Compactness Theorem Research Articles

Nontrivial solutions for a class of Hamiltonian strongly degenerate elliptic system

In this paper, we study the existence of nontrivial to the following Hamiltonian strongly degenerate elliptic system in whole space where are nonnegative functions with and V might unbounded or vanish at infinity. Here is the strongly degenerate operator and the nonlinearity terms are continuous functions and satisfy some assumptions that will be specified later. Due to the unbounded or vanishing of potentials and degeneracy of the operator, some new compact embedding theorems are used in the proof. Our results extend and generalize some existing results.

Compactness of nonlinear integral operators with discontinuous and with singular kernels

This paper establishes compactness of nonlinear integral operators in the space of continuous functions. One result deals with operators whose kernel can have jumps across a finite number of curves, which typically arise from the study of ordinary differential equations with boundary conditions of local or nonlocal type. Several other results deal with operators whose kernels have a singularity, which arise from the study of fractional differential equations. We motivate the study of these integral equations by discussing some initial value problems for fractional differential equations of Caputo and Riemann-Liouville type. We prove a compact embedding theorem for fractional integrals in order to give a new treatment for the singular kernel case.

A new proof of compactness in G(S)BD

Abstract We prove a compactness result in GBD {\operatorname{GBD}} which also provides a new proof of the compactness theorem in GSBD {\operatorname{GSBD}} , due to Chambolle and Crismale. Our proof is based on a Fréchet–Kolmogorov compactness criterion and does not rely on Korn or Poincaré–Korn inequalities.

Long‐Term Dynamic Behavior of a Higher‐Order Coupled Kirchhoff Model with Nonlinear Strong Damping

In this paper, we study the long‐time dynamics problem of a class of higher‐order Kirchhoff coupled systems with nonlinear strong damping. The existence and uniqueness of the solutions of these equations in different spaces are proved by prior estimation and Faedo–Galerkin method; secondly, the family of global attractors of these problems is proved by using the compactness theorem. The results of the Kirchhoff coupled group are promoted through research.

The Long-Term Dynamic Behavior of Solutions to a Class of Generalized Higher-Order Kirchhoff-Type Coupled Wave Equations

In this paper, we study the long-term dynamic behavior of a class of generalized high-order Kirchhoff-type coupled wave equations. Firstly, the existence of uniqueness global solution of this kind of equations in Ek space is proved by prior estimation and Galerkin method; Then, through using Rellich-Kondrachov compact embedding theorem, it is proved that the solution semigroup S(t) has the family of the global attractors Ak in space Ek; Finally, through linearization method, proves that the operator semigroup S(t) Frechet differentiable and the attenuation of linearization problem volume element. Furthermore, we can obtain the finite Hausdorff dimension and Fractal dimension of the family of the global attractors Ak.

Cyclic congruences of slim semimodular lattices and non-finite axiomatizability of some finite structures

We give a new proof of the fact that finite bipartite graphs cannot be axiomatized by finitely many first-order sentences among finite graphs. (This fact is a consequence of a general theorem proved by L. Ham and M. Jackson, and the counterpart of this fact for all bipartite graphs in the class of all graphs is a well-known consequence of the compactness theorem.) Also, to exemplify that our method is applicable in various fields of mathematics, we prove that neither finite simple groups, nor the ordered sets of join-irreducible congruences of slim semimodular lattices can be described by finitely many axioms in the class of finite structures. Since a 2007 result of G. Grätzer and E. Knapp, slim semimodular lattices have constituted the most intensively studied part of lattice theory and they have already led to results even in group theory and geometry. In addition to the non-axiomatizability results mentioned above, we present a new property, called Decomposable Cyclic Elements Property, of the congruence lattices of slim semimodular lattices.

The isoperimetric problemviadirect method in noncompact metric measure spaces with lower Ricci bounds

We establish a structure theorem for minimizing sequences for the isoperimetric problem on noncompact RCD(K, N) spaces (X, d, ℋN). Under the sole (necessary) assumption that the measure of unit balls is uniformly bounded away from zero, we prove that the limit of such a sequence is identified by a finite collection of isoperimetric regions possibly contained in pointed Gromov-Hausdorff limits of the ambient spaceXalong diverging sequences of points. The number of such regions is bounded linearly in terms of the measure of the minimizing sequence. The result follows from a new generalized compactness theorem, which identifies the limit of a sequence of setsEi⊂Xiwith uniformly bounded measure and perimeter, where (Xi,di, ℋN) is an arbitrary sequence of RCD(K,N) spaces. An abstract criterion for a minimizing sequence to converge without losing mass at infinity to an isoperimetric set is also discussed. The latter criterion is new also for smooth Riemannian spaces.

Global weak solutions to the stochastic Ericksen–Leslie system in dimension two

<p style='text-indent:20px;'>We establish the global existence of weak martingale solutions to the simplified stochastic Ericksen–Leslie system modeling the nematic liquid crystal flow driven by Wiener-type noises on the two-dimensional bounded domains. The construction of solutions is based on the convergence of Ginzburg–Landau approximations. To achieve such a convergence, we first utilize the concentration-cancellation method for the Ericksen stress tensor fields based on a Pohozaev type argument, and then the Skorokhod compactness theorem, which is built upon uniform energy estimates.</p>

Generalized Densities on ℝ n and their Applications

Abstract We examine some generalized densities (called (ψ, n)-densities) obtained as a result of strengthening the Lebesgue Density Theorem. It turns out that these notions are the generalizations of superdensity, enhanced density and m-density, and have some applications in the theory of sets of finite perimeter and in Sobolev spaces.

Remarkable Findings in Fundamental Theory of Quantum Mechanics—Matter Wave and Discrete Time in Physics

The relation of matter wave, which is well-known as a hypothesis proposed by de Broglie in 1923, gave basis for establishing the quantum mechanics. After that, experimental results revealed that a micro particle has a wave nature. However, the theoretical validity of the relation itself has never been revealed since his proposal. Theoretical basis that a micro particle has a wave nature has been thus disregarded in the unsolved state. The diffusion equation having been accepted as Fick’s second law was derived from the theory of Markov process in mathematics. It was then revealed that the diffusivity D depends on an angular momentum of a micro particle in a local space. The fact being unable to discriminate between micro particles in a local space resulted in having to accept the existence of minimum time t0 (>0) in the quantum mechanics. Based on t0 and D obtained here, the theoretical validity of relation of matter wave was confirmed. Denying the density theorem in mathematics for time in physics indicates that the probabilistic interpretation is essentially indispensable for understanding the quantum mechanics. The logical necessity of quantum theory itself is thus understandable through introducing t0 into the Newton mechanics. It is remarkable that the value of t0 between 1.14×10-17 s ≤ t0 ≤1.76×10-14 s obtained here is extremely larger than that of the well-known Planck time tp=5.396×10-44 s.

Radial spiky steady states of a flux‐limited Keller–Segel model: Existence, asymptotics, and stability

AbstractThis paper is concerned with the radial stationary problem of a flux‐limited Keller–Segel model derived in a multidimensional bounded domain with Neumann boundary conditions. With the global bifurcation theory and Helly compactness theorem by treating the chemotactic coefficient as a bifurcation parameter, we establish the existence of nonconstant monotone radial stationary solutions and further show that the radial stationary solution will tend to a Dirac delta mass as the chemotactic coefficient tends to infinity. By using the stability criterion of Crandall and Rabinnowitz, we prove the linearized stability of bifurcating stationary solutions near the bifurcation points.

Some compact and non-compact embedding theorems for the function spaces defined by fractional Fourier transform

The fractional Fourier transform is a generalization of the classical Fourier transform through an angular parameter $\alpha $. This transform uses in quantum optics and quantum wave field reconstruction, also its application provides solving some differrential equations which arise in quantum mechanics. The aim of this work is to discuss compact and non-compact embeddings between the spaces $A_{\alpha ,p}^{w,\omega }\left(\mathbb{R}^{d}\right) $ which are the set of functions in ${L_{w}^{1}\left(\mathbb{R}^{d}\right) }$ whose fractional Fourier transform are in ${L_{\omega}^{p}\left(\mathbb{R}^{d}\right) }$. Moreover, some relevant counterexamples are indicated.

Density and connectedness of optimal points with respect to improvement sets

Based on the improvement set E, this paper aims at investigating density and connectedness of the sets of E-optimal points, weak E-optimal points, E-quasi-optimal points, E-Benson proper optimal points, E-super optimal points and E-strictly optimal points. By virtue of the separation theorem for convex sets, we obtain the scalarizaiton results for the sets of E-optimal points, weak E-optimal points, E-Benson proper optimal points, E-super optimal points and E-strictly optimal points. Then we make a new attempt to establish some density theorems for the sets of these optimal points. Finally, we investigate connectedness and arcwise connectedness of the sets of various notions of E-optimality by using the scalarization method and the density theorems.

On the compactness problem for a family of generalized Seiberg–Witten equations in dimension 3

We prove an abstract compactness theorem for a family of generalized Seiberg-Witten equations in dimension three. This result recovers Taubes' compactness theorem for stable flat $\mathbf{P}\mathrm{SL}_2(\mathbf{C})$-connections as well as the compactness theorem for Seiberg-Witten equations with multiple spinors. Furthermore, this result implies a compactness theorem for the ADHM$_{1,2}$ Seiberg-Witten equation, which partially verifies a conjecture by Doan and Walpuski.

Normalizing notations in the Ershov hierarchy

AbstractThe Turing degrees of infinite levels of the Ershov hierarchy were studied by Liu and Peng [8]. In this paper, we continue the study of Turing degrees of infinite levels and lift the study of density property to the levels beyond ω2. In doing so, we rely on notations with some nice properties. We introduce the concept of normalizing notations and generate normalizing notations for higher levels. The generalizations of the weak density theorem and the nondensity theorem are proved for higher levels in the Ershov hierarchy. Furthermore, we also investigate the minimal degrees in the infinite levels of the Ershov hierarchy.

EXCEPTIONAL POINTS FOR DENSITIES GENERATED BY SEQUENCES

In spite of the Lebesgue density theorem, there is a positive δ such that, for every measurable set A⊂ℝ with λ(A)>0 and λ(ℝ\A)>0, there is a point at which both the lower densities of A and of the complement of A are at least δ. The problem of determining the supremum δH of possible values of this δ was studied by V. I. Kolyada, A. Szenes and others, and it was solved by O. Kurka. Lower density of A at x is defined as a lower limit of λ(A∩[x-h,x+h])/2h. Replacing λ(A∩[x-h,x+h])/2h by λ(A∩[x-tn,x+tn])/2tn for a fixed decreasing sequence ⟨t⟩ tending to zero, we obtain a definition of the constant δ⟨t⟩. In our paper we look for an upper bound of all such constants.

LEBESGUE DENSITY AND STATISTICAL CONVERGENCE

The paper presents a generalization of the density point’s notion to the ideal-convergence framework. For an ideal I⊆P(ℕ) (with Fin⊆I), Lebesgue measurable set A⊆ℝ we introduce a definition of a density point of A with respect to I; we prove that the classical approach fits into this generalization (Theorem 4); we construct a family of Cantorlike sets showing that Lebesgue Density Theorem cannot be maximally improved in this direction (Theorem 8).

On the density of unnormalized Tamagawa numbers of orthogonal groups III

This is the final part of a series of three papers. In this part, we finish the proof of the density theorem of the unnormalized Tamagawa numbers of projective special orthogonal groups.

The density theorem for discrete series representations restricted to lattices

This article considers the relation between the spanning properties of lattice orbits of discrete series representations and the associated lattice co-volume. The focus is on the density theorem, which provides a trichotomy characterizing the existence of cyclic vectors and separating vectors, and frames and Riesz sequences. We provide an elementary exposition of the density theorem, that is based solely on basic tools from harmonic analysis, representation theory, and frame theory, and put the results into context by means of examples.

Compact Embedding Theorems for The Space of Functions with Wavelet Transform in Amalgam Spaces

Bu çalışma dalgacık dönüşümü kullanarak 〖 A〗_s (W)_(ω,ϑ)^(p,q,r) (R) uzayını tanımlamak ve ayrıca bu uzayda kapsama, kompakt gömülme teoremlerini incelemek için motive edilmiştir.