Persistent centers of complex systems

On the basis of some works on persistent centers and weakly persistent centers, in this paper we discuss a generalized version of persistent center and weakly persistent center for complex planar differential systems, in which conjugacy of variables may not be required. We give some complex systems which have a persistent center or weakly persistent center at the origin. Then, we find all conditions of persistent center for cubic systems and all conditions of weakly persistent center for complex cubic Lotka–Volterra system. Relations between complex systems and real ones are given concerning persistent centers and weakly persistent centers.     

On polynomial digraphs

Let Φ(x,y) be a bivariate polynomial with complex coefficients. The zeroes of Φ(x,y) are given a combinatorial structure by considering them as arcs of a directed graph G(Φ). This paper studies some relationship between the polynomial Φ(x,y) and the structure of G(Φ).     

Norm Bounds for Ehrhart Polynomial Roots

M. Beck et al. found that the roots of the Ehrhart polynomial of a d-dimensional lattice polytope are bounded above in norm by 1+(d+1)!. We provide an improved bound which is quadratic in d and applies to a larger family of polynomials.     

Chaotic polynomials on Banach spaces

We show the existence of chaotic (in the sense of Devaney) polynomials on Banach spaces of q-summable sequences. Such polynomials P consist of composition of the backward shift with a certain fixed polynomial p of one complex variable on each coordinate. In general we also prove that P is chaotic in the sense of Auslander and Yorke if and only if 0 belongs to the Julia set of p.     

ON THE EXISTENCE OF GLOBAL GENERAL SOLUTIONS OF POLYNOMIAL SYSTEMS

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Implementing an efficient minimum capacity cut algorithm

In this paper, we present an efficient implementation of theO(mn + n ² logn) time algorithm originally proposed by Nagamochi and Ibaraki (1992) for computing the minimum capacity cut of an undirected network. To enhance computation, various ideas are added so that it can contract as many edges as possible in each iteration. To evaluate the performance of the resulting implementation, we conducted extensive computational experiments, and compared the results with that of Padberg and Rinaldi's algorithm (1990), which is currently known as one of the practically fastest programs for this problem. The results indicate that our program is considerably faster than Padberg and Rinaldi's program, and its running time is not significantly affected by the types of the networks being solved.Corresponding author.     

STABILIZATION OF UNITARY GROUPSOVER POLYNOMIAL RINGS

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A constructive approach to nodal splines

In the context of local spline interpolation methods, nodal splines have been introduced as possible fundamental functions by de Villiers and Rohwer in 1988. The corresponding local spline interpolation operator possesses the desirable property of reproducing a large class of polynomials. However, it was remarked that their definition is rather intricate so that it seems desirable to reveal the actual origin of these splines. The real source can be found in the Martensenoperator which can be obtained by two-point Hermite spline interpolation problem posed and proved by Martensen [Darstellung und Entwicklung des Restgliedes der Gregoryschen Quadraturformel mit Hilfe von Spline-Funktionen, Numer. Math. 21(1973)70–80]. On the one hand, we will show how to represent the Hermite Martensen spline recursively and, on the other hand, explicitly in terms of the B-spline by using the famous Marsden identity. Having introduced the Martensenoperator, we will show that the nodal spline interpolation operator can be obtained by a special discretization of the occurring derivatives. We will consider symmetric nodal splines of odd degree that can be obtained by our methods in a natural way.     

On Nondeterminism,Enumeration Reducibility and Polynomial Bounds

Enumeration reducibility is a notion of relative computability between sets of natural numbers where only positive information about the sets is used or produced. Extending e-reducibility to partial functions characterises relative computability between partial functions. We define a polynomial time enumeration reducibility that retains the character of enumeration reducibility and show that it is equivalent to conjunctive non-deterministic polynomial time reducibility. We define the polynomial time e-degrees as the equivalence classes under this reducibility and investigate their structure on the recursive sets, showing in particular that the pe-degrees of the computable sets are dense and do not form a lattice, but that minimal pairs exist. We define a jump operator and use it to produce a characterisation of the polynomial hierarchy.