Relevant bibliographies by topics / Measure metric space

Academic literature on the topic 'Measure metric space'

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Published: 10 December 2022
Last updated: 28 January 2023

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Journal articles on the topic "Measure metric space"

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Wildrick, K., and T. Zürcher. "Space filling with metric measure spaces." Mathematische Zeitschrift 270, no. 1-2 (November 3, 2010): 103–31. http://dx.doi.org/10.1007/s00209-010-0787-1.

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Scheepers, Marion. "Finite powers of strong measure zero sets." Journal of Symbolic Logic 64, no. 3 (September 1999): 1295–306. http://dx.doi.org/10.2307/2586631.

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AbstractIn a previous paper—[17]—we characterized strong measure zero sets of reals in terms of a Ramseyan partition relation on certain subspaces of the Alexandroff duplicate of the unit interval. This framework gave only indirect access to the relevant sets of real numbers. We now work more directly with the sets in question, and since it costs little in additional technicalities, we consider the more general context of metric spaces and prove:1. If a metric space has a covering property of Hurewicz and has strong measure zero, then its product with any strong measure zero metric space is a strong measure zero metric space (Theorem 1 and Lemma 3).2. A subspace X of a σ-compact metric space Y has strong measure zero if, and only if, a certain Ramseyan partition relation holds for Y (Theorem 9).3. A subspace X of a σ-compact metric space Y has strong measure zero in all finite powers if, and only if, a certain Ramseyan partition relation holds for Y (Theorem 12).Then 2 and 3 yield characterizations of strong measure zeroness for σ-totally bounded metric spaces in terms of Ramseyan theorems.
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Heikkinen, Toni, Juha Lehrbäck, Juho Nuutinen, and Heli Tuominen. "Fractional Maximal Functions in Metric Measure Spaces." Analysis and Geometry in Metric Spaces 1 (May 28, 2013): 147–62. http://dx.doi.org/10.2478/agms-2013-0002.

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Abstract We study the mapping properties of fractional maximal operators in Sobolev and Campanato spaces in metric measure spaces. We show that, under certain restrictions on the underlying metric measure space, fractional maximal operators improve the Sobolev regularity of functions and map functions in Campanato spaces to Hölder continuous functions. We also give an example of a space where fractional maximal function of a Lipschitz function fails to be continuous.
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Edgar. "PACKING MEASURE IN GENERAL METRIC SPACE." Real Analysis Exchange 26, no. 2 (2000): 831. http://dx.doi.org/10.2307/44154081.

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Han, Bang-Xian, and Andrea Mondino. "Angles between Curves in Metric Measure Spaces." Analysis and Geometry in Metric Spaces 5, no. 1 (September 2, 2017): 47–68. http://dx.doi.org/10.1515/agms-2017-0003.

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Abstract The goal of the paper is to study the angle between two curves in the framework of metric (and metric measure) spaces. More precisely, we give a new notion of angle between two curves in a metric space. Such a notion has a natural interplay with optimal transportation and is particularly well suited for metric measure spaces satisfying the curvature-dimension condition. Indeed one of the main results is the validity of the cosine formula on RCD*(K, N) metric measure spaces. As a consequence, the new introduced notions are compatible with the corresponding classical ones for Riemannian manifolds, Ricci limit spaces and Alexandrov spaces.
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Yaoyao, Han, and Zhao Kai. "Herz type Hardy spaces on non-homogeneous metric measure space." SCIENTIA SINICA Mathematica 48, no. 10 (October 1, 2018): 1315. http://dx.doi.org/10.1360/n012018-00118.

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Mazurenko, N., and M. Zarichnyi. "Invariant idempotent measures." Carpathian Mathematical Publications 10, no. 1 (July 3, 2018): 172–78. http://dx.doi.org/10.15330/cmp.10.1.172-178.

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The idempotent mathematics is a part of mathematics in which arithmetic operations in the reals are replaced by idempotent operations. In the idempotent mathematics, the notion of idempotent measure (Maslov measure) is a counterpart of the notion of probability measure. The idempotent measures found numerous applications in mathematics and related areas, in particular, the optimization theory, mathematical morphology, and game theory. In this note we introduce the notion of invariant idempotent measure for an iterated function system in a complete metric space. This is an idempotent counterpart of the notion of invariant probability measure defined by Hutchinson. Remark that the notion of invariant idempotent measure was previously considered by the authors for the class of ultrametric spaces. One of the main results is the existence and uniqueness theorem for the invariant idempotent measures in complete metric spaces. Unlikely to the corresponding Hutchinson's result for invariant probability measures, our proof does not rely on metrization of the space of idempotent measures. An analogous result can be also proved for the so-called in-homogeneous idempotent measures in complete metric spaces. Also, our considerations can be extended to the case of the max-min measures in complete metric spaces.
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Aïssaoui, Noureddine. "Strongly nonlinear potential theory on metric spaces." Abstract and Applied Analysis 7, no. 7 (2002): 357–74. http://dx.doi.org/10.1155/s1085337502203024.

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We define Orlicz-Sobolev spaces on an arbitrary metric space with a Borel regular outer measure, and we develop a capacity theory based on these spaces. We study basic properties of capacity and several convergence results. We prove that each Orlicz-Sobolev function has a quasi-continuous representative. We give estimates for the capacity of balls when the measure is doubling. Under additional regularity assumption on the measure, we establish some relations between capacity and Hausdorff measures.
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Xie, Jialiang, Qingguo Li, Shuili Chen, and Huan Huang. "The fuzzy metric space based on fuzzy measure." Open Mathematics 14, no. 1 (January 1, 2016): 603–12. http://dx.doi.org/10.1515/math-2016-0051.

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Abstract In this paper, we study the relation between a fuzzy measure and a fuzzy metric which is induced by the fuzzy measure. We also discuss some basic properties of the constructed fuzzy metric space. In particular, we show that the nonatom of fuzzy measure space can be characterized in the constructed fuzzy metric space.
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Honda, Shouhei. "Bakry-Émery Conditions on Almost Smooth Metric Measure Spaces." Analysis and Geometry in Metric Spaces 6, no. 1 (October 1, 2018): 129–45. http://dx.doi.org/10.1515/agms-2018-0007.

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Abstract In this short note, we give a sufficient condition for almost smooth compact metric measure spaces to satisfy the Bakry-Émery condition BE(K, N). The sufficient condition is satisfied for the glued space of any two (not necessary same dimensional) closed pointed Riemannian manifolds at their base points. This tells us that the BE condition is strictly weaker than the RCD condition even in this setting, and that the local dimension is not constant even if the space satisfies the BE condition with the coincidence between the induced distance by the Cheeger energy and the original distance. In particular, the glued space gives a first example with a Ricci bound from below in the Bakry-Émery sense, whose local dimension is not constant. We also give a necessary and sufficient condition for such spaces to be RCD(K, N) spaces.
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Dissertations / Theses on the topic "Measure metric space"

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Färm, David. "Upper gradients and Sobolev spaces on metric spaces." Thesis, Linköping University, Department of Mathematics, 2006. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-5816.

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The Laplace equation and the related p-Laplace equation are closely associated with Sobolev spaces. During the last 15 years people have been exploring the possibility of solving partial differential equations in general metric spaces by generalizing the concept of Sobolev spaces. One such generalization is the Newtonian space where one uses upper gradients to compensate for the lack of a derivative.

All papers on this topic are written for an audience of fellow researchers and people with graduate level mathematical skills. In this thesis we give an introduction to the Newtonian spaces accessible also for senior undergraduate students with only basic knowledge of functional analysis. We also give an introduction to the tools needed to deal with the Newtonian spaces. This includes measure theory and curves in general metric spaces.

Many of the properties of ordinary Sobolev spaces also apply in the generalized setting of the Newtonian spaces. This thesis includes proofs of the fact that the Newtonian spaces are Banach spaces and that under mild additional assumptions Lipschitz functions are dense there. To make them more accessible, the proofs have been extended with comments and details previously omitted. Examples are given to illustrate new concepts.

This thesis also includes my own result on the capacity associated with Newtonian spaces. This is the theorem that if a set has p-capacity zero, then the capacity of that set is zero for all smaller values of p.

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Estep, Dewey. "Prime End Boundaries of Domains in Metric Spaces and the Dirichlet Problem." University of Cincinnati / OhioLINK, 2015. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1439295199.

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Meizis, Roland [Verfasser], and Anita [Akademischer Betreuer] Winter. "Metric two-level measure spaces : a state space for modeling evolving genealogies in host-parasite systems / Roland Meizis ; Betreuer: Anita Winter." Duisburg, 2019. http://d-nb.info/1191693414/34.

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Malý, Lukáš. "Sobolev-Type Spaces : Properties of Newtonian Functions Based on Quasi-Banach Function Lattices in Metric Spaces." Doctoral thesis, Linköpings universitet, Matematik och tillämpad matematik, 2014. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-105616.

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This thesis consists of four papers and focuses on function spaces related to first-order analysis in abstract metric measure spaces. The classical (i.e., Sobolev) theory in Euclidean spaces makes use of summability of distributional gradients, whose definition depends on the linear structure of Rn. In metric spaces, we can replace the distributional gradients by (weak) upper gradients that control the functions’ behavior along (almost) all rectifiable curves, which gives rise to the so-called Newtonian spaces. The summability condition, considered in the thesis, is expressed using a general Banach function lattice quasi-norm and so an extensive framework is built. Sobolev-type spaces (mainly based on the Lp norm) on metric spaces, and Newtonian spaces in particular, have been under intensive study since the mid-1990s. In Paper I, the elementary theory of Newtonian spaces based on quasi-Banach function lattices is built up. Standard tools such as moduli of curve families and the Sobolev capacity are developed and applied to study the basic properties of Newtonian functions. Summability of a (weak) upper gradient of a function is shown to guarantee the function’s absolute continuity on almost all curves. Moreover, Newtonian spaces are proven complete in this general setting. Paper II investigates the set of all weak upper gradients of a Newtonian function. In particular, existence of minimal weak upper gradients is established. Validity of Lebesgue’s differentiation theorem for the underlying metric measure space ensures that a family of representation formulae for minimal weak upper gradients can be found. Furthermore, the connection between pointwise and norm convergence of a sequence of Newtonian functions is studied. Smooth functions are frequently used as an approximation of Sobolev functions in analysis of partial differential equations. In fact, Lipschitz continuity, which is (unlike -smoothness) well-defined even for functions on metric spaces, often suffices as a regularity condition. Thus, Paper III concentrates on the question when Lipschitz functions provide good approximations of Newtonian functions. As shown in the paper, it suffices that the function lattice quasi-norm is absolutely continuous and a fractional sharp maximal operator satisfies a weak norm estimate, which it does, e.g., in doubling Poincaré spaces if a non-centered maximal operator of Hardy–Littlewood type is locally weakly bounded. Therefore, such a local weak boundedness on rearrangement-invariant spaces is explored as well. Finer qualitative properties of Newtonian functions and the Sobolev capacity get into focus in Paper IV. Under certain hypotheses, Newtonian functions are proven to be quasi-continuous, which yields that the capacity is an outer capacity. Various sufficient conditions for local boundedness and continuity of Newtonian functions are established. Finally, quasi-continuity is applied to discuss density of locally Lipschitz functions in Newtonian spaces on open subsets of doubling Poincaré spaces.
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Enflo, Karin. "Measures of Freedom of Choice." Doctoral thesis, Uppsala universitet, Avdelningen för praktisk filosofi, 2012. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-179078.

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This thesis studies the problem of measuring freedom of choice. It analyzes the concept of freedom of choice, discusses conditions that a measure should satisfy, and introduces a new class of measures that uniquely satisfy ten proposed conditions. The study uses a decision-theoretical model to represent situations of choice and a metric space model to represent differences between options. The first part of the thesis analyzes the concept of freedom of choice. Different conceptions of freedom of choice are categorized into evaluative and non-evaluative, as well as preference-dependent and preference-independent kinds. The main focus is on the three conceptions of freedom of choice as cardinality of choice sets, representativeness of the universal set, and diversity of options, as well as the three conceptions of freedom of rational choice, freedom of eligible choice, and freedom of evaluated choice. The second part discusses the conceptions, together with conditions for a measure and a variety of measures proposed in the literature. The discussion mostly focuses on preference-independent conceptions of freedom of choice, in particular the diversity conception. Different conceptions of diversity are discussed, as well as properties that could affect diversity, such as the cardinality of options, the differences between the options, and the distribution of differences between the options. As a result, the diversity conception is accepted as the proper explication of the concept of freedom of choice. In addition, eight conditions for a measure are accepted. The conditions concern domain-insensitivity, strict monotonicity, no-choice situations, dominance of differences, evenness, symmetry, spread of options, and limited function growth. None of the previously proposed measures satisfy all of these conditions. The third part concerns the construction of a ratio-scale measure that satisfies the accepted conditions. Two conditions are added regarding scale-independence and function growth proportional to cardinality. Lastly, it is shown that only one class of measures satisfy all ten conditions, given an additional assumption that the measures should be analytic functions with non-zero partial derivatives with respect to some function of the differences. These measures are introduced as the Ratio root measures.
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Davtyan, Ashot. "Measure generation in the spaces of planes und lines in R^3." Doctoral thesis, Technische Universitaet Bergakademie Freiberg Universitaetsbibliothek "Georgius Agricola&quot, 2009. http://nbn-resolving.de/urn:nbn:de:swb:105-7072226.

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Das Ziel der Arbeit besteht darin, einen Beitrag zur Entwicklung der kombinatorischen Integralgeometrie zu leisten. In der Arbeit werden Bewertungen (Valuation) in den Räumen der Geraden und Ebenen im $\R^3$ betrachtet, die von Flagfunktionen abhängen. Unter geeigneten Glattheitsvoraussetzungen an die Flagfunktionen werden notwendige und hinreichende Bedingungen gegeben, die die Fortsetzung der entsprechender Bewertung zu einem signierten Maß sichern. Diese integralgeometrischen Untersuchungen führten zu einer Anzahl von interessanten Ergebnissen, speziell bei der Beschreibung von Metriken im Sinne von Hilberts viertem Problem.
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Jones, Rebekah. "A characterization of quasiconformal maps in terms of sets of finite perimeter." University of Cincinnati / OhioLINK, 2019. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1560867563841096.

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Malý, Lukáš. "Newtonian Spaces Based on Quasi-Banach Function Lattices." Licentiate thesis, Linköpings universitet, Matematik och tillämpad matematik, 2012. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-79166.

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The traditional first-order analysis in Euclidean spaces relies on the Sobolev spaces W1,p(Ω), where Ω ⊂ Rn is open and p ∈ [1, ∞].The Sobolev norm is then defined as the sum of Lp norms of a function and its distributional gradient.We generalize the notion of Sobolev spaces in two different ways. First, the underlying function norm will be replaced by the “norm” of a quasi-Banach function lattice. Second, we will investigate functions defined on an abstract metric measure space and that is why the distributional gradients need to be substituted. The thesis consists of two papers. The first one builds up the elementary theory of Newtonian spaces based on quasi-Banach function lattices. These lattices are complete linear spaces of measurable functions with a topology given by a quasinorm satisfying the lattice property. Newtonian spaces are first-order Sobolev-type spaces on abstract metric measure spaces, where the role of weak derivatives is passed on to upper gradients. Tools such asmoduli of curve families and the Sobolev capacity are developed, which allows us to study basic properties of the Newtonian functions.We will see that Newtonian spaces can be equivalently defined using the notion of weak upper gradients, which increases the number of techniques available to study these spaces. The absolute continuity of Newtonian functions along curves and the completeness of Newtonian spaces in this general setting are also established. The second paper in the thesis then continues with investigation of properties of Newtonian spaces based on quasi-Banach function lattices. The set of all weak upper gradients of a Newtonian function is of particular interest.We will prove that minimalweak upper gradients exist in this general setting.Assuming that Lebesgue’s differentiation theoremholds for the underlyingmetricmeasure space,wewill find a family of representation formulae. Furthermore, the connection between pointwise convergence of a sequence of Newtonian functions and its convergence in norm is studied.
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Alleche, Boualem. "Quelques résultats sur la consonance, les multi-applications, et la séquentialité." Rouen, 1996. http://www.theses.fr/1996ROUES027.

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Cette thèse se subdivise en trois parties, la première traite les hyper-espaces. Apres l'introduction et le développement de certains résultats récents sur la consonance, nous introduisons l'idée originale qui consiste à utiliser les mesures de Radon. Nous démontrons que tout espace consonant est pré-Radon. Nous obtenons que la droite de Sorgenfrey n'est pas consonante, que tout ultra-filtre libre sur les entiers regarde comme un sous-espace du Cantor est non consonant, et nous donnons un espace métrisable séparable héréditairement de Baire non consonant. Cette idée a également inspiré un peu plus tard d'autres mathématiciens pour démontrer la non-consonance de l'ensemble des rationnels. Nous généralisons ensuite un résultat obtenu par M. Arab et J. Calbrix sur la coïncidence de la consonance et l'hyperconsonance. Dans la deuxième partie, nous généralisons le théorème de E. Michael de double sélection multivoque et nous obtenons que tout espace Cech-complet sous-métrisable est sélecteur par rapport aux espaces paracompacts. Nous démontrons que la frontière active d'une multi-application S. C. S. D'un espace compact métrisable dans un espace métrisable est, elle aussi, S. C. S. , et nous trouvons le lien que pour les espaces métrisables séparables co-analytiques, la consonance équivaut à être sélecteur par rapport à l'espace de Cantor. Dans la dernière partie, nous étudions les espaces séquentiels. Nous donnons un théorème sur une certaine classe d'applications quotients et nous obtenons que le produit de n copies du fan séquentiel est séquentiel et d'ordre séquentiel égal à n.
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Belili, Nacereddine. "Problèmes des marges et de transport." Rouen, 1998. http://www.theses.fr/1998ROUES022.

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Cette thèse comprend trois parties, Dans la première partie, on donne une synthèse du théorème de dualité relatif au problème des marges, ses diverses applications comme le théorème de Strassen, la caractérisation de l'ordre stochastique et la représentation des métriques minimales. On donne une preuve du théorème de Goldstein basée sur la représentation de la distance de variation totale. Dans la seconde partie, on considère une suite ( µn de probabilités sur Rd convergeant étroitement vers une probabilité µ absolument continue par rapport à la mesure de Lebesgue. On suppose que µn et µ admettent un moment d'ordre p1. On montre l'existence d'une suite de variables aléatoires {(Xn,X)} à valeurs dans Rd × Rd telles que Xn = n(X), où n: Rd -> Rd est c-cycliquement monotone, Xn converge presque-sûrement vers X et où chaque couple (Xn,X) est c-optimal pour ( µn, µ ). Dans la dernière partie, en collaboration avec H. Heinich, nous donnons des propriétés des probabilités qui vérifient la propriété de transport. En particulier, nous examinons le cas des probabilités fortement diffuses. Nous étudions la relation entre la dérivabilité d'une fonction réelle f et le fait que la probabilité L o f*-1 vérifie la propriété de transport, où L est mesure de Lebesgue et la fonction f*(x) := (x,f(x)).
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Books on the topic "Measure metric space"

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Nicola, Gigli, and Savaré Giuseppe, eds. Gradient flows: In metric spaces and in the space of probability measures. Boston: Birkhäuser, 2005.

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Ambrosio, Luigi. Gradient flows: In metric spaces and in the space of probability measures. Basel: Birkhauser, 2004.

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Nicola, Gigli, Savaré Giuseppe, Struwe Michael 1955-, and SpringerLink (Online service), eds. Gradient Flows: In Metric Spaces and in the Space of Probability Measures. Basel: Birkhäuser Basel, 2008.

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Shirali, Satish. A Concise Introduction to Measure Theory. USA: Springer International Publishing AG, 2019.

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Probability measures on metric spaces. Providence, R.I: AMS Chelsea Pub., 2005.

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Fremlin, D. H. Measure-additive coverings and measurable selectors. Warszawa: Państwowe Wydawn. Naukowe, 1987.

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Barlow, Martin T. Stability of parabolic Harnack inequalities on metric measure spaces. Kyoto, Japan: Kyōto Daigaku Sūri Kaiseki Kenkyūjo, 2004.

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1951-, Domínguez Benavides T., and López Acedo G. 1956-, eds. Measures of noncompactness in metric fixed point theory. Basel: Birkhäuser Verlag, 1997.

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Carlsson, Niclas. Markov chains on metric spaces: Invariant measures and asymptotic behaviour. Åbo: Åbo Akademi University Press, 2005.

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Chen, Zhen-Qing. Heat Kernel Estimates for Jump Processes of Mixed Types on Metric Measure Spaces. Kyoto, Japan: Research Institute for Mathematical Sciences, Kyoto University, 2006.

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Book chapters on the topic "Measure metric space"

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van Nes, Akkelies, and Claudia Yamu. "Analysing Linear Spatial Relationships: The Measures of Connectivity, Integration, and Choice." In Introduction to Space Syntax in Urban Studies, 35–86. Cham: Springer International Publishing, 2021. http://dx.doi.org/10.1007/978-3-030-59140-3_2.

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AbstractIn this chapter, we first explain the concept of an axial line and how the axial map is applied in space syntax. We then discuss the static measure of ‘connectivity’ with its ‘one-step’ to ‘n-step’ logic, including its meaning for axialintegration analysis. We further present the segment integration analysis. Using the streetsegment as the basis for analysis allows one to apply three types of distances and three types of radii in space syntax. We then present the most-often used space syntax measures in more depth, namely angularchoice and angular integrationwith metric radius, and introduce the mathematical formulae on how to normalise both measures. Real-life applications illustrate and underpin the usefulness of these measures and their meaning for urban analysis, such as why and how they allow us to identify urban societal processes and their added value at both a citywidescale and a neighbourhoodscale. Finally, we critically reflect on the measures, including their potentials and misfits. Exercises are provided at the end of the chapter.
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Itoh, Tsubasa. "Logarithmic Hölder estimates of 𝑝-harmonic extension operators in a metric measure space." In Complex Analysis and Potential Theory, 163–69. Providence, Rhode Island: American Mathematical Society, 2012. http://dx.doi.org/10.1090/crmp/055/11.

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Sawano, Yoshihiro, Giuseppe Di Fazio, and Denny Ivanal Hakim. "General metric measure spaces." In Morrey Spaces, 247–91. First edition. | Boca Raton : C&H/CRC Press, 2020. | Series: Chapman & Hall/CRC monographs and research notes in mathematics: Chapman and Hall/CRC, 2020. http://dx.doi.org/10.1201/9780429085925-6.

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Sawano, Yoshihiro, Giuseppe Di Fazio, and Denny Ivanal Hakim. "General metric measure spaces." In Morrey Spaces, 247–91. First edition. | Boca Raton : C&H/CRC Press, 2020. | Series: Chapman & Hall/CRC monographs and research notes in mathematics: Chapman and Hall/CRC, 2020. http://dx.doi.org/10.1201/9781003042341-6.

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Sawano, Yoshihiro, Giuseppe Di Fazio, and Denny Ivanal Hakim. "Morrey spaces over metric measure spaces." In Morrey Spaces, 103–42. First edition. | Boca Raton : C&H/CRC Press, 2020. | Series: Chapman & Hall/CRC monographs and research notes in mathematics: Chapman and Hall/CRC, 2020. http://dx.doi.org/10.1201/9781003029076-14.

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Pachl, Jan. "Measures on Complete Metric Spaces." In Uniform Spaces and Measures, 63–79. New York, NY: Springer New York, 2012. http://dx.doi.org/10.1007/978-1-4614-5058-0_6.

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Villani, Cédric. "Convergence of metric-measure spaces." In Grundlehren der mathematischen Wissenschaften, 743–72. Berlin, Heidelberg: Springer Berlin Heidelberg, 2009. http://dx.doi.org/10.1007/978-3-540-71050-9_27.

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Savaré, Giuseppe. "Sobolev Spaces in Extended Metric-Measure Spaces." In Lecture Notes in Mathematics, 117–276. Cham: Springer International Publishing, 2021. http://dx.doi.org/10.1007/978-3-030-84141-6_4.

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Simovici, Dan A., and Chabane Djeraba. "Metric Spaces Topologies and Measures." In Advanced Information and Knowledge Processing, 399–433. London: Springer London, 2014. http://dx.doi.org/10.1007/978-1-4471-6407-4_8.

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Li, Zenghu. "Random Measures on Metric Spaces." In Probability and Its Applications, 1–28. Berlin, Heidelberg: Springer Berlin Heidelberg, 2011. http://dx.doi.org/10.1007/978-3-642-15004-3_1.

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Conference papers on the topic "Measure metric space"

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Argoun, Mohamed B., and Atef Omar Sherif. "A needed metric to measure utilization of indigenous satellite images in developing countries." In 2011 5th International Conference on Recent Advances in Space Technologies (RAST). IEEE, 2011. http://dx.doi.org/10.1109/rast.2011.5966821.

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Zhu, Pengfei, Ren Qi, Qinghua Hu, Qilong Wang, Changqing Zhang, and Liu Yang. "Beyond Similar and Dissimilar Relations : A Kernel Regression Formulation for Metric Learning." In Twenty-Seventh International Joint Conference on Artificial Intelligence {IJCAI-18}. California: International Joint Conferences on Artificial Intelligence Organization, 2018. http://dx.doi.org/10.24963/ijcai.2018/450.

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Most existing metric learning methods focus on learning a similarity or distance measure relying on similar and dissimilar relations between sample pairs. However, pairs of samples cannot be simply identified as similar or dissimilar in many real-world applications, e.g., multi-label learning, label distribution learning or tasks with continuous decision values. To this end, in this paper we propose a novel relation alignment metric learning (RAML) formulation to handle the metric learning problem in those scenarios. Since the relation of two samples can be measured by the difference degree of the decision values, motivated by the consistency of the sample relations in the feature space and decision space, our proposed RAML utilizes the sample relations in the decision space to guide the metric learning in the feature space. Specifically, our RAML method formulates metric learning as a kernel regression problem, which can be efficiently optimized by the standard regression solvers. We carry out several experiments on the single-label classification, multi-label classification, and label distribution learning tasks, to demonstrate that our method achieves favorable performance against the state-of-the-art methods.
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Larochelle, Pierre, and J. Michael McCarthy. "Designing Planar Mechanisms Using a Bi-Invariant Metric in the Image Space of SO (3)." In ASME 1994 Design Technical Conferences collocated with the ASME 1994 International Computers in Engineering Conference and Exhibition and the ASME 1994 8th Annual Database Symposium. American Society of Mechanical Engineers, 1994. http://dx.doi.org/10.1115/detc1994-0198.

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Abstract In this paper we present a technique for using a bi-invariant metric in the image space of spherical displacements for designing planar mechanisms for n (> 5) position rigid body guidance. The goal is to perform the dimensional synthesis of the mechanism such that the distance between the position and orientation of the guided body to each of the n goal positions is minimized. Rather than measure these distances in the plane, we introduce an approximating sphere and identify rotations which are equivalent to the planar displacements to a specified tolerance. We then measure distances between the rigid body and the goal positions using a bi-invariant metric on the image space of SO(3). The optimal linkage is obtained by minimizing this distance over all of the n goal positions. The paper proceeds as follows. First, we approximate planar rigid body displacements with spherical displacements and show that the error induced by such an approximation is of order 1/R2, where R is the radius of the approximating sphere. Second, we use a bi-invariant metric in the image space of spherical displacements to synthesize an optimal spherical 4R mechanism. Finally, we identify the planar 4R mechanism associated with the optimal spherical solution. The result is a planar 4R mechanism that has been optimized for n position rigid body guidance using an approximate bi-invariant metric with an error dependent only upon the radius of the approximating sphere. Numerical results for ten position synthesis of a planar 4R mechanism are presented.
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Wei, Xinpeng, and Xiaoping Du. "A New Robustness Metric for Robust Design Optimization Under Time- and Space-Dependent Uncertainty Through Metamodeling." In ASME 2019 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2019. http://dx.doi.org/10.1115/detc2019-97611.

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Abstract Product performance varies with respect to time and space in many engineering applications. This work discusses how to measure and evaluate the robustness of a product or component when its quality characteristics are functions of random variables, random fields, temporal variables, and spatial variables. At first, the existing time-dependent robustness metric is extended to the present time- and space-dependent problem. The robustness metric is derived using the extreme value of the quality characteristics with respect to temporal and spatial variables for the nominal-the-better type quality characteristics. Then a metamodel-based numerical procedure is developed to evaluate the new robustness metric. The procedure employs a Gaussian Process regression method to estimate the expected quality loss that involves the extreme quality characteristics. The expected quality loss is obtained directly during the regression model building process. Three examples are used to demonstrate the robustness analysis method. The proposed method can be used for robustness assessment during robust design optimization under time- and space-dependent uncertainty.
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Kazerounian, Kazem, and Jahangir Rastegar. "Object Norms: A Class of Coordinate and Metric Independent Norms for Displacements." In ASME 1992 Design Technical Conferences. American Society of Mechanical Engineers, 1992. http://dx.doi.org/10.1115/detc1992-0390.

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Abstract Defining a “norm” that quantifies a measure of “distance” for rigid body displacement in plane or space, is essential to many problems in kinematics, dynamics and control of mechanical and manufacturing systems. The norm is an abstraction of our usual concept of the length. The mapping between different generalized coordinate systems used in the analysis or control of systems may also be derived from this norm. All such norms currently used in different analysis are highly sensitive to 1) coordinate systems defined, and 2) metrics defined. A new class of norms is proposed to quantify the distance in the space. These norms are independent of the coordinate systems or metrics used. Its applications indesign, precision manipulation and control, computational kinematics and dynamics, assembly and part mating in manufacturing, space kinematics and dynamics, shape optimization and motion planning are explored.
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Magalhães, Dimmy, Aurora Pozo, and Roberto Santana. "An empirical comparison of distance/similarity measures for Natural Language Processing." In Encontro Nacional de Inteligência Artificial e Computacional. Sociedade Brasileira de Computação - SBC, 2019. http://dx.doi.org/10.5753/eniac.2019.9328.

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Text Classification is one of the tasks of Natural Language Processing (NLP). In this area, Graph Convolutional Networks (GCN) has achieved values higher than CNN's and other related models. For GCN, the metric that defines the correlation between words in a vector space plays a crucial role in the classification because it determines the weight of the edges between two words (represented by nodes in the graph). In this study, we empirically investigated the impact of thirteen measures of distance/similarity. A representation was built for each document using word embedding from word2vec model. Also, a graph-based representation of five dataset was created for each measure analyzed, where each word is a node in the graph, and each edge is weighted by distance/similarity between words. Finally, each model was run in a simple graph neural network. The results show that, concerning text classification, there is no statistical difference between the analyzed metrics and the Graph Convolution Network. Even with the incorporation of external words or external knowledge, the results were similar to the methods without the incorporation of words. However, the results indicate that some distance metrics behave better than others in relation to context capture, with Euclidean distance reaching the best values or having statistical similarity with the best.
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Bespalov, Dmitriy, Ali Shokoufandeh, William C. Regli, and Wei Sun. "Local Feature Extraction Using Scale-Space Decomposition." In ASME 2004 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2004. http://dx.doi.org/10.1115/detc2004-57702.

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In our recent work we have introduced a framework for extracting features from solid of mechanical artifacts in polyhedral representation based on scale-space feature decomposition [1]. Our approach used recent developments in efficient hierarchical decomposition of metric data using its spectral properties. In that work, through spectral decomposition, we were able to reduce the problem of matching to that of computing a mapping and distance measure between vertex-labeled rooted trees. This work discusses how Scale-Space decomposition frame-work could be extended to extract features from CAD models in polyhedral representation in terms of surface triangulation. First, we give an overview of the Scale-Space decomposition approach that is used to extract these features. Second, we discuss the performance of the technique used to extract features from CAD data in polyhedral representation. Third, we show the feature extraction process on noisy data — CAD models that were constructed using a 3D scanner. Finally, we conclude with discussion of future work.
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Zhao, Qiang, HongTao Wu, and Minghu Zhou. "Generalized Mass Metric and Recursive Momentum Formulation for Dynamics of Multibody Systems." In ASME 2009 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2009. http://dx.doi.org/10.1115/detc2009-86311.

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Generalized mass metric in Riemannian manifold plays a central role in dynamics and control of multibody system (MBS). In this work, two profitable aspects of multibody system dynamics studies, generalized mass metric in Riemannian geometry and recursive momentum formulation, are described. Firstly, we will derive an Adjoint-based expression of Riemannian metric and operator factorization of generalized mass tensor from a general-topology rigid MBS which comprises of a special Euclidian group SE(3) set. The specific expression can help to clearly understand what reasons lead to these components (Riemannian metric) of the generalized mass tensor and how they measure the curves of generalized velocity space. Meanwhile, the power algorithm of MBS is presented based on the Adjoint map of generalized velocity and generalized force. Next, from the generalized momentum definition depending on such Riemannian mass metric, recursive momentum equations of MBS dynamics are developed for progressively more complex systems: open-chains, topological trees, and closed-loop systems. In terms of the relation principle of impulse and momentum, a new method is proposed for describing conservative MBS form a given initial orientation and location to desired final ones without needing to solve motion process.
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Huai, Mengdi, Hongfei Xue, Chenglin Miao, Liuyi Yao, Lu Su, Changyou Chen, and Aidong Zhang. "Deep Metric Learning: The Generalization Analysis and an Adaptive Algorithm." In Twenty-Eighth International Joint Conference on Artificial Intelligence {IJCAI-19}. California: International Joint Conferences on Artificial Intelligence Organization, 2019. http://dx.doi.org/10.24963/ijcai.2019/352.

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As an effective way to learn a distance metric between pairs of samples, deep metric learning (DML) has drawn significant attention in recent years. The key idea of DML is to learn a set of hierarchical nonlinear mappings using deep neural networks, and then project the data samples into a new feature space for comparing or matching. Although DML has achieved practical success in many applications, there is no existing work that theoretically analyzes the generalization error bound for DML, which can measure how good a learned DML model is able to perform on unseen data. In this paper, we try to fill up this research gap and derive the generalization error bound for DML. Additionally, based on the derived generalization bound, we propose a novel DML method (called ADroDML), which can adaptively learn the retention rates for the DML models with dropout in a theoretically justified way. Compared with existing DML works that require predefined retention rates, ADroDML can learn the retention rates in an optimal way and achieve better performance. We also conduct experiments on real-world datasets to verify the findings derived from the generalization error bound and demonstrate the effectiveness of the proposed adaptive DML method.
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Mendez, Mauricio, Gilberto Ochoa-Ruiz, Ivan Garcia, and Andres Mendez-Vazquez. "Finding Significant Features for Few-Shot Learning using Dimensionality Reduction Techniques." In LatinX in AI at Computer Vision and Pattern Recognition Conference 2021. Journal of LatinX in AI Research, 2021. http://dx.doi.org/10.52591/lxai202106213.

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Few-shot learning is a fairly new technique that specialize in problems where we have little amount of data. The goal of this method is to classify categories that hasn’t been seen before with just a handful of samples. Recent approaches, such as metric learning, adopt the meta-learning setting in which we have episodic tasks conformed by support (training) data and query (test) data. Metric learning methods has demonstrated that simple models can achieve good performance, by learning a similarity function to compare the support and the query data. However, the feature space learned by the metric learning may not exploit the information given by a specific few-shot task. In this work, we explore the use of dimension reduction techniques as a way to find task-significant features. We measure the performance of the reduced features by giving a score based on the intra-class and inter-class distance, and select the method in which instances of different classes are distant and instances of the same class are close. This module helps to improve the accuracy performance by allowing the similarity function, given by the metric learning method, to have more discriminative features for the classification.
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Reports on the topic "Measure metric space"

1

Rao, C. R. Differential Metrics in Probability Spaces Based on Entropy and Divergence Measures. Fort Belvoir, VA: Defense Technical Information Center, April 1985. http://dx.doi.org/10.21236/ada160301.

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