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Let X be an infinite Tychonoff space, and Y be a topological subspace of X. In this paper, we study some covering properties of the subspace Cp(Y|X) of Cp(Y) consisting of those functions f ∈ C (Y) which admit a continuous extension to X equipped with the relative topology of Cp(Y). Among other results, we show that (i) Cp(Y|X) has a fundamental bounded resolution if and only if Y is countable; when X is realcompact and Y is closed in X, we have (ii) if Cp(Y|X) admits a resolution of convex compact sets that swallows the local null sequences in Cp(Y|X), then Y is countable and discrete; (iii) if Cp(Y|X) admits a compact resolution that swallows the compact sets, then Y is also countable and discrete, and, as a corollary, we deduce that Cp(Y|X) admits a compact resolution that swallows the compact sets if and only if Cp(Y|X) is a Polish space. We also prove that (iv) for a metrizable space X, Cp(X) is a quasi-(LB)-space if and only if X is σ-compact, and hence for a subspace Y of X, the space Cp(Y|X) is a quasi-(LB)-space. We include some examples and observations that answer natural questions raised in this paper. [ABSTRACT FROM AUTHOR]

If X is a geodesic metric space and x1, x2, x3 ∈ X, a geodesic triangle T = {x1, x2, x3} is the union of the three geodesics [x1x2], [x2x3] and [x3x1] in X. The space X is hyperbolic if there exists a constant δ ≥ 0 such that any side of any geodesic triangle in X is contained in the d-neighborhood of the union of the two other sides. In this paper, we study the hyperbolicity of an important kind of Euclidean graphs called Delaunay triangulations. Furthermore, we characterize the Delaunay triangulations contained in the Euclidean plane that are hyperbolic. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the class of n-quasi-A-(m, q)-isometry operators on a Banach space X, which represents a generalization of the n-quasi-(m, q)-isometry on a Banach space and the nquasi-(A, m)-isometry on a Hilbert space. After giving some basic properties of this class of operators, we study the product and the power of such operators in this class. [ABSTRACT FROM AUTHOR]

In this paper, we construct global solutions to the Cauchy problem on a nonlinear Fokker-Planck equation near Maxwellian with small-amplitude initial data in Sobolev space H x 2 L v 2 by a refined nonlinear energy method. Compared with the results of Liao et al. (Global existence and decay rates of the solutions near Maxwellian for non-linear Fokker-Planck equations, J. Stat. Phys. , 173 (2018), 222–241.), the regularity assumption on the initial data is much weaker. [ABSTRACT FROM AUTHOR]

Let BX be the unit ball of a complex Banach space X, which may be infinite dimensional. The authors characterize the boundedness of the product of the radial derivative operator and the weighted composition operator from the natural Bloch space (or the little Bloch-type space) into the weighted-type space (or the little weighted-type space) on BX. [ABSTRACT FROM AUTHOR]