On Some Extremal Properties of Finite Ultrametric Spaces

We characterize finite ultrametric spaces having the strictly n-ary representing trees and finite ultrametric spaces having the representing trees with injective internal labelings by their extremal properties.     

On Strict Extensions of Nearness Spaces

Strict extensions of nearness spaces are constructed as spaces of round Cauchy filters. Morita's simple extension is identified as the strict extension generated by Morita-generated filters. Carlson's B-completeness is compared with Herrlich completeness and completeness of Morita T-uniformities. The three completeness concepts are shown to be equivalent in regular nearness spaces.     

Combinatorial Optimization Problems in Ultrametric Spaces

We study the solutions of some known combinatorial optimization problems including the minimum matching problem, the minimum spanning tree problem, and the traveling salesman problem in d-dimensional p-adic spaces. It appears that the greedy algorithms yield the optimal solutions of these problems in the ultrametric space, which allows obtaining explicit expressions for the estimates of their averages. We study the asymptotic behavior of these averages as the number of points increases infinitely and find some similarities to the Euclidean case, as well as new, unexpected properties.     

锥超度量空间的不动点理论

给出了锥超度量空间与锥度量空间上Hausdorff度量的定义.并利用球完备的性质在锥超度量空间上证明了有关收缩映射与多值映射的不动点理论.     

The Common Point Theorem for Ultrametric Spaces

In a paper we proved the following fixed point theorem: Every strictly contracting map F:X X of a spherically complete ultrametric space X has a unique fixed point x, that is, F(x) = x. In this paper we study conditions for two self-maps F, G of an ultrametric space to have a common point x, i.e. F(x) = G(x).     

关于Banach空间最小系的扩充

本文就可分Banach空间中元素的最小序列(也称双直交序列)可以扩充到在全空间中完备这一事实,说明在空间不可分情况下,对于由不可数个元素组成的所谓最小系,这种完备性扩充未必可行.此外,还应用最小序列扩充性质给出可分的遗传不可分解空间的一个特征刻画.     

Weak Similarities of Finite Ultrametric and Semimetric Spaces

Weak similarities form a special class of mappings between semimetric spaces. Two semimetric spaces X and Y are weakly similar if there exists a weak similarity Φ: X → Y. We find a structural characteristic of finite ultrametric spaces for which the isomorphism of its representing trees implies a weak similarity of the spaces. We also find conditions under which the Hasse diagrams of balleans of finite semimetric spaces are isomorphic.     

From Isomorphic Rooted Trees to Isometric Ultrametric Spaces

For every finite ultrametric space X we can put in correspondence its representing tree T_X. We found conditions under which the isomorphism of representing trees T_X and T_Y implies the isometricity of ultrametric spaces X and Y having the same range of distances.     

Fixed Points and Stable Balls in Ultrametric Spaces

In [2] we presented a fixed point theorem for strictly contracting mappings of spherically complete ultrametric spaces; in this paper we show how this theorem looks like if we assume only that the mappings are contracting.     

Extensions for Filter Spaces

A family of extensions for a filter space is defined and its properties, including completeness and total boundedness, are investigated.     

On Spaces of Infinitesimal Motions and Three Dimensional Henneberg Extensions

We investigate certain spaces of infinitesimal motions arising naturally in the rigidity theory of bar and joint frameworks. We prove some structure theorems for these spaces and, as a consequence, are able to deduce some special cases of a long standing conjecture of Graver, Tay and Whiteley concerning Henneberg extensions and generically rigid graphs.     

Minimal Extensions in Tensor Product Spaces

LetX,Ybe two separable Banach spaces and letVXandWYbe finite dimensional subspaces. Suppose thatVSX,WZYand letM (S, V),N (Z, W). We will prove that ifαis a reasonable, uniform crossnorm onXYthenλ_MN(V_αW,X_αY)=λ_M(V, X) λ_N(W, Y).Here for any Banach spaceX,VSXandM (S, V)

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Also some applications of the above mentioned result will be presented.     

Normal Extensions of Operators to Krein Spaces

In this paper, it is proved that every bounded linear operator on a Hilbert space has a normal extension to a Krein space. Two criteria for J-subnormality are given. In particular, in order that T be subnormal, it suffices that exp(-\bar \Lambda T^*)exp(\Lambda T) be a positive definite operator function on a bounded infinite subset of complex plane. This improves the condition of Bram [4]. Also it is proved that the local spectral subspaces are closed for J-subnormal operators.     

Heat Kernels for Isotropic-Like Markov Generators on Ultrametric Spaces: a Survey

Let (X, d) be a locally compact separable ultrametric space. Let D be the set of all locally constant functions having compact support. Given a measure m and a symmetric function J(x, y) we consider the linear operator L^Jf(x) = ∫(f(x) ? f(y)) J(x, y)dm(y) defined on the set D. When J(x, y) is isotropic and satisfies certain conditions, the operator (?L^J, D) acts in L²(X,m), is essentially self-adjoint and extends as a self-adjoint Markov generator, its Markov semigroup admits a continuous heat kernel p^J (t, x, y). When J(x, y) is not isotropic but uniformly in x, y is comparable to isotropic function J(x, y) as above the operator (?L^J, D) extends in L²(X,m) as a self-adjointMarkov generator, its Markov semigroup admits a continuous heat kernel p^J(t, x, y), and the function p^J(t, x, y) is uniformly comparable in t, x, y to the function p^J(t, x, y), the heat kernel related to the operator (?L^J,D).     

Interval Extensions of Orders and Temporal Approximation Spaces

Siberian Mathematical Journal - Studying the algorithmic properties of interval extensions of dense linear orders, in particular, the complexity degrees (namely, the $ sSigma $ -degree)...     

L-拟序集上的广义Alexandroff拓扑与广义超度量空间上的广义Alexandroff拓扑

证明[10]中定义的L-拟序集上的广义Alexandroff拓扑是[2]中定义的广义超度量空间上的广义Alexandroff拓扑的推广，并且广义超度量空间中有关广义Alexandroff拓扑的许多性质都可以推广到L-拟序集中。     

Maps on Ultrametric Spaces,Hensel's Lemma,and Differential Equations Over Valued Fields

We give a criterion for maps on ultrametric spaces to be surjective and to preserve spherical completeness. We show how Hensel's Lemma and the multidimensional Hensel's Lemma follow from our result. We give an easy proof that the latter holds in every henselian field. We also prove a basic infinite-dimensional Implicit Function Theorem. Further, we apply the criterion to deduce various versions of Hensel's Lemma for polynomials in several additive operators, and to give a criterion for the existence of integration and solutions of certain differential equations on spherically complete valued differential fields, for both valued D-fields in the sense of Scanlon, and differentially valued fields in the sense of Rosenlicht. We modify the approach so that it also covers logarithmic-exponential power series fields. Finally, we give a criterion for a sum of spherically complete subgroups of a valued abelian group to be spherically complete. This in turn can be used to determine elementary properties of power series fields in positive characteristic.     

On Restrictions and Extensions of the Besov and Triebel-Lizorkin Spaces with Respect to Lipschitz Domains

The restrictions B^s_pq() and F^s_pq() of the Besov and Triebel–Lizorkinspaces of tempered distributions B^s_pq(Rⁿ) and F^s_pq(Rⁿ) to Lipschitzdomains Rⁿ are studied. For general values of parameters (sR,p>0, q>0) a ‘universal’ linear bounded extensionoperator from B^s_pq() and F^s_pq() into the corresponding spaceson Rⁿ is constructed. The construction is based on a new variantof the Calderón reproducing formula with kernels supportedin a fixed cone. Explicit characterizations of the elementsof B^s_pq() and F^s_pq() in terms of their values in are also obtained.     

Borderline Case of Traces and Extensions for Weighted Sobolev Spaces

In this paper,we study the traces and the extensions for weighted Sobolev spaces on upper half spaces when the weights reach to the borderline cases.We first give a full characterization of the existence of trace spaces for these weighted Sobolev spaces,and then study the trace parts and the extension parts between the weighted Sobolev spaces and a new kind of Besov-type spaces(on hyperplanes) which are defined by using integral averages over selected layers of dyadic cubes.