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.     

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.     

AdS3/CFT2 on a Torus in the Sum over Geometries

We investigate the AdS₃/CFT₂ correspondence for the Euclidean AdS₃ space compactified on a solid torus with the CFT field on the regularizing boundary surface in the bulk. Correlation functions corresponding to the bulk theory at a finite temperature tend to the standard CFT correlation functions in the limit of removed regularization. In the sum over geometries in both the regular and the ℤ _N orbifold cases, the two-point correlation function for massless modes transforms into a finite sum of products of the conformal-anticonformal CFT Green's functions up to divergent terms proportional to the volume of the SL(2, ℤ)/ℤ group. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 146, No. 1, pp. 17–30, January, 2006.     

Noncolliding Jacobi processes as limits of Markov chains on the Gelfand–Tsetlin graph

We introduce a stochastic dynamics related to the measures that arise in harmonic analysis on the infinite–dimensional unitary group. Our dynamics is obtained as a limit of a sequence of natural Markov chains on the Gelfand–Tsetlin graph. We compute the finite-dimensional distributions of the limit Markov process, the generator and eigenfunctions of the semigroup related to this process. The limit process can be identified with the Doob h-transform of a family of independent diffusions. The space-time correlation functions of the limit process have a determinantal form. Bibliography: 21 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 360, 2008, pp. 91–123.     

A free boundary problem for $\infty$–Laplace equation

We consider a free boundary problem for the p-Laplacian describing nonlinear potential flow past a convex profile K with prescribed pressure on the free stream line. The main purpose of this paper is to study the limit as of the classical solutions of the problem above, existing under certain convexity assumptions on a(x). We show, as one can expect, that the limit solves the corresponding potential flow problem for the -Laplacian in a certain weak sense, strong enough however, to guarantee uniqueness. We show also that in the special case the limit is given by the distance function. Received: 10 October 2000 / Accepted: 23 February 2001 / Published online: 19 October 2001     

Partially-bent functions

We study a conjecture stated in [6] about the numbers of non-zeros of, respectively, the auto-correlation function and the Walsh transform of the function (–1) ^f(x) , wheref(x) is any boolean function on {0, 1} ⁿ . The result that we obtain leads us to introduce the class of partially-bent functions. We study within these functions the propagation criterion. We characterize those partially-bent functions which are balanced and prove a relation between their number (which is unknown) and the number of non-balanced partially-bent functions on {0, 1} ^n–1. Eventually, we study their correlation immunity.     

The Cascade Algorithm for the Numerical Computation of Refinable Functions

The cascade algorithm is a method which can be used for the computation of refinable functions. We prove here that in general the cascade algorithm will not inherit the accuracy and refinability of the limit refinable function. We provide conditions for which the cascade iteration does indeed preserve accuracy.     

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.     

The effective conductivity of an infinitely interchangeable mixture

We are concerned with the estimation of the effective electrical conductivity of random heterogenous materials. The purpose of this paper is to discuss a property of “statistical symmetry” verified by the symmetric cell materials of Miller. This property will be referred to as infinite interchangeability. The usual way to approach cell materials is through n-point correlation functions. The property of infinite interchangebility permits us to approach cell materials from a completely different point of view. Our main result is a simple algorithm, based on this symmetry property, for computing any coefficient of the perturbation expansion in terms of information from the dilute limit. Specifically, knowledge of the coefficients of the expansion in powers of the volume fraction up to order r allows for computation of the perturbation expansion coefficients up to order (2r + 1). This result, which was previously known for r = 2 in the isotropic case and for r = 1 in the anisotropic case, can also be obtained from the standard correlation function approach, as pointed out by Milton.     

Universality and scaling of correlations between zeros on complex manifolds

We study the limit as N→∞ of the correlations between simultaneous zeros of random sections of the powers L ^N of a positive holomorphic line bundle L over a compact complex manifold M, when distances are rescaled so that the average density of zeros is independent of N. We show that the limit correlation is independent of the line bundle and depends only on the dimension of M and the codimension of the zero sets. We also provide some explicit formulas for pair correlations. In particular, we prove that Hannay’s limit pair correlation function for SU(2) polynomials holds for all compact Riemann surfaces. Oblatum 17-VI-1999 & 31-III-2000?Published online: 5 June 2000     

Calculation of the asymptotics of the two-point correlation function for one-dimensional Bose gas

We consider a quantum field-theoretical model which describes spatially-nonhomogeneous, one-dimensional, repulsive Bose gas in an external harmonic potential. The two-point correlation function is calculated in the framework of functional integration. The corresponding functional integrals are estimated by means of stationary phase approximation. Asymptotic estimates are obtained in the limit as the temperature tends to zero while the volume occupied by the quasi-condensate increases. A power-law behavior is established for the correlation function in this limit. It is shown that the power-law behavior is governed by the critical exponent depending on spatial arguments. Bibliography 32 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 347, 2007, pp. 56–74.     

On the relationship between the moving finite-element procedure and best piecewise L2 fits with adjustable nodes

It is shown that, for a class of time-dependent partial differential equations of the form u_t = ??u, one step of the moving finite-element (MFE) procedure corresponds to one iteration of an algorithm for obtaining best L₂ fits with adjustable nodes to continuous functions. In the steady-state limit the MFE procedure gives the best fit of ??u, with adjustable nodes, to the null function. For first-order partial differential equations, the MFE procedure moves nodes with approximate characteristic nodal speeds. We identify an additional speed component arising directly from the L₂ projection which seeks a best fit in the sense described above. © 1994 John Wiley & Sons, Inc.     

Integral definition of the logarithmic function and the derivative of the exponential function in calculus

Defining the logarithmic function as a definite integral with a variable upper limit, an approach used by some popular calculus textbooks, is problematic. We discuss the disadvantages of such a definition and provide a way to fix the problem. We also consider a definition-based, rigorous derivation of the derivative of the exponential function that is easier, more intuitive, and complies with the standard definitions of the number e, the logarithmic, and the exponential functions.     

Jensen measures and boundary values of plurisubharmonic functions

We study different classes of Jensen measures for plurisubharmonic functions, in particular the relation between Jensen measures for continuous functions and Jensen measures for upper bounded functions. We prove an approximation theorem for plurisubharmonic functions inB-regular domain. This theorem implies that the two classes of Jensen measures coincide inB-regular domains. Conversely we show that if Jensen measures for continuous functions are the same as Jensen measures for upper bounded functions and the domain is hyperconvex, the domain satisfies the same approximation theorem as above. The paper also contains a characterisation in terms of Jensen measures of those continuous functions that are boundary values of a continuous plurisubharmonic function.     

The XX and Ising Limits in Integral Formulas for Finite-Temperature Correlation Functions of the XXZ Chain

We consider a multiple-integral representation for a one-parameter generating function of the finite temperature S^z-S^z correlation functions of the antiferromagnetic spin-1/2 XXZ chain in the XX limit and in the Ising limit. We show how the multiple integrals reduce to single integrals in these limits, thus reproducing known results. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 146, No. 1, pp. 146–160, January, 2006.     

Massless Interacting Scalar Quantum Fields in deSitter Spacetime

We generalize key aspects of gr-qc/1010.5367 (and also gr-qc/1010.5327) to the case of massless lf²ⁿ{\lambda \phi^{2n}} quantum field theory on deSitter spacetime. As in that paper, our key objective is to derive a suitable “Mellin-Barnes-type” representation of deSitter correlation functions in a deSitter-invariant state, which holds to arbitrary orders in perturbation theory, and which incorporates renormalization. The representation is suitable for the study of large distance/time properties of correlation functions. It is arrived at via an analytic continuation from the corresponding objects on the sphere, and, as in the massive case, relies on the use of graph-polynomials and their properties, as well as other tools. However, the perturbation expansion is organized somewhat differently in the massless case, due to the well-known subtleties associated with the “zero-mode” of the quantum field. In particular, the correlation functions do not possess a well-defined limit as the self-coupling constant of the field goes to zero, reflecting the well-known non-existence of a deSitter invariant state in the free massless scalar theory. We establish that generic correlation functions cannot grow more than polynomially in proper time for large time-like separations of the points. Our results thus leave open the possibility of quantum-induced IR-instabilities of deSitter spacetime on very large time-scales.     

Cugliandolo-Kurchan equations for dynamics of Spin-Glasses

We study the Langevin dynamics for the family of spherical p-spin disordered mean-field models of statistical physics. We prove that in the limit of system size N approaching infinity, the empirical state correlation and integrated response functions for these N-dimensional coupled diffusions converge almost surely and uniformly in time, to the non-random unique strong solution of a pair of explicit non-linear integro-differential equations intensively studied by Cugliandolo and Kurchan. Research partially supported by NSF grants #DMS-0406042, #DMS-FRG-0244323     

A generalization of the concept of <Emphasis Type="Italic">q</Emphasis>-fractional integrals

In this paper, we consider the fractional q-integral with variable lower limit of integration. We prove the semigroup property of these integrals, and a formula of Leibniz type. Finally, we evaluate fractional q-integrals of some functions. The consideration of q-exponential function in that sense leads to q-analogs of Mittag-Leffier function.     

Quasi-Profile Log Likelihoods for Unbiased Estimating Functions

This paper presents a new quasi-profile loglikelihood with the standard kind of distributional limit behaviour, for inference about an arbitrary one-dimensional parameter of interest, based on unbiased estimating functions. The new function is obtained by requiring the corresponding quasi-profile score function to have bias and information bias of order O(1). We illustrate the use of the proposed pseudo-likelihood with an application to robust inference in linear models.     

Characterization of Pointwise Hölder Regularity

We study different characterizations of the pointwise Hölder spacesC^s(x₀), including rate of approximation by smooth functions and iterated differences. As an application of our results we study the class of functions that are Hölder exponents and prove that the Hölder exponent of a continuous function is the limit inferior of a sequence of continuous functions, thereby improving a theorem of S. Jaffard.