obtain the solution for the time‐fractional diffusion equation. Using the theory of reproducing kernel, reproducing kernel functions with a polynomial form will be established in polynomial reproducing kernel spaces spanned by the Chebychev basis polynomials. In the numerical method, first the time‐fractional derivative term in the aforementioned equation is approximated by using the finite difference scheme. Then, by the help of collocation method based on reproducing kernel approximation, we will illustrate how to derive the numerical solution in polynomial reproducing kernel space. Finally, to support the accuracy and efficiency of the numerical method, we provide several numerical examples. In numerical experiments, the quality of approximation is calculated by absolute error and discrete error norms. [ABSTRACT FROM AUTHOR]

In this paper, the design of dynamic output feedback control law for polynomial fuzzy systems is studied. The main purpose is to design a tracker control law for fuzzy polynomial systems that are subject to external bounded disturbances. The structure of the control law is considered as fuzzy proportional–integral control. The conditions for deriving the control law are formulated in the form of a sum of square (SOS) feasibility problem. The designed control law will be able to force the system state vector to follow the state vector of a stable reference model in addition to guaranteeing the H∞ performance measure. The simulation results show the efficiency of the proposed method exposed to external disturbance. [ABSTRACT FROM AUTHOR]

The scale effect of mechanical properties is one of the most critical issues in application of the nanostructures. In order to study the scale effect on SH waves in nanoplates, a Legendre polynomial series method combined with the nonlocal theory (N‐LPSA) is developed in this paper. The presented method enables the boundary condition achieved quickly by the derived iterative solution formula of the integral, which contains nonlocal factors and Legendre polynomials. Besides, the formula of escape frequency is deduced analytically to facilitate the study of wave propagation law. Numerical comparisons show the correctness of the N‐LPSA. The results indicate that cut‐off frequencies of some higher‐order modes are higher than their escape frequencies. The anisotropy with the nonlocal effects can lead to the intersection of the modal curves, which does not appear in classical structures. Moreover, the fitting formulas of cut‐off and crossing frequencies are developed to predict the compression and the intersection of the modal curves, respectively. [ABSTRACT FROM AUTHOR]

Attribute‐based encryption (ABE) is a promising management method that enables fine‐grained access control in large‐scale systems. Revocable ABE (RABE) can support a key revocation mechanism in an ABE system. With the advent of the Internet of Things, users may need to delegate their decryption capacity to other devices, which requires that RABE meet a necessary feature called decryption key exposure resistance (DKER). Although many constructions about RABE from bilinear maps have been proposed, the situation of lattice‐based constructions with DKER is less satisfactory. In order to narrow this gap, this paper propose the first lattice‐based RABE with DKER. First, a formal description of RABE with DKER and the corresponding security models is proposed. Subsequently, a lattice‐based RABE scheme without DKER is constructed and it is proved to be selective indistinguishability under chosen‐plaintext attack (IND‐CPA) security based on Learning with Errors (LWE). To achieve DKER, this paper construct a RABE scheme by using the RABE scheme without DKER and a key extension mechanism as its building blocks. Finally, this paper show that this scheme is selective IND‐CPA security, with the DKER based on LWE. [ABSTRACT FROM AUTHOR]

For a class of fourth order gradient flow problems, integration of the scalar auxiliary variable (SAV) time discretization with the penalty‐free discontinuous Galerkin (DG) spatial discretization leads to SAV‐DG schemes. These schemes are linear and shown unconditionally energy stable. However, the reduced linear systems are rather expensive to solve due to the dense coefficient matrices. In this paper, we provide a procedure to pre‐evaluate the auxiliary variable in the piecewise polynomial space. As a result the computational complexity of ON2$$ O\left({\mathcal{N}}^2\right) $$ reduces to O(N)$$ O\left(\mathcal{N}\right) $$ when exploiting the conjugate gradient (CG) solver. This hybrid SAV‐DG method is more efficient and able to deliver satisfactory results of high accuracy. This was also compared with solving the full augmented system of the SAV‐DG schemes. [ABSTRACT FROM AUTHOR]

The higher order degrees are Alexander‐type invariants of complements to an affine plane curve. In this paper, we characterize the vanishing of such invariants for a curve C given as a transversal union of plane curves C′$C^{\prime }$ and C′′$C^{\prime \prime }$ in terms of the finiteness and the vanishing properties of the invariants of C′$C^{\prime }$ and C′′$C^{\prime \prime }$, and whether or not they are irreducible. As a consequence, we prove that the multivariable Alexander polynomial ΔCmulti$\Delta ^{\operatorname{multi}}_C$ is a power of (t−1)$(t-1)$, and we characterize when ΔCmulti=1$\Delta ^{\operatorname{multi}}_C=1$ in terms of the defining equations of C′$C^{\prime }$ and C′′$C^{\prime \prime }$. Our results impose obstructions on the class of groups that can be realized as fundamental groups of complements of a transversal union of curves. [ABSTRACT FROM AUTHOR]

Let Ln8,4 represent a linear octagonal‐quadrilateral network, consisting of n eight‐member rings and n four‐member rings. Such a graph contains a unique pair of opposite edges. The Möbius graph Qn(8, 4) is constructed by reverse identifying these opposite edges, whereas the cylinder graph Qn′(8, 4) identifies the opposite edges in the natural manner. In this paper, the explicit formulas for the Kirchhoff index and complexity of Qn(8, 4) and Qn′(8, 4) are deduced from Laplacian characteristic polynomials using to decomposition theorem and Vieta's theorem. A consequence is the surprising fact that the Kirchhoff index of Qn(8, 4) (resp. Qn′(8, 4)) is approximately a third (resp. half) of its Wiener index as n→∞. [ABSTRACT FROM AUTHOR]

Many destructive methods for measuring residual stresses such as the slitting method require an inverse analysis to solve the problem. The accuracy of the result as well as an uncertainty component (the model uncertainty) depends on the basis functions used in the inverse solution. The use of a series expansion as the basis functions for the inverse solution was analysed in a previous work for the particular case where functions orders grew consecutively. The present work presents a new estimation of the model uncertainty and a new improved methodology to select the final basis functions for the case where the basis is composed of polynomials. Including nonconsecutive polynomial orders in the basis generates a larger space of possible solutions to be evaluated and allows the possibility to include higher‐order polynomials. The paper includes a comparison with two other inverse analyses methodologies applied to synthetically generated data. With the new methodology, the final error is reduced and the uncertainty estimation improved. [ABSTRACT FROM AUTHOR]

An exponential polynomial is a finite linear sum of terms P(z)eQ(z)$P(z)e^{Q(z)}$, where P(z)$P(z)$ and Q(z)$Q(z)$ are polynomials. The early results on the value distribution of exponential polynomials can be traced back to Georg Pólya's paper published in 1920, while the latest results have come out in 2022. Despite of over a century of research work, many intriguing problems on value distribution of exponential polynomials still remain unsolved. The role of exponential polynomials and their quotients in the theories of linear/non‐linear differential equations, oscillation theory and differential‐difference equations will also be discussed. Thirteen open problems are given to motivate the readers for further research in these topics. [ABSTRACT FROM AUTHOR]

This paper considers a continuous three‐phase polynomial regression model with two threshold points for dependent data with heteroscedasticity. We assume the model is polynomial of order zero in the middle regime, and is polynomial of higher orders elsewhere. We denote this model by ℳ2$$ {\mathcal{M}}_2 $$, which includes models with one or no threshold points, denoted by ℳ1$$ {\mathcal{M}}_1 $$ and ℳ0$$ {\mathcal{M}}_0 $$, respectively, as special cases. We provide an ordered iterative least squares (OiLS) method when estimating ℳ2$$ {\mathcal{M}}_2 $$ and establish the consistency of the OiLS estimators under mild conditions. When the underlying model is ℳ1$$ {\mathcal{M}}_1 $$ and is (d0−1)$$ \left({d}_0-1\right) $$th‐order differentiable but not d0$$ {d}_0 $$th‐order differentiable at the threshold point, we further show the Op(N−1/(d0+2))$$ {O}_p\left({N}^{-1/\left({d}_0+2\right)}\right) $$ convergence rate of the OiLS estimators, which can be faster than the Op(N−1/(2d0))$$ {O}_p\left({N}^{-1/\left(2{d}_0\right)}\right) $$ convergence rate given in Feder when d0≥3$$ {d}_0\ge 3 $$. We also apply a model‐selection procedure for selecting ℳκ$$ {\mathcal{M}}_{\kappa } $$; κ=0,1,2$$ \kappa =0,1,2 $$. When the underlying model exists, we establish the selection consistency under the aforementioned conditions. Finally, we conduct simulation experiments to demonstrate the finite‐sample performance of our asymptotic results. [ABSTRACT FROM AUTHOR]

In this paper, we consider the stabilization of the generalized Rao‐Nakra beam equation, which consists of four wave equations for the longitudinal displacements and the shear angle of the top and bottom layers and one Euler–Bernoulli beam equation for the transversal displacement. Dissipative mechanism are provided through viscous damping for two displacements. The location of the viscous damping are divided into two groups, characterized by whether both of the top and bottom layers are directly damped or otherwise. Each group consists of three cases. We obtain the necessary and sufficient conditions for the cases in group 2 to be strongly stable. Furthermore, polynomial stability of certain orders are proved. The cases in group 1 are left for future study. [ABSTRACT FROM AUTHOR]

In this paper, we formulate a numerical method to approximate the solution of two‐dimensional optimal control problem with a fractional parabolic partial differential equation (PDE) constraint in the Caputo type. First, the optimal conditions of the optimal control problems are derived. Then, we discretize the spatial derivatives and time derivatives terms in the optimal conditions by using shifted discrete Legendre polynomials and collocations method. The main idea is simplifying the optimal conditions to a system of algebraic equations. In fact, the main privilege of this new type of discretization is that the numerical solution is directly and globally obtained by solving one efficient algebraic system rather than step‐by‐step process which avoids accumulation and propagation of error. Several examples are tested and numerical results show a good agreement between exact and approximate solutions. [ABSTRACT FROM AUTHOR]