Moment Propagation for Weak Solutions to the Vlasov–Poisson System

We consider weak solutions to the Cauchy problem for the three dimensional Vlasov–Poisson system of equations. We obtain a propagation result for any velocity moment of order > 2 as well as a uniqueness statement in ?³. In the periodic case, we show that velocity moments of order > 14/3 are propagated.     

Cumulant operators and moments of the Itô and Skorohod integrals

We propose a formula for the computation of the moments of all orders of Itô and Skorohod stochastic integrals with respect to Brownian motion, based on cumulant operators defined by the Malliavin calculus. Some characterizations of Gaussian distributions for stochastic integrals are recovered as a consequence.     

A product estimate for Ginzburg–Landau and application to the gradient-flow

We prove a new inequality for the Jacobian (or vorticity) associated to the Ginzburg–Landau energy in any dimension, and give static and dynamical corollaries. We then present a method to prove convergence of gradient-flows of families of energies which Gamma-converge to a limiting energy, which we apply to establish, thanks to the previous dynamical estimate, the limiting dynamical law of a finite number of vortices for the heat-flow of Ginzburg–Landau in dimension 2, with and without magnetic field. To cite this article: E. Sandier, S. Serfaty, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Vanishing integrals for Hall–Littlewood polynomials

It is well known that if one integrates a Schur function indexed by a partition λ over the symplectic (resp. orthogonal) group, the integral vanishes unless all parts of λ have even multiplicity (resp. all parts of λ are even). In a recent paper of Rains and Vazirani, Macdonald polynomial generalizations of these identities and several others were developed and proved using Hecke algebra techniques. However, at q = 0 (the Hall–Littlewood level), these approaches do not work, although one can obtain the results by taking the appropriate limit. In this paper, we develop a direct approach for dealing with this special case. This technique allows us to prove some identities that were not amenable to the Hecke algebra approach. Moreover, we are able to generalize some of the identities by introducing extra parameters. This leads us to a finite-dimensional analog of a recent result of Warnaar, which uses the Rogers–Szegő polynomials to unify some existing summation type formulas for Hall–Littlewood functions.     

Euler’s partition theorem for all moduli and new companions to Rogers–Ramanujan–Andrews–Gordon identities

The Ramanujan Journal - We generalise Euler’s partition theorem involving odd parts and distinct parts for all moduli and provide new companions to...     

Approximation of conjugate differentiable functions by their Abel–Poisson integrals

We obtain the exact values of upper bounds of approximations of classes of periodic conjugate differentiable functions by their Abel–Poisson integrals in uniform and integral metrics. Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 61, No. 1, pp. 73–82, January, 2009.     

Orthogonality relations and Cherednik identities for multivariable Baker–Akhiezer functions

We establish orthogonality relations for the Baker–Akhiezer (BA) eigenfunctions of the Macdonald difference operators. We also obtain a version of Cherednik–Macdonald–Mehta integral for these functions. As a corollary, we give a simple derivation of the norm identity and Cherednik–Macdonald–Mehta integral for Macdonald polynomials. In the appendix written by the first author, we prove a summation formula for BA functions. We also consider more general identities of Cherednik type, which we use to introduce and construct more general, twisted BA functions. This leads to a construction of new quantum integrable models of Macdonald–Ruijsenaars type.     

Wiener–Ingham type inequality for Vilenkin groups and its application to harmonic analysis

An important trigonometric inequality essentially due to Wiener but later on made precise by Ingham concerning the lacunary trigonometric sums \(f(x)=\sum A_ke^{in_kx}\), where \(A_k\)’s are complex numbers, \(n_{-k}=-n_k\) and \(\{n_k\}\) satisfies the small gap condition \((n_{k+1}-n_k)\ge q\ge 1\) for \(k=0,1,2,\ldots \), says that if I is any subinterval of \([-\pi ,\pi ]\) of length \(|I|=2\pi (1+\delta )/q>2\pi /q\) then \(\sum |A_k|^2\le A_{\delta }|I|^{-1}\int _I|f|^2\), \(|A_k|\le A_{\delta }|I|^{-1}\int _I|f|\), wherein \(A_{\delta }\) depends only on \(\delta \). Such an inequality is proved here in the setting of the Vilenkin groups G. The inequality is then applied to generalize the Bernstěin, Szász and Ste?hkin type results concerning the absolute convergence of Fourier series on G.     

Entropy and similarity measure for Atannasov’s interval-valued intuitionistic fuzzy sets and their application

In this study, we first point out the problem of the similarity measure in the literature and then define a new entropy and similarity measure. In order to explore the inter-dependent or interactive characteristics between elements in a set, several Shapley-weighted similarity measures of Atannasov’s interval-valued intuitionistic fuzzy sets are defined by using the well-known Shapley function, which can be seen as an extension of the associated weighted similarity measures. Moreover, if the information about the weights is completely unknown or partially known, models for the optimal fuzzy measures are established, by which the optimal weight vector can be obtained. Finally, an approach to pattern recognition and multi-criteria decision making is developed, and the associated numerical examples are provided to verify the developed methods and demonstrate their practicality and feasibility.     

Moment calculations for piecewise-defined functions: an application to stochastic optimization with coherent risk measures

This work introduces a new analytical approach to the formulation of optimization problems with piecewise-defined (PD) objective functions. First, we introduce a new definition of multivariate PD functions and derive formal results for their continuity and differentiability. Then, we obtain closed-form expressions for the calculation of their moments. We apply these findings to three classes of optimization problems involving coherent risk measures. The method enables one to obtain insights on problem structure and on sensitivity to imprecision at the problem formulation stage, eliminating reliance on ad-hoc post-optimality numerical calculations.     

Weighted restriction type estimates for Grushin operators and application to spectral multipliers and Bochner–Riesz summability

We prove weighted restriction type estimates for Grushin operators. These estimates are then used to prove sharp spectral multiplier theorems as well as Bochner–Riesz summability results with sharp exponent.     

Stabilization and stability for the spherically symmetric Navier–Stokes–Poisson system

We consider global behaviour of viscous compressible flows with spherical symmetry driven by gravitation and an outer pressure, outside a hard core. For a general state function $p=p\left(\rho \right)$, we present global-in-time bounds for solutions with arbitrarily large data. For non-decreasing $p$, the $\omega$-limit set for the density $\rho$ is studied. For increasing $p$, uniqueness and static stability of the stationary solutions (including variational aspects) are investigated. Moreover, stabilization rate bounds toward the statically stable solutions are given and their nonlinear dynamical stability is shown.     

Shift-plethystic trees and Rogers–Ramanujan identities

The Ramanujan Journal - By studying non-commutative series in an infinite alphabet, we introduce shift-plethystic trees and a class of integer compositions as new combinatorial models for the...     

Noncommutative Batalin–Vilkovisky geometry and matrix integrals

I study the new type of supersymmetric matrix models associated with any solution to the quantum master equation of the noncommutative Batalin–Vilkovisky geometry. The asymptotic expansion of the matrix integrals gives homology classes in the Kontsevich compactification of the moduli spaces, which I associated with the solutions to the quantum master equation in my previous paper. I associate with the Bernstein–Leites matrix superalgebra equipped with an odd differentiation, whose square is nonzero, the family of cohomology classes of the compactification. This family is the generating function for the products of the tautological classes. The simplest example of my matrix integrals in the case of dimension zero is a supersymmetric extension of the Kontsevich model of 2-dimensional gravity.     

Two identities for the Bernoulli–Euler numbers

Two identities for the Bernoulli and for the Euler numbers are derived. These identities involve two special cases of central combinatorial numbers. The approach is based on a set of differential identities for the powers of the secant. Generalizations of the Mittag–Leffler series for the secant are introduced and used to obtain closed-form expressions for the coefficients.     

Hyperelliptic integrals and generalized arithmetic–geometric mean

We show how certain determinants of hyperelliptic periods can be computed using a generalized arithmetic-geometric mean iteration, whose initialisation parameters depend only on the position of the ramification points. Special attention is paid to the explicit form of this dependence and the signs occurring in the real domain.