On the traces of Sobolev functions on the boundary of a cusp with a Hölder singularity

For the Sobolev classes W _p ¹ on a “zero” cusp with a Hölder singularity at the vertex, we consider the question of compactness of the embedding of the traces of Sobolev functions into the Lebesgue classes on the boundary of the cusp.     

Exchange the Order of a Successive Integral with Bochner-Martinelli Kernel

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Graphlike Manifolds with the Type of the 1—skeleton of a 4—simplex

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Existence and positivity of a system k-ε with a production term of the Rayleigh-Taylor type

In this paper, we study a modelization of the turbulence which allows us to have a control on the positivity of the kinetic turbulence energy k and the dissipation growth rate of the energy ε. We use for that purpose the maximum principle on a new system modelling the turbulence, written on the variables θ and φ first introduced by Lewandowski (θ = k/ε, φ = ε²/k³).Estimates on θ and φ are given for a turbulent system with a Rayleigh-Taylor type term under a hypothesis of low compressibility of the mean flow, which is more general than the hypothesis of Lewandowski.In a second part, we study a simpler convection diffusion system (the diffusion is a constant) in which there is still the Rayleigh-Taylor term. We show that the presence of this term gives greater solutions of the k, ε system, hence proving that these terms are turbulent kinetic energy production terms.     

Analysis of the one-dimensional Euler–Lagrange equation of continuum mechanics with a Lagrangian of a special form

The equations describing the flow of a one-dimensional continuum in Lagrangian coordinates are studied in this paper by the group analysis method. They are reduced to a single Euler–Lagrange equation which contains two undetermined functions (arbitrary elements). Particular choices of these arbitrary elements correspond to different forms of the shallow water equations, including those with both, a varying bottom and advective impulse transfer effect, and also some other motions of a continuum. A complete group classification of the equations with respect to the arbitrary elements is performed.One advantage of the Lagrangian coordinates consists of the presence of a Lagrangian, so that the equations studied become Euler–Lagrange equations. This allows us to apply Noether’s theorem for constructing conservation laws in Lagrangian coordinates. Not every conservation law in Lagrangian coordinates has a counterpart in Eulerian coordinates, whereas the converse is true. Using Noether’s theorem, conservation laws which can be obtained by the point symmetries are presented, and their analogs in Eulerian coordinates are given, where they exist.     

Reconstruction of the shape of object with near field measurements in a half-plane

We consider a mathematical problem modelling some characteristics of near field optical microscope.We take a monofrequency line source to illuminate a sample with constant index of refraction and use the scattered field data measured near the sample to reconstruct the shape of it. Mixed reciprocity relation and factorization method are applied to solve our problem.Some numerical examples to show the feasibility of the method are presented.     

Properties of the first passage times of the reflected O-U process with a two-sided barrier

In this paper, we give the Laplace transform of the first passage times and obtain the analytic expression of its mean for the reflected Ornstein–Uhlenbeck process with a two-sided barrier for general coefficients.     

Representation of the general solution of an equation of the Cauchy–Riemann type with a supersingular circle and a singular point

For a generalized system of the Cauchy–Riemann type with complex conjugation whose coefficients admit a strong singularity on a circle and a weak singularity at a point, we find an integral representation of the general solution.     

Rate-of-convergence bounds of a finite-difference approximation of the Dirichlet problem for the Helmholtz equation with a coefficient from the “negative” space

Exact difference scheme operators are used to construct a difference scheme that approximates the Dirichlet problem for the equation

Solution of the Thomas–Fermi equation with a convergent approach

The explicit analytic solution of the Thomas–Fermi equation thorough a new kind of analytic technique, namely the homotopy analysis method, was employed by Liao [1]. However, the base functions and the auxiliary linear differential operator chosen were such that the convergence to the exact solution was fairly slow. New base functions and auxiliary linear operator to form a better homotopy are the main concern of the present paper. Optimum convergence control parameter concept is used together with a mathematical proof of the convergence.     

Estimates of the Root Functions of a One-Dimensional Schrödinger Operator with a Strong Boundary Singularity

For any operator defined by the differential operation Lu = ?u″ + q(x)u on the interval G = (0, 1) with complex-valued potential q(x) locally integrable on G and satisfying the inequalities \(\int_{{x_1}}^{{x_2}} {\zeta |(q(\zeta ))|d\zeta \leqslant ln({x_1}/{x_2})} \) and \(\int_{{x_1}}^{{x_2}} {\zeta |(q(1 - \zeta ))|d\zeta \leqslant \gamma ln({x_1}/{x_2})} \) with some constant γ for all sufficiently small 0 < x₁ < x₂, we estimate the norms of root functions in the Lebesgue spaces L _p (G), 1 ≤ p < ∞. We show that for sufficiently small γ these norms satisfy the same estimates asymptotic in the spectral parameter as in the unperturbed case.     

The stability of the viscous shock profiles of the burgers’ equation with a fourth order viscosity

We establish the stability of the viscous shock profiles of the Burgers' equation with fourth order viscosity in a weighted Sobolev space. In the course of our investigation, we develop a new method for dealing with the continuous spectrum of certain types of ordinary and partial differential operators in Hilbert spaces. This method allows us to get results in spaces that are related to L² rather than forcing us to work in spaces related to L^∞.     

On the Absence of a First Order COrrection for the Number of Bound States of a Schrödinger operator with Coulomb Singularity

Contents

1 Introduction 616

2 Regularizaton of the Singular Operator 620

3 Functional Calculus with Projections 621

A The Kutaanheimo–Stiefel Transform and All That 625

B Some Remarks on a Result of Petkov and Robert on the Counting Function in the Context of Compact Group Actions 627

B.1 Introduction and Principal Statements 627

B.2 Sketch of a Proof of Theorem 10 632

B.3 On the Initial Conditions Yielding Periodic Orbits of Regularized and Unregularized Hamiltonians 634

B.4 On the Vanishing of the Second Coefficient 635     

Asymptotics of eigenvalues of the two-dimensional Dirichlet boundary-value problem for the Lamé operator in a domain with a small hole

The Dirichlet boundary-value problem for the eigenvalues of the Lamé operator in a twodimensional bounded domain with a small hole is studied. The asymptotics of the eigenvalue of this boundary-value problem is constructed and justified up to the power of the parameter defining the diameter of the hole.     

Optimal sequencing of a set of positive numbers with the variance of the sequence’s partial sums maximized

We consider the problem of sequencing a set of positive numbers. We try to find the optimal sequence to maximize the variance of its partial sums. The optimal sequence is shown to have a nice structure. It is interesting to note that the symmetric problem which aims at minimizing the variance of the same partial sums is proved to be NP-complete in the literature.     

Expansion in the eigenfunctions of the schrödinger equation with a potential having a periodic asymptotic behavior

A theorem is proved regarding the expansion in the eigenfunctions of the one-dimensional Schrödinger equationL = ?d ^z/dx ²+q(x)(?∞<x<∞)with a potential q(x), satisfying the condition $$\int\limits_0^{ + \infty } {(1 + x^2 )|q(x) - q_ \pm (x)|dx< \infty ,} $$ where q±(x) are periodic functions.     

Asymptotics of Solutions of the Sturm–Liouville Equation with Respect to a Parameter

On a finite segment [0, l], we consider the differential equation

On the Flag f-vector of a Graded Lattice with Nontrivial Homology

It is proved that the Boolean algebra of rank n minimizes the flag f-vector among all graded lattices of rank n, whose proper part has nontrivial top-dimensional homology. The analogous statement for the flag h-vector is conjectured in the Cohen-Macaulay case.