A functional ansatz is developed which gives certain elliptic solutions of the Witten-Dijkgraaf-Verlinde-Verlinde (or WDVV) equation. This is based on the elliptic trilogarithm function introduced by Beilinson and Levin. For this to be a solution results in a number of purely algebraic conditions on the set of vectors that appear in the ansatz, this providing an elliptic version of the idea, in...

All-pass filter design can be generally achieved by solving a system of linear equations.The associatedmatrices involved in the set of linear equations can be further formulated as a Toeplitz-plus-Hankel form such that a matrix inversion is avoided. Consequently, the optimal filter coefficients can be solved by using computationally efficient Levinson algorithms or Cholesky decomposition techni...

By using the trigonometric uniform splines of order 3 with a real tension factor, a numericalmethod is developed for solving a linear second order boundary value problems (2VBP) withDirichlet, Neumann and Cauchy types boundary conditions. The moment at the knots isapproximated by central finite-difference method. The order of convergence of the methodand the theory is illustrated by solving tes...

For a proper cone K ⊂ R and its dual cone K∗ the complementary slackness condition x s = 0 defines an n-dimensional manifold C(K) in the space { (x, s) | x ∈ K, s ∈ K∗ }. When K is a symmetric cone, this fact translates to a set of n bilinear optimality conditions satisfied by every (x, s) ∈ C(K). This proves to be very useful when optimizing over such cones, therefore it is natural to look for...

We study approximative identities In(f)(x) = 1 2 Z ? n(u)f(x ? u) du with positive even trigonometric polynomials n of degree n as kernels. It is well known that in general the approximants In(f) are norm convergent to f with order at most n ?2 and that convergence for the test function f(x) = (2 sin(x=2)) 2 at x = 0 implies norm convergence for a wide class of functions f. We discuss the order...

In this work, we present a simple framework to synthesize human motion. Our main goal is propose methodology tailored for inexperienced users initiate their research in motion simulation and trajectory optimization. The novelties of the work include following. First, trigonometric splines are used instead traditional B-splines discretize generalized coordinates velocities. Second, useful identi...

We consider trigonometric solutions of WDVV equations and derive geometric conditions when a collection of vectors with multiplicities determines such a solution. We incorporate these conditions into the notion of trigonometric Veselov system (∨-system) and we determine all trigonometric ∨-systems with up to five vectors. We show that generalized Calogero-Moser-Sutherland operator admits a fact...

We study cubature formulas for d-dimensional integrals with a high trigonometric degree. To obtain a trigonometric degreè in dimension d, we need about d ` =`! function values if d is large. Only a small number of arithmetical operations is needed to construct the cubature formulas using Smolyak's technique. We also compare diierent methods to obtain formulas with high trigonometric degree. Abs...

We establish a bilinear framework for elliptic soliton solutions which are composed by the Lam\'e-type plane wave factors. $\tau$ functions in Hirota's form derived and vertex operators that generate such presented. Bilinear identities constructed an algorithm to calculate residues equations is formulated. These investigated detail KdV equation sketched KP hierarchy. Degenerations periods of in...

Analogous to the quadratic B-spline curve, a piecewise quadratic trigonometric polynomial curve is presented in this paper. The quadratic trigonometric polynomial curve has C2 continuity, while the quadratic B-spline curve has C1 continuity. The quadratic trigonometric polynomial curve is closer to the given control polygon than the quadratic B-spline curve.