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## Metric Space (metric + space)
## Selected Abstracts## Metric spaces in NMR crystallography CONCEPTS IN MAGNETIC RESONANCE, Issue 4 2009David M. GrantAbstract The anisotropic character of the chemical shift can be measured by experiments that provide shift tensor values and comparing these experimental components, obtained from microcrystalline powders, with 3D nuclear shielding tensor components, calculated with quantum chemistry, yields structural models of the analyzed molecules. The use of a metric tensor for evaluating the mean squared deviations, d2, between two or more tensors provides a statistical way to verify the molecular structure governing the theoretical shielding components. The sensitivity of the method is comparable with diffraction methods for the heavier organic atoms (i.e., C, O, N, etc.) but considerably better for the positions of H atoms. Thus, the method is especially powerful for H-bond structure, the position of water molecules in biomolecular species, and other proton important structural features, etc. Unfortunately, the traditional Cartesian tensor components appear as reducible metric representations and lack the orthogonality of irreducible icosahedral and irreducible spherical tensors, both of which are also easy to normalize. Metrics give weighting factors that carry important statistical significance in a structure determination. Details of the mathematical analysis are presented and examples given to illustrate the reason nuclear magnetic resonance are rapidly assuming an important synergistic relationship with diffraction methods (X-ray, neutron scattering, and high energy synchrotron irradiation). © 2009 Wiley Periodicals, Inc.Concepts Magn Reson Part A 34A: 217,237, 2009. [source] ## Metric spaces and the axiom of choice MLQ- MATHEMATICAL LOGIC QUARTERLY, Issue 5 2003Omar De la CruzAbstract We study conditions for a topological space to be metrizable, properties of metrizable spaces, and the role the axiom of choice plays in these matters. [source] ## Reaching consensus in multiagent decision making INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS, Issue 3 2010Antonio MaturoA group decision making procedure, when developed in a dynamic context, grows from both an evolving knowledge base and the changes in the positions of the components, or experts, of the group. Agreement and compromise go with and drive the steps of the procedure. Let us consider the case that a committee of experts is constituted to take decisions about a subject of social interest. Usually the job ends if a majority of the members of the committee have not too different opinions about the last state of the decision. We intend to clarify the meaning of the statement "have not too different opinions," to define a structure for the concept of consensus. We assume that an external chairman, with complete information about the state of all the components of group, urges or invites decision makers to reach a consensus. The judgements of the experts are represented, in this framework, by points in a metric space, and the consensus is obtained by a dynamical construction of a maximal winning coalition contained in a ball with a fixed and suitably small diameter. This also allows us to deal with the concept of consensus in terms of algorithm. © 2010 Wiley Periodicals, Inc. [source] ## On open-set lattices and some of their applications in semantics INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS, Issue 12 2003Mouw-Ching TjiokIn 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] ## Computational verb systems: Verb numbers INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS, Issue 5 2001Tao YangIn this paper the concepts of (computational) verb number and arithmetical operations of verb numbers are presented. A verb number is a kind of special computational verb, which is derived from structures of (host) verb+(real, interval, fuzzy) number. From the linguistic point of view, a verb number is a computational verb with contexts defined by (real, interval, fuzzy) numbers. Verb numbers can be classified into three basic types based on the outer systems of host verbs. The arithmetic for verb numbers and its rules are presented and proved. If host computational verbs degrade to BE, then verb numbers collapse to real numbers, interval numbers, fuzzy numbers (sets), or other numbers. The set of all verb numbers can be a metric space. The distance between verb numbers can be defined based on the collapses of verb numbers. The cases when verb numbers collapse to triangular fuzzy numbers, trapezoidal fuzzy numbers, and interval numbers are presented and proved. © 2001 John Wiley & Sons, Inc. [source] ## Continuity properties of preference relations MLQ- MATHEMATICAL LOGIC QUARTERLY, Issue 5 2008Marian A. BaroniAbstract Various types of continuity for preference relations on a metric space are examined constructively. In particular, necessary and sufficient conditions are given for an order-dense, strongly extensional preference relation on a complete metric space to be continuous. It is also shown, in the spirit of constructive reverse mathematics, that the continuity of sequentially continuous, order-dense preference relations on complete, separable metric spaces is connected to Ishihara's principleBD -,, and therefore is not provable within Bishop-style constructive mathematics alone. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] ## On countable choice and sequential spaces MLQ- MATHEMATICAL LOGIC QUARTERLY, Issue 2 2008Gonçalo GutierresAbstract Under the axiom of choice, every first countable space is a Fréchet-Urysohn space. Although, in its absence even , may fail to be a sequential space. Our goal in this paper is to discuss under which set-theoretic conditions some topological classes, such as the first countable spaces, the metric spaces, or the subspaces of ,, are classes of Fréchet-Urysohn or sequential spaces. In this context, it is seen that there are metric spaces which are not sequential spaces. This fact raises the question of knowing if the completion of a metric space exists and it is unique. The answer depends on the definition of completion. Among other results it is shown that: every first countable space is a sequential space if and only if the axiom of countable choice holds, the sequential closure is idempotent in , if and only if the axiom of countable choice holds for families of subsets of ,, and every metric space has a unique -completion. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] ## Compactness under constructive scrutiny MLQ- MATHEMATICAL LOGIC QUARTERLY, Issue 6 2004Hajime IshiharaAbstract How are the various classically equivalent definitions of compactness for metric spaces constructively interrelated? This question is addressed with Bishop-style constructive mathematics as the basic system , that is, the underlying logic is the intuitionistic one enriched with the principle of dependent choices. Besides surveying today's knowledge, the consequences and equivalents of several sequential notions of compactness are investigated. For instance, we establish the perhaps unexpected constructive implication that every sequentially compact separable metric space is totally bounded. As a by-product, the fan theorem for detachable bars of the complete binary fan proves to be necessary for the unit interval possessing the Heine-Borel property for coverings by countably many possibly empty open balls. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] ## Two constructive embedding-extension theorems with applications to continuity principles and to Banach-Mazur computability MLQ- MATHEMATICAL LOGIC QUARTERLY, Issue 4-5 2004Andrej BauerAbstract We prove two embedding and extension theorems in the context of the constructive theory of metric spaces. The first states that Cantor space embeds in any inhabited complete separable metric space (CSM) without isolated points, X, in such a way that every sequentially continuous function from Cantor space to , extends to a sequentially continuous function from X to ,. The second asserts an analogous property for Baire space relative to any inhabited locally non-compact CSM. Both results rely on having careful constructive formulations of the concepts involved. As a first application, we derive new relationships between "continuity principles" asserting that all functions between specified metric spaces are pointwise continuous. In particular, we give conditions that imply the failure of the continuity principle "all functions from X to , are continuous", when X is an inhabited CSM without isolated points, and when X is an inhabited locally non-compact CSM. One situation in which the latter case applies is in models based on "domain realizability", in which the failure of the continuity principle for any inhabited locally non-compact CSM, X, generalizes a result previously obtained by Escardó and Streicher in the special case X = C[0, 1]. As a second application, we show that, when the notion of inhabited complete separable metric space without isolated points is interpreted in a recursion-theoretic setting, then, for any such space X, there exists a Banach-Mazur computable function from X to the computable real numbers that is not Markov computable. This generalizes a result obtained by Hertling in the special case that X is the space of computable real numbers. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] ## Consequences of the failure of the axiom of choice in the theory of Lindelöf metric spaces MLQ- MATHEMATICAL LOGIC QUARTERLY, Issue 2 2004Kyriakos KeremedisAbstract We study within the framework of Zermelo-Fraenkel set theory ZF the role that the axiom of choice plays in the theory of Lindelöf metric spaces. We show that in ZF the weak choice principles: (i) Every Lindelöf metric space is separable and (ii) Every Lindelöf metric space is second countable (Forms 340 and 341, respectively, in [10]) are equivalent. We also prove that a Lindelöf metric space is hereditarily separable iff it is hereditarily Lindelöf iff it hold as well the axiom of choice restricted to countable sets and to topologies of Lindelöf metric spaces as the countable union theorem restricted to Lindelöf metric spaces. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] ## The center function on trees NETWORKS: AN INTERNATIONAL JOURNAL, Issue 2 2001F. R. McMorrisAbstract When (X, d) is a finite metric space and , = (x1, ,, xk) , Xk, a central element for , is an element x of X for which max{d (x, xi): i = 1, ,, k} is minimum. The function that returns the set of all central elements for any tuple , is called the center function on X. In this article, the center function on finite trees is characterized. © John Wiley & Sons, Inc. [source] ## Tilings, packings, coverings, and the approximation of functions MATHEMATISCHE NACHRICHTEN, Issue 1 2004Aicke HinrichsAbstract A packing (resp. covering) , of a normed space X consisting of unit balls is called completely saturated (resp. completely reduced) if no finite set of its members can be replaced by a more numerous (resp. less numerous) set of unit balls of X without losing the packing property (resp. covering property) of ,. We show that a normed space X admits completely saturated packings with disjoint closed unit balls as well as completely reduced coverings with open unit balls, provided that there exists a tiling of X with unit balls. Completely reduced coverings by open balls are of interest in the context of an approximation theory for continuous real-valued functions that rests on so-called controllable coverings of compact metric spaces. The close relation between controllable coverings and completely reduced coverings allows an extension of the approximation theory to non-compact spaces. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] ## Continuity properties of preference relations MLQ- MATHEMATICAL LOGIC QUARTERLY, Issue 5 2008Marian A. BaroniAbstract Various types of continuity for preference relations on a metric space are examined constructively. In particular, necessary and sufficient conditions are given for an order-dense, strongly extensional preference relation on a complete metric space to be continuous. It is also shown, in the spirit of constructive reverse mathematics, that the continuity of sequentially continuous, order-dense preference relations on complete, separable metric spaces is connected to Ishihara's principleBD -,, and therefore is not provable within Bishop-style constructive mathematics alone. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] ## On countable choice and sequential spaces MLQ- MATHEMATICAL LOGIC QUARTERLY, Issue 2 2008Gonçalo GutierresAbstract Under the axiom of choice, every first countable space is a Fréchet-Urysohn space. Although, in its absence even , may fail to be a sequential space. Our goal in this paper is to discuss under which set-theoretic conditions some topological classes, such as the first countable spaces, the metric spaces, or the subspaces of ,, are classes of Fréchet-Urysohn or sequential spaces. In this context, it is seen that there are metric spaces which are not sequential spaces. This fact raises the question of knowing if the completion of a metric space exists and it is unique. The answer depends on the definition of completion. Among other results it is shown that: every first countable space is a sequential space if and only if the axiom of countable choice holds, the sequential closure is idempotent in , if and only if the axiom of countable choice holds for families of subsets of ,, and every metric space has a unique -completion. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] ## Effective Borel measurability and reducibility of functions MLQ- MATHEMATICAL LOGIC QUARTERLY, Issue 1 2005Vasco BrattkaAbstract The investigation of computational properties of discontinuous functions is an important concern in computable analysis. One method to deal with this subject is to consider effective variants of Borel measurable functions. We introduce such a notion of Borel computability for single-valued as well as for multi-valued functions by a direct effectivization of the classical definition. On Baire space the finite levels of the resulting hierarchy of functions can be characterized using a notion of reducibility for functions and corresponding complete functions. We use this classification and an effective version of a Selection Theorem of Bhattacharya-Srivastava in order to prove a generalization of the Representation Theorem of Kreitz-Weihrauch for Borel measurable functions on computable metric spaces: such functions are Borel measurable on a certain finite level, if and only if they admit a realizer on Baire space of the same quality. This Representation Theorem enables us to introduce a realizer reducibility for functions on metric spaces and we can extend the completeness result to this reducibility. Besides being very useful by itself, this reducibility leads to a new and effective proof of the Banach-Hausdorff-Lebesgue Theorem which connects Borel measurable functions with the Baire functions. Hence, for certain metric spaces the class of Borel computable functions on a certain level is exactly the class of functions which can be expressed as a limit of a pointwise convergent and computable sequence of functions of the next lower level. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] ## Compactness under constructive scrutiny MLQ- MATHEMATICAL LOGIC QUARTERLY, Issue 6 2004Hajime IshiharaAbstract How are the various classically equivalent definitions of compactness for metric spaces constructively interrelated? This question is addressed with Bishop-style constructive mathematics as the basic system , that is, the underlying logic is the intuitionistic one enriched with the principle of dependent choices. Besides surveying today's knowledge, the consequences and equivalents of several sequential notions of compactness are investigated. For instance, we establish the perhaps unexpected constructive implication that every sequentially compact separable metric space is totally bounded. As a by-product, the fan theorem for detachable bars of the complete binary fan proves to be necessary for the unit interval possessing the Heine-Borel property for coverings by countably many possibly empty open balls. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] ## Two constructive embedding-extension theorems with applications to continuity principles and to Banach-Mazur computability MLQ- MATHEMATICAL LOGIC QUARTERLY, Issue 4-5 2004Andrej BauerAbstract We prove two embedding and extension theorems in the context of the constructive theory of metric spaces. The first states that Cantor space embeds in any inhabited complete separable metric space (CSM) without isolated points, X, in such a way that every sequentially continuous function from Cantor space to , extends to a sequentially continuous function from X to ,. The second asserts an analogous property for Baire space relative to any inhabited locally non-compact CSM. Both results rely on having careful constructive formulations of the concepts involved. As a first application, we derive new relationships between "continuity principles" asserting that all functions between specified metric spaces are pointwise continuous. In particular, we give conditions that imply the failure of the continuity principle "all functions from X to , are continuous", when X is an inhabited CSM without isolated points, and when X is an inhabited locally non-compact CSM. One situation in which the latter case applies is in models based on "domain realizability", in which the failure of the continuity principle for any inhabited locally non-compact CSM, X, generalizes a result previously obtained by Escardó and Streicher in the special case X = C[0, 1]. As a second application, we show that, when the notion of inhabited complete separable metric space without isolated points is interpreted in a recursion-theoretic setting, then, for any such space X, there exists a Banach-Mazur computable function from X to the computable real numbers that is not Markov computable. This generalizes a result obtained by Hertling in the special case that X is the space of computable real numbers. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] ## Consequences of the failure of the axiom of choice in the theory of Lindelöf metric spaces MLQ- MATHEMATICAL LOGIC QUARTERLY, Issue 2 2004Kyriakos KeremedisAbstract We study within the framework of Zermelo-Fraenkel set theory ZF the role that the axiom of choice plays in the theory of Lindelöf metric spaces. We show that in ZF the weak choice principles: (i) Every Lindelöf metric space is separable and (ii) Every Lindelöf metric space is second countable (Forms 340 and 341, respectively, in [10]) are equivalent. We also prove that a Lindelöf metric space is hereditarily separable iff it is hereditarily Lindelöf iff it hold as well the axiom of choice restricted to countable sets and to topologies of Lindelöf metric spaces as the countable union theorem restricted to Lindelöf metric spaces. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] |