Search

Abstract: We study a family of orthogonal polynomials which generalizes a sequence of polynomials considered by L. Carlitz. We show that they are a special case of the Sheffer polynomials and point out some interesting connections with certain Sobolev orthogonal polynomials. [Copyright &y& Elsevier]

Let k be a field of characteristic 0, and let f : kn→kn be a polynomial map with components of the form fi=xi+hi, where the hi are monomials. If the Jacobian determinant of the map f is a nonzero constant, then f is a tame automorphism. If, in addition, each hi is either constant or of degree 2 or more, then f is linearly triangularizable. [Copyright &y& Elsevier]

Abstract: Ahn, Lee, Park and Yoo proved that the best constrained degree reduction of a polynomial f in -norm equals the best weighted Euclidean approximation of the Bernstein–Bézier coefficients of f in a paper, published in the journal, Computer Aided Geometric Design 21 (2) (2004) 181–191. In this note, we point out an error in their paper and give the correct result. [Copyright &y& Elsevier]

Abstract: We recall some bibliography on fast polynomial multiplication related to the recent progress in the paper by Cenk and Ozbudak of 2011. [Copyright &y& Elsevier]

This paper develops a new approach for the design of polynomial-based reference and feedforward control trajectories for the high precision control of flexible servo actuators. First, a robust frequency-domain controller is designed for reference tracking and disturbance rejection. This controller is designed to achieve a desirable loop shape with reasonable bandwidth and stability margins. Then, the polynomial trajectories are derived based on a simplified rigid body representation of the system. To avoid the excitation of the system's resonant modes, this paper proposes the design and implementation of a set of feedforward notch filters. We further investigate the main cause of tracking error resulted from the rigid body approximation, and propose an alternative model for the derivation of the polynomial input trajectories. This model accounts for the cumulative DC gain of the resonant modes ignored in the rigid body approximation. A method for deriving the new polynomial trajectories with the appropriate initial and final time conditions is developed and evaluated for a representative flexible servo system model. Simulation results indicate that the proposed scheme provides significant improvement in the performance of system compared to the conventional methods. • Polynomial feedforward inputs improve settling response of flexible servo systems. • Feedforward notch filters can mitigate high-frequency vibrations during settling. • Accounting for actuator flexibility in the feedforward model reduces settling time. [ABSTRACT FROM AUTHOR]

In this paper, we give a characterization of regular two-graphs of order n in terms of spectral data of two-graphs of order n − 1 and n − 2. Moreover, we prove that a two-graph of order n is regular if and only if all its induced sub-two-graphs with n − 2 vertices have the same characteristic polynomial. [ABSTRACT FROM AUTHOR]

In this paper, we consider the polynomial and exponential convergence rates of the weighted Birkhoff averages of irrational rotations on tori. It is shown that these can be achieved for finite and infinite dimensional tori which correspond to the quasiperiodic and almost periodic dynamical systems respectively, under certain balance between the nonresonant condition and the decay rate of the Fourier coefficients. Diophantine rotations with finite and infinite dimensions are provided as examples. For the first time, we prove the universality of exponential convergence and arbitrary polynomial convergence in the quasiperiodic case and almost periodic case under analyticity respectively. [ABSTRACT FROM AUTHOR]

A well known method to solve the Polynomial Eigenvalue Problem (PEP) is via linearization. That is, transforming the PEP into a generalized linear eigenvalue problem with the same spectral information and solving such linear problem with some of the eigenvalue algorithms available in the literature. Linearizations of matrix polynomials are usually defined using unimodular transformations. In this paper we establish a connection between the standard definition of linearization for matrix polynomials introduced by Gohberg, Lancaster and Rodman and the notion of polynomial system matrix introduced by Rosenbrock. This connection gives new techniques to show that a matrix pencil is a linearization of the corresponding matrix polynomial arising in a PEP. [ABSTRACT FROM AUTHOR]

Graham and Lovász conjectured in 1978 that the sequence of normalized coefficients of the distance characteristic polynomial of a tree of order n is unimodal with the maximum value occurring at ⌊ n 2 ⌋. In this paper we investigate this problem for block graphs. In particular, we prove the unimodality part and we establish the peak for several extremal cases of uniform block graphs with small diameter. [ABSTRACT FROM AUTHOR]

For a vertex v of a graph, the local complementation at v is an operation that replaces the neighborhood of v by its complement graph. Two graphs are locally equivalent if one is obtained from the other by a sequence of local complementations. A graph H is a vertex-minor of a graph G if H is an induced subgraph of a graph locally equivalent to G. Although this concept was introduced in the 1980s, it was not widely known and except for the survey paper of Bouchet published in 1990, there is no comprehensive survey listing all the new developments. We survey classic and recent theorems and conjectures on vertex-minors and related concepts such as circle graphs, cut-rank functions, rank-width, and interlace polynomials. [ABSTRACT FROM AUTHOR]

In this paper, we primarily focus on the eigenvalues of the adjacency matrix and Seidel matrix of chain graphs, referred to as eigenvalues and Seidel eigenvalues of these graphs, respectively. Firstly, we utilize the characteristic polynomial of the adjacency matrix of a chain graph to construct infinite pairs of non-isomorphic cospectral chain graphs. Next, we determine the inertia of the Seidel matrix of a chain graph and establish an interval that does not contain the Seidel eigenvalues of chain graphs. Lastly, we characterize chain graphs with distinct Seidel eigenvalues. [ABSTRACT FROM AUTHOR]

In this paper we provide a tetrahedra-free algorithm to compute low-cardinality quadrature rules with a given degree of polynomial exactness, positive weights and interior nodes on a polyhedral element with arbitrary shape. The key tools are the notion of Tchakaloff discretization set and the solution of moment-matching equations by Lawson-Hanson iterations for NonNegative Least-Squares. Several numerical tests are presented. The method is implemented in Matlab as open-source software. [ABSTRACT FROM AUTHOR]

In this paper we consider Gaussian type quadrature rules for trigonometric polynomials where an even number of nodes is fixed in advance. For an integrable and nonnegative weight function w on the interval E = [ a , a + 2 π) , a ∈ R , these quadrature rules have the following form ∫ E t (x) w (x) d x = ∑ i = 1 2 k a i t (y i) + ∑ i = 1 2 (n + γ) A i t (x i) , t ∈ T 2 (n + γ) + k − 1 , where the nodes y i ∈ E , i = 1 , 2 , ... , 2 k , are fixed and prescribed in advance, γ ∈ { 0 , 1 / 2 } and T n = { cos ⁡ k x , sin ⁡ k x | k = 0 , 1 , ... , n } , n ∈ N. Also, for γ = 1 / 2 , i.e., for the case of quadrature rules for trigonometric polynomials with odd number of nodes, we consider the optimal sets of quadrature rules in the sense of Borges (see [1,13]) for trigonometric polynomials with even number of fixed nodes. Let n = (n 1 , n 2 , ... , n r) , r ∈ N , be a multi-index and let W = (w 1 , w 2 , ... , w r) be a system of weight functions on the interval E = [ a , a + 2 π) , a ∈ R. The optimal set of quadrature rules with respect to (W , n) , with even number of fixed nodes, have the form ∫ E f (x) w m (x) d x ≈ ∑ i = 1 2 k a m , i f (y i) + ∑ i = 1 2 | n | + 1 A m , i f (x i) , m = 1 , 2 , ... r , where | n | = n 1 + n 2 + ⋯ + n r and the nodes y i ∈ E , i = 1 , 2 , ... , 2 k , are fixed and prescribed in advance. For r = 1 the optimal set of quadrature rules reduces to Gaussian quadrature rule for trigonometric polynomials with odd number of nodes. For all mentioned quadrature rules, in addition to the theoretical results, we will present the method for construction and give appropriate numerical examples. [ABSTRACT FROM AUTHOR]

Let F be a field of characteristic zero. By a ⁎-superalgebra we mean an algebra A with graded involution over F. Recently, algebras with graded involution have been extensively studied in PI-theory and the sequence of ⁎-graded codimensions { c n gri (A) } n ≥ 1 has been investigated by several authors. In this paper, we classify varieties generated by unitary ⁎-superalgebras having quadratic growth of ⁎-graded codimensions. As a result we obtain that a unitary ⁎-superalgebra with quadratic growth is T 2 ⁎ -equivalent to a finite direct sum of minimal unitary ⁎-superalgebras with at most quadratic growth, where at least one ⁎-superalgebra of this sum has quadratic growth. Furthermore, we provide a method to determine explicitly the factors of those direct sums. [ABSTRACT FROM AUTHOR]

This paper introduces general methodologies for constructing closed-form solutions to linear constant-coefficient partial differential equations (PDEs) with polynomial right-hand sides in two and three spatial dimensions. Polynomial solutions have recently regained significance in the development of numerical techniques for evaluating volume integral operators and also have potential applications in certain kinds of Trefftz finite element methods. The equations covered in this work include the isotropic and anisotropic Poisson, Helmholtz, Stokes, linearized Navier-Stokes, stationary advection-diffusion, elastostatic equations, as well as the time-harmonic elastodynamic and Maxwell equations. Several solutions to complex PDE systems are obtained by a potential representation and rely on the Helmholtz or Poisson solvers. Some of the cases addressed, namely Stokes flow, Maxwell's equations and linearized Navier-Stokes equations, naturally incorporate divergence constraints on the solution. This article provides a generic pattern whereby solutions are constructed by leveraging solutions of the lowest-order part of the partial differential operator (PDO). With the exception of anisotropic material tensors, no matrix inversion or linear system solution is required to compute the solutions. This work is accompanied by a freely-available Julia library, ElementaryPDESolutions.jl, which implements the proposed methodology in an efficient and user-friendly format. [ABSTRACT FROM AUTHOR]

In Beirão da Veiga et al. 2019, the authors proposed a virtual element method for domains with fixed curved boundary that tends to be locally straight when the discretisation gets finer and finer. In this paper, we properly modify this method so that it can also deal with curved mesh edges which are not necessarily located along boundaries and do not tend to be straight when refining the mesh. To achieve this goal, we assume that curved edges are described by polynomials and we increase the dimension of the virtual element space used in Beirão da Veiga et al. 2019. In the numerical experiments, we compare these two methods. Furthermore, we apply the proposed approach to the benchmark "TEAM 25" problem, an optimal shape design problem in magnetostatics characterised by curved edges. [ABSTRACT FROM AUTHOR]

The concepts of differentiation and integration for matrices are known. As far as each matrix is differentiable, it is not clear a priori whether a given matrix is integrable or not. Recently some progress was obtained for diagonalizable matrices, however general problem remained open. In this paper, we present a full solution of the integrability problem. Namely, we provide necessary and sufficient conditions for a given matrix to be integrable in terms of its characteristic polynomial. Furthermore, we find necessary and sufficient conditions for the existence of integrable and non-integrable matrices with given geometric multiplicities of eigenvalues. Our approach relies on properties of some special classes of polynomials, namely, Shabat polynomials and conservative polynomials, arising in number theory and dynamics. [ABSTRACT FROM AUTHOR]

Interior and exterior polynomials, introduced by Kálmán in Kálmán, (2013), generalized the Tutte polynomial T (x , y) on plane points (1 / x , 1) and (1 , 1 / y) from graphs to hypergraphs. Although the two polynomials were defined under a fixed ordering of hyperedges, they were proved to be independent of the orderings of hyperedges by using techniques of polytopes. Later, the two polynomials were unified to be the Tutte polynomial of polymatroids. The main purpose of this paper is to provide an alternative to Kálmán's proof without using polytopes. Similar to the Tutte's original proof for the Tutte polynomial, our proof is direct and elementary. [ABSTRACT FROM AUTHOR]

Let H be a field with Q ⊆ H ⊆ C , and let p (λ) be a polynomial in H [ λ ] , and let A ∈ H n × n be nonderogatory. In this paper we consider the problem of finding a solution X ∈ H n × n to p (X) = A. A necessary condition for this to be possible is already known from a paper by M.P. Drazin: Exact rational solutions of the matrix equation A = p (X) by linearization. Under an additional condition we provide an explicit construction of such solutions. The similarities and differences with the derogatory case will be discussed as well. One of the tools needed in the paper is a new canonical form, which may be of independent interest. It combines elements of the rational canonical form with elements of the Jordan canonical form. [ABSTRACT FROM AUTHOR]

When analyzing the stability of time-varying delay systems in view of the Lyapunov–Krasovskii functional, a quadratic polynomial function with regard to time-varying delay is always generated. And it is particularly crucial to determine the negativeness of the matrix of such a quadratic form function for obtaining an analysis result expressed in linear matrix inequalities. This paper proposes a method of tangent intersection in the delay interval segmentation, producing the generalized quadratic convex conditions by further utilizing the cross point between every two tangent lines. It reduces the conservatism of the existing conditions remarkably without requiring unexplainable adjustable parameters and additional decision variables. Benefiting from the newly proposed quadratic convex conditions, the novel stability conditions are derived, the superiority of which is demonstrated through several widely used numerical instances and single area power system PI control example. • This paper focuses on stability analysis of continuous time-varying delay systems. • Our interest is to determine the negative definiteness of the quadratic polynomial function. • The main idea is to use the cross points of tangent lines instead of the endpoints in delay subintervals. • Less conservative stability criteria are derived without unexplainable parameters and additional variables. [ABSTRACT FROM AUTHOR]

This paper is the final part in a two-part series investigating toolpath generation for freeform optics manufacturing. Building on the surface generation techniques already presented, this paper introduces a 3D blending surface that has the form of a continuous polynomial curve and is used to smoothly fill in the space between two non-intersecting, polar boundaries and two surfaces of the form Z (X , Y). The proposed blending surface renders it possible to smoothly join two surfaces defined over different domains together, as well as to smooth over the discontinuity created where two surfaces intersect. It is shown that increasing the order of the blending polynomial increases the continuity of the overall piecewise, blended surface. As a corollary, it is possible to increase the overall continuity of a toolpath along its cutting direction. To illustrate the robustness of the proposed surface blending technique, experimental data is presented for the toolpath generation, diamond machining, and measurement of a freeform microlens array. Designed with no exploitable symmetries or simplifications and very difficult to model with traditional CAD software, this surface is comprised of an array of freeform lenslets superimposed and blended onto a freeform base surface. All lenslets are individually tilted about the X and Y -axes such that they are normal to the base surface at their respective locations, and all blends have the same form as the intersection curve between each respective lenslet and base surface. Machined in brass to optical quality, this C 3 continuous surface represents a general case of the proposed surface generation and blending techniques. • An explicit, continuous, polynomial blending function is introduced. • Two freeform surfaces are blended via the area between two non-circular boundaries. • Increasing blending polynomial order increases overall surface continuity. [ABSTRACT FROM AUTHOR]

In this work, published experimental result data of the pulping of tagasaste (Chamaecytisus proliferus L.F.) with soda and anthraquinone (AQ) have been used to develop a model using a neural network. The paper presents the development of a model with a neural network to predict the effects that the operational variables of the pulping reactor (temperature, soda concentration, AQ concentration, time and liquid/solid ratio) have on the properties of the paper sheets of the obtained pulp (brightness, traction index, burst index and tear index). Using a factorial experimental design, the results obtained with the neural network model are compared with those obtained from a polynomial model. The neural network model shows a higher prediction precision that the polynomial model. [Copyright &y& Elsevier]

The independent set polynomial is important in many areas of combinatorics, computer science, and statistical physics. For every integer Δ ≥ 2 , the Shearer threshold is the value λ ⁎ (Δ) = (Δ − 1) Δ − 1 / Δ Δ. It is known that for λ < − λ ⁎ (Δ) , there are graphs G with maximum degree Δ whose independent set polynomial, evaluated at λ , is at most 0. Also, there are no such graphs for any λ > − λ ⁎ (Δ). This paper is motivated by the computational problem of approximating the independent set polynomial when λ < − λ ⁎ (Δ). The key issue in complexity bounds for this problem is "implementation". Informally, an implementation of a real number λ ′ is a graph whose hard-core partition function, evaluated at λ , simulates a vertex-weight of λ ′ in the sense that λ ′ is the ratio between the contribution to the partition function from independent sets containing a certain vertex and the contribution from independent sets that do not contain that vertex. Implementations are the cornerstone of intractability results for the problem of approximately evaluating the independent set polynomial. Our main result is that, for any λ < − λ ⁎ (Δ) , it is possible to implement a set of values that is dense over the reals. The result is tight in the sense that it is not possible to implement a set of values that is dense over the reals for any λ > λ ⁎ (Δ). Our result has already been used in a paper with Bezáková (STOC 2018) to show that it is #P-hard to approximate the evaluation of the independent set polynomial on graphs of degree at most Δ at any value λ < − λ ⁎ (Δ). In the appendix, we give an additional incomparable inapproximability result (strengthening the inapproximability bound to an exponential factor, but weakening the hardness to NP-hardness). [ABSTRACT FROM AUTHOR]

In 2013 Benkart, Lopes and Ondrus introduced and studied in a series of papers the infinite-dimensional unital associative algebra A h generated by elements x , y , which satisfy the relation y x − x y = h for some 0 ≠ h ∈ F [ x ]. In this paper we investigate the standard polynomial identities and minimal identities for certain subspaces of A h over an infinite field of arbitrary characteristic. [ABSTRACT FROM AUTHOR]

Consider a polynomial f (z) = a n z n +... + a 1 z + a 0 of positive coefficients that is stable (in the Hurwitz sense), i.e., every root of f lies in the open left half-plane of C. Due to Garloff and Wagner [J. Math. Anal. Appl. 202 (1996)], the p th Hadamard power of f : f [ p ] (z) : = a n p z n +... + a 1 p z + a 0 p is stable if p is a positive integer number. However, it turns out that f [ p ] does not need to be stable for all real p > 1. A counterexample is known for n = 8 and p = 1.139. On the other hand, f [ p ] is stable for n = 1 , 2 , 3 , 4 , and every p > 1. In this paper we fill the gap by showing that f [ p ] is stable for n = 5 and constructing counterexamples for n ≥ 6. Moreover, by means of Rouché's Theorem, we give some stability conditions for polynomials and two examples that complete and illustrate the results. [ABSTRACT FROM AUTHOR]

In this paper, we establish some relations between irreducible polynomials over a finite field F q and its quadratic extension F q 2 . First we consider a relation between the numbers of irreducible polynomials of a fixed degree over F q and F q 2 , and some relations between self-reciprocal irreducible polynomials over F q and self-conjugate-reciprocal irreducible polynomials over F q 2 . We also obtain formulas for the number and the product of all self-conjugate-reciprocal irreducible monic (SCRIM) polynomials over F q 2 . [ABSTRACT FROM AUTHOR]

In this paper, we design and analyze a parameter-free mixed method of arbitrary polynomial orders for the Stokes equations and linear elasticity problem, where the symmetry of stress is strongly imposed. Equal-order polynomials are exploited for the stress and velocity spaces in which the stress is approximated by discontinuous polynomials. In contrast, the velocity is approximated by an H (div ; Ω) conforming space. As a consequence, the proposed scheme yields divergence-free velocity approximations and is pressure-robust for the Stokes equations. The stable Stokes pair are integrated to facilitate the proof of the inf-sup condition. The comprehensive convergence error estimates for all the variables are explored for the Stokes equations and linear elasticity problem. In particular, we explicitly track the dependence of the error estimates on the physical parameters and illustrate the locking-free property of the proposed scheme, which is non-trivial under the proposed formulation. The judicious balancing of the mixed formulation and the equivalent primal formulation enable us to achieve the robust error estimates for the linear elasticity problem. Several numerical experiments for the Stokes equations and linear elasticity problem are carried out to verify the proposed theories. [ABSTRACT FROM AUTHOR]

A matching in a graph G is a collection of edges of G such that no two of them share a vertex. The number of all matchings in G is called its Hosoya index. In this paper, we compute Hosoya indices of several classes of unbranched polymers made of cycles of the same lengths arranged around a middle path and decorated by attaching to each vertex, a given number of pendent vertices or thorns. We establish linear recurrences satisfied by those numbers and obtain explicit formulas in terms of Fibonacci polynomials and their generalizations. Some possible directions of future research are also indicated. [ABSTRACT FROM AUTHOR]

This paper presents a construction of local C r interpolations on simplicial grids in R d by using piecewise polynomials of degree k ≥ 2 d r + 1. To this end, an interpolative decomposition of the set of corresponding multi-indices is introduced. It is shown that the piecewise polynomial interpolation is of C r continuity provided that the corresponding function is of C 2 d − 1 r continuity. [ABSTRACT FROM AUTHOR]

Let F be any field of characteristic different from two and let E be the Grassmann algebra of an infinite dimensional F -vector space L. In this paper we will provide a condition for a Z 2 -grading on E to behave like the natural Z 2 -grading E c a n. More specifically, our aim is to prove the validity of a weak version of a conjecture presented in [10]. The conjecture poses that every Z 2 -grading on E has at least one non-zero homogeneous element of L. As a consequence, we obtain a characterization of E c a n by means of its Z 2 -graded polynomial identities. Furthermore we construct a Z 2 -grading on E that gives a negative answer to the conjecture. [ABSTRACT FROM AUTHOR]

This paper proves that the characteristic polynomial is a complete unitary invariant for pairs of projection matrices. Some special cases involving three or more projections are also considered. [ABSTRACT FROM AUTHOR]

In this paper we study identities and images of polynomials on null-filiform Leibniz algebras. If L n is an n -dimensional null-filiform Leibniz algebra, we exhibit a finite minimal basis for Id (L n) , the polynomial identities of L n , and we explicitly compute the images of multihomogeneous polynomials on L n. We present necessary and sufficient conditions for the image of a multihomogeneous polynomial f to be a subspace of L n. For the particular case of multilinear polynomials, we prove that the image is always a vector space, showing that the analogue of the L'vov-Kaplansky conjecture holds for L n. We also prove similar results for an analog of null-filiform Leibniz algebras in the infinite-dimensional case. [ABSTRACT FROM AUTHOR]

This paper is concerned with approximate I -optimal designs for full polynomial models over the unit ball. The designs have support on an appropriate set of spheres concentric or coincident with the boundary of the unit ball and place weights on the uniform distributions over those spheres. The result is stated in a single sentence in the paper by Galil and Kiefer (1977) but has not been revisited since. The proof indicated by these authors is formalized and the requisite designs are constructed using an approach which emanates from Euclidean design theory. Comparisons of the approximate I -optimal designs with their D -optimal counterparts are made and a Pareto approach to obtaining a design which is a compromise between D - and I -efficiency is introduced. Examples which reinforce the findings are presented throughout. • Formulation of approximate I -optimal designs for full polynomial models over the unit ball. • Construction of the approximate I -optimal designs based on methods gleaned from Euclidean design theory. • A comparison of the approximate I -optimal designs with their D -optimal counterparts. [ABSTRACT FROM AUTHOR]

In this paper, the practical stabilization of highly nonlinear Takagi–Sugeno fuzzy complex networks with Markovian jump (HT-SFNM) is investigated via aperiodically intermittent discrete-time observation control (AID-TOC). Compared to existing articles on highly nonlinear complex networks, this paper considers both Takagi–Sugeno fuzzy rules and Markovian jump for the first time, and uses AID-TOC to solve its stability problem. Furthermore, more general polynomial growth conditions are adopted instead of the conventional linear growth conditions, which makes the model more universal. Finally, two numerical examples are presented to illustrate the effectiveness of the theoretical results. [ABSTRACT FROM AUTHOR]

This paper describes an implementation and tests of a blending scheme for regularly sampled data interpolation, and in particular studies the order of approximation for the method. This particular implementation is a special case of an earlier scheme by Fang for fitting a parametric surface to interpolate the vertices of a closed polyhedron with n -sided faces, where a surface patch is constructed for each face of the polyhedron, and neighbouring faces can meet with a user specified order of continuity. The specialization described in this paper considers functions of the form z = f (x , y) with the patches meeting with C 2 continuity. This restriction allows for investigation of order of approximation, and it is shown that the functional version of Fang's scheme has polynomial precision. • Analyzed and tested a functional form of a parametric data interpolation scheme. • Proved that this functional scheme has the polynomial precision of subsurfaces constructed for initial step of the scheme. • Gave several surface constructions for the initial step withe polynomial precision of different degrees. [ABSTRACT FROM AUTHOR]

The roots of the closure polynomial associated with a given mechanism determine its assembly modes. In the case of 6R closed-loop mechanisms, these polynomials are usually expressed in the half-angle tangent of one of its joints. In this paper, we derive closure polynomials of 6R robots in terms of distances, not angles. The use of a distance-based formulation provides, in general, a fundamental advantage since it leads to closure conditions without requiring neither variable eliminations nor variable substitutions. We restrict our attention, though, to robots with coplanar consecutive joint axes, i.e. , robots whose consecutive axes intersect at either proper or improper points. We show that this particular arrangement of joints does not result in a reduction in the maximum number of the inverse kinematic solutions with respect to the general case. Moreover, this family of robots include broadly used offset-wrist arms. For instance, in this paper, we obtain closure polynomials for robots such as the FANUC CRX-10iA/L, the UR10e, and the KUKA LBR iiwa R800 robot in generic form (i.e. , as a function of their end-effector locations). • The inverse kinematics of lobster arms is solved using distance geometry. • Closure polynomials are obtained in terms of distances. No angles are involved. • Neither variable eliminations nor variable substitutions are required. • The method is exemplified on a wide variety of cobots. [ABSTRACT FROM AUTHOR]

Let G be a group and F an infinite field. Assume that A is a finite dimensional F -algebra with an elementary G -grading. In this paper, we study the graded identities satisfied by the tensor product grading on the F -algebra A ⊗ C , where C is an H -graded colour β -commutative algebra. More precisely, under a technical condition, we provide a basis for the T G -ideal of graded polynomial identities of A ⊗ C , up to graded monomial identities. Furthermore, the F -algebra of upper block-triangular matrices U T (d 1 , ... , d n) , as well as the matrix algebra M n (F) , with an elementary grading such that the neutral component corresponds to its diagonal, are studied. As a consequence of our results, a basis for the graded identities, up to graded monomial identities of degrees ≤ 2 d − 1 , for M d (E) and M q (F) ⊗ U T (d 1 , ... , d n) , with a tensor product grading, is exhibited. In this latter case, d = d 1 + ... + d n. Here E denotes the infinite dimensional Grassmann algebra with its natural Z 2 -grading, and the grading on M q (F) is Pauli grading. The results presented in this paper generalize results from [14] and from other papers which were obtained for fields of characteristic zero. [ABSTRACT FROM AUTHOR]

We design a virtual element method for the numerical treatment of the two-dimensional parabolic variational inequality problem on unstructured polygonal meshes. Due to the expected low regularity of the exact solution, the virtual element method is based on the lowest-order virtual element space that contains the subspace of the linear polynomials defined on each element. The connection between the nonnegativity of the virtual element functions and the nonnegativity of the degrees of freedom, i.e., the values at the mesh vertices, is established by applying the Maximum and Minimum Principle Theorem. The mass matrix is computed through an approximate L 2 polynomial projection, whose properties are carefully investigated in the paper. We prove the well-posedness of the resulting scheme in two different ways that reveal the contractive nature of the VEM and its connection with the minimization of quadratic functionals. The convergence analysis requires the existence of a nonnegative quasi-interpolation operator, whose construction is also discussed in the paper. The variational crime introduced by the virtual element setting produces five error terms that we control by estimating a suitable upper bound. Numerical experiments confirm the theoretical convergence rate for the refinement in space and time on three different mesh families including distorted squares, nonconvex elements, and Voronoi tesselations. [ABSTRACT FROM AUTHOR]

Given a possibly singular matrix polynomial P (z) , we study how the eigenvalues, eigenvectors, root polynomials, minimal indices, and minimal bases of the pencils in the vector space DL (P) introduced in Mackey, Mackey, Mehl, and Mehrmann [SIAM J. Matrix Anal. Appl. 28(4), 971-1004, 2006] are related to those of P (z). If P (z) is regular, it is known that those pencils in DL (P) satisfying the generic assumptions in the so-called eigenvalue exclusion theorem are strong linearizations for P (z). This property and the block-symmetric structure of the pencils in DL (P) have made these linearizations among the most influential for the theoretical and numerical treatment of structured regular matrix polynomials. However, it is also known that, if P (z) is singular, then none of the pencils in DL (P) is a linearization for P (z). In this paper, we prove that despite this fact a generalization of the eigenvalue exclusion theorem holds for any singular matrix polynomial P (z) and that such a generalization allows us to recover all the relevant quantities of P (z) from any pencil in DL (P) satisfying the eigenvalue exclusion hypothesis. Our proof of this general theorem relies heavily on the representation of the pencils in DL (P) via Bézoutians by Nakatsukasa, Noferini and Townsend [SIAM J. Matrix Anal. Appl. 38(1), 181-209, 2015]. [ABSTRACT FROM AUTHOR]

In this paper, we propose an effective numerical method to solve the one- and two-dimensional time-fractional convection-diffusion equations based on the Caputo derivative. The presented approach employs a hybrid method that combines Lucas and Fibonacci polynomials with the Caputo derivative definition. The main objective is to transform the problem into a time-discrete form utilizing the Caputo derivative technique and then approximate the function's derivative using Fibonacci polynomials. To evaluate the efficiency and accuracy of the proposed technique, we apply it to one- and two-dimensional problems and compare the results with the exact as well as with existing methods in recent literature. The comparison demonstrates that the proposed approach is highly efficient, accurate and ease to implement. [ABSTRACT FROM AUTHOR]

A graph G is divisible by a graph H if the characteristic polynomial of G is divisible by that of H. In this paper, a necessary and sufficient condition for recursive graphs to be divisible by a path is used to show that the H-shape graph P 2 , 2 ; n − 4 2 , n − 7 , known to be (for n large enough) the minimizer of the spectral radius among the graphs of order n and diameter n − 5 , is determined by its adjacency spectrum if and only if n ≠ 10 , 13 , 15. [ABSTRACT FROM AUTHOR]

Let A be a finite dimensional associative algebra with derivations over a field of characteristic zero, i.e., an algebra whose structure is enriched by the action of a Lie algebra L by derivations, and let c n L (A) , n ≥ 1 , be its differential codimension sequence. Such sequence is exponentially bounded and exp L ⁡ (A) = lim n → ∞ ⁡ c n L (A) n is an integer that can be computed, called differential PI-exponent of A. In this paper we prove that for any Lie algebra L , exp L ⁡ (A) coincides with exp ⁡ (A) , the ordinary PI-exponent of A. Furthermore, in case L is a solvable Lie algebra, we apply such result to classify varieties of L -algebras of almost polynomial growth, i.e., varieties of exponential growth such that any proper subvariety has polynomial growth. [ABSTRACT FROM AUTHOR]

Lei, Yeh and Zhang put forward the anti-forcing number a f (G , M) for a perfect matching M in a graph G , which is the minimum number of edges of G not in M whose deletion results in a subgraph with a unique perfect matching M. The anti-forcing numbers of all perfect matchings form the anti-forcing spectrum of G. The anti-forcing polynomial A f (G , x) of G is a counting polynomial for classifying perfect matchings possessing the same anti-forcing number in G. In this paper, we deduce recurrence formula of the anti-forcing polynomial and continuity of the anti-forcing spectrum for catacondensed hexagonal systems. [ABSTRACT FROM AUTHOR]

In this paper we consider images of (ordinary) noncommutative polynomials on matrix algebras endowed with a graded structure. We give necessary and sufficient conditions to verify that some multilinear polynomial is a central polynomial, or a trace zero polynomial, and we use this approach to present an equivalent statement to the Lvov-Kaplansky conjecture. [ABSTRACT FROM AUTHOR]

In this paper, we present a general method to construct q -analogues and other generalizations of Pascal-like matrices. Our matrices are obtained as functions of strictly lower triangular matrices and include several types of generalized Pascal-like matrices and matrices related with modified Hermite polynomials of two variables and other polynomial sequences. We find explicit expressions for products, powers, and inverses of the matrices and also some factorization formulas using this method. [ABSTRACT FROM AUTHOR]