Study of the higher-order asymptotic behavior of quantum field expansions in the theory of two-dimensional fully developed turbulence

We use a simplified (0+1)-dimensional theory to develop approaches for studying the higher-order asymptotic behavior of quantum field expansions in the two-dimensional theory of fully developed turbulence. We consider the asymptotic behavior of the correlation function in the small-time limit in the theory of fully developed turbulence and derive and investigate the stationarity equation. We show that the perturbation series in this limit has a finite convergence radius.     

Velocity fields with power-law spectra for modeling turbulent flows

We consider a generalization of homogeneous and isotropic Çinlar velocity fields to capture power-law spectra. The random velocity field is non-Gaussian with a representation motivated by Lagrangian and Eulerian observations. A wide range of turbulent flows can be generated by varying the stochastic parameters of the model. The velocity field being a functional version of Poisson shot-noise is constructed as the superposition of eddies randomized through their types and arrival times. We introduce a dependence between the eddy types which are spatial parameters and the decay parameter which is temporal. As a result, long-range correlation in space and a power-law spectrum previously used with Ornstein–Uhlenbeck velocity fields are achieved. We show that a corresponding power-law form for the probability distribution of the eddy diameter is sufficient for this result. The parameters of the probability distribution are further specified in view of Kolmogorov theory of the inertial scales. In particular, ∣k∣^−5/3 scaling of the spectrum is obtained. In the diffusive limit, we show that the parameters governing the decay and the arrival rate, and the speed of rotation of an eddy increase while its diameter decreases. That is, the eddies arrive fast, decay fast, and rotate fast with a small radius for a Brownian limit.     

Analytic study on angular fields of HRR solution

By using the constructing function method, a systematic and strict analysis is carried out on the angular distribution field near a crack tip in a power-law hardening material and the analytic solution is provided for HRR problem. In addition, the equivalence of H equation and RR equation is proved. The present analytic solutions for HRR problem can reduce to (he well-known Williams solution in the limit case ofN→1 (orn→1) and Prandtl solution in the limit case ofN→0 (orn→∞). It is particularly interesting that from the deformation theory of plasticity one obtains the Prandtl solution based on the increatment theory of plasticity. Project supported by the National Natural Science Foundation of China (Grant No. 19132022).     

Density Functional Theory and Optimal Transportation with Coulomb Cost

We present here novel insight into exchange‐correlation functionals in density functional theory, based on the viewpoint of optimal transport. We show that in the case of two electrons and in the semiclassical limit, the exact exchange‐correlation functional reduces to a very interesting functional that depends on an optimal transport map T associated with a given density ρ. The limit problem has been suggested, on grounds of formal arguments, in the physics literature, but it appears that it has not hitherto been interpreted as an optimal transport problem. Since the above limit is strongly correlated, the limit functional yields insight into electron correlations. We prove the existence and uniqueness of such an optimal map for any number of electrons and each ρ and determine the map explicitly in the case when ρ is radially symmetric. © 2012 Wiley Periodicals, Inc.     

The Value Function of Singularly Perturbed Control Systems

The limit as ɛ→ 0 of the value function of a singularly perturbed optimal control problem is characterized. Under general conditions it is shown that limit value functions exist and solve in a viscosity sense a Hamilton—Jacobi equation. The Hamiltonian of this equation is generated by an infinite horizon optimization on the fast time scale. In particular, the limit Hamiltonian and the limit Hamilton—Jacobi equation are applicable in cases where the reduction of order, namely setting ɛ = 0 , does not yield an optimal behavior. Accepted 18 November 1999     

Extremal dependence measure and extremogram: the regularly varying case

The dependence of large values in a stochastic process is an important topic in risk, insurance and finance. The idea of risk contagion is based on the idea of large value dependence. The Gaussian copula notoriously fails to capture this phenomenon. Two notions in a process or vector context which summarize extremal dependence in a function comparable to a correlation function are the extremal dependence measure (EDM) and the extremogram. We review these ideas and compare the two tools and end with a central limit theorem for a natural estimator of the EDM which allows drawing confidence bands comparable to those provided by Bartlett’s formula in a classical context of sample correlation functions.     

Consequences of weak Allee effect on prey in the May–Holling–Tanner predator–prey model

In this work, a modified Holling–Tanner predator–prey model is analyzed, considering important aspects describing the interaction such as the predator growth function is of a logistic type; a weak Allee effect acting in the prey growth function, and the functional response is of hyperbolic type. Making a change of variables and time rescaling, we obtain a polynomial differential equations system topologically equivalent to the original one in which the non‐hyperbolic equilibrium point (0,0) is an attractor for all parameter values. An important consequence of this property is the existence of a separatrix curve dividing the behavior of trajectories in the phase plane, and the system exhibits the bistability phenomenon, because the trajectories can have different ω ? limit sets; as example, the origin (0,0) or a stable limit cycle surrounding an unstable positive equilibrium point. We show that, under certain parameter conditions, a positive equilibrium may undergo saddle‐node, Hopf, and Bogdanov–Takens bifurcations; the existence of a homoclinic curve on the phase plane is also proved, which breaks in an unstable limit cycle. Some simulations to reinforce our results are also shown. Copyright © 2015 John Wiley & Sons, Ltd.     

Sharp-interface limit of the Allen-Cahn action functional in one space dimension

We analyze the sharp-interface limit of the action minimization problem for the stochastically perturbed Allen-Cahn equation in one space dimension. The action is a deterministic functional which is linked to the behavior of the stochastic process in the small noise limit. Previously, heuristic arguments and numerical results have suggested that the limiting action should “count” two competing costs: the cost to nucleate interfaces and the cost to propagate them. In addition, constructions have been used to derive an upper bound for the minimal action which was proved optimal on the level of scaling. In this paper, we prove that for d = 1, the upper bound achieved by the constructions is in fact sharp. Furthermore, we derive a lower bound for the functional itself, which is in agreement with the heuristic picture. To do so, we characterize the sharp-interface limit of the space-time energy measures. The proof relies on an extension of earlier results for the related elliptic problem. Mathematics Subject Classification (2000) 49J45, 35R60, 60F10     

Central limit theorems for non-linear functionals of Gaussian fields

In the present paper it is shown that the central limit theorem holds for some non-linear functionals of stationary Gaussian fields if the correlation function of the underlying field tends fast enough to zero. The results are formulated in terms of the Hermite rank of the functional and of the rate of the correlation function. Then we show an example when the limit field is self-similar and Gaussian but not necessarily consisting of independent elements.     

Ising limit of a Heisenberg XXZ magnet and some temperature correlation functions

We consider the Heisenberg spin-1/2 XXZ magnet in the case where the anisotropy parameter tends to infinity (the so-called Ising limit). We find the temperature correlation function of a ferromagnetic string above the ground state. Our approach to calculating correlation functions is based on expressing the wave function in the considered limit in terms of Schur symmetric functions. We show that the asymptotic amplitude of the above correlation function at low temperatures is proportional to the squared number of strict plane partitions in a box.     

The Long-Distance Asymptotics for the Correlation Function of the Third Spin Components in the XX0 Model with Interaction Through Two Sites

The XX0 chain in the external magnetic field directed along the z axis is discussed. The Hamiltonian describing the exchange interaction between two and four neighboring sites of the chain is constructed. An integral representation for the equal-time temperature correlation function of the third spin components is given. From the long-distance asymptotic of this correlation function, the correlation radius is calculated and its low-temperature behavior is studied. Bibliography: 11 titles.     

Gaussian fluctuations of connectivities in the subcritical regime of percolation

Summary We consider thed-dimensional Bernoulli bond percolation model and prove the following results for allp _c : (1) The leading power-law correction to exponential decay of the connectivity function between the origin and the point (L, 0, ..., 0) isL ^–(d–1)/2 . (2) The correlation length, (p) is real analytic. (3) Conditioned on the existence of a path between the origin and the point (L, 0, ..., 0), the hitting distribution of the cluster in the intermediate planes,x ₁ =qL,0, obeys a multidimensional local limit theorem. Furthermore, for the two-dimensional percolation system, we prove the absence of a roughening transition: For allp>p _c , the finite-volume conditional measures, defined by requiring the existence of a dual path between opposing faces of the boundary, converge—in the infinite-volume limit—to the standard Bernoulli measure.Work supported in part by G.N.A.F.A. (C.N.R.)Work supported in part by NSF Grant No. DMS-88-06552     

Blow-up solutions to a system of nonlinear Volterra equations

A system of nonlinear Volterra integral equations with convolution kernels is considered. Estimates are given for the blow-up time when conditions are such that the solution is known to become unbounded in finite time. For two examples that arise in combustion problems, numerical estimates of blow-up time are presented. Additionally, the asymptotic behavior of the blow-up solution in the key limit is established for the power-law and exponential nonlinearity cases.     

Limit shapes and the complex Burgers equation

In this paper we study surfaces in R ³ that arise as limit shapes in random surface models related to planar dimers. These limit shapes are surface tension minimizers, that is, they minimize a functional of the form ∫σ(∇h) dx dy among all Lipschitz functions h taking given values on the boundary of the domain. The surface tension σ has singularities and is not strictly convex, which leads to formation of facets and edges in the limit shapes. We find a change of variables that reduces the Euler–Lagrange equation for the variational problem to the complex inviscid Burgers equation (complex Hopf equation). The equation can thus be solved in terms of an arbitrary holomorphic function, which is somewhat similar in spirit to Weierstrass parametrization of minimal surfaces. We further show that for a natural dense set of boundary conditions, the holomorphic function in question is, in fact, algebraic. The tools of algebraic geometry can thus be brought in to study the minimizers and, especially, the formation of their singularities. This is illustrated by several explicitly computed examples.     

Uniform convergence and sequence of maps on a compact metric space with some chaotic properties

Recently, C. Tain and G. Chen introduced a new concept of sequence of time invariant function. In this paper we try to investigate the chaotic behavior of the uniform limit function f : X → X of a sequence of continuous topologically transitive (in strongly successive way) functions fn : X → X, where X is a compact interval. Surprisingly, we find that the uniform limit function is chaotic in the sense of Devaney. Lastly, we give an example to show that the denseness property of Devaney’s definition is lost on the limit function.     

Multifractal detrended fluctuation analysis of analog random multiplicative processes

We investigate non-Gaussian statistical properties of stationary stochastic signals generated by an analog circuit that simulates a random multiplicative process with weak additive noise. The random noises are originated by thermal shot noise and avalanche processes, while the multiplicative process is generated by a fully analog circuit. The resulting signal describes stochastic time series of current interest in several areas such as turbulence, finance, biology and environment, which exhibit power-law distributions. Specifically, we study the correlation properties of the signal by employing a detrended fluctuation analysis and explore its multifractal nature. The singularity spectrum is obtained and analyzed as a function of the control circuit parameter that tunes the asymptotic power-law form of the probability distribution function.     

Multi-variable translation equation which arises from homothety

In many regular cases, there exists a (properly defined) limit of iterations of a function in several real variables, and this limit satisfies the functional equation _boxclose)=((xz)(1-z)/z){(1-z)\phi({\bf x})=\phi(\phi({\bf x}z)(1-z)/z)}; here z is a scalar and x is a vector. This is a special case of a well-known translation equation. In this paper we present a complete solution to this functional equation when f{\phi} is a continuous function on a single point compactification of a 2-dimensional real vector space. It appears that, up to conjugation by a homogeneous continuous function, there are exactly four solutions. Further, in a 1-dimensional case we present a solution with no regularity assumptions on f{\phi}.     

Total gluon propagator in light-cone gauge and the transversality problem of the polarization operator in the infrared region. II

A three-gluon vertex that depends on a gauge vectorn and is free of kinematic and spurion (np)^–1 singularities in the physical region is constructed. It is shown that in the case of integer power-law asymptotic behavior of the gluon propagator there is separation of the singular contributions of the single- and two-loop diagrams in the Dyson—Schwinger equation. Necessary conditions for realization ofp ^–4 asymptotic behavior are obtained, and it is shown that an important role in their fulfillment is played by the parts of the three-gluon vertex that depend on the gauge vector. A solution of these conditions is found; when they are fulfilled, they lead to infrared asymptotic behaviors of the complete Dyson—Schwinger equation and its single-loop approximation that are mutually compatible.Tbilisi State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 100, No. 1, pp. 14–23, July, 1994.