We consider a billiard system inside a ring formed by two ellipses with common foci F_1 and F_2 affected by Coulomb forces, condensed in F_1 and F_2 with charges γ_1 and γ_2 , respectively. The reflection from the boundary of the domain is supposed to be essentially elastic. This billiard is Liouville integrable in terms of piecewise smoothness. An explicit formula of the additional first integral was found. We study the topology of the Liouville foliation of this system in several cases: γ_1<0,γ_2=0 (Kepler case); γ_1>0,γ_2=0 ; γ_1=γ_2 . For each of these cases, Fomenko and Fomenko–Zieschang invariants are calculated.
Luminosity points of sets are studied; these are the points for which the Kolmogorov (solar) condition for a best approximant holds. We obtain sufficient conditions for a given point of a set to be a luminosity point of this set. The results are formulated in terms of points of monotone connectedness of the set. We consider the characteristics of luminosity points which involve individual characteristics of both the space and the set (points of monotone connectedness of a set and rotund points of the unit sphere of the space). Prox-monotone path-connected sets are introduced; for such a set M , if a point y is a nearest point from this set for some point x∉ M , then y is a point of monotone connectedness of M . We also show that each boundedly compact prox-monotone path-connected set (prox-Menger connected set) is a sun. Applications to the space c_0 are also given.
In the paper, theorems on the decomposition of the Wiener measure with respect to the orbits of the action of the diffeomorphism group are proved.
The spreading of a droplet of incompressible fluid over a substrate under the action of capillary forces and gravity is studied within a phase-field framework. An asymptotic analysis shows that the solution of the phase-field system (the regularized problem) converges to the solution of the corresponding limit problem when a regularization parameter tends to zero. The theoretical results are illustrated by numerical simulations for hydrophobic and hydrophilic substrates.
A system of two differential equations with nonlinear delayed feedback containing a large parameter is considered. The nonlinearity is assumed to be compactly supported, i.e., it takes the zero value outside a certain domain of variation of the argument. The construction of asymptotics of solutions for sufficiently large values of the parameter appearing in the original system is investigated. Conditions for the existence of various types of relaxation oscillations with asymptotically large amplitude are identified. An asymptotics of the corresponding solutions is constructed, and it is shown that their dynamical properties can be complicated. The research methodology is based on the application of a special large parameter method using the reduction of the study of the original system to the analysis of constructed finite-dimensional mappings. The possibility of obtaining asymptotic formulas for solutions of the original system is due to the fact that the significant influence of nonlinearity occurs only on relatively small time intervals while, on the “main” time interval, the behavior of the solutions is determined by a linear system of differential equations.
The properties of the operator describing flexural waves are described. This operator is defined by a fourth-order differential expression with a singular point. A Hilbert space is constructed, and the domain of the operator and the Gelfand triple are found. For a model operator corresponding to a quadratic profile, the spectrum and corresponding (distributional) eigenfunctions are explicitly calculated.
The structure of norm continuous representations of some locally compact groups in a Hilbert spaces is refined.
In L_2(ℝ^d) , we consider a self-adjoint bounded operator 𝔸_ε(t) , t≥ 0 , ε >0 , of the form (𝔸_ε(t) u) (𝐱) = ε^-d-2∫_ℝ^d a((𝐱 - 𝐲)/ ε) μ(𝐱 /ε, 𝐲 /ε, t/ε^2) ( u(𝐱) - u(𝐲) ) d𝐲. It is assumed that a(·)∈ L_1(ℝ^d) is a nonnegative function such that ∫_ℝ^d | 𝐱 |^3 a(𝐱) d𝐱<∞ and a(-𝐱) = a(𝐱) ; the function μ(𝐱,𝐲,t) is measurable and periodic in all variables, moreover, μ(𝐱,𝐲,t) = μ(𝐲,𝐱,t) and 0< μ_- ⩽μ(𝐱,𝐲,t) ⩽μ_+< ∞ . Let u_ε(𝐱,t) , 𝐱∈ℝ^d , t ≥ 0 , be the solution of the Cauchy problem ∂_t u_ε(𝐱,t) = - 𝔸_ε(t) u_ε(𝐱,t), u_ε(𝐱,0) = φ(𝐱), where φ∈ L_2(ℝ^d) . We show that, for a chosen t>0 , the solution u_ε(·,t) converges in L_2(ℝ^d) , as ε→ 0 , to the solution u_0(·,t) of the homogenized problem ∂_t u_0(𝐱,t)= - 𝔸^0 u_0(𝐱,t), u_0(𝐱,0) = φ(𝐱), where 𝔸^0= - div g^0 ∇ and g^0 is a positive definite effective matrix. The following error estimate holds: u_ε(·,t) - u_0(·,t)_L_2(ℝ^d)≤C ε/(t + ε^2)^1/2φ_L_2(ℝ^d), t ≥ 0, ε >0.
In this paper, we investigate the Chafee–Infante equation in a locally periodic perforated domain with a small parameter ε>0 characterizing the diameter of the cavities and the distance between them. The asymptotic behavior of the problem depends on the scaling of the boundary effects, characterized by the parameter ε^θ in Fourier condition on the boundary of the cavities, leading to two distinct regimes. In contrast to the critical case ( θ=1 ), in which the condition at the boundary of the cavities gives a potential term and new coefficients of the elliptic operator to the homogenized equation, in the subcritical and supercritical cases the situation changes radically. The subcritical case ( θ > 1 ): The influence of boundary conditions on perforations does not contribute to the potential term but still changes the coefficients of the homogenized equation. We prove that both trajectory and global attractors of the original problem converge to those of the homogenized problem. The supercritical case ( θ < 1 ): In this case, the solutions degenerate as the small parameter tends to zero. It has been proven that the attractors also tend to zero in the corresponding topology. In the subcritical case, the homogenized limit is rigorously derived via auxiliary cell problems and asymptotic expansions. Appropriate functional spaces with weak topology are introduced, and the existence of trajectory attractors is established. In the supercritical case, we use integral estimates to prove the degeneration of solutions and attractors.
The purpose of this paper is to carry out a comprehensive analysis of the properties of degenerate bivariate Appell polynomials by making extensive use of algebraic and probabilistic techniques. The determinant expressions for the degenerate cosine-Appell and sine-Appell polynomials and for their conjugate forms are established by employing several recursive formulas. Further, we focus on exploring probabilistic properties, in which there associated a family of degenerate bivariate Appell polynomials to every random variable χ with some exponential moments e^χ u , which provides some powerful tools for a deeper understanding of the behavior and properties of degenerate bivariate Appell polynomials. Examples are framed which introduces a random variable whose mass function is given in terms of degenerate bivariate Bernoulli polynomials. Some expressions for the expectation E[χ] of the random variable associated with the degenerate bivariate Bernoulli polynomials are derived. It is demonstrated that the probabilistic approach offers interesting new results and developments for these polynomials.
In a general three-dimensional domain we consider a second order differential operator with variable coefficients subject to an arbitrary boundary condition; the operator is supposed to be m -sectorial. We show how to add a point interaction supported by a given curve Γ with a complex-valued coupling function to this operator. We provide a rigorous way for defining such an operator with point interaction and study its basic properties. Our approach is based on the main idea used to define the Laplacian in the three-dimensional space on a given curve, but we modify this idea quite essentially avoiding at the same time any implicit Green functions. Our definition provides an effective tool for studying general operators in three-dimensional domains with point interactions on given curves.
This paper derives the normal ordering expansion for the operator (a^†a+r)_n,λ , using two methods: the action of the generalized displacement operator on number states and the determination of a recurrence relation for expansion coefficients. Here a^† and a are respectively the boson creation and annihilation operators. We also deduce the inverse relation for this expansion. Applying coherent state techniques, we establish identities for the diagonal matrix elements ⟨ z| (a^† a+r)_k,λ|z⟩ , which directly yield a Dobinski-like formula for the degenerate r -Bell numbers. Furthermore, we present a new recurrence relation for these combinatorial numbers.
We study solar properties of monotone path-connected and Menger-connected boundedly weakly compact sets in normed linear spaces. We show that any Menger-connected boundedly weakly compact subset of a separable normed space is a sun. We also prove that any nonempty monotone path-connected weakly compact subset of normed linear space is a sun. For boundedly weakly compact subsets of the space C(Q,ℝ) , we obtain a criterion of monotone path-connectedness in terms of solarity. Namely, if M is a nonempty boundedly weakly compact set in C(Q) , then M is a sun if and only if M is monotone path-connected. We show that, given a monotone path-connected subset M of a rotund space X , if x∉ M has a unique nearest point in M , then x is a solar point for M . As a corollary, we prove that in a rotund space every proximinal monotonously path-connected set is a Chebyshev sun.
The classical Atiyah–Bott formula (1967) expresses the Lefschetz number of a geometric endomorphism of an elliptic complex of (pseudo)differential operators on a closed smooth manifold as the sum of contributions of fixed points of the corresponding diffeomorphism, assuming that all fixed points are nondegenerate. These contributions explicitly depend only on the endomorphism itself but not on the operators forming the complex. In 1999, one of the authors, together with B.-W. Schulze, B. Yu. Sternin, and V. E. Shatalov, generalized this formula to the case of manifolds with conical singularities (or, equivalently, with cylindrical ends), where the contributions of interior fixed points are supplemented by those of fixed cylindrical ends (which already depend on the operators of the complex themselves). In recent decades, a number of papers have appeared in the literature devoted to elliptic theory on manifolds with periodic ends. The present paper gives a Lefschetz formula on such manifolds; in the special case of manifolds with cylindrical ends, it strengthens previously obtained results.
This paper studies the properties of the inertia tensor for tops in 3-dimensional (pseudo-) Euclidean space and for plates on the Lobachevskii plane. A canonical form is described to which one can reduce the (pseudo-)Euclidean inner product and the inertia operator of any top by changing the basis. All tops whose inertia operators are nondiagonalizable (every top of this kind lies in a plane tangent to the isotropic cone) are described. All realizable triples (J_1,J_2,J_3) of principal moments of inertia are described in terms of triangle inequalities and their pseudo-Euclidean analogs. A criterion for the realizability of such a triple by a plate on the Lobachevskii plane is obtained. All realizable pairs of triples (J_1,J_2,J_3) of principal moments of inertia and coordinates of the center of mass (c_1,c_2,c_3) of a top in the principal axes of inertia are described in both the diagonalizable and nondiagonalizable cases. In all cases, it is shown that a body consisting of no more than six points can be taken as the realizing top. Eigenvalues of all degenerate inertia operators are described. As an application, realizable examples of integrable tops in 3-dimensional pseudo-Euclidean space (the Euler and Lagrange tops) are described.
We establish a correspondence between the semi-infinite and infinite Volterra lattices with a finite logarithmic Hamiltonian and certain classes of even probability measures. In doing so, we apply the inverse spectral theory of Jacobi operators and the theory of orthogonal polynomials.
The Lefschetz number of an endomorphism of an elliptic complex is expressed in terms of regularized traces of the operators defining the endomorphism. This result is obtained under certain conditions on the wavefront sets of the operators in question. In the particular case of geometric endomorphisms of the complex, we obtain the classical Atiyah–Bott formula. As an application, we compute the Lefschetz numbers of nonlocal elliptic operators associated with an action of a finite group on a closed smooth manifold. For the de Rham complex, this gives a formula for the Lefschetz number in the cohomology of the orbit space in terms of fixed points.
We address the problem of identification of branched coverings (continuous open surjections p Y→ X of Hausdorff spaces with uniformly bounded number of pre-images) with faithful unital positive conditional expectations E C(Y)→ C(X) topologically of index-finite type. Caused by recent progress (A. Chirvasitu, 2024) in the field and detection of an issue in our old paper, we revisit it to make some advances.
This is the third part of our work dealing with spectral analysis for the Hamiltonian of three identical one-dimensional quantum particles. It deals with positive energies and the corresponding generalized eigenfunctions. The asymptotic behavior of the generalized eigenfunctions is studied and interpreted in terms of the diffraction theory.
