A new symmetric interior penalty discontinuous Galerkin formulation for the Serre–Green–Naghdi equations

In this work, we investigate the construction of a new discontinuous Galerkin discrete formulation to approximate the solution of Serre–Green–Naghdi (SGN) equations in the one-dimensional horizontal framework. Such equations describe the time evolution of shallow water free surface flows in the fully nonlinear and weakly dispersive asymptotic approximation regime. A new non-conforming discrete formulation belonging to the family of symmetric interior penalty discontinuous Galerkin methods is introduced to accurately approximate the solutions of the second order elliptic operator occurring in the SGN equations. We show that the corresponding discrete bilinear form enjoys some consistency and coercivity properties, thus ensuring that the corresponding discrete problem is well-posed. The resulting global discrete formulation is then validated through an extended set of benchmarks, including convergence studies and comparisons with data taken from experiments.     

Fillmore–Springer–Cnops Construction Implemented in <Literal><Emphasis FontCategory="SansSerif">GiNaC</Emphasis></Literal>

This is an implementation of the Fillmore–Springer–Cnops construction (FSCc) based on the Clifford algebra capacities [10] of the GiNaC computer algebra system. FSCc linearises the linear-fraction action of the M?bius group. This turns to be very useful in several theoretical and applied fields including engineering. The core of this realisation of FSCc is done for an arbitrary dimension, while a subclass for two dimensional cycles add some 2D-specific routines including a visualisation to PostScript files through the MetaPost or Asymptote software. This library is a backbone of many result published in [9], which serve as illustrations of its usage. It can be ported (with various level of required changes) to other CAS with Clifford algebras capabilities.     

Comments on “Embedded pair of extended Runge–Kutta–Nyström type methods for perturbed oscillators”

In a recent paper Fang [Embedded pair of extended Runge–Kutta–Nyström type methods for perturbed oscillators, Appl. Math. Modell. (2009), doi:10.1016/j.apm.2009.12.004] considered the embedded pair of extended Runge–Kutta–Nyström type methods for perturbed oscillators and analyzed numerical stability and phase properties of the methods. The authors claimed that their methods are based on the order conditions of extended Runge–Kutta–Nyström type methods presented by Yang et al. [Extended RKN-type methods for numerical integration of perturbed oscillators, Comput. Phys. Commun. 180 (2009) 1777–1794]. However, some careless mistakes have been made in that paper. For this reason we will make some comments on that paper.     

Spatially homogeneous solutions of the Vlasov–Nordström–Fokker–Planck system

The Vlasov–Nordström–Fokker–Planck system describes the evolution of self-gravitating matter experiencing collisions with a fixed background of particles in the framework of a relativistic scalar theory of gravitation. We study the spatially-homogeneous system and prove global existence and uniqueness of solutions for the corresponding initial value problem in three momentum dimensions. Additionally, we study the long time asymptotic behavior of the system and prove that even in the absence of friction, solutions possess a non-trivial asymptotic profile. An exact formula for the long time limit of the particle density is derived in the ultra-relativistic case.     

Mean field dynamics of fermions and the time-dependent Hartree–Fock equation

The time-dependent Hartree–Fock equations are derived from the N-body linear Schrödinger equation with the mean-field scaling in the limit N→+∞ and for initial data that are close to Slater determinants. Only the case of bounded, symmetric binary interaction potentials is treated in this work. We prove that, as N→+∞, the first partial trace of the N-body density operator approaches the solution of the time-dependent Hartree–Fock equations (in operator form) in the sense of the trace norm.     

Applications aux sommes elliptiques d'Apostol–Dedekind–Zagier

In this Note, we give two applications to our work [Bayad, C. R. Acad. Sci. Paris, Ser. I 339 (2004); DOI: 10.1016/j.crma.2004.07.018] concerning multiple elliptic Apostol–Dedekind–Zagier sums. These elliptic sums are defined by means of certain Jacobi modular forms of two variables $D_{\tau }(z\text{;}\phi )$. When $\text{Im}(\tau )\to \infty$, these elliptic sums give the classical Apostol–Dedekind–Zagier multiple sums [Apostol, Duke Math. J. 17 (1950) 147–157, Pacific. J. Math 2 (1952) 1–9; Zagier, Math. Ann, 202 (1973) 149–172]. We give a reciprocity law for these sums. To cite this article: A. Bayad, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

A Radon–Nikodým theorem for Fréchet measures

We apply results in operator space theory to the setting of multidimensional measure theory. Using the extended Haagerup tensor product of Effros and Ruan, we derive a Radon–Nikodým theorem for bimeasures and then extend the result to general Fréchet measures (scalar-valued polymeasures). We also prove a measure-theoretic Grothendieck inequality, provide a characterization of the injective tensor product of two spaces of Lebesgue integrable functions, and discuss the possibility of a bounded convergence theorem for Fréchet measures.     

Sharp reversed Hardy–Littlewood–Sobolev inequality on <Emphasis Type="Bold">R</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

A damped nonlinear wave equation with a degenerate and nonlocal damping term is considered. Well-posedness results are discussed, as well as the exponential stability of the solutions. The degeneracy of the damping term is the novelty of this stability approach.     

A One-Parametric Method for Determining Parameters in the Schwarz–Christoffel Integral

We propose a method for determining parameters in the Schwarz–Christoffel integral. The desired mapping embeds into a one-parametric family of conformal mappings of the upper half-plane onto the family of polygons which was obtained by shifting one or several vertices of some initial polygon with angle preservation. We consider the case when the family of polygons and the initial polygon have the same number of vertices; the case when the family of polygons has two mobile vertices coinciding at the initial moment and not coinciding with other vertices; and the other case that the family of polygons is a polygon with mobile cut. The problem of finding the parameters of a family of mappings is reduced to integrating some system of ordinary differential equations.

     

Monge–Ampère measures on subvarieties

In this article we address the question whether the complex Monge–Ampère equation is solvable for measures with large singular part. We prove that under some conditions there is no solution when the right-hand side is carried by a smooth subvariety in Cⁿ${\mathbb{C}}^n$ of dimension k<n$k<n$.     

Cramér–Karhunen–Loève representation and harmonic principal component analysis of functional time series

We develop a doubly spectral representation of a stationary functional time series, and study the properties of its empirical version. The representation decomposes the time series into an integral of uncorrelated frequency components (Cramér representation), each of which is in turn expanded in a Karhunen–Loève series. The construction is based on the spectral density operator, the functional analogue of the spectral density matrix, whose eigenvalues and eigenfunctions at different frequencies provide the building blocks of the representation. By truncating the representation at a finite level, we obtain a harmonic principal component analysis of the time series, an optimal finite dimensional reduction of the time series that captures both the temporal dynamics of the process, as well as the within-curve dynamics. Empirical versions of the decompositions are introduced, and a rigorous analysis of their large-sample behaviour is provided, that does not require any prior structural assumptions such as linearity or Gaussianity of the functional time series, but rather hinges on Brillinger-type mixing conditions involving cumulants.     

Bishop–Phelps–Bollobás property for certain spaces of operators

We characterize the Banach spaces Y   for which certain subspaces of operators from _L1(μ)$L_1(\mu )$ into Y have the Bishop–Phelps–Bollobás property in terms of a geometric property of Y, namely AHSP. This characterization applies to the spaces of compact and weakly compact operators. New examples of Banach spaces Y with AHSP are provided. We also obtain that certain ideals of Asplund operators satisfy the Bishop–Phelps–Bollobás property.     

A Bishop–Phelps–Bollobás type theorem for uniform algebras

This paper is devoted to showing that Asplund operators with range in a uniform Banach algebra have the Bishop–Phelps–Bollobás property, i.e., they are approximated by norm attaining Asplund operators at the same time that a point where the approximated operator almost attains its norm is approximated by a point at which the approximating operator attains it. To prove this result we use the weak^∗${}^{\ast }$-to-norm fragmentability of weak^∗${}^{\ast }$-compact subsets of the dual of Asplund spaces and we need to observe a Urysohn type result producing peak complex-valued functions in uniform algebras that are small outside a given open set and whose image is inside a Stolz region.     

Marcus–Lushnikov processes,Smoluchowski’s and Flory’s models

The Marcus–Lushnikov process is a finite stochastic particle system in which each particle is entirely characterized by its mass. Each pair of particles with masses x$x$ and y$y$ merges into a single particle at a given rate K(x,y)$K(x,y)$. We consider a strongly gelling   kernel behaving as K(x,y)=x^αy+xy^α$K(x,y)=x^{\alpha }y+x{y}^{\alpha }$ for some α∈(0,1]$\alpha \in (0,1]$. In such a case, it is well-known that gelation occurs, that is, giant particles emerge. Then two possible models for hydrodynamic limits of the Marcus–Lushnikov process arise: the Smoluchowski equation, in which the giant particles are inert, and the Flory equation, in which the giant particles interact with finite ones.     

Codimension-<Emphasis Type="Italic">p</Emphasis> Paley–Wiener theorems

We obtain the generalized codimension-p Cauchy–Kovalevsky extension of the exponential function in R ^m =R ^p ⊕R ^q , where p>1, , and prove the corresponding codimension-p Paley–Wiener theorems.     

Non existence of <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">2</Emphasis></Superscript>–compact solutions of the Kadomtsev–Petviashvili II equation

We prove that there is no nontrivial solution of the Kadomtsev–Petviashvili II equation (KP II equation) which is L² compact (i.e. uniformly localized in L² norm) and travel to the right in the x variable. This result extends the previous work of de Bouard and Saut [3] stating that there is no traveling wave solution for the KP II equation. The proof uses a monotonicity property of the L² mass for solutions of the KP II equation (similar to the one for the KdV equation [12], [14]) and two virial type relations. The result still holds for some natural generalizations of the KP II equation (general nonlinearity, higher dispersion) and does not rely on the integrability of the equation. Mathematics Subject Classification (2000):35Q53, 35B05     

Fisher–KPP equation on the Heisenberg group

In this paper, we consider the Fisher–KPP equation on the Heisenberg group. We discuss the existence of global solutions, asymptotic behavior of global solutions and blow-up solutions. Moreover, we extend the obtained results to the time-fractional Fisher–KPP equation on the Heisenberg group.