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Abstract: An identity by Chaundy and Bullard writes () as a sum of two truncated binomial series. In a paper which appeared in 2008 in Indag. Math. the authors surveyed many aspects of this identity. In the present paper we discuss much earlier occurrences of this identity in works by Hering (1868), de Moivre (1738) and de Montmort (1713). A relationship with Krawtchouk polynomials in work by Greville (1966) is also discussed. [Copyright &y& Elsevier]

Abstract This paper is devoted to the mathematical and numerical analysis of a mixed-mixed PDE system describing the stress-assisted diffusion of a solute into an elastic material. The equations of elastostatics are written in mixed form using stress, rotation and displacements, whereas the diffusion equation is also set in a mixed three-field form, solving for the solute concentration, for its gradient, and for the diffusive flux. This setting simplifies the treatment of the nonlinearity in the stress-assisted diffusion term. The analysis of existence and uniqueness of weak solutions to the coupled problem follows as combination of Schauder and Banach fixed-point theorems together with the Babuška–Brezzi and Lax–Milgram theories. Concerning numerical discretization, we propose two families of finite element methods, based on either PEERS or Arnold–Falk–Winther elements for elasticity, and a Raviart–Thomas and piecewise polynomial triplet approximating the mixed diffusion equation. We prove the well-posedness of the discrete problems, and derive optimal error bounds using a Strang inequality. We further confirm the accuracy and performance of our methods through computational tests. [ABSTRACT FROM AUTHOR]

Abstract: The main object of this paper is to investigate several general families of hypergeometric polynomials and their associated multiple integral representations. By suitably specializing our main results, the corresponding integral representations are deduced for such familiar classes of hypergeometric polynomials as (for example) the generalized Bedient polynomials and the generalized Cesàro polynomials. Each of the integral representations, which are derived in this paper, may be viewed also as a linearization relationship for the product of two different members of the associated family of hypergeometric polynomials. [Copyright &y& Elsevier]

Abstract: Let G be a simple undirected graph with the characteristic polynomial of its Laplacian matrix , . Aleksandar Ilić [A. Ilić, Trees with minimal Laplacian coefficients, Comput. Math. Appl. 59 (2010) 2776–2783] identified n-vertex trees with given matching number q which simultaneously minimize all Laplacian coefficients. In this paper, we give another proof of this result. Generalizing the approach in the above paper, we determine n-vertex trees with given matching number q which have the second minimal Laplacian coefficients. We also identify the n-vertex trees with a perfect matching having the largest and the second largest Laplacian coefficients, respectively. Extremal values on some indices, such as Wiener index, modified hyper-Wiener index, Laplacian-like energy, incidence energy, of n-vertex trees with matching number q are obtained in this paper. [Copyright &y& Elsevier]

Abstract: Complex financial instruments with multiple state variables often have no analytical formulas and therefore must be priced by numerical methods, like lattice ones. For pricing convertible bonds and many other interest rate-sensitive products, research has focused on bivariate lattices for models with two state variables: stock price and interest rate. This paper shows that, unfortunately, when the interest rate component allows rates to grow in magnitude without bounds, those lattices generate invalid transition probabilities. As the overwhelming majority of stochastic interest rate models share this property, a solution to the problem becomes important. This paper presents the first bivariate lattice that guarantees valid probabilities. The proposed bivariate lattice grows (super)polynomially in size if the interest rate model allows rates to grow (super)polynomially. Furthermore, we show that any valid constant-degree bivariate lattice must grow superpolynomially in size with log-normal interest rate models, which form a very popular class of interest rate models. Therefore, our bivariate lattice can be said to be optimal. [Copyright &y& Elsevier]

Abstract: Let be a non-trivial, loopless graph and for each non-trivial subgraph of , let . The graph is 1-balanced if , the maximum among , taken over all non-trivial subgraphs of , is attained when . This quantity is called the fractional arboricity of the graph . The value appears in a paper by Picard and Queyranne and has been studied extensively by Catlin, Grossman, Hobbs and Lai. The quantity measures how much a given graph differs from being 1-balanced. In this paper, we describe a systematic method of modifying a given graph to obtain a 1-balanced graph on the same number of vertices and edges. We obtain this by a sequence of iterations; each iteration re-defining one end-vertex of an edge in the given graph. After each iteration, either the value of the new graph formed is less than that of the graph from the previous iteration or the size of the maximal -achieving subgraph of the new graph is smaller than that of the graph in the previous iteration. We show that our algorithm is polynomial in time complexity. Further ways to decrease the number of iterations are also discussed. [Copyright &y& Elsevier]

Abstract: In this paper, we present a simple and elegant procedure for the construction of iterative methods for the simultaneous determination of (simple or multiple) zeros of an algebraic polynomial. This procedure is based on the application of a special type of functions, called Weierstrass’ functions, to suitable zero-finding methods for a single zero. For demonstration, using this approach we derive many known iterative methods in a simpler way compared with original derivations, as well as some new methods. Aside from the presented methodology in developing zero-finding methods, the paper offers a short review of simultaneous methods including some historical notes. [Copyright &y& Elsevier]

In this paper, the output tracking control issues of polynomial-fuzzy-model-based (PFMB) systems equipped with mismatched interval type-2 (IT2) membership functions are investigated. The output-feedback IT2 polynomial fuzzy controller connected with the nonlinear plant in a closed loop drives the system states of the nonlinear plant to track those of a stable reference model. The system stability is investigated based on the Lyapunov stability theory under the sum-of-squares (SOS)-based analysis approach and the SOS-based stability conditions are derived subjecting to an H ∞ performance. In addition, the fuzzy controller does not need to share the same membership functions with the plant. Moreover, the information of membership functions is included in the analysis to facilitate the analysis and relax the stability conditions. Numerical and experimental examples are presented to verify the effectiveness of the proposed tracking control approach. [ABSTRACT FROM AUTHOR]

Solving polynomial systems with noise over F 2 is a fundamental problem in computer science, especially in cryptanalysis. ISBS is a new method for solving this problem based on the idea of incrementally solving the noisy polynomial systems and backtracking all the possible noises, and it has better performance than other methods in solving some problems generated from cryptanalysis. In this paper, some further researches on ISBS are presented. The structure and size of the search tree of ISBS are theoretically analyzed. Then two major improvements, artificial noise-bound strategy and s -direction approach, are proposed. Based on these improvements, a modified ISBS algorithm is implemented, and the experiments of solving the Cold Boot key recovery problems of the block cipher Serpent with symmetric noise, show that this modified algorithm is more efficient than the original one. [ABSTRACT FROM AUTHOR]

In this paper, we consider the mixed finite element method of the plane elasticity equations based on the Hellinger–Reissner variational principle. Low order mixed finite element spaces are used to approximate the stress and the displacement. Based on the local polynomial projection stabilization method, we present a stabilization scheme for the pairs to overcome the lack of the inf–sup condition. The stability is proved and the error estimate is derived. At last, two numerical examples are implemented to test the stability and effectiveness of the proposed method. These pairs are very convenient in the numerical implementation for the mixed form of the plane elasticity equations. [ABSTRACT FROM AUTHOR]

J.-B. Meilhan and the second author showed that any Milnor μ ¯ -invariant of length between 3 and 2 k + 1 can be represented as a combination of HOMFLYPT polynomial of knots obtained by certain band sum of the link components, if all μ ¯ -invariants of length ≤ k vanish. They also showed that their formula does not hold for length 2 k + 2 . In this paper, we improve their formula to give the μ ¯ -invariants of length 2 k + 2 by adding correction terms. The correction terms can be given by a combination of HOMFLYPT polynomial of knots determined by μ ¯ -invariants of length k + 1 . In particular, for any 4-component link the μ ¯ -invariants of length 4 are given by our formula, since all μ ¯ -invariants of length 1 vanish. [ABSTRACT FROM AUTHOR]

In this paper, by using multivariate divided difference (Rabut, 2001) to approximate the partial derivative and the idea of the superposition (Waldron, 2009), we modify a multiquadric quasi-interpolation operator (Ling, 2004) based on a dimension-splitting technique with the property of linear reproducing to gridded data on multi-dimensional spaces, such that a family of proposed multivariate multiquadric quasi-interpolation operators Φ r + 1 has the property of r + 1 ( r ∈ Z , r ⩾ 0 ) degree polynomial reproducing and converges up to a rate of r + 2 . In addition, the proposed quasi-interpolation operator only demands information of location points rather than the derivatives of the function approximated. Moreover, we give the approximation error of our quasi-interpolation operator. Finally, some numerical experiments are shown to confirm the approximation capacity of our quasi-interpolation operator. [ABSTRACT FROM AUTHOR]

Abstract: Quasi-Minimal Residual (QMR)-type methods have been developed for improving the oscillations in the residual norms of the Bi-Conjugate Gradient (Bi-CG) method and the hybrid Bi-CG methods. The IDRstab method, which combines the Induced Dimension Reduction (IDR)(s) method with higher-order stabilizing polynomials, is often more effective than IDR(s) and the hybrid Bi-CG methods, but the residual norms oscillate. In this paper, we design a QMR variant of IDRstab by using a residual smoothing technique in order to obtain smoother convergence behavior. Numerical experiments show the effectiveness of our QMR variant of IDRstab. [Copyright &y& Elsevier]

Abstract: In Suzuki (1998) [7] Suzuki gave a classification of association schemes with multiple Q-polynomial structures, allowing for one exceptional case which has five classes. In this paper, we rule out the existence of this case. Hence Suzukiʼs theorem mirrors exactly the well-known counterpart for association schemes with multiple P-polynomial structures, a result due to Eiichi Bannai and Etsuko Bannai in 1980. [Copyright &y& Elsevier]

Abstract: Although being powerful, the differential transform method (DTM) yet suffers from a drawback which is the lack of a systematic methodology for derivation of the differential transforms for nonlinear expressions. In the current paper, it is shown that this defect can be overcome with the help of the Adomian polynomials perfectly. The application of the proposed technique in treatment of nonlinear differential equations is well illustrated by a number of examples. In addition, the transformed analogues of some frequent nonlinearities are presented. [Copyright &y& Elsevier]

Abstract: This paper deals with the error analysis of generalized polynomial chaos (gPC) for nonlinear random ordinary differential equations. The analysis shows that the global error mainly relies on the projection error and the numerical error. For the deterministic systems obtained from the gPC method, a kind of numerical approach with error analysis is given. At last, a numerical experiment is carried out to support the theoretical results. [Copyright &y& Elsevier]

Abstract: The development of new classes of linearizations of square matrix polynomials that generalize the classical first and second Frobenius companion forms has attracted much attention in the last decade. Research in this area has two main goals: finding linearizations that retain whatever structure the original polynomial might possess, and improving properties that are essential for accurate numerical computation, such as eigenvalue condition numbers and backward errors. However, all recent progress on linearizations has been restricted to square matrix polynomials. Since rectangular polynomials arise in many applications, it is natural to investigate if the new classes of linearizations can be extended to rectangular polynomials. In this paper, the family of Fiedler linearizations is extended from square to rectangular matrix polynomials, and it is shown that minimal indices and bases of polynomials can be recovered from those of any linearization in this class via the same simple procedures developed previously for square polynomials. Fiedler linearizations are one of the most important classes of linearizations introduced in recent years, but their generalization to rectangular polynomials is nontrivial, and requires a completely different approach to the one used in the square case. To the best of our knowledge, this is the first class of new linearizations that has been generalized to rectangular polynomials. [Copyright &y& Elsevier]

Abstract: In this paper the combined integral method is applied to a simple one-dimensional ablation problem. One of the drawbacks of heat balance integral methods is how to choose the approximating function. It is common to use a polynomial form but even then it is not clear what the power of the highest order term should be. Previous studies have determined exponents either from exact solutions or from expansions valid over short time scales; neither approach is satisfactory nor very accurate for larger times. We combine the heat balance and refined integral methods to determine this exponent as part of the solution process, and conclude that it is in fact time-dependent in the ablation stage. From comparing the approximate solutions with numerical and exact analytical solutions whenever possible, we show that this new method greatly improves the accuracy on standard methods, without overcomplicating the method. [Copyright &y& Elsevier]

Abstract: In this paper, by using a new non-polynomial parameters cubic spline in space direction and compact finite difference in time direction, we get a class of new high accuracy scheme of O(τ 4 + h 2) and O(τ 4 + h 4) for solving telegraph equation if we suitably choose the cubic spline parameters. Meanwhile, stability condition of the difference scheme has been carried out. Finally, numerical examples are used to illustrate the efficiency of the new difference scheme. [Copyright &y& Elsevier]

Abstract: Let be an infinite field and let where the entries . In this paper we establish necessary and sufficient conditions under which it is possible to prescribe some entries of ,so that the pair is completely controllable. [Copyright &y& Elsevier]

Abstract: In this paper, center conditions and bifurcations of limit cycles for a class of cubic polynomial system in which the origin is a nilpotent singular point are studied. A recursive formula is derived to compute quasi-Lyapunov constant. Using the computer algebra system Mathematica, the first seven quasi-Lyapunov constants of the system are deduced. At the same time, the conditions for the origin to be a center and 7-order fine focus are derived respectively. A cubic polynomial system that bifurcates seven limit cycles enclosing the origin (node) is constructed. [Copyright &y& Elsevier]

Abstract: In this paper, the problem of center conditions and bifurcation of limit cycles at the infinity for a class of cubic systems are investigated. The method is based on a homeomorphic transformation of the infinity into the origin, the first 21 singular point quantities are obtained by computer algebra system Mathematica, the conditions of the origin to be a center and a 21st order fine focus are derived, respectively. Correspondingly, we construct a cubic system which can bifurcate seven limit cycles from the infinity by a small perturbation of parameters. At the end, we study the isochronous center conditions at the infinity for the cubic system. [Copyright &y& Elsevier]

Abstract: Descartes’ rule of signs yields an upper bound for the number of positive and negative real roots of a given polynomial. The fundamental theorem of algebra implies a similar property; every real polynomial of degree n ⩾1 has at most n real zeroes. In this paper, we describe axiomatically function families possessing one or another of these properties. The resulting families include, at least, all polynomial functions and sums of exponential functions. As an application of our approach, we consider, among other things, a method for identifying certain type of bases for the Euclidean space. [Copyright &y& Elsevier]

Abstract: In this paper, we consider a strongly damped wave equation with fractional damping on part of its boundary and also with an internal source. Under some appropriate assumptions on the parameters, and with certain initial data, a blow-up result with positive initial energy is established. [Copyright &y& Elsevier]

Abstract: In this paper we derive an approximation property of four-point interpolatory curve subdivision, based on local cubic polynomial fitting. We show that when the scheme is used to generate a limit curve that interpolates given irregularly spaced points, sampled from a curve in any space dimension with a bounded fourth derivative, and the chosen parameterization is chordal, the accuracy is fourth order as the mesh size goes to zero. In contrast, uniform and centripetal parameterizations yield only second order. [Copyright &y& Elsevier]

Abstract: In this paper, the higher-order tangent numbers and higher-order secant numbers, and , have been studied in detail. Several known results regarding and have been brought together along with many new results and insights and they all have been proved in a simple and unified manner. In particular, it is shown that the higher-order tangent numbers constitute a special class of the partial multivariate Bell polynomials and that can be computed from the knowledge of . In addition, a simple explicit formula involving a double finite sum is deduced for the numbers and it is shown that are linear combinations of the classical tangent numbers . [Copyright &y& Elsevier]

Abstract: Let be a graph of order and the Laplacian characteristic polynomial of . Zhou and Gutman proved that among all trees of order , the th coefficient is largest when the tree is a path and is smallest for a star. In this paper, for two given positive integers and , we characterize the trees with a given bipartition which have the minimal and second minimal Laplacian coefficients. [Copyright &y& Elsevier]

Abstract: This paper explores the problems of generalized center conditions and integrability of resonant infinity for a complex polynomial differential system. The method is based on converting resonant infinity into an elementary singular point by a homeomorphic transformation. The calculation of generalized singular point quantities is an effective way of finding necessary conditions for integrable systems. A new recursive algorithm for computing generalized singular point quantities at the origin of the transformed system is derived. At the same time, a necessary and sufficient condition for resonant infinity to be a generalized complex center is presented. As an application of the new recursive algorithm, the generalized center conditions for resonant infinity for a class of cubic systems are discussed. To the best of our knowledge, this is the first time that the generalized center problem has been considered for resonant infinity. [Copyright &y& Elsevier]

Abstract: The well-known Faà di Bruno formula for higher derivatives of a composite function plays an important role in combinatorics. In this paper we obtain a unified approach to some recurrence sequences for the complete Bell polynomials by using Faà di Bruno’s formula. Furthermore, we obtain an inverse relation of Lah numbers of two types, and by using this relation we obtain some new and interesting combinatorial identities. [Copyright &y& Elsevier]

Abstract: In this paper, we show the applicability of the first integral method, which is based on the ring theory of commutative algebra, to the regularized long-wave Burgers equation and the Gilson–Pickering equation under a parameter condition. Our method provides polynomial first integrals for autonomous planar systems. Through the established first integrals, exact traveling wave solutions are derived in a concise manner. [Copyright &y& Elsevier]

Abstract: In this paper we consider identical parallel machines scheduling problems with a deteriorating maintenance activity. In this model, each machine has a deteriorating maintenance activity, that is, delaying the maintenance increases the time required to perform it. We need to make a decision on when to schedule the rate-modifying activities and the sequence of jobs to minimize some objective function. We concentrate on two goals separately, namely, minimizing the total absolute differences in completion times (TADC) and the total absolute differences in waiting times (TADW). We show that the problems remain polynomially solvable under the proposed model. [Copyright &y& Elsevier]

Abstract: This paper is dedicated to the stability analysis of a class of uncertain distributed delay systems, the kernel of which can be modeled as a polynomial function of the delay. The results are constructed by rewriting the system as an uncertain interconnected model. Appropriate robust control tools, i.e. quadratic separation, are then used to address the stability issue. To this end, some relations that highlight relevant characteristics of the delayed term are added to the interconnected model leading then to the conservatism reduction. Finally, numerical examples show the effectiveness of the proposed method. [Copyright &y& Elsevier]

Abstract: This paper describes a simple and efficient approach to the Troesch’s problem. In this approach, the hyperbolic nonlinear term in the equation is first converted into polynomial nonlinear terms by variable transformation, and a simple shooting method is then used directly to solve this transformed problem. The calculated results are in excellent agreement with those obtained by other analytical and numerical methods. [Copyright &y& Elsevier]

Abstract: A numerical elimination method is presented in this paper for floating-point computation in polynomial algebra. The method is designed to calculate one or more polynomials in an elimination ideal by a sequence of matrix rank/kernel computation. The method is reliable in numerical computation with verifiable stability and a sensitivity measurement. Computational experiment shows that the method possesses significant advantages over classical resultant computation in numerical stability and in producing eliminant polynomials with lower degrees and fewer extraneous factors. The elimination algorithm combined with an approximate GCD finder appears to be effective in solving polynomial systems for positive dimensional solutions. [Copyright &y& Elsevier]

Abstract: Sum of squares (SOS) decompositions for nonnegative polynomials are usually computed numerically, using convex optimization solvers. Although the underlying floating point methods in principle allow for numerical approximations of arbitrary precision, the computed solutions will never be exact. In many applications such as geometric theorem proving, it is of interest to obtain solutions that can be exactly verified. In this paper, we present a numeric–symbolic method that exploits the efficiency of numerical techniques to obtain an approximate solution, which is then used as a starting point for the computation of an exact rational result. We show that under a strict feasibility assumption, an approximate solution of the semidefinite program is sufficient to obtain a rational decomposition, and quantify the relation between the numerical error versus the rounding tolerance needed. Furthermore, we present an implementation of our method for the computer algebra system Macaulay 2. [Copyright &y& Elsevier]

Abstract: Equations representing data in prismatic array are useful for interpreting experiments involving three independent parameters. This paper illustrates two sets of four polynomial equations for the nine-point cubical array. The equations are exact on trilinear numbers and their second and third powers. The equations are potentially useful alternatives to the trilinear equation when they do not generate complex-number coefficients. [Copyright &y& Elsevier]

Abstract: In this paper, using Lagrange interpolation with special nodes, we construct an iteration for simultaneously finding all zeros of an algebraic polynomial with high order convergence. The convergence analysis, initial conditions and numerical examples are given. [Copyright &y& Elsevier]

In this paper, two weighted essentially nonoscillatory (ENO) schemes are presented on triangular meshes. By combining quadratic polynomials with weights on the ENO stencil, we construct a scheme with second-order accuracy and another scheme with third-order accuracy. Numerical results show the accuracy and stability of the weighted ENO schemes and resolution for discontinuity. [Copyright &y& Elsevier]

In this paper we investigate the inner products in the space of rational functions and the space of Mu¨ntz polynomials. We prove that the orthogonality of a sequence in one of mentioned spaces can be induced by the orthogonality of the corresponding sequence in another space. Finally, we point to several special cases, i.e., some very different classes of well–known functions we represent on a unique way. [Copyright &y& Elsevier]