Compactness Theorem Research Articles

Some compactness theorems via m-Bakry–Émery and m-modified Ricci curvatures with negative m

Stimulated by S. Ohta and W. Wylie, we establish some compactness theorems for complete Riemannian manifolds via m-Bakry–Émery and m-modified Ricci curvatures with negative m. Our results may be considered as generalizations of the classical compactness theorems via Ricci curvature due to S.B. Myers, W. Ambrose, G.J. Galloway, and J. Cheeger, M. Gromov, and M. Taylor, and relax some previous compactness criteria for complete Riemannian manifolds via m-Bakry–Émery and m-modified Ricci curvatures obtained when m is a positive constant or infinity.

Boxes, extended boxes and sets of positive upper density in the Euclidean space

AbstractWe prove that sets with positive upper Banach density in sufficiently large dimensions contain congruent copies of all sufficiently large dilates of three specific higher-dimensional patterns. These patterns are: 2n vertices of a fixed n-dimensional rectangular box, the same vertices extended with n points completing three-term arithmetic progressions, and the same vertices extended with n points completing three-point corners. Our results provide common generalizations of several Euclidean density theorems from the literature.

Castelnuovo's bound and rigidity in almost complex geometry

This article is concerned with the question of whether an energy bound implies a genus bound for pseudo-holomorphic curves in almost complex manifolds. After reviewing what is known in dimensions other than six, we establish a new result in this direction in dimension six; in particular, for symplectic Calabi–Yau 3–folds. The proof relies on compactness and regularity theorems for pseudo-holomorphic currents.

On convergence and compactness in variation with shift of discrete probability laws

We consider a class of discrete distribution functions, whose characteristic functions are separated from zero, i. e. their absolute values are greater than positive constant on the real line. The class is rather wide, because it contains discrete infinitely divisible distribution functions, functions of lattice distributions, whose characteristic functions have no zeroes on the real line, and also distribution functions with a jump greater than 1/2. Recently the authors showed that characteristic functions of elements of this class admit the Lévy-Khinchine type representations with non-monotonic spectral function. Thus our class is included in the set of so called quasi-infinitely divisible distribution functions. Using these representation the authors also obtained limit and compactness theorems with convergence in variation for the sequences from this class. This note is devoted to similar results concerning convergence and compactness but with weakened convergence in variation. Replacing of type of convergence notably expands applicability of the results.

Geometry of Lagrangian self-shrinking tori and applications to the piecewise Lagrangian mean curvature flow

We study geometric properties of the Lagrangian self-shrinking tori in $\mathbb R^4$. When the area is bounded above uniformly, we prove that the entropy for the Lagrangian self-shrinking tori can only take finitely many values; this is done by deriving a {\L}ojasiewicz-Simon type gradient inequality for the branched conformal self-shrinking tori and then combining with the compactness theorem in \cite{CMa}. When the area bound is small, we show that any Lagrangian self-shrinking torus in $\mathbb R^4$ with small area is embedded with uniform curvature estimates, and the space of such tori is compact. Using the finiteness of entropy values, we construct a piecewise Lagrangian mean curvature flow for Lagrangian immersed tori in $\mathbb R^4$, along which the Lagrangian condition is preserved, area is decreasing, and the type I singularities that are compact with a fixed area upper bound can be perturbed away in finite steps. This is a Lagrangian version of the construction for embedded surfaces in $\mathbb R^3$ in \cite{CM}.

The Family of Global Attractors for a Generalized Kirchhoff Equations

This paper deals with the existence and uniqueness of solutions of generalized Kirchhoff equations and the family of global attractors for the equation and its dimension estimation. First, the stress term of Kirchhoff equation is properly assumed. When certain conditions are met between the order m and the degree p of Banach space , the existence and uniqueness of the solution of equation are obtained by a prior estimation and Galerkin’s method; Then, the bounded absorption set is obtained by prior estimation, and it is proved that the solution semigroup generated by the equation has a family of global attractors in phase space by using Rellich-Kondrachov compact embedding theorem. Further, the equation is linearized and rewritten into a first-order variational equation, and it is proved that the solution semigroup is Fréchet differentiable on ; Finally, the upper bound of Hausdorff dimension and Fractal dimension of is estimated, and the Hausdorff dimension and Fractal dimension are finite.

Global Attractors and Their Dimension Estimates for a Class of Generalized Kirchhoff Equations

In this paper, we studied the long-time properties of solutions of generalized Kirchhoff-type equation with strongly damped terms. Firstly, appropriate assumptions are made for the nonlinear source term g (u) and Kirchhoff stress term M (s) in the equation, and the existence and uniqueness of the solution are proved by using uniform prior estimates of time and Galerkin’s finite element method. Then, abounded absorption set B0k is obtained by prior estimation, and the Rellich-kondrachov’s compact embedding theorem is used to prove that the solution semigroup S (t) generated by the equation has a family of the global attractor Ak in the phase space . Finally, linearize the equation and verify that the semigroups are Frechet diifferentiable on Ek. Then, the upper boundary estimation of the Hausdorff dimension and Fractal dimension of a family of the global attractor Ak was obtained.

Gibbs Measure for the Higher Order Modified Camassa-Holm Equation

This paper is devoted to constructing a globally rough solution for the higher order modified Camassa-Holm equation with randomization on initial data and periodic boundary condition. Motivated by the works of Thomann and Tzvetkov (Nonlinearity, 23 (2010), 2771–2791), Tzvetkov (Probab. Theory Relat. Fields, 146 (2010), 4679–4714), Burq, Thomann and Tzvetkov (Ann. Fac. Sci. Toulouse Math., 27 (2018), 527–597), the authors first construct the Borel measure of Gibbs type in the Sobolev spaces with lower regularity, and then establish the existence of global solution to the equation with the helps of Prokhorov compactness theorem, Skorokhod convergence theorem and Gibbs measure.

Global Existence of a Virus Infection Model with Saturated Chemotaxis

In this paper, a virus infection model with saturated chemotaxis is formulated and analyzed, where the chemotactic sensitivity for chemotactic movements of the cells is described. This model contains three state variables namely the population density of uninfected cells, the population density of infected cells and the concentration of virus particles, respectively. By virtue of regularized approximation technique and fixed point theorem, the local solvability of the regularized system corresponding to the original system is established. Then by extracting a suitable sequence along which the respective approximate solutions approach a limit in convenient topologies, with addition of Gagliardo-Nirenberg interpolation inequality as well as Lp-estimate techniques, we show that the original system describing the virus infection model exists at least one global weak solution. To illustrate the application of our theoretical results, an optimal control problem of the epidemic system is considered, where the admissible control domain is assumed to be a bounded closed convex subset. With the help of Aubin compactness theorem and lower semicontinuous of the cost functional, the existence of the optimal pair is proved. Our results generalize and improve partial previously known ones, and moreover, we first prove that the optimal control problem has at least one optimal pair.

A density theorem for asymptotically hyperbolic initial data

A density theorem for asymptotically hyperbolic initial data

Equidistributed periodic orbits of $C^\infty$-generic three-dimensional Reeb flows

We prove that, for a $C^\infty$-generic contact form $\lambda$ adapted to a given contact distribution on a closed three-manifold, there exists a sequence of periodic Reeb orbits which is equidistributed with respect to $d\lambda$. This is a quantitative refinement of the $C^\infty$-generic density theorem for three-dimensional Reeb flows, which was previously proved by the author. The proof is based on the volume theorem in embedded contact homology (ECH) by Cristofaro-Gardiner, Hutchings, Ramos, and inspired by the argument of Marques-Neves-Song, who proved a similar equidistribution result for minimal hypersurfaces. We also discuss a question about generic behavior of periodic Reeb orbits representing ECH homology classes, and give a partial affirmative answer to a toy model version of this question which concerns boundaries of star-shaped toric domains.

Discrete-to-continuum limits of planar disclinations

In materials science, wedge disclinations are defects caused by angular mismatches in the crystallographic lattice. To describe such disclinations, we introduce an atomistic model in planar domains. This model is given by a nearest-neighbor-type energy for the atomic bonds with an additional term to penalize change in volume. We enforce the appearance of disclinations by means of a special boundary condition. Our main result is the discrete-to-continuum limit of this energy as the lattice size tends to zero. Our proof relies on energy relaxation methods. The main mathematical novelty of our proof is a density theorem for the special boundary condition. In addition to our limit theorem, we construct examples of planar disclinations as solutions to numerical minimization of the model and show that classical results for wedge disclinations are recovered by our analysis.

Theoretical Evidence for Wave Nature of Micro Particle and New Theory of Its Collective Motion in Material

Since a material is composed of micro particles, investigating behavior of those particles is essentially dominant for materials science. The diffusivity of diffusion equation is relevant to not only a collective motion of micro particles but also a motion of single particle. An elementary process of diffusion was thus theoretically investigated in a local space and time. As a result, the investigation concluded that the wave nature of micro particle results from denying the mathematical density theorem of a real time in the Newton mechanics. In other words, the basic theory of quantum mechanics is established in accordance with the cause-and-effect relationship in the Newton mechanics, for the first time, regardless of the de Broglie hypothesis. In relation to the collective motion of micro particles, the new diffusion theory was also reasonably established using the universal expression of diffusivity obtained here. In the present paper, the new findings indispensable for the fundamental knowledge in physics are thus systematically discussed in accordance with the theoretical frame in physics.

Formalization of the Equivalence among Completeness Theorems of Real Number in Coq

The formalization of mathematics based on theorem prover becomes increasingly important in mathematics and computer science, and, particularly, formalizing fundamental mathematical theories becomes especially essential. In this paper, we describe the formalization in Coq of eight very representative completeness theorems of real numbers. These theorems include the Dedekind fundamental theorem, Supremum theorem, Monotone convergence theorem, Nested interval theorem, Finite cover theorem, Accumulation point theorem, Sequential compactness theorem, and Cauchy completeness theorem. We formalize the real number theory strictly following Landau’s Foundations of Analysis where the Dedekind fundamental theorem can be proved. We extend this system and complete the related notions and properties for finiteness and sequence. We prove these theorems in turn from Dedekind fundamental theorem, and finally prove the Dedekind fundamental theorem by the Cauchy completeness theorem. The full details of formal proof are checked by the proof assistant Coq, which embodies the characteristics of reliability and interactivity. This work can lay the foundation for many applications, especially in calculus and topology.

On the reconstruction via Urysohn-Chlodovsky operators

Abstract The main first goal of this work is to introduce an Urysohn type Chlodovsky operators defined on positive real axis by using the Urysohn type interpolation of the given function f and bounded on every finite subinterval. The basis used in this construction are the Fréchet and Prenter Density Theorems together with Urysohn type operator values instead of the rational sampling values of the function. Afterwards, we will state some convergence results, which are generalization and extension of the theory of classical interpolation of functions to operators.

Nondiscrete parabolic characters of the free group F2: supergroup density and Nielsen classes in the complement of the Riley slice

A parabolic representation of the free group $F_2$ is one in which the images of both generators are parabolic elements of $PSL(2,\IC)$. The Riley slice is a closed subset ${\cal R}\subset \IC$ which is a model for the parabolic, discrete and faithful characters of $F_2$. The complement of the Riley slice is a bounded Jordan domain within which there are isolated points, accumulating only at the boundary, corresponding to parabolic discrete and faithful representations of rigid subgroups of $PSL(2,\IC)$. Recent work of Aimi, Akiyoshi, Lee, Oshika, Parker, Lee, Sakai, Sakuma \& Yoshida, have topologically identified all these groups. Here we give the first identified substantive properties of the nondiscrete representations and prove a supergroup density theorem: given any irreducible parabolic representation $\rho_*:F_2\to PSL(2,\IC)$ whatsoever, any non-discrete parabolic representation $\rho_0$ has an arbitrarily small perturbation $\rho_\epsilon$ so that $\rho_\epsilon(F_2)$ contains a conjugate of $\rho_*(F_2)$ as a proper subgroup. This implies that if $\Gamma_*$ is any nonelementary group generated by two parabolic elements (discrete or otherwise) and $\gamma_0$ is any point in the complement of the Riley slice, then in any neighbourhood of $\gamma$ there is a point corresponding to a nonelementary group generated by two parabolics with a conjugate of $\Gamma_*$ as a proper subgroup. Using these ideas we then show that there are nondiscrete parabolic representations with an arbitrarily large number of distinct Nielsen classes of parabolic generators.

Müntz-Szász type theorems for the density of the span of powers of functions

The aim of this paper is to establish density properties in Lp spaces of the span of powers of functions {ψλ:λ∈Λ}, Λ⊂N in the spirit of the Müntz-Szász Theorem. As density is almost never achieved, we further investigate the density of powers and a modulation of powers {ψλ,ψλeiαt:λ∈Λ}. Finally, we establish a Müntz-Szász Theorem for density of translates of powers of cosines {cosλ⁡(t−θ1),cosλ⁡(t−θ2):λ∈Λ}. Under some arithmetic restrictions on θ1−θ2, we show that density is equivalent to a Müntz-Szász condition on Λ and we conjecture that those arithmetic restrictions are not needed. Some links are also established with the recently introduced concept of Heisenberg Uniqueness Pairs.

Uniform Continuity of Fractal Interpolation Function

In order to research analysis properties of fractal interpolation function generated by the iterated function system defined by affine transformation, the continuity of fractal interpolation function is proved by the continuous definition of function and the uniform continuity of fractal interpolation function is proved by the definition of uniform continuity and compactness theorem of sequence of numbers or finite covering theorem in this paper. The result shows that the fractal interpolation function is uniformly continuous in a closed interval which is from the abscissa of the first interpolation point to that of the last one.

Corrigendum to: On the Speed of Convergence in the Strong Density Theorem

This note corrects two misleading typographical errors in On the speed of convergence in the strong density theorem by P. Georgopoulos and C. Cryllakis, Real Anal. Exch. 44(1) 2019.

Pseudo-holomorphic curves: A very quick overview

AbstractThis is a review article on pseudo-holomorphic curves which attempts at touching all the main analytical results. The goal is to make a user friendly introduction which is accessible to those without an analytical background. Indeed, the major accomplishment of this review is probably its short length.Nothing in here is original and can be found in more detailed accounts such as [6] and [8]. The exposition of the compactness theorem is somewhat different from that in the standard references and parts of it are imported from harmonic map theory [7], [5]. The references used are listed, but of course any mistake is my own fault.