Degree 2k Research Articles

Rational period functions and indefinite binary quadratic forms, II

In I-4] Eichler introduced a generalized abelian integral, now called an Eichler integral, obtained as a (2k l)-fold integral of an entire modular form of positive weight 2k. Under the action of the modular group, the transformation equation of the Eichler integral contains a polynomial of degree 2k 2 which is called the period polynomial, by analogy with the abelian integral. The coefficients of the period polynomial are, essentially, the periods of the corresponding modular form. This is the basis of the beautiful results of Eichler and Shimura [12] on Eichler cohomology. In [7] Knopp further generalized Eichler integrals by introducing modular integrals with period functions that are now rational functions as opposed to being polynomials. In a sequence of papers I-7, 8] Knopp developed the theory of modular integrals. Parson and Rosen 1-11] provided additional examples; but it was Choie I-3] who finally settled the existence question by showing how to construct rational period functions for any integral weight. The object of this paper is to begin the classification of rational period functions of positive weight. This is accomplished by looking at possible irreducible systems of poles, as defined in Sect. 4. If the irreducible system of poles exhibits algebraic symmetry, that is, if the poles occur in conjugate pairs, then the rational period function of weight 2k, k odd, whose pole set is the irreducible system of poles is of the form

Limit Cycles Close to Infinity of Certain Non-Linear Differential Equations

AbstractThrough successive radial perturbations of a certain planar Hamiltonian polynomial vector field of degree 2K + 1, we obtain a least K limit cycles containing (2K + 1)2 singularities.

Cycles containing matchings and pairwise compatible euler tours

AbstractLet M be a matching in a graph G such that d(x) + d(y) ≥ |G| for all pairs of independent vertices x and y of G that are incident with M. We determine a necessary and sufficient condition for M to be contained in a cycle of G. This extends results of Häggkvist and Berman, and implies that if M is a 1‐factor of G and |G|  0 (mod 4), then M is contained in a Hamilton cycle of G. We use our results to deduce that an eulerian graph of minimum degree 2k contains k pairwise compatible Euler tours.

Minimal cubature formulae of degree 2k−1 for two classical functionals

Minimal cubature formulae of degree 2k−1 for two classical functionals

Controllability of linear time-invariant systems

In this paper it is shown that for any controllable linear time-invariant system, an arbitrary initial point can be driven to an arbitrary final point by means of an input, polynomial in t of degree 2k— I, with k the controllability index of the system. Also, the state trajectory is polynomial in time and of the same degree. The bound on the degree is an improvement with respect to similar results in the literature. The proof of this result is rather direct compared to the involved proof in the literature of a weaker theorem. An extension of this result considering sinusoidal inputs and state trajectories is provided.

Optimal addition of knots to cubature formulae for planar regions

A method is described to add knots to a cubature formula of degree 2k−1 for an integral over a symmetric region, to obtain a cubature formula of degree 2k+1. This method is used to construct cubature formulae for the square, the circle, the hexagon and the entire plane.

On high-order error criteria for system identification

A new family of algorithms has been considered for the purpose of identifying FIR (finite impulse response) systems by means of minimizing the sum of the output error to the degree 2K, K = 1, 2, . . . . A heuristic argument is presented to the effect that utilization of high-order error criteria (K > 1) will yield, in certain cases, considerable improvement in the performance (lower estimation error can be expected for the same amount of data). A simple approximate expression is derived which allows the choice of the most suitable value of K for the given application.

Construction of fully symmetric numerical integration formulas of fully symmetric numerical integration formulas

The paper develops a construction for finding fully symmetric integration formulas of arbitrary degree 2k+1 inn-space such that the number of evaluation points isO((2n)k)/k!),n → ∞. Formulas of degrees 3, 5, 7, 9, are relatively simple and are presented in detail. The method has been tested by obtaining some special formulas of degrees 7, 9 and 11 but these are not presented here.