Mayer–Vietoris sequence for generating families in diffeological spaces

We prove a version of the Mayer–Vietoris sequence for De Rham differential forms in diffeological spaces. It is based on the notion of a generating family instead of that of a covering by open subsets.     

A Discrete-Time Clark–Ocone Formula and its Application to an Error Analysis

In this paper, we will establish a discrete-time version of Clark(–Ocone–Haussmann) formula, which can be seen as an asymptotic expansion in a weak sense. The formula is applied to the estimation of the error caused by the martingale representation. Throughout, we use another distribution theory with respect to Gaussian rather than Lebesgue measure, which can be seen as a discrete Malliavin calculus.     

Estimation of the Density of Hypoelliptic Diffusion Processes with Application to an Extended Itô's Formula

We prove a uniform bound for the density, p _t (x), of the solution at time t(0, 1] of a 1-dimensional stochastic differential equation, under hypoellipticity conditions. A similar bound is obtained for an expression involving the distributional derivative (with respect to x) of p _t (x). These results are applied to extend the Itô formula to the composition of a function (satisfying slight regularity conditions) with a hypoelliptic diffusion process in the spirit of the work of Föllmer et al. ⁽⁵⁾     

Applications of an Explicit Formula for the Generalized Euler Numbers

The authors establish an explicit formula for the generalized Euler NumbersE2n^（x）, and obtain some identities and congruences involving the higher＇order Euler numbers, Stirling numbers, the central factorial numbers and the values of the Riemann zeta-function.     

A Stereological Version of the Gauss–Bonnet Formula

We give a stereological version of the Gauss–Bonnet formula in order to compute the Euler characteristic of a domain with boundary in a smooth orientable surface in ³, by looking at contacts with a 'sweeping' plane.     

A Two-Level Procedure for the Global Optimum Design of Composite Modular Structures—Application to the Design of an Aircraft Wing

This work concerns a two-level procedure for the global optimum design of composite modular structures. The case-study considered is the least weight design of a stiffened wing-box for an aircraft structure. The method is based on the use of the polar formalism and on a genetic algorithm. In the first level of the procedure, the optimal structure is designed as it was composed by a single equivalent layer, while a laminate realizing the optimal structure is found in the second level. The method is able to automatically find the optimal number of modules, no simplifying assumptions are used, and it can be easily generalized to other problems. The work is divided into two parts: the theoretical formulation in this first part, the genetic procedure and some numerical examples in the second one.     

A Two-Level Procedure for the Global Optimum Design of Composite Modular Structures—Application to the Design of an Aircraft Wing

This work concerns a two-level procedure for the global optimum design of composite modular structures. The case-study considered is the least weight design of a stiffened wing-box for an aircraft structure. The method is based on the use of the polar formalism and on a genetic algorithm. In the first level of the procedure, the optimal structure is designed as composed by a single equivalent layer, while a laminate realizing the optimal structure is found in the second level. The method is able to automatically find the optimal number of modules; no simplifying assumptions are used and it can be easily generalized to other problems. The work is divided into two parts: the theoretical formulation in the first part, the genetic procedure and some numerical examples in this second one.     

Extension of the Ahiezer–Kac Determinant Formula to the Case of Real-Valued Symbols with Two Real Zeros

The Fredholm determinant asymptotics for self-adjoint convolution operators on finite intervals with real symbols vanishing on the real axis is studied. Explicit formulae are obtained in the case where the symbol satisfies the generalized zero index condition and has only two simple zeros of analytic type. These formulae are direct extensions of the Ahiezer–Kac–Szegö limit theorem which, in particular, take into account the oscillating character of the asymptotics.     

A Free Boundary Problem for an Elliptic–Parabolic System: Application to a Model of Tumor Growth

Abstract

In this article, we study a free boundary problem for a system of two partial differential equations, one parabolic and other elliptic. The system models the growth of a tumor with arbitrary initial shape. We establish the existence and uniqueness of a solution for some time interval. In the special case where we only have the elliptic equation, the problem coincides with the Hele–Shaw problem.     

An Alexandrov–Fenchel-Type Inequality in Hyperbolic Space with an Application to a Penrose Inequality

We prove a sharp Alexandrov–Fenchel-type inequality for star-shaped, strictly mean convex hypersurfaces in hyperbolic n-space, n ≥ 3. The argument uses two new monotone quantities along the inverse mean curvature flow. As an application we establish, in any dimension, an optimal Penrose inequality for asymptotically hyperbolic graphs carrying a minimal horizon, with the equality occurring if and only if the graph is an anti-de Sitter–Schwarzschild solution. This sharpens previous results by Dahl–Gicquaud–Sakovich and settles, for this class of initial data sets, the conjectured Penrose inequality for time-symmetric space–times with negative cosmological constant. We also explain how our methods can be easily adapted to derive an optimal Penrose inequality for asymptotically locally hyperbolic graphs in any dimension n ≥ 3. When the horizon has the topology of a compact surface of genus at least one, this provides an affirmative answer, for this class of initial data sets, to a question posed by Gibbons, Chru?ciel and Simon on the validity of a Penrose-type inequality for exotic black holes.     

Titchmarsh–Weyl Formula for the Spectral Density of a Class of Jacobi Matrices in the Critical Case

Functional Analysis and Its Applications - We consider a class of Jacobi matrices with unbounded entries in the so-called critical (double root, Jordan block) case. We prove a formula which relates...     

Application of the homotopy analysis method to the Poisson–Boltzmann equation for semiconductor devices

This paper describes the application of a recently developed analytic approach known as the homotopy analysis method to derive an approximate solution to the nonlinear Poisson–Boltzmann equation for semiconductor devices. Specifically, this paper presents an analytic solution to potential distribution in a DG-MOSFET (Double Gate-Metal Oxide Semiconductor Field Effect Transistor). The DG-MOSFET represents one of the most advanced device structures in semiconductor technology and is a primary focus of modeling efforts in the semiconductor industry.     

A Geometric Uncertainty Principle with an Application to Pleijel’s Estimate

Let \({\Omega \subset \mathbb{R}^2}\) be an open, bounded domain and \({\Omega = \bigcup_{i = 1}^{N} \Omega_{i}}\) be a partition. Denote the Fraenkel asymmetry by \({0 \leq \mathcal{A}(\Omega_i) \leq 2}\) and write $$D(\Omega_i) := \frac{|\Omega_{i}| - {\rm min}_{1 \leq j \leq N}{|\Omega_{j}|}}{|\Omega_{i}|}$$ with \({0 \leq D(\Omega_{i}) \leq 1}\) . For N sufficiently large depending only on \({\Omega}\) , there is an uncertainty principle $$\left(\sum_{i=1}^{N}{\frac{|\Omega_{i}|}{|\Omega|}{\mathcal{A}}(\Omega_i)}\right) + \left(\sum_{i=1}^{N}{\frac{|\Omega_i|}{|\Omega|}D(\Omega_i)}\right) \geq \frac{1}{60000}.$$ The statement remains true in dimensions \({n \geq 3}\) for some constant \({c_{n} > 0}\) . As an application, we give an (unspecified) improvement of Pleijel’s estimate on the number of nodal domains of a Laplacian eigenfunction and an improved inequality for a spectral partition problem.     

A Generalization of the Mean Size Formula of Wavelet Packets in Lp

The purpose of this paper is to investigate the mean size formula of wavelet packets in Lp for 0 〈 p ≤ ∞. We generalize a mean size formula of wavelet packets given in terms of the p-norm joint spectral radius and we also give some asymptotic formulas for the Lp-norm or quasi-norm on the subdivision trees. All results will be given in the general setting,     

On the Krein–Rechtman Theorem in the Presence of Multiple Points

Mathematical Notes -     

Uniform Estimates for the Fourier Transform of Surface Carried Measures in ?3 and an Application to Fourier Restriction

Let S be a hypersurface in \BbbR³{\Bbb{R}}^{3} which is the graph of a smooth, finite type function φ, and let μ=ρ  dσ be a surface carried measure on S, where dσ denotes the surface element on S and ρ a smooth density with sufficiently small support. We derive uniform estimates for the Fourier transform [^(m)]\hat{\mu} of μ, which are sharp except for the case where the principal face of the Newton polyhedron of φ, when expressed in adapted coordinates, is unbounded. As an application, we prove a sharp L ^p -L ² Fourier restriction theorem for S in the case where the original coordinates are adapted to φ. This improves on earlier joint work with M. Kempe.     

Application of the trigonal curve to the Blaszak–Marciniak lattice hierarchy

We develop a method for constructing algebro-geometric solutions of the Blaszak–Marciniak (BM) lattice hierarchy based on the theory of trigonal curves. We first derive the BM lattice hierarchy associated with a discrete (3×3)-matrix spectral problem using Lenard recurrence relations. Using the characteristic polynomial of the Lax matrix for the BM lattice hierarchy, we introduce a trigonal curve with two infinite points, which we use to establish the associated Dubrovin-type equations. We then study the asymptotic properties of the algebraic function carrying the data of the divisor and the Baker–Akhiezer function near the two infinite points on the trigonal curve. We finally obtain algebro-geometric solutions of the entire BM lattice hierarchy in terms of the Riemann theta function.