Absolute fixed point sets for multi-valued maps

The notion of a multi-valued absolute fixed point set (MAFS) will be defined and characterized in the setting of set-valued maps with images containing multiple components.

     

Field theory of the two-dimensional Ising model: Equivalence to the free particle one-dimensional Dirac equation

The Schultz-Mattis-Lieb fermion formulation of the two-dimensional Ising model is simplified by means of long-wavelength approximations which become exact in the critical region. The resulting continuum theory has a Hamiltonian density which is shown to be identical, to within a perfect derivative, to that of free, spinless particles satisfying the one-dimensional Dirac equation. Filling the negative-energy single-particle states of momentumq and mass gives an integral over the single-particle energies -( ²+k²)^1/2. Because varies linearly with the temperature, differentiating twice gives Onsager's logarithmic singularity in the specific heat.Work supported in part by the Office of Naval Research.     

A note on occupation times of random walks

In this note I show that some asymptotic results on the average occupancy time of an interval derived for lattice random walks with negative exponential transition probabilities are true for all random walks whose transition probabilities have a finite variance. The proof is based on the continuum limit.     

Clones from creatures

We show that (consistently) there is a clone on a countable set such that the interval of clones above is linearly ordered and has no coatoms.

     

The aqueous Raman optical activity spectra of 4(R)‐hydroxyproline: theory and experiment

Vibrational Raman optical activity (ROA) spectra have been measured for aqueous solutions of 4(R)‐hydroxyproline at three different pH values and are compared with theoretical results calculated for several conformations of anionic, cationic and zwitterionic 4(R)‐hydroxyproline using density functional theory (DFT) and the polarizable continuum model (PCM). The experimental ROA bands have been ascribed to the normal modes by comparison of the experimental and calculated vibrational frequencies and ROA intensities. Overall, using PCM for geometry optimization and force field calculations gives simulated Raman and ROA spectra that agree with the main features of the experimental spectra, whereas using PCM also in the calculations of optical tensors seems more problematic. Copyright © 2010 John Wiley & Sons, Ltd.     

Topology and topological sequence entropy

Let X be a compact metric space and T:X-→X be continuous.Let h*(T)be the supremum of topological sequence entropies of T over all the subsequences of Z+and S(X)be the set of the values h*(T)for all the continuous maps T on X.It is known that{0}■S(X)■{0,log 2,log 3,...}∪{∞}.Only three possibilities for S(X)have been observed so far,namely S(X)={0},S(X)={0,log 2,∞}and S(X)={0,log 2,log 3,...}∪{∞}.In this paper we completely solve the problem of finding all possibilities for S(X)by showing that in fact for every set{0}?A?{0,log 2,log 3,...}∪{∞}there exists a one-dimensional continuum XAwith S(XA)=A.In the construction of XAwe use Cook continua.This is apparently the first application of these very rigid continua in dynamics.We further show that the same result is true if one considers only homeomorphisms rather than continuous maps.The problem for group actions is also addressed.For some class of group actions(by homeomorphisms)we provide an analogous result,but in full generality this problem remains open.     

Ab initio calculation of the 4He(e, e'd )d reaction

The two-body knock-out reaction ⁴He(e, e'd )d is calculated at various momentum transfers. The full four-nucleon dynamics is taken into account microscopically both in the initial and the final states. As NN interaction the central MT-I/III potential is used. The calculation shows a strong reduction of the coincidence cross-section due to the final-state interaction. Nonetheless, the theoretical results exhibit a considerable overestimation of the experimental cross-section at lower momentum transfer. Comparisons with other, less complete, calculations suggest that consideration of a more realistic ground state might not be sufficient for a good agreement with experiment, rather a more realistic final-state interaction could play an essential role.     

Wave features of a hyperbolic prey–predator model

A hyperbolic predator–prey model is proposed within the context of extended thermodynamics. The nature of the steady state solutions for the uniform and non‐uniform perturbations are analyzed. The existence of smooth traveling wave‐like solutions, related to the invasion of the predator population into a prey‐only state is discussed. Validation of the model in point is also accomplished by searching for numerical solutions of the system, which also points out limit cycles in the populations. Copyright © 2010 John Wiley & Sons, Ltd.     

Continuum model for dislocation dynamics in a slip plane

In this paper, we present a continuum model for dislocation dynamics in a slip plane, which accurately incorporates both the long-range interaction and the local line tension effect of dislocations. Unlike the continuum models in the literature using dislocation densities, we use the disregistry across the slip plane to represent the continuous distribution of dislocations in the slip plane, which has the advantage of including the orientation dependence of dislocations in a very simple way. The continuum dislocation dynamics model is validated by linear instability analysis of a uniform dislocation array to small perturbations and comparisons of the results with those of the discrete dislocation dynamics model. Numerical examples for the evolution of distributions of dislocations and plastic slips in a slip plane are presented.     

Continuous Correspondence of Conservation Laws of the Semi-discrete AKNS System

In this paper we investigate the semi-discrete Ablowitz–Kaup–Newell–Segur (sdAKNS) hierarchy, and specifically their Lax pairs and infinitely many conservation laws, as well as the corresponding continuum limits. The infinitely many conserved densities derived from the Ablowitz-Ladik spectral problem are trivial, in the sense that all of them are shown to reduce to the first conserved density of the AKNS hierarchy in the continuum limit. We derive new and nontrivial infinitely many conservation laws for the sdAKNS hierarchy, and also the explicit combinatorial relations between the known conservation laws and our new ones. By performing a uniform continuum limit, the new conservation laws of the sdAKNS system are then matched with their counterparts of the continuous AKNS system.