Some new linear relations for even degree polynomial splines on a uniform mesh

This paper is concerned with even degree polynomial splines, degreeN, defined on a uniform mesh. There are well-known linear relations among the spline values atN successive knots and thepth derivatives of the spline at the sameN knots. Some new linear relations are developed which connectN–1 spline values withN–1 derivative values. The leading terms of the truncation errors of these new linear relations are found in terms of Bernoulli numbers.     

On the number of critical periods for planar polynomial systems of arbitrary degree

We construct a class of planar systems of arbitrary degree n having a reversible center at the origin and such that the number of critical periods on its period annulus grows quadratically with n. As far as we know, the previous results on this subject gave systems having linear growth.     

Polynomial first integrals for weight-homogeneous planar polynomial differential systems of weight degree 3

Our main result is the classification of all weight-homogeneous planar polynomial differential systems of weight degree 3 having a polynomial first integral.     

Non-isochronicity of the center at the origin in polynomial Hamiltonian systems with even degree nonlinearities

In 2002 X. Jarque and J. Villadelprat proved that no center in a planar polynomial Hamiltonian system of degree 4 is isochronous and raised a question: Is there a planar polynomial Hamiltonian system of even degree which has an isochronous center? In this paper we give a criterion for non-isochronicity of the center at the origin of planar polynomial Hamiltonian systems. Moreover, the orders of weak centers are determined. Our results answer a weak version of the question, proving that there is no planar polynomial Hamiltonian system with only even degree nonlinearities having an isochronous center at the origin.     

Centers of quasi-homogeneous polynomial planar systems

In this paper we determine the centers of quasi-homogeneous polynomial planar vector fields of degree 0, 1, 2, 3 and 4. In addition, in every case we make a study of the reversibility and the analytical integrability of each one of the above centers. We find polynomial centers which are neither orbitally reversible nor analytically integrable, this is a new scenario in respect to the one of non-degenerate and nilpotent centers.     

A topological spline system approach to even degree polynomial splines

The existence of minimum norm properties for even degree polynomial splines, analogous to the. ones known for odd degree splines, is investigated within the framework of the theory of

topological spline systems. It is shown that such properties cannot exist for even degree splines interpolating functions halfway between the partition points. For another class of even

degree spline functions, however, which hterpolate the local integrals of given functions with respect to the partitions, the seeked minimum norm properties can be proved. This is carried out

by first investigating a generalized problem within the theory of spline systems and then deriving corresponding conclusions. As a corollary the existence of spline systems with respect to differential operators of fractional degree is obtained.     

The degree of the E-characteristic polynomial of an even order tensor

The E-characteristic polynomial of an even order supersymmetric tensor is a useful tool in determining the positive definiteness of an even degree multivariate form. In this paper, for an even order tensor, we first establish the formula of its E-characteristic polynomial by using the classical Macaulay formula of resultants, then give an upper bound for the degree of that E-characteristic polynomial. Examples illustrate that this bound is attainable in some low order and dimensional cases.     

Reduction of integrable planar polynomial differential systems

The integrability problem consists in finding the class of functions, a first integral of a given planar polynomial differential system must belong to. We recall the characterization of systems which admit a Darboux, elementary, Liouvillian or Weierstrass first integral. The reduction problem of an integrable planar system consists in finding the class of functions, a map that reduces the original system (transforms into a simple system or equation) must belong to. We identify the class of functions of this map for polynomial, rational, Darboux, elementary, Liouvillian and Weierstrass integrable systems.     

Isochronicity conditions for some planar polynomial systems

We study the isochronicity of centers at O∈R² for systems , , where A,B∈R[x,y], which can be reduced to the Liénard type equation. Using the so-called C-algorithm we have found 27 new multiparameter isochronous centers.     

Isochronicity conditions for some planar polynomial systems II

We study the isochronicity of centers at O∈R² for systems where A,B∈R[x,y], which can be reduced to the Liénard type equation. When deg(A)?4 and deg(B)?4, using the so-called C-algorithm we found 36 new multiparameter families of isochronous centers. For a large class of isochronous centers we provide an explicit general formula for linearization. This paper is a direct continuation of a previous one with the same title [Islam Boussaada, A. Raouf Chouikha, Jean-Marie Strelcyn, Isochronicity conditions for some planar polynomial systems, Bull. Sci. Math. 135 (1) (2011) 89–112], but it can be read independently.     

Maximum size of a planar graph with given degree and even diameter

We offer the exact solution of the degree-diameter problem for planar graphs in the case of even diameter 2d and large degree Δ≥6(12d+1). New graph examples are constructed improving the lower bounds for Δ≥5.     

图的特征多项式的若干性质

图的特征多项式有许多性质,本文给出了特征多项式的指数表达式,特征多项式的导数的几个不同表达式以及高阶导数的图论意义。     

Research on the properties of some planar polynomial differential equations

In this article, we use the new method of reflecting function to study the behavior of solutions of nonlinear time-vary differential equations, and give the sufficient conditions for these equations which have the reflecting function in the form of linear and fractional. We applied the obtained results to discuss the qualitative behavior of solutions of the higher degree polynomial differential systems and derive the sufficient conditions for a critical point to be a center.     

Some properties of polynomial subgroup growth groups

A PSG group is one in which the number of subgroups of given index is bounded by a fixed power of this index. The finitely generated PSG groups are known. Here we prove some properties of such groups which need not be finitely generated. We derive, e.g., restrictions on the chief factors (Theorem 1) and on the number of generators of subgroups (Theorem 5). To Wolf Prize laureate John Thompson Partially supported by a BSF grant.     

The recovery of even polynomial potentials

We consider the problem of reconstructing an even polynomial potential from one set of spectral data of a Sturm-Liouville problem. We show that we can recover an even polynomial of degree 2m from m+1 given Taylor coefficients of the characteristic function whose zeros are the eigenvalues of one spectrum. The idea here is to represent the solution as a power series and identify the unknown coefficients from the characteristic function. We then compute these coefficients by solving a nonlinear algebraic system, and provide numerical examples at the end. Because of its algebraic nature, the method applies also to non self-adjoint problems.     

Some properties of optimal spacing in polynomial estimation

Summary It is shown that the choice of x values that minimizes the generalized variance of the estimates of the coefficients in polynomial regression also minimizes the maximum of the generalized variance of the estimates of an arbitrary number of ordinates on the regression curve. This optimum spacing is also optimal with respect to the estimation of the coefficients of the first derivative of the polynomial regression function. This research was partially supported by mean of an Air Force Research Grant while the author was on sabbatical leave in Japan.     

Some basic properties of planar Singer groups

A planar Singer group is a collineation group of a finite (in this article) projective plane acting regularly on the points of the plane. Theorem 1 gives a characterization of abelian planar Singer groups. This leads to a necessary and sufficient condition for an inner automorphism to be a multiplier. The Sylow 2-structure of a multiplier group and some of its consequences are given in Theorem 3. One important result in studying multipliers of an abelian Singer group is the existence of a common fixed line. We extend this to an arbitrary planar Singer group in Theorem 4. Theorem 5 studies the order of an abelian group of multiplers. If this order equals to the order of the plane plus 1, then the number of points of the plane is a prime. If this order is odd, then it is at most the planar order plus 1.Partially supported by a NSA grant.     

Uniform isochronous center of a class of higher degree polynomial differential systems

In this paper,we give the necessary and sufficient conditions for a class of higher degree polynomial systems to have a uniform isochronous center.At the same time,we prove that for this system the composition conjecture is correct.