Note on 2-rational fields

We give an alternative computation of the Galois group of the maximal 2-ramified and complexified pro-2-extension of any 2-rational number field (Theorem 2), a particular case of results of Movahhedi-Nguyen Quang Do. This short Note is motivated by the paper [J. Jossey, Galois 2-extensions unramified outside 2, J. Number Theory 124 (2007) 42-76] and, at this occasion, we bring into focus some classical technics of abelian ?-ramification which, unfortunately, are often ignored, especially those developed by J.-F. Jaulent with the ?-adic class field theory, and by G. Gras in his book on class field theory, and which considerably simplify the study of such subjects; for instance, our proof of Theorem 2 generalizes the purpose of Jossey's paper in such a way using a result of Herfort-Zalesskii. This Note is mainly an attempt of clarification about ?-rationality.     

Existence result of second-order differential equations with integral boundary conditions at resonance

By using Mawhin's continuation theorem, some sufficient conditions for the existence of solution for a class of second-order differential equations with integral boundary conditions at resonance are established, which are complement of previously known results. The interesting point is that we shall deal with the case dimKerL=2, which will cause some difficulties in constructing the projector Q. Since all the existence results obtained in previous papers are for the case dimKerL=1, our work is new.     

Nash equilibria in all-optical networks

We consider the problem of routing a number of communication requests in WDM (wavelength division multiplexing) all-optical networks from the standpoint of game theory. If we view each routing request (pair of source-target nodes) as a player, then a strategy consists of a path from the source to the target and a frequency (color). To reflect the restriction that two requests must not use the same frequency on the same edge, conflicting strategies are assigned a prohibitively high cost.Under this formulation, we consider several natural cost functions, each one reflecting a different aspect of restriction in the available bandwidth. For each cost function we examine the problem of the existence of pure Nash equilibria, the complexity of recognizing and computing them and finally, the problem in which we are given a Nash equilibrium and we are asked to find a better one in the sense that the total bandwidth used is less. As it turns out some of these problems are tractable and others are NP-hard.     

Equality of Lifshitz and van Hove exponents on amenable Cayley graphs

We study the low energy asymptotics of periodic and random Laplace operators on Cayley graphs of amenable, finitely generated groups. For the periodic operator the asymptotics is characterised by the van Hove exponent or zeroth Novikov–Shubin invariant. The random model we consider is given in terms of an adjacency Laplacian on site or edge percolation subgraphs of the Cayley graph. The asymptotic behaviour of the spectral distribution is exponential, characterised by the Lifshitz exponent. We show that for the adjacency Laplacian the two invariants/exponents coincide. The result holds also for more general symmetric transition operators. For combinatorial Laplacians one has a different universal behaviour of the low energy asymptotics of the spectral distribution function, which can be actually established on quasi-transitive graphs without an amenability assumption. The latter result holds also for long range bond percolation models.     

Stable invariant manifolds for parabolic dynamics

We consider nonautonomous equations v^′=A(t)v in a Banach space that exhibit stable and unstable behaviors with respect to arbitrary growth rates ecρ(t) for some function ρ(t). This corresponds to the existence of a “generalized” exponential dichotomy, which is known to be robust. When ρ(t)≠t this behavior can be described as a type of parabolic dynamics. We consider the general case of nonuniform exponential dichotomies, for which the Lyapunov stability is not uniform. We show that for any sufficiently small perturbation f of a “generalized” exponential dichotomy there is a stable invariant manifold for the perturbed equation v^′=A(t)v+f(t,v). We also consider the case of exponential contractions, which allow a simpler treatment, and we show that they persist under sufficiently small nonlinear perturbations.     

Grey dynamic input-output analysis

A system whose information is partially known and partially unknown is named as a grey system and it is general situation in reality. On the basis of grey number/matrix and their covered operation, some definitions of grey functions are enumerated. Combining grey system theory with traditional dynamic input-output analysis, we propose the grey dynamic input-output analysis. It has two types, i.e., the continuous and the disperse. The most important results are the computational formulas of the matrix-covered set of inverse grey investment coefficient matrix. By using them, we can forecast and control the economic system under the uncertain situation. For the calculation of the matrix-covered set of inverse grey matrices, we obtain the desired formulas, which can be applied to general grey matrices. The formulas can greatly promote the development of grey theory and enrich the applied fields of uncertain mathematics. The modified case illustrates our method.     

Value sharing of meromorphic functions and their derivatives

In this paper, we estimate the size of ρn's in the famous L. Zalcman's lemma. With it, we obtain some uniqueness theorems for meromorphic functions f and f^′ when they share two transcendental meromorphic functions.     

Generalised energy conservation law for wave equations with variable propagation speed

We investigate the long time behaviour of the L²-energy of solutions to wave equations with variable speed of propagation. The novelty of the approach is the combination of estimates for higher order derivatives of the coefficient with a stabilisation property.     

A note on a paper of I.D. Arand?elovi? on asymptotic contractions

W.A. Kirk [W.A. Kirk, Fixed points of asymptotic contractions, J. Math. Anal. Appl. 277 (2003) 645-650] defined the notion of an asymptotic contraction on a metric space and using ultrapower techniques he gave a nonconstructive proof of an asymptotic version of the Boyd-Wong fixed point theorem. Subsequently, I.D. Arand?elovi? [I.D. Arand?elovi?, On a fixed point theorem of Kirk, J. Math. Anal. Appl. 301 (2005) 384-385] established somewhat more general version of Kirk's result and he gave an elementary proof of it. However, our purpose is to show that there is an error in this proof and, moreover, Arand?elovi?'s theorem is false. We also explain how to correct this result.     

Class field theory for a product of curves over a local field

We prove that the kernel of the reciprocity map for a product of curves over a p-adic field with split semi-stable reduction is divisible. We also consider the K ₁ of a product of curves over a number field.