Identification Of Metrics Research Articles

Intuitionistic-distribution numbers

In this paper, we first give the definition of the intuitionistic fuzzy numbers, then we define intuitionistic-distribution numbers and discuss their main properties. At last, we discuss the applications of the intuitionistic-distribution fuzzy number in the intuitionistic generalized pseudo-probability metric spaces and the intuitionistic pseudo-metric spaces.

A CHARACTERIZATION OF NONLINEAR OPEN MAPPINGS BETWEEN PSEUDOMETRIZABLE TOPOLOGICAL VECTOR SPACES

We prove a new characterization of arbitrary (including nonlinear) open maps between pseudometrizable topological vector spaces. We use it to give a new proof that nonconstant analytic functions are open.

INTUITIONISTIC FUZZY QUASI-METRIC AND PSEUDO-METRIC SPACES

In this paper, we propose a new definition of intuitionistic fuzzy quasi-metric and pseudo-metric spaces based on intuitionistic fuzzy points. We prove some properties of intuitionistic fuzzy quasi- metric and pseudo-metric spaces, and show that every intuitionistic fuzzy pseudo-metric space is intu- itionistic fuzzy regular and intuitionistic fuzzy completely normal and hence intuitionistic fuzzy normal. These are the intuitionistic fuzzy generalization of the corresponding properties of fuzzy quasi-metric and pseudo- metric spaces.

Estimates for the distribution of the supremum of Θ-pre-Gaussian random processes

We consider some classes of pre-Gaussian random processes. We obtain several theorems on estimation of distribution of the supremum of these processes both on pseudometric space (T, ρ) and on ℝ.

Randomization of classical inference patterns and its application

By means of randomization, the concept of D-randomized truth degree of formulas in two-valued propositional logic is introduced, and it is proved that the set of values of D-randomized truth degree of formulas has no isolated point in [0,1]. The concepts of D-logic pseudo-metric and D-logic metric space are also introduced and it is proved that there is no isolated point in the space. The new built D-randomized concepts are extensions of the corresponding concepts in quantified logic. Moreover, it is proved that the basic logic connectives are continuous operators in D-logic metric space. Lastly, three different types of approximate reasoning patterns are proposed.

MetricMap: An Embedding Technique for Processing Distance-Based Queries in Metric Spaces

In this paper, we present an embedding technique, called MetricMap, which is capable of estimating distances in a pseudometric space. Given a database of objects and a distance function for the objects, which is a pseudometric, we map the objects to vectors in a pseudo-Euclidean space with a reasonably low dimension while preserving the distance between two objects approximately. Such an embedding technique can be used as an approximate oracle to process a broad class of distance-based queries. It is also adaptable to data mining applications such as data clustering and classification. We present the theory underlying MetricMap and conduct experiments to compare MetricMap with other methods including MVP-tree and M-tree in processing the distance-based queries. Experimental results on both protein and RNA data show the good performance and the superiority of MetricMap over the other methods.

Continuity and Boundedness of Infinitely Divisible Processes: A Poisson Point Process Approach

Sufficient conditions for boundedness and continuity are obtained for stochastically continuous infinitely divisible processes, without Gaussian component, {Y(t),t ∈T}, where T is a compact metric space or pseudo-metric space. Such processes have a version given by Y(t)=X(t)+b(t),t∈T where b is a deterministic drift function and $$X(t) = \int_S f(t,s) \left[N(ds)-(|f(t,s)|\vee 1)^{-1}\nu(ds)\right].$$ Here N is a Poisson random measure on a Borel space S with σ−finite mean measure ν, and $$f{:} T \times S \mapsto R$$ is a measurable deterministic function. Let τ: T2 → R+ be a continuous pseudo–metric on T. Define the τ-Lipschitz norm of the sections of f by $$ \|f\|_{\tau} (s)=D^{-1}f(t_0,s) + \sup\limits_{u,v\in T} {{|f(u,s)-f(v,s)|}\over{\tau(u,v)}}$$ for some t0 ∈T, where D is the diameter of (T,τ). The sufficient conditions for boundedness and continuity of X are given in terms of the measure $$\nu,\||f\||_\tau$$ and majorizing measure and or metric entropy conditions determined by τ. They are applied to stochastic integrals of the form $$Y(t) = \int_S g(t,s)M(ds)\quad t\in T,$$ where M is a zero-mean, independently scattered, infinitely divisible random measure without Gaussian component. Several examples are given which show that in many cases the conditions obtained are quite sharp. In addition to obtaining conditions for continuity and boundedness, bounds are obtained for the weak and strong L p norms of $$\sup_{t\in T}|X(t)|$$ and $$\sup_{\tau (t,u)\leqslant\delta,t,u \in T}|X(t) - X(u)|$$ for all $$0<\delta \leqslant D$$ . These results depend on inequalities for moments and related functions of the weak and strong $$\ell^p$$ norms of sequences {x j }, which are the events of Poisson point process M on R+ and are given in terms of the intensity measure of M. These results are of independent interest.

The Large Deviation Principle for Stochastic Processes. II

We discuss the large deviation principle of stochastic processes as random elements of $l_{\infty}(T)$. We show that the large deviation principle in $l_{\infty}(T)$ is equivalent to the large deviation principle of the finite dimensional distributions plus an exponential asymptotic equicontinuity condition with respect to a pseudometric which makes T a totally bounded pseudometric space. This result allows us to obtain necessary and sufficient conditions for the large deviation principle of different types of stochastic processes. We discuss the large deviation principle of Gaussian and Poisson processes. As an application, we determine the integrability of the iterated fractional Brownian motion.

The Large Deviation Principle for Stochastic Processes. Part I

We discuss the large deviation principle of stochastic processes as random elements of~$l_{\infty}(T)$. We show that the large deviation principle in $l_{\infty}(T)$ is equivalent to the large deviation principle of the finite dimensional distributions plus an exponential asymptotic equicontinuity condition with respect to a pseudometric, which makes~T a totally bounded pseudometric space. This result allows us to obtain necessary and sufficient conditions for the large deviation principle of different types of stochastic processes. We discuss the large deviation principle of Gaussian and Poisson processes. As an application, we determine the integrability of the iterated fractional Brownian motion.

An extension of Sugeno integral

In this paper, we extend the concept of Sugeno fuzzy integral from nonnegative fuzzy measurable functions to extended real-valued fuzzy measurable functions and discuss the lost genuine properties for this extension; several necessary and sufficient conditions of absolute ( S)-integrability for extended real-valued fuzzy measurable functions are given. Moreover, the space ( S( μ), ρ(.,.)) of all fuzzy measurable functions will be proved to be a pseudo-metric space under a necessary and sufficient condition. Finally, as an application of this extension the Pettis integral will be established for this kind of fuzzy integral.

The Cartesian Closed Topological Hull of the Category of Approach Uniform Spaces

The category AUnif of approach uniform spaces and uniform contractions properly combines uniform spaces and extended pseudo-metric spaces but (like Unif) lacks convenience, such as cartesian closedness. This paper therefore considers its cartesian closed topological hull, which is first described as a subcategory of SAULim, the category of semi-approach uniform limit spaces and uniform contractions. This hull is then also given a description inside the topological universe hull of AUnif and is shown to be a reasonable generalization of the corresponding hull of Unif. Furthermore, some referencing notes are provided with respect to similar results that can be obtained when starting from qAUnif (where symmetry assumptions are omitted).

Solve[order/topology == quasi-metric/x, x

In the study of the semantics of programming languages, the qualitative framework using partially ordered sets and the quantitative framework using pseudo-metric spaces have existed separately for years. Smyth however noticed that both concepts can be unified by means of quasi-metric spaces.Recent literature concerning these “quantitative domains”, lacks the canonicity which is so typical for the relationship between topological techniques and theoretical computer science in the classical settings mentioned above. On the one hand, this yields the use of structures which could be considered “ad hoc” from a categorical point of view, such as continuity spaces by Flagg and Kopperman. On the other hand, this yields “incomplete structures”, which essentially belong to one of both classical settings, such as the generalized Scott topology by Bonsangue e.a.We shall discuss a natural generalization of the symbiosis between ordered sets and topology to an analogous relationship between quasi-metric spaces and approach spaces. Approach spaces seem to be an important tool in the study of certain aspects concerning quantitative domains.

The Baire Category Theorem and choice

The status of the Baire Category Theorem in ZF (i.e., Zermelo–Fraenkel set theory without the Axiom of Choice) is investigated. Typical results: 1. The Baire Category Theorem holds for compact pseudometric spaces. 2. The Axiom of Countable Choice is equivalent to the Baire Category Theorem for countable products of compact pseudometric spaces. 3. The Axiom of Dependent Choice is equivalent to the Baire Category Theorem for countable products of compact Hausdorff spaces. 4. The Baire Category Theorem for B -compact regular spaces is equivalent to the conjunction of the Axiom of Dependent Choice and the Weak Ultrafilter Theorem .

On Weierstrass compact pseudometric spaces and a weak form of the axiom of choice

We show that the countable multiple choice axiom CMC is equivalent to the assertion: Weierstrass compact pseudometric spaces are compact.

On the existence of doubling measures with certain regularity properties

Given a compact pseudo-metric space, we associate to it upper and lower dimensions, depending only on the pseudo-metric. Then we construct a doubling measure for which the measure of a dilated ball is closely related to these dimensions.

On the LP space of observables

A space of fuzzy sets M is considered as a base of a quantum structure model (fuzzy quantum logic, see Riečan and Neubrunn, Integral, Measure and Ordering, Kluwer, Dordrecht, 1997). A state is a morphism from M to the unit interval, an observable is a morphism from Borel sets to M. The subspace L P of observables is studied and it is proved that L P considered with the corresponding pseudometric is a complete pseudometric space.

The applications of interval-valued fuzzy numbers and interval-distribution numbers

In this paper, we first define interval-valued fuzzy numbers and interval-distribution numbers, and then their extended operations are given. Next, we discuss some applications of generalized pseudo-probability metric spaces and pseudo-metric spaces.

Countable choice and pseudometric spaces

In the realm of pseudometric spaces the role of choice principles is investigated. In particular it is shown that in ZF (i.e., Zermelo-Fraenkel set theory without the axiom of choice) the axiom of countable choice is not only sufficient but also necessary to establish each of the following results: 1. 1. separable ↔ countable base, 2. 2. separable ↔ Lindelöf, 3. 3. separable ↔ topologically totally bounded, 4. 4. compact → separable, 5. 5. separability is hereditary, 6. 6. the Baire Category Theorem for complete spaces with countable base, 7. 7. the Baire Category Theorem for complete, totally bounded spaces, 8. 8. compact ↔ sequentially compact, 9. 9. compact ↔ (totally bounded and complete), 10. 10. sequentially compact ↔ (totally bounded and complete), 11. 11. Weierstraβ compact ↔ (totally bounded and complete).

Further criteria for the indecomposability of finite pseudometric spaces

We continue the study of indecomposable finite (consisting of a finite number of points) pseudometric spaces (i.e., spaces whose only decomposition into a sum is the division of all distances in equal proportion). We prove that the indecomposability property is invariant under the following operation: connect two disjoint points by an additional simple chain, which is the inverted copy of the shortest path connecting these points. The indecomposability of the spaces presented by the graphsK m,n (m ≥ 2,n ≥ 3) with edges of equal length is also proved.

Decomposition of finite pseudometric spaces

Here we define decomposable pseudometrics. A pseudometric is decomposable if it can be represented as the sum of two pseudometrics that are obtained in a way other than the multiplication all distances by a positive factor. We consider spaces consisting ofn points. We prove that there exist a finite number of indecomposable pseudometrics (that is, a basis) such that any pseudometric is a linear combination of basic pseudometrics with nonnegative coefficients. Forn ≤ 7, the basic pseudometrics are listed. A decomposability test is derived for finite pseudometric spaces. We also establish some other conditions of decomposability and indecomposability.