C(k)-solvability near the characteristic set for a class of planar complex vector fields of infinite type

This paper deals with semi-global C(k)-solvability of complex vector fields of the form L = partial derivative/partial derivative t + x(r) (a(x) + ib(x))partial derivative/partial derivative x, r >= 1, defined on Omega(epsilon) = (-epsilon, epsilon) x S(1), epsilon > 0, where a and b are C(infinity) real-valued functions in (-epsilon, epsilon). It is shown that the interplay between the order of vanishing of the functions a and b at x = 0 influences the C(k)-solvability at Sigma = {0} x S(1). When r = 1, it is permitted that the functions a and b of L depend on the x and t variables, that is, L = partial derivative/partial derivative t + x(a(x, t) + ib(x, t))partial derivative/partial derivative x, where (x, t) is an element of Omega(epsilon).

Gevrey solvability near the characteristic set for a class of planar complex vector fields of infinite type

We study the Gevrey solvability of a class of complex vector fields, defined on Omega(epsilon) = (-epsilon, epsilon) x S(1), given by L = partial derivative/partial derivative t + (a(x) + ib(x))partial derivative/partial derivative x, b not equivalent to 0, near the characteristic set Sigma = {0} x S(1). We show that the interplay between the order of vanishing of the functions a and b at x = 0 plays a role in the Gevrey solvability. (C) 2008 Elsevier Inc. All rights reserved.

Approach to Equilibrium for a Class of Random Quantum Models of Infinite Range

We consider random generalizations of a quantum model of infinite range introduced by Emch and Radin. The generalizations allow a neat extension from the class l (1) of absolutely summable lattice potentials to the optimal class l (2) of square summable potentials first considered by Khanin and Sinai and generalised by van Enter and van Hemmen. The approach to equilibrium in the case of a Gaussian distribution is proved to be faster than for a Bernoulli distribution for both short-range and long-range lattice potentials. While exponential decay to equilibrium is excluded in the nonrandom l (1) case, it is proved to occur for both short and long range potentials for Gaussian distributions, and for potentials of class l (2) in the Bernoulli case. Open problems are discussed.

Loops, Chains, Sheets, and Networks from Variable Coordination of Cu(hfac)(2) with a Flexibly Hinged Aminoxyl Radical Ligand

One pair of reactants, Cu(hfac)(2) = M and the hinge-flexible radical ligand 5-(3-N-tert-butyl-N-aminoxylphenyl)pyrimidine (3PPN = L), yields a diverse set of five coordination complexes: a cyclic loop M(2)L(1) dimer; a 1:1 cocrystal between an M(2)L(2) loop and an ML(2) fragment; a ID chain of M(2)L(2) loops linked by M; two 2D M(3)L(2) networks of (M-L)(n) chains crosslinked by M with different repeat length pitches; a 3D M(3)L(2) network of M(2)L(2) loops cross-linking (M-L)(n)-type chains with connectivity different from those in the 2D networks. Most of the higher dimensional complexes exhibit reversible, temperature-dependent spin-state conversion of high-temperature paramagnetic states to lower magnetic moment states having antiferromagnetic exchange within Cu-ON bonds upon cooling, with accompanying bond contraction. The 3D complex also exhibited antiferromagnetic exchange between Cu(II) ions linked in chains through pyrimidine rings.

Perfect Simulation of a Coupling Achieving the (d)over-bar-distance Between Ordered Pairs of Binary Chains of Infinite Order

We explicitly construct a stationary coupling attaining Ornstein`s (d) over bar -distance between ordered pairs of binary chains of infinite order. Our main tool is a representation of the transition probabilities of the coupled bivariate chain of infinite order as a countable mixture of Markov transition probabilities of increasing order. Under suitable conditions on the loss of memory of the chains, this representation implies that the coupled chain can be represented as a concatenation of i.i.d. sequences of bivariate finite random strings of symbols. The perfect simulation algorithm is based on the fact that we can identify the first regeneration point to the left of the origin almost surely.

Involutions of RA Loops

Let L be an RA loop, that is, a loop whose loop ring over any coefficient ring R is an alternative, but not associative, ring. Let l bar right arrow l(theta) denote an involution on L and extend it linearly to the loop ring RL. An element alpha is an element of RL is symmetric if alpha(theta) = alpha and skew-symmetric if alpha(theta) = -alpha. In this paper, we show that there exists an involution making the symmetric elements of RL commute if and only if the characteristic of R is 2 or theta is the canonical involution on L, and an involution making the skew-symmetric elements of RL commute if and only if the characteristic of R is 2 or 4.

Sylow`s theorem for Moufang loops

For finite Moufang loops, we prove an analog of the first Sylow theorem giving a criterion for the existence of a p-Sylow subloop. We also find the maximal order of p-subloops in the Moufang loops that do not possess p-Sylow subloops. (c) 2009 Elsevier Inc. All rights reserved.

Algebraic Bol loops

In this paper, we study the category of algebraic Bol loops over an algebraically closed field of definition. On the one hand, we apply techniques from the theory of algebraic groups in order to prove structural theorems for this category. On the other hand, we present some examples showing that these loops lack some nice properties of algebraic groups; for example, we construct local algebraic Bol loops which are not birationally equivalent to global algebraic loops.

Commutative Moufang loops and alternative algebras

We describe bases of free commutative Moufang loop with seven generators and calculate the order of this loop. (c) 2011 Published by Elsevier Inc.

Birkhoffian Systems in Infinite Dimensional Manifolds

We generalize the theory of Kobayashi and Oliva (On the Birkhoff Approach to Classical Mechanics. Resenhas do Instituto de Matematica e Estatistica da Universidade de Sao Paulo, 2003) to infinite dimensional Banach manifolds with a view towards applications in partial differential equations.