A two-component geodesic equation on a space of constant positive curvature

We propose a new two-component geodesic equation with the unusual property that the underlying space has constant positive curvature. In the special case of one space dimension, the equation reduces to the two-component Hunter–Saxton equation.     

Three-element zoom lens with fixed distance between focal points

This work deals with a theoretical analysis of zoom lenses with a fixed distance between focal points. Equations are derived for the primary (paraxial) design of the basic parameters of a three-element zoom lens. It is shown that the number of optical elements for such a lens must be larger than two.     

Dependence of the average to-node distance on the node degree for random graphs and growing networks

In a connected graph, nodes can be characterised locally (with their degree k) or globally (e.g. with their average length path to other nodes). Here we investigate how depends on k. The numerical algorithm based on the construction of the distance matrix is applied to random graphs and the growing networks: the scale-free ones and the exponential ones. The results are relevant for search strategies in different networks.Received: 15 June 2004, Published online: 21 October 2004PACS: 02.10.Ox Combinatorics; graph theory - 05.10.-a Computational methods in statistical physics and nonlinear dynamics     

Multicritical points in a system with random single-ion anisotropy

A system with two competing anisotropies, one of which is random, is considered. Phase diagram in the molecular field approximation with two bicritical points, and a region of metastability is constructed. The renormalization group analysis of the critical behaviour for such a system is carried out. Instead of one of the bicritical points a tetracritical point described by the decoupled fixed point is found.     

Perturbations of geodesic flows on surface of constant negative curvature and their mixing properties

We consider one parameter analytic hamiltonian perturbations of the geodesic flows on surfaces of constant negative curvature. We find two different necessary and sufficient conditions for the canonical equivalence of the perturbed flows and the non-perturbed ones. One condition says that the Hamilton-Jacobi equation (introduced in this work) for the conjugation problem should admit a solution as a formal power series (not necessarily convergent) in the perturbation parameter. The alternative condition is based on the identification of a complete set of invariants for the canonical conjugation problem. The relation with the similar problems arising in the KAM theory of the perturbations of quasi periodic hamiltonian motions is briefly discussed. As a byproduct of our analysis we obtain some results on the Livscic, Guillemin, Kazhdan equation and on the Fourier series for the SL(2, ) group. We also prove that the analytic functions on the phase space for the geodesic flow of unit speed have a mixing property (with respect to the geodesic flow and to the invariant volume measure) which is exponential with a universal exponent, independent on the particular function, equal to the curvature of the surface divided by 2. This result is contrasted with the slow mixing rates that the same functions show under the horocyclic flow: in this case we find that the decay rate is the inverse of the time (up to logarithms).Part of this work was performed while the first and third authors were in residence at the Institute for Mathematics and its Applications at the University of Minnesota, Minneapolis, MN 55455, USASupported by the Mathematics Dept. of Princeton University of by Stiftung Volkswagenwerk through IHES, and IMA     

Effect of a random field conjugate to a non-critical variable on phase transitions

We consider the phase transition in an anisotropic system with a random field conjugate to a non-critical variable. It is shown that in this case a sufficiently weak random field acts like random potentials.     

A random walk on a random channel with absorbing barriers

We consider random walks on one-dimensional random channels between two absorbing barriers. The problem can perhaps be used to model the diffusion of a molecule in a “random” membrane, the molecule traversing a random channel formed by the constituent membrane molecules. We are able to analytically follow the transition from diffusive to non-diffusive behavior as the minimum number of channel segments required to traverse the membrane increases.     

A note on correlations between eigenvalues of a random matrix

Dyson's method is adopted here for the so called Gaussian ensembles. Incidently this confirms the long cherished belief that the statistical properties of a small number of eigenvalues is the same for the two kinds of ensembles, the circular and the Gaussian ones.     

Generation of random fields defined on a cluster of points when the random field has a given site-to-site correlation

A computational scheme for generating a random field defined on a cluster of points with the random field having a site-to-site correlation function close to some desired correlation function is presented. It is assumed that the quantity to be simulated either is, or may be related to, a random quantity having a joint normal distribution with the same variance at each site in the cluster and with a correlation coefficient between sites depending only on the separation of the sites. The scheme can be used for any random quantity which is defined at lattice sites in a cluster and may be applied, for example, to both exchange fields and AB-alloys.     

A concept of the mass center of a system of material points in the constant curvature spaces

This article demonstrates that in the Lobatchevsky space and on a sphere of arbitrary dimensions, the concept of the mass center of a system of mass points can be correctly defined. Presented are: a uniform geometric construction for defining the mass center; hyperbolic and spheric lever rules; the theorem of uniqueness for determining the mass center in these spaces. Among the compact manifolds, only the sphere possesses this property.     

End-to-end distance on contour loops of random Gaussian surfaces

A self-consistent field theory that describes a part of a contour loop of a random Gaussian surface as a trajectory interacting with itself is constructed. The exponent nu characterizing the end-to-end distance is obtained by a Flory argument. The result is compared with different previous derivations and is found to agree with that of Kondev and Henley over most of the range of the roughening exponent of the random surface.     

A restricted random walk on a one-dimensional random chain

It is found that under some special conditions the inverse velocity for a restricted random walk diverges even though it is finite for the corresponding unrestricted walk. This leads to an anomalous t^z (0<z<1) behaviour for the mean distance travelled in time t.     

基于离散随机介质平均散射声功率的无损测温方法

生物组织中超声散射回波的平均功率与其温度有关.该理论包含:(1)生物组织近似为离散随机介质.(2)离散随机介质的平均散射功率与组织的衰减系数α和声速c有关。对于似水生物组织(如:猪肝),当温度升高时,衰减系数的减小使平均散射声功率增加,但其影响大小与时间窗△t有关,而声速的增大则使平均散射声功率减小。实验结果表明:生物组织的平均散射声功率随温度变化趋势明显,该结果提供一种肿瘤热疗无损伤检测温度的新方法。     

A connection between a system of random walks and rumor transmission

We establish a relationship between the phenomenon of rumor transmission on a population and a probabilistic model of interacting particles on the complete graph. More precisely, we consider variations of the Maki–Thompson epidemic model and the “frog model” of random walks, which were introduced in the scientific literature independently and in different contexts. We analyze the Markov chains which describe these models, and show a coupling between them. Our connection shows how the propagation of a rumor in a closed homogeneously mixing population can be described by a system of random walks on the complete graph. Additionally, we discuss further applications of the random walk model which are relevant to the modeling of different biological dynamics.     

A random walker on a ratchet

We analyze a model for a walker moving on a ratchet potential. This model is motivated by the properties of transport of motor proteins, like kinesin and myosin. The walker consists of two feet that are represented as two particles coupled nonlinearly through a bistable potential. In contrast to linear coupling, the bistable potential admits a richer dynamics, where the ordering of the particles can alternate during the walking. The transitions between the two stable states on the bistable potential correspond to a walking with alternating particles. We distinguish between two main walking styles: alternating and no alternating, resembling the hand-over-hand and the inchworm walking in motor proteins, respectively. When the equilibrium distance between the two particles divided by the periodicity of the ratchet is an integer, we obtain a maximum for the current, indicating optimal transport.     

A simple calculation for the average number of steps to trapping in lattice random walks

We show that the asymptotic results for the average number of steps to trapping at an irreversible trapping site on aD-dimensional finite lattice can be obtained from the generating function for random walks on aninfinite perfect lattice. This introduces a significant simplification into such calculations. An interesting corollary of these calculations is the conclusion that a random walker traverses, on the average, all the distinct nontrapping lattice sites before arriving on the trapping site.This work was supported in part by NSF Grants MPS72-04363-A03 and CHE75-20624.     

Expansion of the distance between two points in spheroidal functions as applied to problems of mathematical physics

A complete set of biharmonic functions is constructed in spheroidal coordinates. The distance between two points and its inverse value are expanded into a double series in terms of such functions. Possible applications in the theory of elasticity, in astrophysics, and other fields of mathematical physics are pointed out.     

Depinning of a discrete elastic string from a two-dimensional random array of weak pinning points

The present work is essentially concerned with the development of statistical theory for the low temperature dislocation glide in concentrated solid solutions where atom-sized obstacles impede plastic flow. In connection with such a problem, we compute analytically the external force required to drag an elastic string along a discrete two-dimensional square lattice, where some obstacles have been randomly distributed. Some numerical simulations allow us to demonstrate the remarkable agreement between simulations and theory for an obstacle density ranging from 1% to 50% and for lattices with different aspect ratios. The theory proves efficient on the condition that the obstacle-chain interaction remains sufficiently weak compared to the string stiffness.