Relevant bibliographies by topics / Linear metric space

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  1. Journal articles
  2. Dissertations / Theses
  3. Books
  4. Book chapters
  5. Conference papers

Academic literature on the topic 'Linear metric space'

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Published: 24 April 2022

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Journal articles on the topic "Linear metric space"

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KRÖN, BERNHARD, JÖRG LEHNERT, NORBERT SEIFTER, and ELMAR TEUFL. "LINEAR AND PROJECTIVE BOUNDARY OF NILPOTENT GROUPS." Glasgow Mathematical Journal 57, no. 3 (December 22, 2014): 591–632. http://dx.doi.org/10.1017/s0017089514000512.

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AbstractWe define a pseudometric on the set of all unbounded subsets of a metric space. The Kolmogorov quotient of this pseudometric space is a complete metric space. The definition of the pseudometric is guided by the principle that two unbounded subsets have distance 0 whenever they stay sublinearly close. Based on this pseudometric we introduce and study a general concept of boundaries of metric spaces. Such a boundary is the closure of a subset in the Kolmogorov quotient determined by an arbitrarily chosen family of unbounded subsets. Our interest lies in those boundaries which we get by choosing unbounded cyclic sub(semi)groups of a finitely generated group (or more general of a compactly generated, locally compact Hausdorff group). We show that these boundaries are quasi-isometric invariants and determine them in the case of nilpotent groups as a disjoint union of certain spheres (or projective spaces). In addition we apply this concept to vertex-transitive graphs with polynomial growth and to random walks on nilpotent groups.
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Michael, George. "A metric linear space is an open cone." Kyoto Journal of Mathematics 52, no. 4 (2012): 833–38. http://dx.doi.org/10.1215/21562261-1728893.

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Pluzhnikova, Elena Aleksandrovna, Tatyana Vladimirovna Zhukovskaya, and Yuriy Anatol’evich Moiseev. "ON SETS OF METRIC REGULARITY OF MAPPINGS IN SPACES WITH VECTOR-VALUED METRIC." Tambov University Reports. Series: Natural and Technical Sciences, no. 123 (2018): 547–54. http://dx.doi.org/10.20310/1810-0198-2018-23-123-547-554.

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Spaces with vector-valued metric are considered. The values of a vectorvalued metric are elements of a cone in some linear normed space. The concept of the set of metric regularity for mapping in spaces with vector-valued metric is formulated. A statement on the stability of the set of metric regularity of a given mapping for its Lipschitz perturbations in spaces with vector-valued metric is obtained.
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Rohen, Yumnam, Tatjana Dosenovic, and Stojan Radenovic. "A note on the paper "A fixed point theorems in Sb-metric spaces"." Filomat 31, no. 11 (2017): 3335–46. http://dx.doi.org/10.2298/fil1711335r.

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Very recently, N. Souayan and N. Mlaiki [Nazir Souayan and Nabil Mlaiki, A fixed point theorem in Sb-metric spaces, J. Math. Comput. Sci. 16 (2016), 131-139] and S. Sedghi et al. [S. Sedghi, A. Gholidahneb, T. Dosenovic, J. Esfahani, S. Radenovic, Common fixed point of four maps in Sb-metric spaces, to appear in J. Linear Topol. Algebra], introduced the concept of Sb-metric space as a generalization of S-metric space. In this paper, we modified the definition of Sb-metric introduced by Souayan and Mlaiki and prove some coupled common fixed point theorems in Sb-metric space. We also present an example to confirm our theoretical results.
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Holá, Ľubica. "The Attouch-Wets topology and a characterisation of normable linear spaces." Bulletin of the Australian Mathematical Society 44, no. 1 (August 1991): 11–18. http://dx.doi.org/10.1017/s0004972700029415.

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Let X and Y be metric spaces and C(X, Y) be the space of all continuous functions from X to Y. If X is a locally connected space, the compact-open topology on C(X, Y) is weaker than the Attouch-Wets topology on C(X, Y). The result is applied on the space of continuous linear functions. Let X be a locally convex topological linear space metrisable with an invariant metric and X* be a continuous dual. X is normable if and only if the strong topology on X* and the Attouch-Wets topology coincide.
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Roy, Kushal, Hossein Alaeidizaji, Mantu Saha, Babak Mohammadi, and Vahid Parvaneh. "Some Fixed-Point Theorems over a Generalized F -Metric Space." Advances in Mathematical Physics 2021 (May 5, 2021): 1–7. http://dx.doi.org/10.1155/2021/5570653.

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In this article, the concept of sequential F -metric spaces has been introduced as a generalization of usual metric spaces, b -metric spaces, J S -metric spaces, and mainly F -metric spaces. Some topological properties of such spaces have been discussed here. By considering this notion, we prove fixed-point theorems for some classes of contractive mappings over such spaces. Examples have been given in order to examine the validity of the underlying space and in support of our fixed-point theorems. Moreover, our fixed-point theorem is applied to obtain solution of a system of linear algebraic equations.
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HYTÖNEN, TUOMAS, DACHUN YANG, and DONGYONG YANG. "The Hardy space H1 on non-homogeneous metric spaces." Mathematical Proceedings of the Cambridge Philosophical Society 153, no. 1 (December 8, 2011): 9–31. http://dx.doi.org/10.1017/s0305004111000776.

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AbstractLet (, d, μ) be a metric measure space and satisfy the so-called upper doubling condition and the geometrical doubling condition. We introduce the atomic Hardy space H1(μ) and prove that its dual space is the known space RBMO(μ) in this context. Using this duality, we establish a criterion for the boundedness of linear operators from H1(μ) to any Banach space. As an application of this criterion, we obtain the boundedness of Calderón–Zygmund operators from H1(μ) to L1(μ).
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Larotonda, Gabriel. "Metric geometry of infinite-dimensional Lie groups and their homogeneous spaces." Forum Mathematicum 31, no. 6 (November 1, 2019): 1567–605. http://dx.doi.org/10.1515/forum-2019-0127.

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AbstractWe study the geometry of Lie groups G with a continuous Finsler metric, in presence of a subgroup K such that the metric is right-invariant for the action of K. We present a systematic study of the metric and geodesic structure of homogeneous spaces M obtained by the quotient {M\simeq G/K}. Of particular interest are left-invariant metrics of G which are then bi-invariant for the action of K. We then focus on the geodesic structure of groups K that admit bi-invariant metrics, proving that one-parameter groups are short paths for those metrics, and characterizing all other short paths. We provide applications of the results obtained, in two settings: manifolds of Banach space linear operators, and groups of maps from compact manifolds.
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Constantini, Camillo, and Wieslaw Kubís. "Paths in hyperspaces." Applied General Topology 4, no. 2 (October 1, 2003): 377. http://dx.doi.org/10.4995/agt.2003.2040.

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<p>We prove that the hyperspace of closed bounded sets with the Hausdor_ topology, over an almost convex metric space, is an absolute retract. Dense subspaces of normed linear spaces are examples of, not necessarily connected, almost convex metric spaces. We give some necessary conditions for the path-wise connectedness of the Hausdorff metric topology on closed bounded sets. Finally, we describe properties of a separable metric space, under which its hyperspace with the Wijsman topology is path-wise connected.</p>
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Zhang, Zihou, and Chunyan Liu. "The Representations and Continuity of the Metric Projections on Two Classes of Half-Spaces in Banach Spaces." Abstract and Applied Analysis 2014 (2014): 1–5. http://dx.doi.org/10.1155/2014/908676.

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We show a necessary and sufficient condition for the existence of metric projection on a class of half-spaceKx0*,c={x∈X:x*(x)≤c}in Banach space. Two representations of metric projectionsPKx0*,candPKx0,care given, respectively, whereKx0,cstands for dual half-space ofKx0*,cin dual spaceX*. By these representations, a series of continuity results of the metric projectionsPKx0*,candPKx0,care given. We also provide the characterization that a metric projection is a linear bounded operator.
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Dissertations / Theses on the topic "Linear metric space"

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Hjelm, Andersson Hampus. "Classification of second order symmetric tensors in the Lorentz metric." Thesis, Linköpings universitet, Matematiska institutionen, 2010. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-57197.

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This bachelor thesis shows a way to classify second order symmetric tensors in the Lorentz metric. Some basic prerequisite about indefinite and definite algebra is introduced, such as the Jordan form, indefinite inner products, the Segre type, and the Minkowski space. There are also some results concerning the invariant 2-spaces of a symmetric tensor and a different approach on how to classify second order symmetric tensor.
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Peske, Wendy Ann. "A topological approach to nonlinear analysis." CSUSB ScholarWorks, 2005. https://scholarworks.lib.csusb.edu/etd-project/2779.

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A topological approach to nonlinear analysis allows for strikingly beautiful proofs and simplified calculations. This topological approach employs many of the ideas of continuous topology, including convergence, compactness, metrization, complete metric spaces, uniform spaces and function spaces. This thesis illustrates using the topological approach in proving the Cauchy-Peano Existence theorem. The topological proof utilizes the ideas of complete metric spaces, Ascoli-Arzela theorem, topological properties in Euclidean n-space and normed linear spaces, and the extension of Brouwer's fixed point theorem to Schauder's fixed point theorem, and Picard's theorem.
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Van, Zyl Augustinus Johannes. "Metrical aspects of the complexification of tensor products and tensor norms." Thesis, Pretoria : [s.n.], 2009. http://upetd.up.ac.za/thesis/available/etd-07142009-180520.

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Santos, Vagner dos. "ANÁLISE DE ESTRUTURAS PERIÓDICAS E EROSÃO NO ESPAÇO DE PARÂMETROS DE SISTEMAS NÃO-LINEARES." UNIVERSIDADE ESTADUAL DE PONTA GROSSA, 2015. http://tede2.uepg.br/jspui/handle/prefix/837.

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In this work we investigated the parameter space of a system consisting of two oscillators coupled in a master-slave configuration. In order to do so we employed Lyapunov exponent's diagrams in the parameter space, the distribution of the finite-time Lyapunov exponents and its positive fraction. We were able to show that, when the slave system is coupled to the master, the shrimp-shaped periodic structures that previously existed begins to be eroded from the outside, and that the erosion progresses with the increase in the intensity of the coupling. We also showed that in the region where the erosion takes place the second Lyapunov exponent exhibits a bimodal distribution with a maximum close to zero and the other close to 0:1. By plotting the points in the phase space that belong to each maximum we were able to identify two kinds of attractors, a limit cicle and a chaotic attractor, in which the slave system intermittently moves.
Neste trabalho foi investigado o espaço de parâmetros de um sistema formado por dois osciladores acoplados em uma configuração mestre-escravo. Como ferramenta de análise utilizamos diagramas de expoente de Lyapunov no espaço de parâmetros, a distribuição a tempo finito do expoente de Lyapunov e sua fração positiva. Mostramos que quando o sistema escravo é acoplado ao mestre as estruturas periódicas em formato de camarão existentes anteriormente começam a ser erodidas de fora para dentro, e que essa erosão aumenta com o aumento da intensidade do acoplamento. Mostramos também que na região em que ocorre a erosão o segundo expoente de Lyapunov apresenta uma distribuição bimodal com um máximo próximo a zero e outro próximo a 0; 1. Plotando os pontos no espaço de fase pertencentes a cada um dos máximos encontramos dois tipos de atratores, um ciclo limite e um atrator caótico, nos quais o sistema escravo transita de forma intermitente.
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Kuo, Tian-Yuan, and 郭添源. "Some Fixed Point Theorems in Metric Linear Spaces and Menger Spaces." Thesis, 1993. http://ndltd.ncl.edu.tw/handle/08988699053162712207.

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Toloane, Ellen Mohau. "A study of Monoidal t-norm based Logic." Thesis, 2014.

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The logical system MTL (for Monoidal t-norm Logic) is a formalism of the logic of left-continuous t-norms, which are operations that arise in the study of fuzzy sets and fuzzy logic. The objective is to investigate the important results on MTL and collect them together in a coherent form. The main results considered will be the completeness results for the logic with respect to MTL-algebras, MTL-chains (linearly ordered MTL-algebras) and standard MTL-algebras (left-continuous t-norm algebras). Completeness of MTL with respect to standard MTL-algebras means that MTL is indeed the logic of left-continuous t-norms. The logical system BL (for Basic Logic) is an axiomatic extension of MTL; we will consider the same completeness results for BL; that is we will show that BL is complete with respect to BL-algebras, BL-chains and standard BL-algebras (continuous t-norm algebras). Completeness of BL with respect to standard BL-algebras means that BL is the logic of continuous t-norms.
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Books on the topic "Linear metric space"

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Metrics on the phase space and non-selfadjoint pseudo-differential operators. Basel: Birkhäuser, 2010.

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Metric linear spaces. Dordrecht, Holland: D. Reidel, 1985.

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Linear operators in spaces with an indefinite metric. Chichester [England]: Wiley, 1989.

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Optimization on metric and normed spaces. New York: Springer, 2010.

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Verheul, E. R. Multimedians in metric and normed spaces. Amsterdam, the Netherlands: Centrum voor Wiskunde en Informatica, 1993.

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Introduction to the analysis of metric spaces. Cambridge [Cambridgeshire]: Cambridge University Press, 1987.

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Beer, Gerald Alan. Topologies on closed and closed convex sets. Dordrecht: Kluwer Academic Publishers, 1993.

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Linear approximations in convex metric spaces and the application in the mixture theory of probability theory. Singapore: World Scientific, 1993.

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Spaces of constant curvature. 6th ed. Providence, R.I: AMS Chelsea Pub., 2011.

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Ibragimov, Zair. Topics in several complex variables: First USA-Uzbekistan Conference on Analysis and Mathematical Physics, May 20-23, 2014, California State University, Fullerton, California. Providence, Rhode Island: American Mathematical Society, 2016.

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Book chapters on the topic "Linear metric space"

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Zapała, A. M. "Brownian sheets in a locally pseudoconvex metric linear space." In Lecture Notes in Mathematics, 406–24. Berlin, Heidelberg: Springer Berlin Heidelberg, 1989. http://dx.doi.org/10.1007/bfb0083409.

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Iokhvidov, E. I. "Some characteristics of a linear manifold in a Kreĭn space and their applications." In Contributions to Operator Theory in Spaces with an Indefinite Metric, 253–57. Basel: Birkhäuser Basel, 1998. http://dx.doi.org/10.1007/978-3-0348-8812-7_12.

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Bechberger, Lucas, and Kai-Uwe Kühnberger. "Generalizing Psychological Similarity Spaces to Unseen Stimuli." In Language, Cognition, and Mind, 11–36. Cham: Springer International Publishing, 2021. http://dx.doi.org/10.1007/978-3-030-69823-2_2.

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AbstractThe cognitive framework of conceptual spaces proposes to represent concepts as regions in psychological similarity spaces. These similarity spaces are typically obtained through multidimensional scaling (MDS), which converts human dissimilarity ratings for a fixed set of stimuli into a spatial representation. One can distinguish metric MDS (which assumes that the dissimilarity ratings are interval or ratio scaled) from nonmetric MDS (which only assumes an ordinal scale). In our first study, we show that despite its additional assumptions, metric MDS does not necessarily yield better solutions than nonmetric MDS. In this chapter, we furthermore propose to learn a mapping from raw stimuli into the similarity space using artificial neural networks (ANNs) in order to generalize the similarity space to unseen inputs. In our second study, we show that a linear regression from the activation vectors of a convolutional ANN to similarity spaces obtained by MDS can be successful and that the results are sensitive to the number of dimensions of the similarity space.
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Johnson, Christopher, and Haitao Wang. "A Linear-Time Algorithm for Radius-Optimally Augmenting Paths in a Metric Space." In Lecture Notes in Computer Science, 466–80. Cham: Springer International Publishing, 2019. http://dx.doi.org/10.1007/978-3-030-24766-9_34.

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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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Roman, Steven. "Metric Spaces." In Advanced Linear Algebra, 301–24. New York, NY: Springer New York, 2008. http://dx.doi.org/10.1007/978-0-387-72831-5_12.

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Roman, Steven. "Metric Spaces." In Advanced Linear Algebra, 239–61. New York, NY: Springer New York, 1992. http://dx.doi.org/10.1007/978-1-4757-2178-2_13.

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Lantsman, M. H. "Metric Spaces." In Asymptotics of Linear Differential Equations, 14–37. Dordrecht: Springer Netherlands, 2001. http://dx.doi.org/10.1007/978-94-015-9797-5_2.

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Roman, Steven. "Metric Vector Spaces." In Advanced Linear Algebra, 205–37. New York, NY: Springer New York, 1992. http://dx.doi.org/10.1007/978-1-4757-2178-2_12.

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Malkowsky, Eberhard, and Vladimir Rakočević. "Linear Metric Spaces." In Advanced Functional Analysis, 67–98. Boca Raton, Florida : CRC Press, [2019]: CRC Press, 2019. http://dx.doi.org/10.1201/9780429442599-3.

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Conference papers on the topic "Linear metric space"

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Smith, Timothy, and Kevin Silva. "High Sea State Linear Seakeeping Applicability Metric." In SNAME 30th American Towing Tank Conference. SNAME, 2017. http://dx.doi.org/10.5957/attc-2017-0040.

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Linear seakeeping predictions are attractive for design space exploration and preliminary or simple case motion estimates due to calculation speed and relatively simple input requirements. Linear seakeeping theory is an established prediction method with well-known assumptions. One of these assumptions is the assumption that the motions are small. The validity of this assumption is investigated by comparisons with a body exact nonlinear seakeeping code over a range of significant wave heights. A modern naval destroyer and a generic tumblehome ship are examined over a range of speeds, wave headings, and sea significant wave heights. A comparison between linear and nonlinear seakeeping results for the two hull forms show range of linear behavior for different geometries. A general metric based on relative motion is proposed to quantify the validity of the assumption and indicate up to what point linear seakeeping is appropriate including effect of hull form, speed, and relative wave heading.
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Lee, Dongha, Chanyoung Park, Hyunjun Ju, Junyoung Hwang, and Hwanjo Yu. "Action Space Learning for Heterogeneous User Behavior Prediction." 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.2019/392.

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Users' behaviors observed in many web-based applications are usually heterogeneous, so modeling their behaviors considering the interplay among multiple types of actions is important. However, recent collaborative filtering (CF) methods based on a metric learning approach cannot learn multiple types of user actions, because they are developed for only a single type of user actions. This paper proposes a novel metric learning method, called METAS, to jointly model heterogeneous user behaviors. Specifically, it learns two distinct spaces: 1) action space which captures the relations among all observed and unobserved actions, and 2) entity space which captures high-level similarities among users and among items. Each action vector in the action space is computed using a non-linear function and its corresponding entity vectors in the entity space. In addition, METAS adopts an efficient triplet mining algorithm to effectively speed up the convergence of metric learning. Experimental results show that METAS outperforms the state-of-the-art methods in predicting users' heterogeneous actions, and its entity space represents the user-user and item-item similarities more clearly than the space trained by the other methods.
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Min, Martin Renqiang, Hongyu Guo, and Dongjin Song. "Exemplar-centered Supervised Shallow Parametric Data Embedding." In Twenty-Sixth International Joint Conference on Artificial Intelligence. California: International Joint Conferences on Artificial Intelligence Organization, 2017. http://dx.doi.org/10.24963/ijcai.2017/345.

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Metric learning methods for dimensionality reduction in combination with k-Nearest Neighbors (kNN) have been extensively deployed in many classification, data embedding, and information retrieval applications. However, most of these approaches involve pairwise training data comparisons, and thus have quadratic computational complexity with respect to the size of training set, preventing them from scaling to fairly big datasets. Moreover, during testing, comparing test data against all the training data points is also expensive in terms of both computational cost and resources required. Furthermore, previous metrics are either too constrained or too expressive to be well learned. To effectively solve these issues, we present an exemplar-centered supervised shallow parametric data embedding model, using a Maximally Collapsing Metric Learning (MCML) objective. Our strategy learns a shallow high-order parametric embedding function and compares training/test data only with learned or precomputed exemplars, resulting in a cost function with linear computational complexity for both training and testing. We also empirically demonstrate, using several benchmark datasets, that for classification in two-dimensional embedding space, our approach not only gains speedup of kNN by hundreds of times, but also outperforms state-of-the-art supervised embedding approaches.
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Ge, Q. J., Ping Zhao, Anurag Purwar, and Xiangyun Li. "A Novel Approach to Algebraic Fitting of a Pencil of Quadrics for Planar 4R Motion Synthesis." In ASME 2012 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2012. http://dx.doi.org/10.1115/detc2012-71190.

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The use of the image space of planar displacements for planar motion approximation is a well studied subject. While the constraint manifolds associated with planar four-bar linkages are algebraic, geometric (or normal) distances have been used as default metric for least-squares fitting of these algebraic manifolds. This paper studies the problem of using algebraic distance for least-squares fitting of quadrics defining the constraint manifolds associated with Planar 4R mechanisms. It shows that the problem can be solved by fitting a pencil of quadrics with linear coefficients to a set of image points of a given one dimensional set of displacements. This linear formulation leads to a simple and fast algorithm for kinematic synthesis in the image space.
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Louca, Loucas S., and Evagoras Xydas. "Model Reduction of Modal Representations." In ASME 2006 International Mechanical Engineering Congress and Exposition. ASMEDC, 2006. http://dx.doi.org/10.1115/imece2006-14097.

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Dynamic analysis is extensively used to study the behavior of continuous and lumped parameter linear systems. In addition to the physical space, analyses can also be performed in the modal space where useful frequency information of the system can be extracted. More specifically, modal analysis can be used for the analysis and controller design of dynamic systems, where reduction of model complexity without degrading the accuracy is often required for the efficient use of the model. The reduction of modal models has been extensively studied and many reduction techniques are available. The majority of these techniques use frequency as the metric to determine the reduced model. This paper describes a new method for calculating the modal power of lumped parameter systems with the use of the bond graph representation, which is developed through a power conserving modal decomposition. This method is then used to reduce the size of the model. This technique is based on the previously developed Model Order Reduction Algorithm (MORA), which uses an energy-based metric to generate a series of proper reduced models. An example is provided to demonstrate the calculation of the modal power and the elimination of unimportant modes or modal elements using MORA.
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Maißer, P. "Differential-Geometric Methods in Multibody Dynamics and Control." In ASME 2005 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2005. http://dx.doi.org/10.1115/detc2005-84860.

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This paper presents a differential-geometric approach to the multibody system dynamics regarded as a point dynamics in a n-dimensional configuration space Rn. This configuration space becomes a Riemannian space Vn the metric of which is defined by the kinetic energy of the multibody system (MBS). Hence, all concepts and statements of the Riemannian geometry can be used to study the dynamics of MBS. One of the key points is to set up the non-linear Lagrangian motion equations of tree-like MBS as well as of constrained mechanical systems, the perturbed equations of motion, and the motion equations of hybrid MBS in a derivative-free manner. Based on this approach transformation properties can be investigated for application in real-time simulation, control theory, Hamilton mechanics, the construction of first integrals, stability etc. Finally, a general Lyapunov-stable force control law for underactuated systems is given that demonstrates the power of the approach in high-performance sports applications.
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Chelidze, David, Joseph P. Cusumano, and Anindya Chatterjee. "Failure Prognosis Using Nonlinear Short-Time Prediction and Multi-Time Scale Recursive Estimation." In ASME 2001 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2001. http://dx.doi.org/10.1115/detc2001/vib-21407.

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Abstract In this paper we describe a new, general purpose machinery diagnostic/prognostic algorithm for tracking and predicting evolving damage using only available “macroscopic” observable quantities. The damage is viewed as occurring in a hierarchical dynamical system consisting of a directly observable, “fast” subsystem coupled with a hidden, “slow” subsystem describing damage evolution. This method provides damage diagnostics and failure prognostics requiring only the measurements from the fast subsystem and a model of the slow subsystem. Damage tracking is accomplished by a two-time-scale modeling strategy based on phase space reconstruction using the measured fast-time data. Short-time predictive models are constructed using the reconstructed phase space of the reference (undamaged) fast subsystem. Later, fast-time data for the damaged system is collected and used to estimate the short-time reference model prediction error, or a tracking function. An average value of the tracking function over a given data record is used as a tracking metric, or measure of the current damage state. Recursive, nonlinear filtering is used to estimate the actual damage state based on the tracking metric input. Estimates of remaining useful life are obtained recursively using a linear Kalman filter. This method is applied to an experimental nonlinear oscillator containing a beam with a crack which propagates to complete failure. We demonstrate the ability to track the evolving damage state of the beam using only strain time series data. We also give accurate predictions of remaining useful life, in real time, beginning well in advance of the final complete fracture at the end of experiment.
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8

Chelidze, David. "Multidimensional Hidden Slow Variable Tracking in a Hierarchical Dynamical System." In ASME 2003 International Mechanical Engineering Congress and Exposition. ASMEDC, 2003. http://dx.doi.org/10.1115/imece2003-41421.

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In this paper, we present a novel method for multidimensional damage identification based on a dynamical systems approach to damage evolution. This approach does not depend on the knowledge of particular damage physics, and is appropriate for systems where damage evolves on much slower time scale then the directly observable dynamics. In an experimental context, the phase space reconstruction and locally linear models are used to quantify small distortions occurring in a dynamical system’s phase space due to damage accumulation. These measurements are then related to the drifts in damage variables. A mathematical model of a harmonically driven cantilever beam in a force field of two battery-powered electromagnets is used to demonstrate validity of the method. It is explicitly demonstrated that an affine projection of the described damage metric accurately tracks the two competing damage processes. For practical damage identification purposes, the tracking data is analyzed using the proper orthogonal decomposition (POD) and optimal tracking (OT) methods. Both methods correctly identify the two dominant damage modes. However, the OT is more impervious to changes in fast-time dynamics and provides a significantly better signal-to-noise ratio. The OT-based damage observer is demonstrated to be within a linear transformation from the actual damage states.
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Shukla, Amit. "Classification of Nonlinear Dynamics of Human Posture Using Support Vector Machines." In ASME 2013 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2013. http://dx.doi.org/10.1115/detc2013-13609.

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Human postural control can be modeled as a complex, non-linear multi-sensory and multi actuation system. Postural control requires a balance between various sensory inputs and the corresponding response in a motor control framework. Various motor control disorders result in a propensity for fall. In this paper, a classification scheme using support vector machines (SVM) is investigated to classify individuals between healthy and those prone to balance disorders including falls. First, a computational model is described to illustrate the proposed approach. It is shown that using SVM, in conjunction with L2 Norm associated with the phase space trajectories, it is possible to classify fallers and non-fallers. This metric can then be utilized for clinical application upon further evaluation. This may benefit clinical evaluation of Parkinson’s subjects by complementing the widely used UPDRS scale.
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Luo, Yong, Yonggang Wen, Tongliang Liu, and Dacheng Tao. "General Heterogeneous Transfer Distance Metric Learning via Knowledge Fragments Transfer." In Twenty-Sixth International Joint Conference on Artificial Intelligence. California: International Joint Conferences on Artificial Intelligence Organization, 2017. http://dx.doi.org/10.24963/ijcai.2017/341.

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Transfer learning aims to improve the performance of target learning task by leveraging information (or transferring knowledge) from other related tasks. Recently, transfer distance metric learning (TDML) has attracted lots of interests, but most of these methods assume that feature representations for the source and target learning tasks are the same. Hence, they are not suitable for the applications, in which the data are from heterogeneous domains (feature spaces, modalities and even semantics). Although some existing heterogeneous transfer learning (HTL) approaches is able to handle such domains, they lack flexibility in real-world applications, and the learned transformations are often restricted to be linear. We therefore develop a general and flexible heterogeneous TDML (HTDML) framework based on the knowledge fragment transfer strategy. In the proposed HTDML, any (linear or nonlinear) distance metric learning algorithms can be employed to learn the source metric beforehand. Then a set of knowledge fragments are extracted from the pre-learned source metric to help target metric learning. In addition, either linear or nonlinear distance metric can be learned for the target domain. Extensive experiments on both scene classification and object recognition demonstrate superiority of the proposed method.
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