Cayley graphs of finitely generated groups

There does not exist a Borel choice of generators for each finitely generated group which has the property that isomorphic groups are assigned isomorphic Cayley graphs.

     

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Some global results for eigenvalue problems governed by a variational inequality

In this paper, we use the Leray-Schauder degree theory to obtain some information about the structure of the solution set of a large class of eigenvalue problems governed by a variational inequality. Applications are given to the unilateral postbuckling of a thin elastic plate.     

Bayesian Probabilities and the Histories Algebra

We attempt a justification of a generalisation of the consistent histories programme using a notion of probability that is valid for all complete sets of history propositions. This consists of introducing Cox's axioms of probability theory and showing that our candidate notion of probability obeys them. We also give a generalisation of Bayes' theorem and comment upon how Bayesianism should be useful for the quantum gravity/cosmology programmes. PACS: 02.50.Cw;03.65.Ta;04.60.-m.     

On disjoint configurations in infinite graphs

For a graph A and a positive integer n, let nA denote the union of n disjoint copies of A; similarly, the union of ?₀ disjoint copies of A is referred to as ?₀A. It is shown that there exist (connected) graphs A and G such that nA is a minor of G for all n??, but ?₀A is not a minor of G. This supplements previous examples showing that analogous statements are true if, instead of minors, isomorphic embeddings or topological minors are considered. The construction of A and G is based on the fact that there exist (infinite) graphs G₁, G₂,… such that G_i is not a minor of G_j for all i ≠ j. In contrast to previous examples concerning isomorphic embeddings and topological minors, the graphs A and G presented here are not locally finite. The following conjecture is suggested: for each locally finite connected graph A and each graph G, if nA is a minor of G for all n ? ?, then ?₀A is a minor of G, too. If true, this would be a far‐reaching generalization of a classical result of R. Halin on families of disjoint one‐way infinite paths in graphs. © 2002 Wiley Periodicals, Inc. J Graph Theory 39: 222–229, 2002; DOI 10.1002/jgt.10016     

Dynamic NMR study of the flexibility of 2-amino-3-aroyl-4,6-diaryl-pyrylium salts

Dynamic NMR investigations of a number of 2-amino-3-aroyl-4,6-diaryl-pyrylium salts were carried out. The barrier to rotation of the partial C, N double bond was determined and proved to be in the range of 62 to 63 kJ/mol. Quantum chemical calculations of bond orders and electron densities of the different atoms in the molecules show the distinct double bond character of the exocyclic C, N bond. This is in agreement with the relatively high barrier to rotation. By quantum chemical ab initio 3-21G calculations, the dynamic behaviour of this kind of compounds was simulated; two pairs (image and mirror image) of ground state conformations, in coincidence with the experiment, were obtained. Received: 10 May 1996 / Revised: 1 July 1996 / Accepted: 4 July 1996     

H‐self‐adjoint matrices. Application to radiosity

In computer graphics, in the radiosity context, a linear system Φx=b must be solved and there exists a diagonal positive matrix H such that H Φ is symmetric. In this article, we extend this property to complex matrices: we are interested in matrices which lead to Hermitian matrices under premultiplication by a Hermitian positive‐definite matrix H. We shall prove that these matrices are self‐adjoint with respect to a particular innerproduct defined on ?ⁿ. As a result, like Hermitian matrices, they have real eigenvalues and they are diagonalizable. We shall also show how to extend the Courant–Fisher theorem to this class of matrices. Finally, we shall give a new preconditioning matrix which really improves the convergence speed of the conjugate gradient method used for solving the radiosity problem. Copyright © 2002 John Wiley & Sons, Ltd.     

Universal ZN-graded differential calculus

We investigate the propeties of differential algebras generated by an operator d satisfying the property d^N = 0 instead of d² = 0 as in the usual case. Several examples of realizations of such differential algebras are given, either in the context of Z_N-graded N × N matrix algebras, or as a generalized differential calculus on manifolds.     

Influence of trans-cis photoisomerizations of solubilized compounds on transition temperatures in the system cetyltrimethylammonium bromide-water: Light-induced phase transitions

Transition temperatures (TN1) from the nematic lyotropic liquid-crystalline phase to the isotropic phase were measured for the system cetyltrimethylammonium bromide (CTAB) water in the presence of small amounts of 3-stilbene carboxylic acid (3SC), 4-stilbene carboxylic acid (4SC) and Δ2/2'-bi-(2H-l,4-benzothiazine) (BT). TNI, increases as a function of trans-3SC or trans-4SC concentration, ranging from 01 to 08 wt %, by up to 12°C. A further increase in TNI between 2 and 5°C can be achieved by photochemically converting the solubilized trans stilbene derivatives to the cis isomers. Irradiation of a trans-3SC containing sample at a temperature just above TNl leads to a light-induced phase transition to the lyotropic liquid-crystalline phase. Solubilization of trans-BT causes a slight decrease of TNI while photoisomerization to cis-BT increases T_NI by 1°C.