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Open Set (open + set)

Distribution by Scientific Domains

Mathematics and Statistics	66%

Selected Abstracts

On open-set lattices and some of their applications in semantics

INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS, Issue 12 2003
Mouw-Ching Tjiok
In this article, we present the theory of Kripke semantics, along with the mathematical framework and applications of Kripke semantics. We take the Kripke-Sato approach to define the knowledge operator in relation to Hintikka's possible worlds model, which is an application of the semantics of intuitionistic logic and modal logic. The applications are interesting from the viewpoint of agent interactives and process interaction. We propose (i) an application of possible worlds semantics, which enables the evaluation of the truth value of a conditional sentence without explicitly defining the operator "," (implication), through clustering on the space of events (worlds) using the notion of neighborhood; and (ii) a semantical approach to treat discrete dynamic process using Kripke-Beth semantics. Starting from the topological approach, we define the measure-theoretical machinery, in particular, we adopt the methods developed in stochastic process,mainly the martingale,to our semantics; this involves some Boolean algebraic (BA) manipulations. The clustering on the space of events (worlds), using the notion of neighborhood, enables us to define an accessibility relation that is necessary for the evaluation of the conditional sentence. Our approach is by taking the neighborhood as an open set and looking at topological properties using metric space, in particular, the so-called ,-ball; then, we can perform the implication by computing Euclidean distance, whenever we introduce a certain enumerative scheme to transform the semantic objects into mathematical objects. Thus, this method provides an approach to quantify semantic notions. Combining with modal operators Ki operating on E set, it provides a more-computable way to recognize the "indistinguishability" in some applications, e.g., electronic catalogue. Because semantics used in this context is a local matter, we also propose the application of sheaf theory for passing local information to global information. By looking at Kripke interpretation as a function with values in an open-set lattice ,,U, which is formed by stepwise verification process, we obtain a topological space structure. Now, using the measure-theoretical approach by taking the Borel set and Borel function in defining measurable functions, this can be extended to treat the dynamical aspect of processes; from the stochastic process, considered as a family of random variables over a measure space (the probability space triple), we draw two strong parallels between Kripke semantics and stochastic process (mainly martingales): first, the strong affinity of Kripke-Beth path semantics and time path of the process; and second, the treatment of time as parametrization to the dynamic process using the technique of filtration, adapted process, and progressive process. The technique provides very effective manipulation of BA in the form of random variables and ,-subalgebra under the cover of measurable functions. This enables us to adopt the computational algorithms obtained for stochastic processes to path semantics. Besides, using the technique of measurable functions, we indeed obtain an intrinsic way to introduce the notion of time sequence. © 2003 Wiley Periodicals, Inc. [source]

A remark on the pressure for the Navier,Stokes flows in 2-D straight channel with an obstacle

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 8 2004
H. Morimoto
Abstract Let T=,×(-1,1) and &ℴ,,2 be a smoothly bounded open set, closure of which is contained in T. We consider the stationary Navier,Stokes flows in . In general, the pressure is determined up to a constant. Since , has two extremities, we want to know if we can choose the constant same. We study the behaviour of the pressure at the infinity in , and give a relation between the velocity and the pressure difference. Copyright © 2004 John Wiley & Sons, Ltd. [source]

Extension of bounded vector-valued functions

MATHEMATISCHE NACHRICHTEN, Issue 5 2009
Leonhard Frerick
Abstract In this paper we consider extensions of bounded vector-valued holomorphic (or harmonic or pluriharmonic) functions defined on subsets of an open set , , ,N. The results are based on the description of vector-valued functions as operators. As an application we prove a vector-valued version of Blaschke's theorem (© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]

Singular operators in variable spaces Lp (·)(,, ,) with oscillating weights

MATHEMATISCHE NACHRICHTEN, Issue 9-10 2007
Vakhtang Kokilashvili
Abstract We study the boundedness of singular Calderón,Zygmund type operators in the spaces Lp (·)(,, ,) over a bounded open set in ,n with the weight , (x) = wk(|x , xk|), xk , , where wk has the property that wk(r) , , where is a certain Zygmund-type class. The boundedness of the singular Cauchy integral operator S, along a Carleson curve , is also considered in the spaces Lp (·)(,, ,) with similar weights. The weight functions wk may oscillate between two power functions with different exponents. It is assumed that the exponent p (·) satisfies the Dini,Lipschitz condition. The final statement on the boundedness is given in terms of the index numbers of the functions wk (similar in a sense to the Boyd indices for the Young functions defining Orlicz spaces). (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]

I,Time, Topology and Physical Geometry

ARISTOTELIAN SOCIETY SUPPLEMENTARY VOLUME, Issue 1 2010
Tim Maudlin
The standard mathematical account of the sub-metrical geometry of a space employs topology, whose foundational concept is the open set. This proves to be an unhappy choice for discrete spaces, and offers no insight into the physical origin of geometrical structure. I outline an alternative, the Theory of Linear Structures, whose foundational concept is the line. Application to Relativistic space-time reveals that the whole geometry of space-time derives from temporal structure. In this sense, instead of spatializing time, Relativity temporalizes space. [source]

Existence of an asymptotic velocity and implications for the asymptotic behavior in the direction of the singularity in T3 -Gowdy

COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 7 2006
Hans Ringström
This is the first of two papers that together prove strong cosmic censorship in T3 -Gowdy space-times. In the end, we prove that there is a set of initial data, open with respect to the C2 × C1 topology and dense with respect to the C, topology, such that the corresponding space-times have the following properties: Given an inextendible causal geodesic, one direction is complete and the other is incomplete; the Kretschmann scalar, i.e., the Riemann tensor contracted with itself, blows up in the incomplete direction. In fact, it is possible to give a very detailed description of the asymptotic behavior in the direction of the singularity for the generic solutions. In this paper, we shall, however, focus on the concept of asymptotic velocity. Under the symmetry assumptions made here, Einstein's equations reduce to a wave map equation with a constraint. The target of the wave map is the hyperbolic plane. There is a natural concept of kinetic and potential energy density; perhaps the most important result of this paper is that the limit of the potential energy as one lets time tend to the singularity for a fixed spatial point is 0 and that the limit exists for the kinetic energy. We define the asymptotic velocity v, to be the nonnegative square root of the limit of the kinetic energy density. The asymptotic velocity has some very important properties. In particular, curvature blowup and the existence of smooth expansions of the solutions close to the singularity can be characterized by the behavior of v,. It also has properties such that if 0 < v,(,0) < 1, then v, is smooth in a neighborhood of ,0. Furthermore, if v,(,0) > 1 and v, is continuous at ,0, then v, is smooth in a neighborhood of ,0. Finally, we show that the map from initial data to the asymptotic velocity is continuous under certain circumstances and that what will in the end constitute the generic set of solutions is an open set with respect to C2 × C1 topology on initial data. © 2005 Wiley Periodicals, Inc. [source]