A unified approach to fuzzy pattern recognition

Goldfarb has proposed a unified approach to pattern recognition. In Goldfarb's approach the data from a pseudometric space are isometrically embedded in a pseudo-Euclidean space. The data representation is built for every particular data set. The resulting data space will be data dependent. The aim of this paper is to extend Goldfarb's approach for fuzzy clustering. A fuzzy clustering procedure for a pseudometric data set is given. A generalisation of this algorithm to obtain the cluster substructure of a fuzzy class is also proposed.     

Divided powers and composition in integro-differential algebras

An integro-differential algebra of arbitrary characteristic is given the structure of a uniform topological space, such that the ring operations as well as the derivation (= differentiation operator) and Rota–Baxter operator (= integral operator) are uniformly continuous. Using topological techniques and the central notion of divided powers, this allows one to introduce a composition for (topologically) complete integro-differential algebras; this generalizes the series case, viz. meaning formal power series in characteristic zero and Hurwitz series in positive characteristic. The canonical Hausdorff completion for pseudometric spaces is shown to produce complete integro-differential algebras.The setting of complete integro-differential algebras allows us to describe exponential and logarithmic elements in a way that reflects the “integro-differential properties” known from analysis. Finally, we prove also that any complete integro-differential algebra is saturated, in the sense that every (monic) linear differential equation possesses a regular fundamental system of solutions.While the paper focuses on the commutative case, many results are given for the general case of (possibly noncommutative) rings, whenever this does not require substantial modifications.     

Hofer’s metric on the space of Lagrangian submanifolds and wrapped Floer homology

In this paper, we study the relationship between wrapped Floer homology and displaceability of a Lagrangian submanifold which we call vanishing theorem of wrapped Floer homology. We also use this theorem to study Hofer’s pseudometric on the space of Lagrangian submanifolds. We prove an inequality, the Lagrangian version of the inequality of Gromov width and displacement energy, which is called energy-capacity inequality.     

Sigma-compactness of metric boolean algebras and uniform convergence of frequencies to probabilities

The topological properties of a pseudometric space defined by a measure are investigated. Criteria of compactness and σ-compactness of this space are proved. A new sufficient condition for the uniform convergence (over an event class) of frequencies to probabilities is proved as a corollary.     

Pre-Schwarz导数意义下区域之间的距离

Lehto曾用Schwarz导数定义了边界多于一点的两个单连通区域的Mbius等价类之间的"距离",并猜测它是一个距离.但最近Bozin和Markovic否定了这一猜想.一个自然的问题就是:在Pre-Schwarz导数意义相应情况如何?用Pre-Schwarz导数给出了边界多于一点的两个单连通区域的仿射等价类之间的"距离",并证明了这样定义的"距离"是一个伪距离,即使将其限制在由具有解析边界的单连通区域的仿射等价类空间上也是如此.     

Metric Axiomatics of Euclidean Space @forename = A. A.

A version of axioms of Euclidean space based on a single initial notion, namely on the notion of distance, is considered. The notions of straight line and plane are introduced in terms of distance. Thus, Euclidean space is regarded as a metric space with metric satisfying the corresponding axioms. Bibliography: 3 titles.     

Stability analysis of stochastic programs with second order dominance constraints

In this paper we present a stability analysis of a stochastic optimization problem with stochastic second order dominance constraints. We consider a perturbation of the underlying probability measure in the space of regular measures equipped with pseudometric discrepancy distance (Römisch in Stochastic Programming. Elsevier, Amsterdam, pp 483–554, 2003). By exploiting a result on error bounds in semi-infinite programming due to Gugat (Math Program Ser B 88:255–275, 2000), we show under the Slater constraint qualification that the optimal value function is Lipschitz continuous and the optimal solution set mapping is upper semicontinuous with respect to the perturbation of the probability measure. In particular, we consider the case when the probability measure is approximated by an empirical probability measure and show an exponential rate of convergence of the sequence of optimal solutions obtained from solving the approximation problem. The analysis is extended to the stationary points.     

The Ultrametric Space of Plane Branches

We study properties of the space of irreducible germs of plane curves (branches), seen as an ultrametric space. We provide various geometrical methods to measure the distance between two branches and to compare distances between branches, in terms of topological invariants of the singularity which comprises some of the branches. We show that, in spite of being very close to the notion of intersection multiplicity between two germs, this notion of distance behaves very differently. For instance, any value in [0, 1] ∩ ? is attained as the distance between a fixed branch and some other branch, in contrast with the fact that the semigroup of the fixed branch has gaps. We also present results that lead to interpret this distance as a sort of geometric distance between the topological equivalence or equisingularity classes of branches.     

On the Metrizability of Fuzzy Neighborhood Spaces

A fuzzy neighborhood space is said to be pseudometrizable if, and only if, the associated fuzzy topology is induced by a probabilistic pseudometric under. By the aid of the concept of quasi-inverse functions it will be shown that a fuzzy neighborhood space is pseudometrizable if, and only if, every associated level-topology is pseudometrizable and a family of generating pseudometrics is monotone decreasing.     

Transportation distances and noise sensitivity of multiplicative Lévy SDE with applications

This article assesses the distance between the laws of stochastic differential equations with multiplicative Lévy noise on path space in terms of their characteristics. The notion of transportation distance on the set of Lévy kernels introduced by Kosenkova and Kulik yields a natural and statistically tractable upper bound on the noise sensitivity. This extends recent results for the additive case in terms of coupling distances to the multiplicative case. The strength of this notion is shown in a statistical implementation for simulations and the example of a benchmark time series in paleoclimate.     

Multiobjective optimization in a pseudometric objective space as applied to a general model of business activities

It is shown that finding the equivalence set for solving multiobjective discrete optimization problems is advantageous over finding the set of Pareto optimal decisions. An example of a set of key parameters characterizing the economic efficiency of a commercial firm is proposed, and a mathematical model of its activities is constructed. In contrast to the classical problem of finding the maximum profit for any business, this study deals with a multiobjective optimization problem. A method for solving inverse multiobjective problems in a multidimensional pseudometric space is proposed for finding the best project of firm’s activities. The solution of a particular problem of this type is presented.     

Distance Types in Banach Spaces

We introduce and study the notion of a distance type, on a Banach space, defined by a nested sequence of convex sets. Among other things, we show that there always exist distance types that are not types in the classical sense. Then, we recover the notion of the flat nested sequence of Milman and Milman and show that distance types defined by flat nested sequences coincide with the bidual types of Farmaki. These results are applied to show that a flat nested sequence of convex sets is Wijsman convergent to the intersection of their weak*-closures in bidual space.     

Navigation in tree spaces

The orientable cover of the moduli space of real genus zero algebraic curves with marked points is a compact aspherical manifold tiled by associahedra, which resolves the singularities of the space of phylogenetic trees. The resolution maps planar metric trees to their underlying abstract representatives, collapsing and folding an explicit geometric decomposition of the moduli space into cubes, endowing the resolving space with an interesting canonical pseudometric. Indeed, the given map can be reinterpreted as relating the real and the tropical versions of the Deligne–Knudsen–Mumford compactification of the moduli space of Riemann spheres.     

关于概率赋范线性空间上的线性算子的一致收敛

本文引入了概率赋范线性空间上线性算子的一致收敛和可完全刻划这种收敛的算子间的概率距离概念,并利用这些概念获得了算子连续和算子列一致收敛的本质特征,及其连续性和全连续性对于一致收敛极限运算的封闭性.     

Topological structures of complex belief systems

The concepts of substantive beliefs and derived beliefs are defined, a set of substantive beliefs S like open set and the neighborhood of an element substantive belief. A semantic operation of conjunction is defined with a structure of an Abelian group. Mathematical structures exist such as poset beliefs and join‐semilattttice beliefs. A metric space of beliefs and the distance of belief depending on the believer are defined. The concepts of closed and opened ball are defined. S′ is defined as subgroup of the metric space of beliefs Σ and S′ is a totally limited set. The term s is defined (substantive belief) in terms of closing of S′. It is deduced that Σ is paracompact due to Stone's Theorem. The pseudometric space of beliefs is defined to show how the metric of the nonbelieving subject has a topological space like a nonmaterial abstract ideal space formed in the mind of the believing subject, fulfilling the conditions of Kuratowski axioms of closure. To establish patterns of materialization of beliefs we are going to consider that these have defined mathematical structures. This will allow us to understand better cultural processes of text, architecture, norms, and education that are forms or the materialization of an ideology. This materialization is the conversion by means of certain mathematical correspondences, of an abstract set whose elements are beliefs or ideas, in an impure set whose elements are material or energetic. Text is a materialization of ideology. © 2013 Wiley Periodicals, Inc. Complexity 19: 46–62, 2013     

Extreme values of operator norms in spaces with equivalent norms

The paper is concerned with various characteristics of spaces (L, ‖·‖) with equivalent norms and of operators on these spaces. In particular, the notion of distance between equivalent norms in the space L is introduced and the relationship between norms of the space and norms of operators in L is studied. Bibliography: 4 titles. Translated fromObchyslyuval'na ta Prykladna Matematyka, No. 80, 1996, pp. 25–32.     

Codimension and pseudometric in co-Heyting algebras

In this paper, we introduce a notion of dimension and codimension for every element of a bounded distributive lattice L. These notions prove to have a good behavior when L is a co-Heyting algebra. In this case the codimension gives rise to a pseudometric on L which satisfies the ultrametric triangle inequality. We prove that the Hausdorff completion of L with respect to this pseudometric is precisely the projective limit of all its finite dimensional quotients. This completion has some familiar metric properties, such as the convergence of every monotonic sequence in a compact subset. It coincides with the profinite completion of L if and only if it is compact or equivalently if every finite dimensional quotient of L is finite. In this case we say that L is precompact. If L is precompact and Hausdorff, it inherits many of the remarkable properties of its completion, specially those regarding the join/meet irreducible elements. Since every finitely presented co-Heyting algebra is precompact Hausdorff, all the results we prove on the algebraic structure of the latter apply in particular to the former. As an application, we obtain the existence for every positive integers n, d of a term t _{n, d} such that in every co-Heyting algebra generated by an n-tuple a, t _{n, d} (a) is precisely the maximal element of codimension d.     

An Approximate Nerve Theorem

The nerve theorem relates the topological type of a suitably nice space with the nerve of a good cover of that space. It has many variants, such as to consider acyclic covers and numerous applications in topology including applied and computational topology. The goal of this paper is to relax the notion of a good cover to an approximately good cover, or more precisely, we introduce the notion of an \(\varepsilon \)-acyclic cover. We use persistent homology to make this rigorous and prove tight bounds between the persistent homology of a space endowed with a function and the persistent homology of the nerve of an \(\varepsilon \)-acyclic cover of the space. Our approximations are stated in terms of interleaving distance between persistence modules. Using the Mayer–Vietoris spectral sequence, we prove upper bounds on the interleaving distance between the persistence module of the underlying space and the persistence module of the nerve of the cover. To prove the best possible bound, we must introduce special cases of interleavings between persistence modules called left and right interleavings. Finally, we provide examples which achieve the bound proving the lower bound and tightness of the result.     

Smooth kinematic-type manifolds

We propose a general approach for describing different causality-type relations on smooth manifolds. The causality structure can be defined either axiomatically (by a cone in the tangent space) or by a pseudometric with the signature (+−...−) or (+−...−0...0). In the latter case, the manifold acquires the structure of a fibered space with “absolute simultaneity” fibers. The smooth structure (atlas) of the manifold is directly related to its causal structure. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 119, No. 2, pp. 264–281, May, 1999.