Analysis Mathematica最新文献

The systems of nonlinear differential equations of certain types can be simplified to matrix forms. Two types of matrix differential equations will be considered in the paper, one is Fermat type matrix differential equation

where n = 2 and n = 3, another is Malmquist type matrix differential equation

, where α (≠ 0), β, γ are constants. By solving the systems of nonlinear differential equations, we obtain some properties on the meromorphic matrix solutions of the above matrix differential equations. In addition, we also consider two types of nonlinear differential equations, one of them is called Bi-Fermat differential equation.

We observe that some self-similar measures that we call essentially of finite type satisfy countable measure type condition. We make use of this condition to set up a framework to obtain a precise analog of Weyl’s asymptotic formula for the eigenvalue counting function of Laplacians defined by measures, emphasizing on one-dimensional self-similar measures with overlaps. As an application of our result, we obtain an analog of a semi-classical asymptotic formula for the number of negative eigenvalues of fractal Schrödinger operators as the parameter tends to infinity.

We consider the Hankel multidimensional operator defined on]0, +∞[ⁿ by

where (alpha = ({alpha _1},{alpha _2}, ldots ,{alpha _n}) in ] - {1 over 2}, + infty {[^n}). We give the most important harmonic analysis results related to the operator Δ_α (translation operators τ_x, convolution product * and Hankel transform ℌ_α).

Using harmonic analysis results, we study spaces of Sobolev type for which we make explicit kernels reproducing. Next, we define and study the gaussian convolution ({{cal G}^t}), t > 0, associated with the Hankel multidimensinal operator Δ_α. This transformation generalizes the classical gaussian transformation. We establish the most important properties of this transformation. In particular, we show that the gaussian transformation solves the heat equation, that is

In the second part of this work, we prove the existence and uniqueness of the extremal function associated with the gaussian transformation. We express this function using the reproducing kernels and we prove the important estimates for this extremal function.

Let E be a Banach space. For a topological space X, let ({{cal C}_b}(X,E)) be the space of all bounded continuous E-valued functions on X, and let ({{cal C}_K}(X,E)) be the subspace of ({{cal C}_b}(X,E)) consisting of all functions having a pre-compact image in E. We show that ({{cal C}_K}(X,E)) is isometrically isomorphic to the injective tensor product ({{cal C}_b}(X){{hat otimes}_varepsilon}E), and that ({{cal C}_b}(X,E) = {{cal C}_b}(X){{hat otimes}_varepsilon}E) if and only if E is finite dimensional. Next, we consider the space Lip(X, E) of E-valued Lipschitz operators on a metric space (X, d) and its subspace Lip_K(X, E) of Lipschitz compact operators. Utilizing the results on ({{cal C}_b}(X,E)), we prove that Lip_K(X, E) is isometrically isomorphic to a tensor product ({rm{Lip}}(X){{hat otimes}_alpha}E), and that ({rm{Lip}}(X,E) = {rm{Lip}}(X){{hat otimes}_alpha}E) if and only if E is finite dimensional. Finally, we consider the space D¹(X, E) of continuously differentiable functions on a perfect compact plane set X and show that, under certain conditions, D¹(X, E) is isometrically isomorphic to a tensor product ({D^1}(X){hat otimes _beta}E).

In 2008, Takahashi introduced the James type constants. We discuss here the pointwise James type constant: for all x ∈ X, ∥x∥ = 1,

We show that in almost transitive Banach spaces, the map x ∈ X, ∥x∥ = 1 ↦ J(x, X, t) is constant. As a consequence and having in mind the Mazur’s rotation problem, we prove that for almost transitive Banach spaces, the condition (J(x,X,t) = sqrt 2 ) for some unit vector x ∈ X implies that X is Hilbert.

(μ; ν)-Hankel operators between separable Hilbert spaces were introduced and studied recently (A. Mirotin and E. Kuzmenkova, μ-Hankel operators on Hilbert spaces, Opuscula Math., 41 (2021), 881–899). This paper is devoted to generalization of (μ; ν)-Hankel operators to the case of (non-separable in general) Hardy spaces over compact and connected Abelian groups. In this setting bounded (μ; ν)-Hankel operators are fully described under some natural conditions. Examples of integral operators are also considered.

In this paper we are concerned with the asymptotic behavior of

the trace of the pseudoinverse of the Laplacian matrix related with the square lattice, as n → ∞. The method we developed for such sums in former papers depends on the use of Taylor approximations for the summands. It was shown that the error term depends on whether the Taylor polynomial used is of degree two or higher. Here we carry this out for the square lattice with a fourth degree Taylor polynomial and thereby obtain a result with an improved error term which is perhaps the most precise one can hope for.

A space of universal disposition is a Banach space which has certain natural extension properties for isometric embeddings of Banach spaces belonging to a specific class. We study spaces of universal disposition for non-archimedean Banach spaces. In particular, we introduce the classification of non-archimedean Banach spaces depending on the cardinality of maximal orthogonal sets, which can be viewed as a kind of special density and characterize spaces of universal disposition for each distinguished class.

Every graph of bounded degree endowed with the counting measure satisfies a local version of L^p-Poincaré inequality, p ∈ [1, ∞]. We show that on graphs which are trees the Poincaré constant grows at least exponentially with the radius of balls. On the other hand, we prove that, surprisingly, trees endowed with a flow measure support a global version of L^p-Poincaré inequality, despite the fact that they are nondoubling measures of exponential growth.