Relevant bibliographies by topics / Embedding dimension

Contents

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

Academic literature on the topic 'Embedding dimension'

Author: Grafiati
Published: 4 June 2021
Last updated: 1 February 2022

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Journal articles on the topic "Embedding dimension"

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Sivakumar, B., R. Berndtsson, J. Olsson, K. Jinno, and A. Kawamura. "Dynamics of monthly rainfall-runoff process at the Gota basin: A search for chaos." Hydrology and Earth System Sciences 4, no. 3 (September 30, 2000): 407–17. http://dx.doi.org/10.5194/hess-4-407-2000.

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Abstract. Sivakumar et al. (2000a), by employing the correlation dimension method, provided preliminary evidence of the existence of chaos in the monthly rainfall-runoff process at the Gota basin in Sweden. The present study verifies and supports the earlier results and strengthens such evidence. The study analyses the monthly rainfall, runoff and runoff coefficient series using the nonlinear prediction method, and the presence of chaos is investigated through an inverse approach, i.e. identifying chaos from the results of the prediction. The presence of an optimal embedding dimension (the embedding dimension with the best prediction accuracy) for each of the three series indicates the existence of chaos in the rainfall-runoff process, providing additional support to the results obtained using the correlation dimension method. The reasonably good predictions achieved, particularly for the runoff series, suggest that the dynamics of the rainfall-runoff process could be understood from a chaotic perspective. The predictions are also consistent with the correlation dimension results obtained in the earlier study, i.e. higher prediction accuracy for series with a lower dimension and vice-versa, so that the correlation dimension method can indeed be used as a preliminary indicator of chaos. However, the optimal embedding dimensions obtained from the prediction method are considerably less than the minimum dimensions essential to embed the attractor, as obtained by the correlation dimension method. A possible explanation for this could be the presence of noise in the series, since the effects of noise at higher embedding dimensions could be significantly greater than that at lower embedding dimensions. Keywords: Rainfall-runoff; runoff coefficient; chaos; phase-space; correlation dimension; nonlinear prediction; noise
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Aleksić, Zoran. "Estimating the embedding dimension." Physica D: Nonlinear Phenomena 52, no. 2-3 (September 1991): 362–68. http://dx.doi.org/10.1016/0167-2789(91)90132-s.

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Grines, V. Z., E. Ya Gurevich, and O. V. Pochinka. "On Embedding of the Morse-Smale Diffeomorphisms in a Topological Flow." Contemporary Mathematics. Fundamental Directions 66, no. 2 (December 15, 2020): 160–81. http://dx.doi.org/10.22363/2413-3639-2020-66-2-160-181.

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This review presents the results of recent years on solving of the Palis problem on finding necessary and sufficient conditions for the embedding of Morse-Smale cascades in topological flows. To date, the problem has been solved by Palis for Morse-Smale diffeomorphisms given on manifolds of dimension two. The result for the circle is a trivial exercise. In dimensions three and higher new effects arise related to the possibility of wild embeddings of closures of invariant manifolds of saddle periodic points that leads to additional obstacles for Morse-Smale diffeomorphisms to embed in topological flows. The progress achieved in solving of Paliss problem in dimension three is associated with the recently obtained complete topological classification of Morse-Smale diffeomorphisms on three-dimensional manifolds and the introduction of new invariants describing the embedding of separatrices of saddle periodic points in a supporting manifold. The transition to a higher dimension requires the latest results from the topology of manifolds. The necessary topological information, which plays key roles in the proofs, is also presented in the survey.
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HAMBLY, B. M., and T. KUMAGAI. "ASYMPTOTICS FOR THE SPECTRAL AND WALK DIMENSION AS FRACTALS APPROACH EUCLIDEAN SPACE." Fractals 10, no. 04 (December 2002): 403–12. http://dx.doi.org/10.1142/s0218348x02001270.

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We discuss the behavior of the dynamic dimension exponents for families of fractals based on the Sierpinski gasket and carpet. As the length scale factor for the family tends to infinity, the lattice approximations to the fractals look more like the tetrahedral or cubic lattice in Euclidean space and the fractal dimension converges to that of the embedding space. However, in the Sierpinski gasket case, the spectral dimension converges to two for all dimensions. In two dimensions, we prove a conjecture made in the physics literature concerning the rate of convergence. On the other hand, for natural families of Sierpinski carpets, the spectral dimension converges to the dimension of the embedding Euclidean space. In general, we demonstrate that for both cases of finitely and infinitely ramified fractals, a variety of asymptotic values for the spectral dimension can be achieved.
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SATHER-WAGSTAFF, SEAN. "EMBEDDING MODULES OF FINITE HOMOLOGICAL DIMENSION." Glasgow Mathematical Journal 55, no. 1 (August 2, 2012): 85–96. http://dx.doi.org/10.1017/s0017089512000353.

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AbstractThis paper builds on work of Hochster and Yao that provides nice embeddings for finitely generated modules of finite G-dimension, finite projective dimension or locally finite injective dimension. We extend these results by providing similar embeddings in the relative setting, that is, for certain modules of finite GC-dimension, finite C-projective dimension, locally finite C-injective dimension or locally finite C-injective dimension where C is a semidualizing module. Along the way, we extend some results for modules of finite homological dimension to modules of locally finite homological dimension in the relative setting.
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Chapman, S. T., P. A. García-Sánchez, D. Llena, and J. Marshall. "Elements in a Numerical Semigroup with Factorizations of the Same Length." Canadian Mathematical Bulletin 54, no. 1 (March 1, 2011): 39–43. http://dx.doi.org/10.4153/cmb-2010-068-3.

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AbstractQuestions concerning the lengths of factorizations into irreducible elements in numerical monoids have gained much attention in the recent literature. In this note, we show that a numerical monoid has an element with two different irreducible factorizations of the same length if and only if its embedding dimension is greater than two. We find formulas in embedding dimension three for the smallest element with two different irreducible factorizations of the same length and the largest element whose different irreducible factorizations all have distinct lengths. We show that these formulas do not naturally extend to higher embedding dimensions.
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Gács, Peter. "Clairvoyant embedding in one dimension." Random Structures & Algorithms 47, no. 3 (June 13, 2014): 520–60. http://dx.doi.org/10.1002/rsa.20551.

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Lequain, Yves. "Embedding dimension in local rings." Communications in Algebra 18, no. 11 (January 1990): 3923–31. http://dx.doi.org/10.1080/00927879008824117.

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Rosales, J. C., and P. A. Garc�a-S�anchez. "Numerical semigroups with embedding dimension three." Archiv der Mathematik 83, no. 6 (December 2004): 488–96. http://dx.doi.org/10.1007/s00013-004-1149-1.

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Mees, A. I., P. E. Rapp, and L. S. Jennings. "Singular-value decomposition and embedding dimension." Physical Review A 36, no. 1 (July 1, 1987): 340–46. http://dx.doi.org/10.1103/physreva.36.340.

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Dissertations / Theses on the topic "Embedding dimension"

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Rosa, Renata Martins da. "Dimensão de mergulho (embedding dimension) de anéis locais." reponame:Biblioteca Digital de Teses e Dissertações da UFRGS, 1992. http://hdl.handle.net/10183/127096.

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Neste trabalho é mostrado que para um anel local de Cohrn-Macaulay e para. um anel local cujo corpo residual seja algebricament.c fechado e cujo anel gnKluado associado seja um domínio, vale que a dimensão ele mergulho ( embedcling dimension) é menor ou igual a sua dimensão de I~rull mais sua multiplicidade menos um. Na segunda parte do trabalho vemos que, dados dois inteiros m ≥ 1, d ≥ 2 e um inteiro arbitrário e ≥ dm, existe um anel local com multiplicidade, dimensão de Krull e dimensão de mergulho iguais a m, d e e respectiYamente.
In this work we show that. for a Cohen-Macaulay ring and for a local ring with algebraically closed residual field and associated graded ring a domain. we hav: the embedding dimension is less than or equal t.o t.he Erull climension plus the multiplicity mines one. Secondly, we see that, given integers m ≥ 1, d ≥ 2 anel given an arbitrary integer e ≥ dm , there exists a local ring with multiplicity Krull dimcnsion d and embedding dimension t.
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El, Khoury Sabine. "A class of Gorenstein Artin algebras of embedding dimension four." Diss., Columbia, Mo. : University of Missouri-Columbia, 2007. http://hdl.handle.net/10355/5930.

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Thesis (Ph. D.)--University of Missouri-Columbia, 2007.
The entire dissertation/thesis text is included in the research.pdf file; the official abstract appears in the short.pdf file (which also appears in the research.pdf); a non-technical general description, or public abstract, appears in the public.pdf file. Title from title screen of research.pdf file (viewed on March 20, 2009) Vita. Includes bibliographical references.
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Chen, Huiyuan. "Dimension Reduction for Network Analysis with an Application to Drug Discovery." Case Western Reserve University School of Graduate Studies / OhioLINK, 2020. http://rave.ohiolink.edu/etdc/view?acc_num=case1598303654048332.

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Liang, Zhiyu. "Eigen-analysis of kernel operators for nonlinear dimension reduction and discrimination." The Ohio State University, 2014. http://rave.ohiolink.edu/etdc/view?acc_num=osu1388676476.

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Boland, Josephine Anne. "Embedding a civic engagement dimension within the higher education curriculum : a study of policy, process and practice in Ireland." Thesis, University of Edinburgh, 2008. http://hdl.handle.net/1842/5804.

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As the civic role of higher education attracts renewed critical attention, the idea of engagement has come to the fore. Civic engagement, as espoused in many institutional missions, encompasses a diversity of goals, strategies and activities. Latterly, these have included particular approaches to teaching and learning. This research examines the process of embedding a civic engagement dimension within the higher education curriculum in Ireland. I use the term ‘pedagogy for civic engagement’as a generic term for a range of academic practices –variously referred to as ‘service learning’or ‘community based learning’–which share an explicit civic focus. Academic practice serves as the central focus with attention to pertinent aspects of the prevailing context. Using a multi-site case study conducted in the spirit of naturalistic enquiry, I examine four cases of this curriculum innovation, drawn from the university and institute of technology sectors in Ireland, with unstructured interviews and documents as the main sources of data. I interrogate the underpinning rationale for ‘pedagogy for civic engagement’–as gleaned from the literature, the policy context and the case studies –exploring implicit conceptions in relation to knowledge, curriculum, civil society, community and the purpose of higher education. The study draws its empirical data from those responsible for implementing this pedagogy –the ‘embedders’–and a range of other actors. Interviews were carried out with academic staff, project directors, educational developers, academic managers and leaders. Key actors from the national policy context and from the international field of civic engagement also participated in the study. Four orientations to civic engagement are identified, revealing the multifaceted rationale. I explore the process of operationalising the pedagogy and the factors impacting on academics’capacity and willingness to embed it. While the study does not directly examine the experience of students and community partners their role within the process, as perceived by academic staff and others, is problematised. The implications of the putative unresolved epistemology of this pedagogy are explored in light of how participants conceive of and practice it. Academics’ambivalence about the place of values in higher education emerges as a theme and the issue of agency recurs. I explore how the pedagogy may be conceived of in terms of the teaching, research and service roles of academics and consider how it may be positioned within an institution. Opportunities for alignment are identified at a number of levels from constructive alignment within the curriculum to alignment with national strategic priorities. I explore the unrealised potential of the Irish National Framework of Qualifications –specifically the ‘insight’dimension –as a means of enabling and legitimising the pedagogy, in light of the prominence afforded to the principle of subsidiarity in Irish higher education policy. The localised way in which these practices have been adopted and adapted underlines the significance of context and culture. ‘Pedagogy for civic engagement’as a concept and as a practice challenges a range of assumptions and traditional practices, raising fundamental questions regarding the role and purpose of higher education –and not just in contemporary Ireland.
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Fontes, Nuno Ricardo Moura. "Sistemas dinâmicos, análise numérica de séries temporais e aplicações às finanças." Master's thesis, Instituto Superior de Economia e Gestão, 2013. http://hdl.handle.net/10400.5/6454.

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Mestrado em Matemática Financeira
Taken's theorem (1981) shows how the series of measurements from a given system can be used to reconstruct the original system's underlying dynamic process. In this work we start from this point and build a bridge between theoretical results and its practical application. Several algorithms are presented and then rebuilt in an effort to reach a middle ground between computer resources optimization and output accuracy. Among these algorithms, the biggest emphasis is put on the correlation dimension algorithm by Grassberger and Procaccia which allows for the deduction of the system's embedding dimension. The results derived are then used to build a forecast approach inspired by the analogues method. The purpose of this work is to show there is potential for dynamical systems' modelling tools to be used in financial markets, especially for intra-day purposes where decision and computational times need to be very small.
O teorema de Takens (1981) mostra como uma série de medições obtidas de um dado sistema podem ser usadas para reconstruir o sistema dinâmico original. Neste trabalho, parte-se deste teorema e constrói-se a ponte entre conceitos teóricos e a sua aplicação numérica. Vários algoritmos são apresentados e depois reconstruídos com o objetivo de se atingir um compromisso entre otimização de recursos computacionais e rigor nos resultados. Entre esses algoritmos, a maior ênfase é colocada no do cálculo do integral de correlação de Grassberger-Procaccia que permite a dedução da dimensão de imersão de um dado sistema. Os resultados obtidos são usados na construção de um modelo de previsão inspirado pela abordagem dos pontos análogos, ou método dos análogos. O objetivo deste trabalho é mostrar que existe potencial na aplicação de ferramentas de modelação de sistemas dinâmicos caóticos no mercado financeiro, em especial em transações intra-diárias onde tempos de decisão e computação têm de ser muito reduzidos.
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Grant, Elyot. "Dimension reduction algorithms for near-optimal low-dimensional embeddings and compressive sensing." Thesis, Massachusetts Institute of Technology, 2013. http://hdl.handle.net/1721.1/84869.

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Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2013.
Cataloged from PDF version of thesis.
Includes bibliographical references (pages 41-42).
In this thesis, we establish theoretical guarantees for several dimension reduction algorithms developed for applications in compressive sensing and signal processing. In each instance, the input is a point or set of points in d-dimensional Euclidean space, and the goal is to find a linear function from Rd into Rk , where k << d, such that the resulting embedding of the input pointset into k-dimensional Euclidean space has various desirable properties. We focus on two classes of theoretical results: -- First, we examine linear embeddings of arbitrary pointsets with the aim of minimizing distortion. We present an exhaustive-search-based algorithm that yields a k-dimensional linear embedding with distortion at most ... is the smallest possible distortion over all orthonormal embeddings into k dimensions. This PTAS-like result transcends lower bounds for well-known embedding teclhniques such as the Johnson-Lindenstrauss transform. -- Next, motivated by compressive sensing of images, we examine linear embeddings of datasets containing points that are sparse in the pixel basis, with the goal of recoving a nearly-optimal sparse approximation to the original data. We present several algorithms that achieve strong recovery guarantees using the near-optimal bound of measurements, while also being highly "local" so that they can be implemented more easily in physical devices. We also present some impossibility results concerning the existence of such embeddings with stronger locality properties.
by Elyot Grant.
S.M.
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Moon, Gordon Euhyun. "Parallel Algorithms for Machine Learning." The Ohio State University, 2019. http://rave.ohiolink.edu/etdc/view?acc_num=osu1561980674706558.

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Melo, Givanildo Donizeti de [UNESP]. "Sobre a dimensão do quadrado de um espaço métrico compacto X de dimensão n e o conjunto dos mergulhos de X em R2n." Universidade Estadual Paulista (UNESP), 2016. http://hdl.handle.net/11449/138318.

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Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)
Neste trabalho nós estudamos o seguinte resultado: para um espaço métrico compacto X, de dimensão n, o subespaço dos mergulhos de X em R2n é denso no espaço das funções contínuas de X em R2n se, e somente se, dim(X x X)<2n. A demonstração apresentada é aquela dada por J. Krasinkiewicz e por S. Spiez.
In this work we study the following result: given a compact metric space X of dimension n, the subspace consisting of all embeddings of X into R2n is dense in the space of all continuous maps of X into R2n if and only if dim(X x X)<2n. The presented proof is the one given by J. Krasinkiewicz e por S. Spiez.
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Weerasekara, Aruna Bandara. "Electrical and Optical Characterization of Group III-V Heterostructures with Emphasis on Terahertz Devices." Digital Archive @ GSU, 2007. http://digitalarchive.gsu.edu/phy_astr_diss/16.

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Electrical and optical characterizations of heterostructures and thin films based on group III-V compound semiconductors are presented. Optical properties of GaMnN thin films grown by Metalorganic Chemical Vapor Deposition (MOCVD) on GaN/Sapphire templates were investigated using IR reflection spectroscopy. Experimental reflection spectra were fitted using a non - linear fitting algorithm, and the high frequency dielectric constant (ε∞), optical phonon frequencies of E1(TO) and E1(LO), and their oscillator strengths (S) and broadening constants (Γ) were obtained for GaMnN thin films with different Mn fraction. The high frequency dielectric constant (ε∞) of InN thin films grown by the high pressure chemical vapor deposition (HPCVD) method was also investigated by IR reflection spectroscopy and the average was found to vary between 7.0 - 8.6. The mobility of free carriers in InN thin films was calculated using the damping constant of the plasma oscillator. The terahertz detection capability of n-type GaAs/AlGaAs Heterojunction Interfacial Workfunction Internal Photoemission (HEIWIP) structures was demonstrated. A threshold frequency of 3.2 THz (93 µm) with a peak responsivity of 6.5 A/W at 7.1 THz was obtained using a 0.7 µm thick 1E18 cm−3 n - type doped GaAs emitter layer and a 1 µm thick undoped Al(0.04)Ga(0.96)As barrier layer. Using n - type doped GaAs emitter layers, the possibility of obtaining small workfunctions (∆) required for terahertz detectors has been successfully demonstrated. In addition, the possibility of using GaN (GaMnN) and InN materials for terahertz detection was investigated and a possible GaN base terahertz detector design is presented. The non - linear behavior of the Inter Pulse Time Intervals (IPTI) of neuron - like electric pulses triggered externally in a GaAs/InGaAs Multi Quantum Well (MQW) structure at low temperature (~10 K) was investigated. It was found that a grouping behavior of IPTIs exists at slow triggering pulse rates. Furthermore, the calculated correlation dimension reveals that the dimensionality of the system is higher than the average dimension found in most of the natural systems. Finally, an investigation of terahertz radiation efect on biological system is reported.
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Books on the topic "Embedding dimension"

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Dimensions, embeddings, and attractors. Cambridge: Cambridge University Press, 2011.

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Elmar, Winkelnkemper, ed. High-dimensional knot theory: Algebraic surgery in codimension 2. Berlin: Springer, 1998.

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Gould, Jeremy David. Embeddings, dimension groups and presentations of AF algebras, and the index of subfactors. [s.l.]: typescript, 1989.

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Two-dimensional mesh embedding for Galerkin B-spline methods. [Washington, DC]: National Aeronautics and Space Administration, 1995.

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deLancey, Moser Robert, and United States. National Aeronautics and Space Administration., eds. Two-dimensional mesh embedding for Galerkin B-spline methods. [Washington, DC]: National Aeronautics and Space Administration, 1995.

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deLancey, Moser Robert, and United States. National Aeronautics and Space Administration., eds. Two-dimensional mesh embedding for Galerkin B-spline methods. [Washington, DC]: National Aeronautics and Space Administration, 1995.

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deLancey, Moser Robert, and United States. National Aeronautics and Space Administration., eds. Two-dimensional mesh embedding for Galerkin B-spline methods. [Washington, DC]: National Aeronautics and Space Administration, 1995.

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Behrens, Stefan, Boldizsar Kalmar, Min Hoon Kim, Mark Powell, and Arunima Ray, eds. The Disc Embedding Theorem. Oxford University Press, 2021. http://dx.doi.org/10.1093/oso/9780198841319.001.0001.

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The disc embedding theorem provides a detailed proof of the eponymous theorem in 4-manifold topology. The theorem, due to Michael Freedman, underpins virtually all of our understanding of 4-manifolds in the topological category. Most famously, this includes the 4-dimensional topological Poincaré conjecture. Combined with the concurrent work of Simon Donaldson, the theorem reveals a remarkable disparity between the topological and smooth categories for 4-manifolds. A thorough exposition of Freedman’s proof of the disc embedding theorem is given, with many new details. A self-contained account of decomposition space theory, a beautiful but outmoded branch of topology that produces non-differentiable homeomorphisms between manifolds, is provided. Techniques from decomposition space theory are used to show that an object produced by an infinite, iterative process, which we call a skyscraper, is homeomorphic to a thickened disc, relative to its boundary. A stand-alone interlude explains the disc embedding theorem’s key role in smoothing theory, the existence of exotic smooth structures on Euclidean space, and all known homeomorphism classifications of 4-manifolds via surgery theory and the s-cobordism theorem. The book is written to be accessible to graduate students working on 4-manifolds, as well as researchers in related areas. It contains over a hundred professionally rendered figures.
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United States. National Aeronautics and Space Administration. and Mississippi State University. Dept. of Aerophysics and Aerospace Engineering., eds. Adaptive grid embedding for the two-dimensional flux-split Euler equations. Mississippi State, Miss: Mississippi State University, Dept. of Aerospace Engineering, 1990.

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Five-dimensional Physics: Classical And Quantum Consequences of Kaluza-klein Cosmology. World Scientific Publishing Company, 2006.

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Book chapters on the topic "Embedding dimension"

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Daverman, Robert, and Gerard Venema. "Engulfing, cellularity, and embedding dimension." In Embeddings in Manifolds, 97–143. Providence, Rhode Island: American Mathematical Society, 2009. http://dx.doi.org/10.1090/gsm/106/04.

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Orlando, Giuseppe, Ruedi Stoop, and Giovanni Taglialatela. "Embedding Dimension and Mutual Information." In Nonlinearities in Economics, 105–8. Cham: Springer International Publishing, 2021. http://dx.doi.org/10.1007/978-3-030-70982-2_7.

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Guègan, Dominique, and Francesco Lisi. "Predictive dimension: an alternative definition to embedding dimension." In COMPSTAT, 319–24. Heidelberg: Physica-Verlag HD, 2000. http://dx.doi.org/10.1007/978-3-642-57678-2_40.

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Carroll, T. L., and J. M. Byers. "Calculating Embedding Dimension with Confidence Estimates." In Understanding Complex Systems, 211–23. Cham: Springer International Publishing, 2019. http://dx.doi.org/10.1007/978-3-030-10892-2_21.

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Rosales, J. C., and P. A. García-Sánchez. "Numerical semigroups with embedding dimension three." In Numerical Semigroups, 137–54. New York, NY: Springer New York, 2009. http://dx.doi.org/10.1007/978-1-4419-0160-6_10.

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Rosales, J. C., and P. A. García-Sánchez. "Numerical semigroups with maximal embedding dimension." In Numerical Semigroups, 19–32. New York, NY: Springer New York, 2009. http://dx.doi.org/10.1007/978-1-4419-0160-6_3.

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Maugeri, Nicola, and Giuseppe Zito. "Embedding Dimension of a Good Semigroup." In Numerical Semigroups, 197–230. Cham: Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-40822-0_13.

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Coornaert, Michel. "Applications of Mean Dimension to Embedding Problems." In Universitext, 139–55. Cham: Springer International Publishing, 2015. http://dx.doi.org/10.1007/978-3-319-19794-4_8.

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Faragó, András. "Low Distortion Metric Embedding into Constant Dimension." In Lecture Notes in Computer Science, 114–23. Berlin, Heidelberg: Springer Berlin Heidelberg, 2011. http://dx.doi.org/10.1007/978-3-642-20877-5_12.

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Cao, Liangyue. "Determining Minimum Embedding Dimension from Scalar Time Series." In Modelling and Forecasting Financial Data, 43–60. Boston, MA: Springer US, 2002. http://dx.doi.org/10.1007/978-1-4615-0931-8_3.

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Conference papers on the topic "Embedding dimension"

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Luo, Gongxu, Jianxin Li, Hao Peng, Carl Yang, Lichao Sun, Philip S. Yu, and Lifang He. "Graph Entropy Guided Node Embedding Dimension Selection for Graph Neural Networks." In Thirtieth International Joint Conference on Artificial Intelligence {IJCAI-21}. California: International Joint Conferences on Artificial Intelligence Organization, 2021. http://dx.doi.org/10.24963/ijcai.2021/381.

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Graph representation learning has achieved great success in many areas, including e-commerce, chemistry, biology, etc. However, the fundamental problem of choosing the appropriate dimension of node embedding for a given graph still remains unsolved. The commonly used strategies for Node Embedding Dimension Selection (NEDS) based on grid search or empirical knowledge suffer from heavy computation and poor model performance. In this paper, we revisit NEDS from the perspective of minimum entropy principle. Subsequently, we propose a novel Minimum Graph Entropy (MinGE) algorithm for NEDS with graph data. To be specific, MinGE considers both feature entropy and structure entropy on graphs, which are carefully designed according to the characteristics of the rich information in them. The feature entropy, which assumes the embeddings of adjacent nodes to be more similar, connects node features and link topology on graphs. The structure entropy takes the normalized degree as basic unit to further measure the higher-order structure of graphs. Based on them, we design MinGE to directly calculate the ideal node embedding dimension for any graph. Finally, comprehensive experiments with popular Graph Neural Networks (GNNs) on benchmark datasets demonstrate the effectiveness and generalizability of our proposed MinGE.
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Kaplan, Daniel T. "Model-independent technique for determining the embedding dimension." In SPIE's 1993 International Symposium on Optics, Imaging, and Instrumentation, edited by Louis M. Pecora. SPIE, 1993. http://dx.doi.org/10.1117/12.162676.

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"NEIGHBORHOOD FUNCTION DESIGN FOR EMBEDDING IN REDUCED DIMENSION." In International Conference on Neural Computation Theory and Applications. SciTePress - Science and and Technology Publications, 2011. http://dx.doi.org/10.5220/0003681201900195.

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Zhao, Xiangyu, Haochen Liu, Hui Liu, Jiliang Tang, Weiwei Guo, Jun Shi, Sida Wang, Huiji Gao, and Bo Long. "AutoDim: Field-aware Embedding Dimension Searchin Recommender Systems." In WWW '21: The Web Conference 2021. New York, NY, USA: ACM, 2021. http://dx.doi.org/10.1145/3442381.3450124.

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Chelidze, David. "Statistical Characterization of Nearest Neighbors to Reliably Estimate Minimum Embedding Dimension." In ASME 2014 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2014. http://dx.doi.org/10.1115/detc2014-34746.

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False nearest neighbors (FNN) is one of the essential methods used in estimating the minimally sufficient embedding dimension in delay coordinate embedding of deterministic time series. Its use for stochastic and noisy deterministic time series is problematic and erroneously indicates a finite embedding dimension. Various modifications to the original method have been proposed to mitigate this problem, but those are still not reliable for noisy time series. Nearest neighbor statistics are studied for uncorrelated random time series and contrasted with the deterministic statistics. A new FNN metric is constructed and its performance is evaluated for deterministic, stochastic, and random time series. The results are also contrasted with surrogate data analysis and show that the new metric is robust to noise. It also clearly identifies random time series as not having a finite embedding dimension and provides information about the deterministic part of stochastic processes. The new metric can also be used for differentiating between chaotic and random time series.
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Wang, Yu. "Single Training Dimension Selection for Word Embedding with PCA." In Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Processing (EMNLP-IJCNLP). Stroudsburg, PA, USA: Association for Computational Linguistics, 2019. http://dx.doi.org/10.18653/v1/d19-1369.

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Liu, Biying, Guangyuan Liu, and Zhaofang Yang. "Analysis of Affective State from Galvanic Skin Response Using Correlation Dimension and Embedding Dimension." In 2012 5th International Symposium on Computational Intelligence and Design (ISCID). IEEE, 2012. http://dx.doi.org/10.1109/iscid.2012.10.

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Zhang, Miao, Huiqi Li, and Steven Su. "High Dimensional Bayesian Optimization via Supervised Dimension Reduction." 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.2019/596.

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Bayesian optimization (BO) has been broadly applied to computational expensive problems, but it is still challenging to extend BO to high dimensions. Existing works are usually under strict assumption of an additive or a linear embedding structure for objective functions. This paper directly introduces a supervised dimension reduction method, Sliced Inverse Regression (SIR), to high dimensional Bayesian optimization, which could effectively learn the intrinsic sub-structure of objective function during the optimization. Furthermore, a kernel trick is developed to reduce computational complexity and learn nonlinear subset of the unknowing function when applying SIR to extremely high dimensional BO. We present several computational benefits and derive theoretical regret bounds of our algorithm. Extensive experiments on synthetic examples and two real applications demonstrate the superiority of our algorithms for high dimensional Bayesian optimization.
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Zhao, Ning, Junjie Shen, Yuhe Liu, and Xiaolin Ma. "Analysis of Embedding Dimension of Built-in sensor Nonlinear Output." In 2018 IEEE 4th Information Technology and Mechatronics Engineering Conference (ITOEC). IEEE, 2018. http://dx.doi.org/10.1109/itoec.2018.8740584.

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Xing, Changyou, and Ming Chen. "Research on Optimizing Embedding Space Dimension in Network Coordinate System." In 2009 Eighth IEEE/ACIS International Conference on Computer and Information Science. IEEE, 2009. http://dx.doi.org/10.1109/icis.2009.63.

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Reports on the topic "Embedding dimension"

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Krauthgamer, Robert, Nathan Linial, and Avner Magen. Metric Embeddings - Beyond One-Dimensional Distortion. Fort Belvoir, VA: Defense Technical Information Center, May 2002. http://dx.doi.org/10.21236/ada619309.

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