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In this paper, we study the large N behavior of the smallest eigenvalue λ N of the (N + 1) × (N + 1) Hankel matrix, H N = (μ j + k) 0 ≤ j , k ≤ N , generated by the γ dependent Jacobi weight w (z , γ) = e − γ z z α (1 − z) β , z ∈ [ 0 , 1 ] , γ ∈ R , α > − 1 , β > − 1. Applying the arguments of Szegö, Widom and Wilf, we obtain the asymptotic representation of the orthonormal polynomials P N (z) , z ∈ C ﹨ [ 0 , 1 ] , with the weight w (z , γ) = e − γ z z α (1 − z) β. Using the polynomials P N (z) , we obtain the theoretical expression of λ N , for large N. We also display the smallest eigenvalue λ N for sufficiently large N , computed numerically. [ABSTRACT FROM AUTHOR]

In the present paper, we introduce Δ ω -Laguerre based Appell polynomials. The first main aim of the paper is to investigate their main properties including explicit representation, determinantal form, recurrence relation, lowering operators (L O) , integro-partial raising operator (I P R O) and integro-partial difference equation (I P D E). The second aim is to give the corresponding results, which are known or new, to the usual Laguerre based Appell polynomials which can be obtain immediately from the limiting case ω → 0. Furthermore, we introduce Δ ω -Laguerre-Carlitz Bernoulli, Δ ω -Laguerre-Carlitz Euler, Δ ω -Laguerre-Miller-Lee, Δ ω -Laguerre-Boole polynomials and Δ ω -Laguerre-Bernoulli polynomials of the second kind as particular cases of Δ ω -Laguerre based Appell polynomials. The corresponding main results have also been exhibited for these new families of polynomials. [ABSTRACT FROM AUTHOR]

This paper studies the general decay synchronization of a class of bidirectional associative memory neural networks with time-varying delays. First, a useful lemma which generalizes the classical exponential synchronization and polynomial synchronization is introduced. Then by using this lemma, some simple sufficient criteria ensuring the general decay synchronization of considered bidirectional associative memory neural networks are obtained via designing a novel nonlinear feedback controller and using some inequality techniques. Finally, two numerical examples are provided to demonstrate the feasibility of the established theoretical results. The results of this paper are general since the classical polynomial synchronization and exponential synchronization can be seen the special cases of general decay synchronization. [ABSTRACT FROM AUTHOR]

This paper establishes convergence rates for discrete-time Markov chains on a countable state space that are stochastically ordered starting from a stationary distribution under perturbation. We investigate the explicit criteria to obtain the ordinary ergodicity, geometric ergodicity and polynomial ergodicity for the embedded M/G/1 queue under perturbation. The explicit geometric convergence rates for the original system and the system under perturbation are caculated. Our bounds in the geometric case and polynomial case are closely connected to the first hitting times. Two examples are provided to illustrate our result. [ABSTRACT FROM AUTHOR]

Let G = (V , E) be a graph. The weak rupture degree of G is defined as r w (G) = max { ω (G − X) − | X | − m e (G − X) : ω (G − X) > 1 } , where the maximum is taken over all X , the subset of V (G), ω (G − X) is the number of components in G − X , and m e (G − X) is the size (edge number) of a largest component in G − X. This is an important parameter to quantitatively describe the invulnerability of networks. In this paper, based on a study of relationship between network structure and the weak rupture degree, a polynomial algorithm for computing the weak rupture degree of trees is given. [ABSTRACT FROM AUTHOR]

In this paper we employ the Euclidean division for polynomials to recover uniquely a Jacobi matrix in terms of the mixed spectral data consisting of its partial entries and the information given on its full spectrum together with a subset of eigenvalues of its truncated matrix obtained by deleting the last row and last column, or its rank-one modification matrix modified by adding a constant to the last element. A necessary and sufficient condition is provided for the existence of the inverse problem. A numerical algorithm and a numerical example are given. [ABSTRACT FROM AUTHOR]

• We give a combinatorial explanation of the weighted enumeration of rooted spanning forests of a graph. • We give a combinatorial explanation of the characteristic polynomial of a graph. • We give a combinatorial explanation of the weighted Laplacian det (L + d i a g (x 1 , x 2 , ... , x n)) of a graph G. Assume that G is a graph with n vertices v 1 , v 2 , ... , v n. Given a rooted spanning forest F of G , which has s components and roots v k 1 , v k 2 , ... , v k s , define the weight ω (F) of F to be ∏ i = 1 s x k i , where x 1 , x 2 , ... , x n are n independent commuting variables. In this paper, we give combinatorial explanations of the weighted enumeration of rooted spanning forests of G , the Laplacian characteristic polynomial of G , and the weighted Laplacian det [ L + d i a g (x 1 , x 2 , ... , x n) ] , where L is the Laplacian matrix of G and d i a g (x 1 , x 2 , ... , x n) is a diagonal matrix. As applications, we count rooted spanning forests in the almost complete bipartite graph, and we also give a new proof of a formula on the number of rooted spanning forests in a complete multipartite graph K a 1 , a 2 , ... , a s each of which has b i roots in the i -th part for i = 1 , 2 , ... , s. [ABSTRACT FROM AUTHOR]

• Derive a new formula related to signless Laplacian characteristic polynomial of a graph with a special strtucture. • Characterize the unique graph with the maximum signless Laplacian spectral radius among all blow-ups of a tree. • Determine the minimum signless Laplacian spectral radius of balanced blow-ups of a tree. For a positive integer vector a = (a 1 , a 2 , ... , a k) and a graph T with V (T) = { v 1 , v 2 , ... , v k } , the a -blow-up of T , denoted by T (a) , is the graph that arises from T by replacing every vertex v i of T with an a i -clique K a i (1 ≤ i ≤ k), where K a j is adjacent to K a s in T (a) if and only if v j is adjacent to v s in T. When all a i (1 ≤ i ≤ k) are equal to some positive integer a , the corresponding blow-up is called balanced. In this paper, the extremal graphs with the maximum signless Laplacian index among all a -blow-ups of trees with k vertices are first characterized, and then the minimum signless Laplacian index among all balanced blow-ups of trees with k vertices is determined. [ABSTRACT FROM AUTHOR]

In this paper, an iteration process, referred to in short as MMP, will be considered. This iteration is related to finding the maximum modulus of a complex polynomial over a unit disc on the complex plane creating intriguing images. Kalantari calls these images polynomiographs independently from whether they are generated by the root finding or maximum modulus finding process applied to any polynomial. We show that the images can be easily modified using different MMP methods (pseudo-Newton, MMP-Householder, methods from the MMP-Basic, MMP-Parametric Basic or MMP-Euler–Schröder Families of Iterations) with various kinds of non-standard iterations. Such images are interesting from three points of views: scientific, educational and artistic. We present the results of experiments showing automatically generated non-trivial images obtained for different modifications of root finding MMP-methods. The colouring by iteration reveals the dynamic behaviour of the used root finding process and its speed of convergence. The results of the present paper extend Kalantari’s recent results in finding the maximum modulus of a complex polynomial based on Newton’s process with the Picard iteration to other MMP-processes with various non-standard iterations. [ABSTRACT FROM AUTHOR]

This paper deals with the stability analysis of Linear Time-Invariant (LTI) systems in the presence of a bounded and time-invariant delay parameter. The paper extends the theory of characteristic quasi-polynomial to derive equivalent stability conditions. Based on the quasi-polynomial, a two-variate polynomial is defined and it is proved that system stability is equivalently guaranteed if all roots of the new polynomial are not allocated in a certain region of the right half plane (RHP). Moreover, the algorithm uses the proposed equivalent stability conditions to develop a set of implementable stability conditions based on the exposed edges theorem. The algorithm is applied to three simulation examples that admit some novel results for these systems. The first example shows the step-by-step procedure for the proposed algorithm. The second and the third ones compare various aspects of the proposed algorithm with some existing methods. Two hundred and fifty delay systems are randomly generated and the proposed algorithm is compared with them in terms of conservativeness and computational burden. The results reveal the superiority of the proposed method over the existing ones. [ABSTRACT FROM AUTHOR]

• Novel Results on Partial Hosoya Polynomials: An Application in Chemistry • Submitted to Appl. Math. Comput. • In the paper several mistype errors were corrected and now the paper is complete. This article deal with an investigation of certain properties of partial Hosoya polynomial and then computing this new polynomial for some family of well-known graphs. Also, we verify some results concerning the zeros location of this polynomial in a ring shaped region. In this way, we include not only new results as special cases, but also improve the results due to Dehmer et al. [12] as a particular case. In continuing, the unique positive root of a new version of partial Hosoya polynomial has been evaluated. Finally, we explore some functional measures based on the complex zeros of partial Hosoya polynomials and then we compute the Pearson correlation between these measures and well-known topological indices. Our results show that these new measures have a linear correlation with the Wiener index. [ABSTRACT FROM AUTHOR]

Abstract Let A (G) be an adjacency matrix of a graph G. Then the polynomial π (G , x) = per (x I − A (G)) is called the permanental polynomial of G , and the permanental sum of G is the sum of the absolute values of the coefficients of π (G, x). In this paper, the second largest and second smallest permanental sums among connected unicyclic graphs and the corresponding extremal graphs are determined. In addition, we show that the computation of permanental sum is # P -complete. [ABSTRACT FROM AUTHOR]

Abstract This paper is concerned with the construction of efficient preconditioning method for systems arising from implicit Runge–Kutta discretizations of time-dependent PDEs. A polynomial preconditioner is proposed based on the Kronecker product splitting iteration method introduced by Chen (BIT 54:607-721, 2014). Some useful properties on the spectrum of the preconditioned matrix are established. Numerical experiments concerning discrete solution of fractional diffusion equations are presented to show the effectiveness of the proposed polynomial preconditioner. [ABSTRACT FROM AUTHOR]

Abstract For a graph G with adjacency matrix A (G) and degree-diagonal matrix D (G), Cvetković et al introduced a bivariate polynomial ϕ G (x , t) = d e t (x I − (A (G) − t D (G))) , where I is the identity matrix. The polynomial ϕ G (x, t) not only generalizes the characteristic polynomials of some well-known matrices related to G , such as the adjacency, the Laplacian matrices, but also has an elegant combinatorial interpretation as being equivalent to the Bartholdi zeta function. Let G = H [ G 1 , G 2 , ... , G k ] be the generalized join graph of G 1 , G 2 , ... , G k determined by graph H. In this paper, we first give a decomposition formula for ϕ G (x, t). The decomposition formula provides us a new method to construct infinitely many pairs of non-regular ϕ -cospectral graphs. Then, as applications, explicit expressions for ϕ G (x, t) of some special kinds of graphs are given. [ABSTRACT FROM AUTHOR]

Abstract A k -path vertex cover (VCP k) is a vertex set C of graph G such that every path of G on k vertices has at least one vertex in C. Because of its background in keeping data integrality of a network, minimum VCP k problem (MinVCP k) has attracted a lot of researches in recent years. This paper studies the minimum weight connected VCP k problem (MinWCVCP k), in which every vertex has a weight and the VCP k found by the algorithm induces a connected subgraph and has the minimum weight. It is known that MinWCVCP k is set-cover-hard. We present two polynomial-time approximation algorithms for MinWCVCP 3. The first one is a greedy algorithm achieving approximation ratio 3ln n. The difficulty lies in its analysis dealing with a non-submodular potential function. The second algorithm is a 2-stage one, finding a VCP 3 in the first stage and then adding more vertices for connection. We show that its approximation ratio is at most ln δ max + 4 + ln 2 , where δ max is the maximum degree of the graph. Considering the inapproximability of this problem, this ratio is asymptotically tight. [ABSTRACT FROM AUTHOR]

Abstract Recently, the 2D-Laguerre–Konhauser polynomials were introduced in [1]. In the present paper, first of all, we propose another bivariate polynomial family which is bi-orthonormal with the 2D-Laguerre–Konhauser polynomials. Then, we consider a convolution integral equation with 2D-Laguerre–Konhauser polynomials in the kernel and we obtain its solution by introducing a new family of bivariate Mittag-Leffler functions. Furthermore, we introduce a double (fractional) integral operator including bivariate Mittag-Leffler functions in the kernel. This integral operator includes the double Riemann–Liouville fractional integral operator. We investigate its transformation properties on the continuous function and Lebesgue summable function spaces. Also, the semigroup property of the operator is investigated. We further study some miscelenenous properties of 2D-Laguerre–Konhauser polynomials and bivariate Mittag-Leffler functions such as generating function, Schläfli's integral represantation. Finally, we approximate to the image of any bivariate continuous function under the action of the proposed double integral operators. [ABSTRACT FROM AUTHOR]

Abstract This paper develops the partially truncated Euler–Maruyama method for a class of highly nonlinear pantograph stochastic differential equations under the generalized Khasminskii-type conditions. The order of Lp -convergence is obtained. Moreover, some almost sure polynomial stability and mean square polynomial stability criteria are established for the numerical solution. Numerical examples are provided to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

Abstract In this paper, multivariate polynomials in the Bernstein basis over a box (tensorial Bernstein representation) are considered. A new matrix method for the computation of the polynomial coefficients with respect to the Bernstein basis, the so-called Bernstein coefficients, are presented and compared with existing methods. Also matrix methods for the calculation of the Bernstein coefficients over subboxes generated by subdivision of the original box are proposed. All the methods solely use matrix operations such as multiplication, transposition, and reshaping; some of them rely also on the bidiagonal factorization of the lower triangular Pascal matrix or the factorization of this matrix by a Toeplitz matrix. In the case that the coefficients of the polynomial are due to uncertainties and can be represented in the form of intervals it is shown that the developed methods can be extended to compute the set of the Bernstein coefficients of all members of the polynomial family. [ABSTRACT FROM AUTHOR]

Abstract An independent set in a graph G is a set of pairwise non-adjacent vertices. Let i k (G) denote the number of independent sets of cardinality k in G. Then, its generating function I (G ; x) = ∑ k = 0 α (G) i k (G) x k is called the independence polynomial of G (Gutman and Harary, 1983). Alavi et al. (1987) conjectured that the independence polynomial of any tree or forest is unimodal. This conjecture is still open. In this paper, after obtaining recurrence relations and giving factorizations of independence polynomials for certain classes of trees, we prove the log-concavity of their independence polynomials. Thus, our results confirm the conjecture of Alavi et al. for some special cases. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we consider the asymptotic behavior of solutions to the linear spatially homogeneous Boltzmann equation for hard potentials without angular cutoff. We obtain an optimal rate of exponential convergence towards equilibrium in a L 1-space with a polynomial weight. Our strategy is taking advantage of a spectral gap estimate in the Hilbert space L 2 (μ − 1 2 ) and a quantitative spectral mapping theorem developed by Gualdani et al. (2017). [ABSTRACT FROM AUTHOR]

Abstract In this paper, the error and stability analysis of the method of fundamental solution (MFS) is explored for biharmonic equations. The bounds of errors are derived for the fundamental solutions r 2ln r in bounded simply-connected domains, and the polynomial convergence rates are obtained for certain smooth solutions. The bounds of condition number are also derived to show the exponential growth rates for disk domains. Numerical experiments are carried out to support the above analysis, which is the first time to provide the rigorous analysis of the MFS using r 2ln r for biharmonic equations. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we study the adaptive fashion of the Ciarlet–Raviart mixed method for biharmonic equation/eigenvalue problem with simply supported boundary condition in R d. We propose an a posteriori error indicator of the Ciarlet–Raviart approximate solution for the biharmonic equation and an a posteriori error indicator of the Ciarlet–Raviart approximate eigenfuction, and prove the reliability and efficiency of the indicators. We also give an a posteriori error indicator for the approximate eigenvalue and prove its reliability. We design an adaptive Ciarlet–Raviart mixed method with piecewise polynomials of degree less than or equal to m , and numerical experiments show that numerical eigenvalues obtained by the method can achieve the optimal convergence order O (d o f − 2 m d ) (d = 2 , m = 2 , 3 ; d = 3 , m = 3). [ABSTRACT FROM AUTHOR]

The fundamental purpose of the present paper is to apply an effective numerical algorithm based on the mixture of homotopy analysis technique, Sumudu transform approach and homotopy polynomials to obtain the approximate solution of a nonlinear fractional Drinfeld–Sokolov–Wilson equation. The nonlinear Drinfeld–Sokolov–Wilson equation naturally occurs in dispersive water waves. The uniqueness and convergence analysis are shown for the suggested technique. The convergence of the solution is fixed and managed by auxiliary parameter ℏ. The numerical results are shown graphically. Results obtained by the application of the technique disclose that the suggested scheme is very accurate, flexible, effective and simple to use. [ABSTRACT FROM AUTHOR]

The characteristic polynomial of a graph is the characteristic polynomial of its adjacency matrix. Finding efficient algorithms for computing characteristic polynomial of graphs is an active area of research and for some graph classes, like threshold graphs, there exist very fast algorithms which exploit combinatorial structure of the graphs. In this paper, we put forward some novel ideas based on divisor technique to obtain fast algorithms for computing the characteristic polynomial of threshold and chain graphs. [ABSTRACT FROM AUTHOR]

In this paper, we consider several scheduling problems on a serial-batch machine for scheduling jobs with or without precedence relations. Under the serial-batch setting, the jobs in a batch are processed in succession and are removed until the last job in this batch finishes its processing. Thus, the processing time of a batch is equal to the sum of processing times of jobs in the batch. When a new batch starts, a constant setup time is required for the machine. The objectives of the problems involve minimizing makespan and a maximum cost. For these problems, we either present polynomial-time algorithms to generate all Pareto optimal points and find a corresponding Pareto optimal schedule for each Pareto optimal point, or give the strong NP-hardness proof. Experimentation results show that the proposed algorithms for the considered problems are very efficient. [ABSTRACT FROM AUTHOR]

A dominating set S of a graph is a metric-locating-dominating set if each vertex of the graph is uniquely distinguished by its distances from the elements of S , and the minimum cardinality of such a set is called the metric-location-domination number. In this paper, we undertake a study that, in general graphs and specific families, relates metric-locating-dominating sets to other special sets: resolving sets, dominating sets, locating-dominating sets and doubly resolving sets. We first characterize the extremal trees of the bounds that naturally involve metric-location-domination number, metric dimension and domination number. Then, we prove that there is no polynomial upper bound on the location-domination number in terms of the metric-location-domination number, thus extending a result of Henning and Oellermann. Finally, we show different methods to transform metric-locating-dominating sets into locating-dominating sets and doubly resolving sets. Our methods produce new bounds on the minimum cardinalities of all those sets, some of them concerning parameters that have not been related so far. [ABSTRACT FROM AUTHOR]

In our first paper we showed how Gauss's method for determining the initial position and velocity vectors from two inertial position vectors at two times in an idealized Keplerian two-body elliptical orbit can be made more robust and efficient by replacing functional iteration with Newton–Raphson iteration. To do this we split the orbit determination algorithm into two sub-algorithms, the x -iteration to find the zero of the fixed-point function F( x ) when the true anomaly angular difference between the two vectors is large and the y -iteration to find the zero of the fixed-point function G ( y ) when the angular difference is small. We derived necessity and sufficiency conditions for the existence and uniqueness of these zeros and conjectured that both functions were concave. In this paper we first simplify the necessity and sufficiency conditions. We then prove that both F and G are indeed concave. Existence and uniqueness of the zero of G and its concavity then imply quadratic convergence of the Newton–Raphson iterative sequence from an arbitrary starting point in its domain. Quadratic convergence to the zero of F ( x ) is also the case in its domain except for a small neighborhood of x = 1 , corresponding to the true anomaly difference angle between the position vectors in a small neighborhood of 180°. [ABSTRACT FROM AUTHOR]

• We propose the Gauss–Legendre polynomials. • We also propose the Gauss–Legendre curve as the barycentric combination of the control points with the weights given by the Gauss–Legendre polynomials. • We discuss the shape control of polynomial curves using the Gauss–Legendre curves and analyze basic properties of the Gauss–Legendre polynomials. The Gauss–Legendre (GL) polygon was recently introduced for the shape control of Pythagorean hodograph curves. In this paper, we consider the GL polygon of general polynomial curves. The GL polygon with n + 1 control points determines a polynomial curve of degree n as a barycentric combination of the control points. We identify the weight functions of this barycentric combination and define the GL polynomials, which form a basis of the polynomial space like the Bernstein polynomial basis. We investigate various properties of the GL polynomials such as the partition of unity property, symmetry, endpoint interpolation, and the critical values in comparison with the Bernstein polynomials. We also present the definite integral and higher derivatives of the GL polynomials. We then discuss the shape control of polynomial curves using the GL polygon. We claim that the design process of high degree polynomial curves using the GL polygon is much easier and more predictable than if the curve is given in the Bernstein–Bézier form. This is supported by some neat illustrative examples. [ABSTRACT FROM AUTHOR]

• For any connected graph G, α (G) ≤ m G [ 0 , n − α (G) ]. • We classify graph where the equality holds for α (G) = 3 or n − 3. • When α (G) = 3 , we used the result of Zhang and Luo. • When α (G) = n − 3 , we give a numerical criterion using the characteristic polynomial of graphs. For a graph G , let α (G) be the independence number of G , let L (G) be the Laplacian matrix of G , and let m G I be the number of eigenvalues of L (G) in the interval I. Ahanjideh, Akbari, Fakharan and Trevisan proved that α (G) ≤ m G [ 0 , n − α (G) ] if G is an n -vertex connected graph. Choi, Moon and Park characterized graphs with α (G) = m G [ 0 , n − α (G) ] for α (G) = 2 and α (G) = n − 2. In this paper, we give a characterization for α (G) = 3 and α (G) = n − 3. [ABSTRACT FROM AUTHOR]

This paper aims to provide a natural generalization and unification of a series of multiple integral representations for special classes of hypergeometric polynomials recently obtained by several authors. This generalization is obtained by considering a very large family of hypergeometric polynomials. The multiple integral representations given in this paper may be viewed as linearization relationship for the product of two different members of the associated family of hypergeometric polynomials. [ABSTRACT FROM AUTHOR]

The paper concerns the decompositions of polynomials onto iterated integrals. This is a continuation of our previous paper (Lorenc, submitted for publication), in which the existence of such decomposition for the Faulhaber polynomials is proven. In the current paper we prove the basic theorem (Theorem 2) presenting the necessary and sufficient conditions for the existence of such decomposition. We discuss these conditions in the wider context of theory of the real and complex polynomials. A number of exemplary decompositions onto iterated integrals of the known classical kinds of polynomials are also presented. [ABSTRACT FROM AUTHOR]

A novel class of quasi-polynomials orthogonal with respect to the fractional integration operator has been developed in this paper. The related Gaussian quadrature formulas for numerical evaluation of fractional order integrals have also been proposed. By allowing the commensurate order of quasi-polynomials to vary independently of the integration order, a family of fractional quadrature formulas has been developed for each fractional integration order, including novel quadrature formulas for numerical approximation of classical, integer order integrals. A distinct feature of the proposed quadratures is high computational efficiency and flexibility, as will be demonstrated in the paper. As auxiliary results, the paper also presents methods for Lagrangian and Hermitean quasi-polynomial interpolation and Hermitean fractional quadratures. The development is illustrated by numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, we apply the weak Galerkin (WG) finite element method to the Darcy–Stokes equations. This method provides accurate approximations for the velocity and the pressure variables. General polygonal or polyhedral partitions can be applied in this method. The finite element space which is made up of piecewise polynomials is easy to be constructed. These advantages make the weak Galerkin finite element method efficient and highly flexible. Optimal rates of convergence for the velocity function u and the pressure function p are established in corresponding norms. In addition, the convergence rates are independent of the viscosity parameter ϵ. Several numerical experiments are provided to illustrate the robustness, flexibility and validity of the weak Galerkin finite element method. [ABSTRACT FROM AUTHOR]

Let G be a graph and A ( G ) the adjacency matrix of G . The polynomial π ( G , x ) = per ( x I − A ( G ) ) is called the permanental polynomial of G , and the permanental sum of G is the summation of the absolute values of the coefficients of π ( G, x ). In this paper, we investigate properties of permanental sum of a graph, prove recursive formulas to compute the permanental sum of a graph, and show that the ordering of graphs with respect to permanental sum. Furthermore, we determine the upper and lower bounds of permanental sum of unicyclic graphs, and the corresponding extremal unicyclic graphs are also determined. [ABSTRACT FROM AUTHOR]

This paper presents a procedure to detect unreliable computer simulations of recursive functions. The proposed method calculates a lower bound error which is derived from two different pseudo-orbits based on interval extensions. The interval extensions are generated by taking into account the associative property of multiplication, which keeps the same error bound. We have tested our approach on the logistic map using many different programming languages and simulation packages, including Matlab, Scilab, Octave, Fortran and C. In all cases, the number of iterates is significantly lower than that considered reliable in the existing literature. We have also used the lower bound error on the logistic map and on the polynomial NARMAX for the Rössler equations to estimate the largest Lyapunov exponent, which determines the critical simulation time that guarantees the reliability of the simulation. [ABSTRACT FROM AUTHOR]

In this paper, the problem of finding numerical solution for a system of Cauchy type singular integral equations of first kind with index zero is considered. The analytic solution of such system is known. But it is of limited use as it is a nontrivial task to use it practically due to the presence of singularity in the known solution itself. Therefore, a residual based Galerkin method is proposed with Legendre polynomials as basis functions to find its numerical solution. The proposed method converts the system of Cauchy type singular integral equations into a system of linear algebraic equations which can be solved easily. Further, Hadamard conditions of well-posedness are established for system of Cauchy singular integral equations as well as for system of linear algebraic equations which is obtained as a result of approximation of system of singular integral equations with Cauchy kernel. The theoretical error bound is derived which can be used to obtain any desired accuracy in the approximate solution of system of Cauchy singular integral equations. The derived theoretical error bound is also validated with the help of numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, the performance of a parametric family including Newton’s and Traub’s schemes on multiple roots is analyzed. The local order of convergence on nonlinear equations with multiple roots is studied as well as the dynamical behavior in terms of the damping parameter on cubic polynomials with multiple roots. The fixed and critical points, and the associated parameter plane are some of the characteristic dynamical features of the family which are obtained in this work. From the analysis of these elements we identify members of the family of methods with good numerical properties in terms of stability and efficiency both for finding the simple and multiple roots, and also other ones with very unstable behavior. [ABSTRACT FROM AUTHOR]

Let G be a connected graph with vertex set V ( G ) and edge set E ( G ). Let T ( G ) be the diagonal matrix of vertex transmissions of G and D ( G ) be the distance matrix of G . The distance Laplacian matrix of G is defined as L ( G ) = T ( G ) − D ( G ) . The distance signless Laplacian matrix of G is defined as Q ( G ) = T ( G ) + D ( G ) . In this paper, we show that the complements of path and cycle are determined by their distance (signless) Laplacian spectra. [ABSTRACT FROM AUTHOR]

In the present paper, a modification of hat functions (MHFs) has been considered for solving a class of nonlinear fractional integro-differential equations with weakly singular kernels, numerically. The fractional order operational matrix of integration is introduced. We provide an error estimation for the approximation of a function by a series of MHFs. To suggest a numerical method, the main problem is converted to an equivalent Volterra integral equation of the second kind and operational matrices of MHFs are used to reduce the problem to the solution of bivariate polynomial equations. Finally, illustrative examples are provided to confirm the accuracy and validity of the proposed method. [ABSTRACT FROM AUTHOR]

The aim of the present paper is to introduce generalized Szász–Mirakyan operators including Brenke type polynomials and investigate their approximation properties. We obtain convergence properties of our operators with the help of Korovkin’s theorem and the order of convergence by using a classical approach, the second modulus of continuity and Peetre’s K -functional. We also give asymptotic formula and the convergence of the derivatives for these operators. Furthermore, an example of Szász–Mirakyan operators including Gould–Hopper polynomials is presented. In the end, we show graphical representation. [ABSTRACT FROM AUTHOR]

In this paper, we perform a further investigation for the Bernoulli polynomials of the second kind. By making use of the generating function methods and summation transform techniques, we establish a higher-order convolution identity for the Bernoulli polynomials of the second kind. We also present some illustrative special cases as well as immediate consequences of the main result. [ABSTRACT FROM AUTHOR]

In this paper, a parametric family of seventh-order of iterative method to solve systems of nonlinear equations is presented. Its local convergence is studied and quadratic polynomials are used to investigate its dynamical behavior. The study of the fixed and critical points of the rational function associated to this class allows us to obtain regions of the complex plane where the method is stable. By depicting parameter planes and dynamical planes we obtain complementary information of the analytical results. These results are used to solve some nonlinear problems. [ABSTRACT FROM AUTHOR]

The energy of a graph is defined as the sum of the absolute values of its eigenvalues. In 1940 Coulson obtained an important integral formula which makes it possible to calculate the energy of a graph without knowing its spectrum. Recently several Coulson-type integral formulas have been obtained for various energies and some other invariants of graphs based on eigenvalues. For a complex polynomial ϕ ( z ) = ∑ k = 0 n a k z n − k = a 0 ∏ k = 1 n ( z − z k ) of degree n and a real number α , the general energy of ϕ ( z ), denoted by E α ( ϕ ), is defined as ∑ z k ≠ 0 | z k | α when there exists k 0 ∈ { 1 , 2 , … , n } such that z k 0 ≠ 0 , and 0 when z 1 = ⋯ = z n = 0 . In this paper we give Coulson-type integral formulas for the general energy of polynomials whose roots are all real numbers in the case that α ∈ Q . As a consequence of this result, we obtain an integral formula for the 2 l -th spectral moment of a graph. Furthermore, we show that our formulas hold when α is an irrational number with 0 < | α | < 2 and do not hold with | α | > 2. [ABSTRACT FROM AUTHOR]

Birkhoff (or lacunary) interpolation is an extension of polynomial interpolation that appears when observation gives irregular information about some function and its derivatives. A Birkhoff interpolation problem is not always solvable even in the appropriate polynomial or rational space. In this paper we split up the initial problem in subproblems having a unique polynomial solution and use multinode rational basis functions in order to obtain a global interpolant. [ABSTRACT FROM AUTHOR]

The study of limit cycles for Liénard system is very important not only in theoretical studies but also in applications. In this paper, we study the number of limit cycles for a class of Liénard systems with restoring forces of seventh degree. Let H ( n, m ) denote the maximum number of limit cycles bifurcated from the generalized Liénard system x ˙ = y , y ˙ = − g ( x ) − f ( x ) y , where f ( x ) and g ( x ) are polynomials in x and deg f = n , det g = m . We greatly improve the existing results of H ( n, m ) for m = 7 , n = 4 and m = 7 , n = 2 n ¯ with 4 ≤ n ¯ ≤ 20 . [ABSTRACT FROM AUTHOR]

In this paper, multivariate polynomials in the Bernstein basis over a simplex (simplicial Bernstein representation) are considered. Two matrix methods for the computation of the polynomial coefficients with respect to the Bernstein basis, the so-called Bernstein coefficients, are presented. Also matrix methods for the calculation of the Bernstein coefficients over subsimplices generated by subdivision of the standard simplex are proposed and compared with the use of the de Casteljau algorithm. The evaluation of a multivariate polynomial in the power and in the Bernstein basis is considered as well. All the methods solely use matrix operations such as multiplication, transposition, and reshaping; some of them rely also on the bidiagonal factorization of the lower triangular Pascal matrix or the factorization of this matrix by a Toeplitz matrix. The latter one enables the use of the Fast Fourier Transform hereby reducing the amount of arithmetic operations. [ABSTRACT FROM AUTHOR]

In this paper we examine semidefinite linear programming approximations to a class of semi-infinite convex polynomial optimization problems, where the index sets are described in terms of convex quadratic inequalities. We present a convergent hierarchy of semidefinite linear programming relaxations under a mild well-posedness assumption. We also provide additional conditions under which the hierarchy exhibits finite convergence. These results are derived by first establishing characterizations of optimality which can equivalently be reformulated as linear matrix inequalities. A separation theorem of convex analysis and a sum-of-squares polynomial representation of positivity of real algebraic geometry together with a special variable transformation play key roles in achieving the results. Finally, as applications, we present convergent relaxations for a broad class of robust convex polynomial optimization problems. [ABSTRACT FROM AUTHOR]

In this paper we propose some different strategies to approximate hypersingular integrals ∫ = 0 + ∞ G ( x ) ( x − t ) p + 1 d x , where p is a positive integer, t > 0 and the integral is understood in the Hadamard finite part sense. Hadamard Finite Part integrals (shortly FP integrals), regarded as p th derivative of Cauchy principal value integrals, are of interest in the solution of hypersingular BIE, which model many different kind of Physical and Engineering problems (see [1] and the references therein, [2], [3, 4]). The procedure we employ here is based on a simple tool like the “truncated” Gaussian rule (see [5]), conveniently modified to remove numerical cancellation. We will consider functions G having different decays at infinity. The method is shown to be numerically stable and convergent and some error estimates in suitable Zygmund-type spaces are proved. Finally, some numerical tests which confirm the efficiency of the proposed procedures are presented. [ABSTRACT FROM AUTHOR]

In this paper, we present non-uniform subdivision schemes with variable parameter sequences. A locally different tension parameter is set at each edge of the initial control polygon to control locally the shape of the resulting curve such that the scheme becomes non-uniform. Due to the variable parameters, the scheme can reproduce locally different analytic curves such as conics, Lissajous, trigonometric and catenary curves. Hence blending curves including such analytic components can be successfully generated. We discuss the convergence and smoothness of the proposed non-uniform schemes and present some numerical results to demonstrate their advantages in geometric modeling. Furthermore, as an application, we propose a chamfering algorithm which can be used in designing automobile and mechanical products. [ABSTRACT FROM AUTHOR]

In this paper, a non-stationary combined subdivision scheme is presented, which can unify several existing non-stationary approximating and interpolatory subdivision schemes. This scheme is obtained by generalizing the connection between the approximating and interpolatory schemes in the stationary case, first formalized by Maillot & Stam using a push-back operator, to the non-stationary case. For such a combined scheme, we investigate its C l convergence and exponential polynomial generation/reproduction property and get that it can reach C 4 degree of smoothness and generate/reproduce certain exponential polynomials with suitable choices of the parameters. Besides, we give a more generalized combined scheme for the purpose of generating and reproducing more general exponential polynomials. The performance of our new schemes is illustrated by several numerical examples. [ABSTRACT FROM AUTHOR]