Статті в журналах: "Banach manifold with uniform structure"

1

Bogdanskii, Yu V., and E. V. Moravetskaya. "Surface Measures on Banach Manifolds with Uniform Structure." Ukrainian Mathematical Journal 69, no. 8 (December 3, 2017): 1196–219. http://dx.doi.org/10.1007/s11253-017-1425-4.

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2

Moravets’ka, K. V. "Differentiability of Borel Measures Along Vector Fields on Banach Manifolds with Uniform Structure." Ukrainian Mathematical Journal 68, no. 10 (March 2017): 1552–73. http://dx.doi.org/10.1007/s11253-017-1312-z.

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3

Bogdanskii, Yu V., and E. V. Moravetskaya. "Transitivity of the Surface Measures on Banach Manifolds with Uniform Structures." Ukrainian Mathematical Journal 69, no. 10 (March 2018): 1507–19. http://dx.doi.org/10.1007/s11253-018-1452-9.

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4

Halakatti, S. C. P., and Archana Halijol. "Fuzzy smooth homotopy on fuzzy Banach manifold." Journal of Advanced Studies in Topology 8, no. 2 (October 14, 2017): 117. http://dx.doi.org/10.20454/jast.2017.1293.

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In this paper the structure of fuzzy Banach manifolds has been enriched by inducing three different equivalence relations on fuzzy Banach atlases. Also a Network fuzzy Banach manifold has been defined admitting two equivalence relation and two group structures. Further we define fuzzy smooth homotopy on fuzzy Banach manifold and the study has been extended for three different fuzzy path connectedness inducing equivalence relations and fundamental group structure which is invariant under fuzzy homeomorphism.

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5

Kalton, N. J. "The uniform structure of Banach spaces." Mathematische Annalen 354, no. 4 (December 1, 2011): 1247–88. http://dx.doi.org/10.1007/s00208-011-0743-3.

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6

Kalton, N. J. "Uniform homeomorphisms of Banach spaces and asymptotic structure." Transactions of the American Mathematical Society 365, no. 2 (September 13, 2012): 1051–79. http://dx.doi.org/10.1090/s0002-9947-2012-05665-0.

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7

Finster, Felix, and Magdalena Lottner. "Banach manifold structure and infinite-dimensional analysis for causal fermion systems." Annals of Global Analysis and Geometry 60, no. 2 (May 31, 2021): 313–54. http://dx.doi.org/10.1007/s10455-021-09775-4.

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AbstractA mathematical framework is developed for the analysis of causal fermion systems in the infinite-dimensional setting. It is shown that the regular spacetime point operators form a Banach manifold endowed with a canonical Fréchet-smooth Riemannian metric. The so-called expedient differential calculus is introduced with the purpose of treating derivatives of functions on Banach spaces which are differentiable only in certain directions. A chain rule is proven for Hölder continuous functions which are differentiable on expedient subspaces. These results are made applicable to causal fermion systems by proving that the causal Lagrangian is Hölder continuous. Moreover, Hölder continuity is analyzed for the integrated causal Lagrangian.

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8

Mishra, Pradip. "Darboux chart on projective limit of weak symplectic Banach manifold." International Journal of Geometric Methods in Modern Physics 12, no. 07 (July 10, 2015): 1550072. http://dx.doi.org/10.1142/s0219887815500723.

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Suppose M be the projective limit of weak symplectic Banach manifolds {(Mi, ϕij)}i, j∈ℕ, where Mi are modeled over reflexive Banach space and σ is compatible with the projective system (defined in the article). We associate to each point x ∈ M, a Fréchet space Hx. We prove that if Hx are locally identical, then with certain smoothness and boundedness condition, there exists a Darboux chart for the weak symplectic structure.

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9

Smith, Mark A., and Barry Turett. "Some examples concerning normal and uniform normal structure in Banach spaces." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 48, no. 2 (April 1990): 223–34. http://dx.doi.org/10.1017/s1446788700035655.

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AbstractExamples are given that show the following: (1) normal structure need not be inherited by quotient spaces; (2) uniform normal structure is not a self-dual property; and (3) no degree of k–uniform rotundity need be present in a space with uniform normal structure.

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10

Eichhorn, Jürgen. "The Banach manifold structure of the space of metrics on noncompact manifolds." Differential Geometry and its Applications 1, no. 2 (September 1991): 89–108. http://dx.doi.org/10.1016/0926-2245(91)90024-4.

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11

Dhompongsa, S., P. Piraisangjun, and S. Saejung. "Generalised Jordan-von Neumann constants and uniform normal structure." Bulletin of the Australian Mathematical Society 67, no. 2 (April 2003): 225–40. http://dx.doi.org/10.1017/s0004972700033694.

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12

Saejung, Satit, and Ji Gao. "On Semi-Uniform Kadec-Klee Banach Spaces." Abstract and Applied Analysis 2010 (2010): 1–12. http://dx.doi.org/10.1155/2010/652521.

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Inspired by the concept ofU-spaces introduced by Lau, (1978), we introduced the class of semi-uniform Kadec-Klee spaces, which is a uniform version of semi-Kadec-Klee spaces studied by Vlasov, (1972). This class of spaces is a wider subclass of spaces with weak normal structure and hence generalizes many known results in the literature. We give a characterization for a certain direct sum of Banach spaces to be semi-uniform Kadec-Klee and use this result to construct a semi-uniform Kadec-Klee space which is not uniform Kadec-Klee. At the end of the paper, we give a remark concerning the uniformly alternative convexity or smoothness introduced by Kadets et al., (1997).

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13

Gao, Ji, and Ka-Sing Lau. "On two classes of Banach spaces with uniform normal structure." Studia Mathematica 99, no. 1 (1991): 41–56. http://dx.doi.org/10.4064/sm-99-1-41-56.

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14

Zuo, Zhan-fei, and Chun-lei Tang. "Schäffer-type constant and uniform normal structure in Banach spaces." Annals of Functional Analysis 7, no. 3 (August 2016): 452–61. http://dx.doi.org/10.1215/20088752-3605636.

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15

Yagisita, Hiroki. "Finite-dimensional complex manifolds on commutative Banach algebras and continuous families of compact complex manifolds." Complex Manifolds 6, no. 1 (January 1, 2019): 228–64. http://dx.doi.org/10.1515/coma-2019-0012.

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AbstractLet Γ(M) be the set of all global continuous cross sections of a continuous family M of compact complex manifolds on a compact Hausdorff space X. In this paper, we introduce a C(X)-manifold structure on Γ(M). Especially, if X is contractible, then Γ(M) is a finite-dimensional C(X)-manifold. Here, C(X) denotes the Banach algebra of all complex-valued continuous functions on X.

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16

Suri, Ali, and Somaye Rastegarzadeh. "Complete lift of vector fields and sprays to T∞M." International Journal of Geometric Methods in Modern Physics 12, no. 10 (October 25, 2015): 1550113. http://dx.doi.org/10.1142/s0219887815501133.

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In this paper for a given Banach, possibly infinite dimensional, manifold M we focus on the geometry of its iterated tangent bundle TrM, r ∈ ℕ ∪ {∞}. First we endow TrM with a canonical atlas using that of M. Then the concepts of vertical and complete lifts for functions and vector fields on TrM are defined which they will play a pivotal role in our next studies i.e. complete lift of (semi)sprays. Afterward we supply T∞M with a generalized Fréchet manifold structure and we will show that any vector field or (semi)spray on M, can be lifted to a vector field or (semi)spray on T∞M. Then, despite of the natural difficulties with non-Banach modeled manifolds, we will discuss about the ordinary differential equations on T∞M including integral curves, flows and geodesics. Finally, as an example, we apply our results to the infinite-dimensional case of manifold of closed curves.

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17

Kutzarova, D., E. Maluta, and S. Prus. "On some p−estimates for Banach spaces." Bulletin of the Australian Mathematical Society 48, no. 2 (October 1993): 187–94. http://dx.doi.org/10.1017/s000497270001563x.

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18

Josué Vieira, Francisca Leidmar, Luiza Helena Félix de Andrade, Rui Facundo Vigelis, and Charles Casimiro Cavalcante. "A Deformed Exponential Statistical Manifold." Entropy 21, no. 5 (May 15, 2019): 496. http://dx.doi.org/10.3390/e21050496.

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Consider μ a probability measure and P μ the set of μ -equivalent strictly positive probability densities. To endow P μ with a structure of a C ∞ -Banach manifold we use the φ -connection by an open arc, where φ is a deformed exponential function which assumes zero until a certain point and from then on is strictly increasing. This deformed exponential function has as particular cases the q-deformed exponential and κ -exponential functions. Moreover, we find the tangent space of P μ at a point p, and as a consequence the tangent bundle of P μ . We define a divergence using the q-exponential function and we prove that this divergence is related to the q-divergence already known from the literature. We also show that q-exponential and κ -exponential functions can be used to generalize of Rényi divergence.

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19

Budzyńska, Monika, Tadeusz Kuczumow, and Simeon Reich. "Uniform asymptotic normal structure, the uniform semi-Opial property and fixed points of asymptotically regular uniformly lipschitzian semigroups. Part I." Abstract and Applied Analysis 3, no. 1-2 (1998): 133–51. http://dx.doi.org/10.1155/s1085337598000475.

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In this paper we introduce the uniform asymptotic normal structure and the uniform semi-Opial properties of Banach spaces. This part is devoted to a study of the spaces with these properties. We also compare them with those spaces which have uniform normal structure and with spaces withWCS(X)>1.

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20

Radnell, David, Eric Schippers, and Wolfgang Staubach. "A Hilbert manifold structure on the Weil–Petersson class Teichmüller space of bordered Riemann surfaces." Communications in Contemporary Mathematics 17, no. 04 (June 22, 2015): 1550016. http://dx.doi.org/10.1142/s0219199715500169.

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We consider bordered Riemann surfaces which are biholomorphic to compact Riemann surfaces of genus g with n regions biholomorphic to the disk removed. We define a refined Teichmüller space of such Riemann surfaces (which we refer to as the WP-class Teichmüller space) and demonstrate that in the case that 2g + 2 - n > 0, this refined Teichmüller space is a Hilbert manifold. The inclusion map from the refined Teichmüller space into the usual Teichmüller space (which is a Banach manifold) is holomorphic. We also show that the rigged moduli space of Riemann surfaces with non-overlapping holomorphic maps, appearing in conformal field theory, is a complex Hilbert manifold. This result requires an analytic reformulation of the moduli space, by enlarging the set of non-overlapping mappings to a class of maps intermediate between analytically extendible maps and quasiconformally extendible maps. Finally, we show that the rigged moduli space is the quotient of the refined Teichmüller space by a properly discontinuous group of biholomorphisms.

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21

BORZELLINO, JOSEPH E., and VICTOR BRUNSDEN. "THE STRATIFIED STRUCTURE OF SPACES OF SMOOTH ORBIFOLD MAPPINGS." Communications in Contemporary Mathematics 15, no. 05 (September 30, 2013): 1350018. http://dx.doi.org/10.1142/s0219199713500181.

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We consider four notions of maps between smooth C∞ orbifolds [Formula: see text], [Formula: see text] with [Formula: see text] compact (without boundary). We show that one of these notions is natural and necessary in order to uniquely define the notion of orbibundle pullback. For the notion of complete orbifold map, we show that the corresponding set of Cr maps between [Formula: see text] and [Formula: see text] with the Cr topology carries the structure of a smooth C∞ Banach (r finite)/Fréchet (r = ∞) manifold. For the notion of complete reduced orbifold map, the corresponding set of Cr maps between [Formula: see text] and [Formula: see text] with the Cr topology carries the structure of a smooth C∞ Banach (r finite)/Fréchet (r = ∞) orbifold. The remaining two notions carry a stratified structure: The Cr orbifold maps between [Formula: see text] and [Formula: see text] is locally a stratified space with strata modeled on smooth C∞ Banach (r finite)/Fréchet (r = ∞) manifolds while the set of Cr reduced orbifold maps between [Formula: see text] and [Formula: see text] locally has the structure of a stratified space with strata modeled on smooth C∞ Banach (r finite)/Fréchet (r = ∞) orbifolds. Furthermore, we give the explicit relationship between these notions of orbifold map. Applying our results to the special case of orbifold diffeomorphism groups, we show that they inherit the structure of C∞ Banach (r finite)/Fréchet (r = ∞) manifolds. In fact, for r finite they are topological groups, and for r = ∞ they are convenient Fréchet Lie groups.

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22

Ciaglia, F. M., A. Ibort, J. Jost, and G. Marmo. "Manifolds of classical probability distributions and quantum density operators in infinite dimensions." Information Geometry 2, no. 2 (October 24, 2019): 231–71. http://dx.doi.org/10.1007/s41884-019-00022-1.

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Abstract The manifold structure of subsets of classical probability distributions and quantum density operators in infinite dimensions is investigated in the context of $$C^{*}$$C∗-algebras and actions of Banach-Lie groups. Specificaly, classical probability distributions and quantum density operators may be both described as states (in the functional analytic sense) on a given $$C^{*}$$C∗-algebra $$\mathscr {A}$$A which is Abelian for Classical states, and non-Abelian for Quantum states. In this contribution, the space of states $$\mathscr {S}$$S of a possibly infinite-dimensional, unital $$C^{*}$$C∗-algebra $$\mathscr {A}$$A is partitioned into the disjoint union of the orbits of an action of the group $$\mathscr {G}$$G of invertible elements of $$\mathscr {A}$$A. Then, we prove that the orbits through density operators on an infinite-dimensional, separable Hilbert space $$\mathcal {H}$$H are smooth, homogeneous Banach manifolds of $$\mathscr {G}=\mathcal {GL}(\mathcal {H})$$G=GL(H), and, when $$\mathscr {A}$$A admits a faithful tracial state $$\tau $$τ like it happens in the Classical case when we consider probability distributions with full support, we prove that the orbit through $$\tau $$τ is a smooth, homogeneous Banach manifold for $$\mathscr {G}$$G.

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23

Saejung, Satit. "Sufficient conditions for uniform normal structure of Banach spaces and their duals." Journal of Mathematical Analysis and Applications 330, no. 1 (June 2007): 597–604. http://dx.doi.org/10.1016/j.jmaa.2006.07.087.

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24

Chen, Bingren, Zhijian Yang, Qi Liu, and Yongjin Li. "Some New James Type Geometric Constants in Banach Spaces." Symmetry 14, no. 2 (February 18, 2022): 405. http://dx.doi.org/10.3390/sym14020405.

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We will introduce four new geometric constants closely related to the James constant J(X), which have symmetric structure, along with a discussion on the relationships among them and some other well-known geometric constants via several inequalities, together with the calculation of several values on some specific spaces. In addition, we will characterize geometric properties of J1(X), such as uniform non-squareness and uniformly normal structure.

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25

Radhakrishnan, M., and S. Rajesh. "Existence ofixed points for pointwise eventually asymptotically nonexpansive mappings." Applied General Topology 20, no. 1 (April 1, 2019): 119. http://dx.doi.org/10.4995/agt.2019.10360.

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<p>Kirk introduced the notion of pointwise eventually asymptotically non-expansive mappings and proved that uniformly convex Banach spaces have the fixed point property for pointwise eventually asymptotically non expansive maps. Further, Kirk raised the following question: “Does a Banach space X have the fixed point property for pointwise eventually asymptotically nonexpansive mappings when ever X has the fixed point property for nonexpansive mappings?”. In this paper, we prove that a Banach space X has the fixed point property for pointwise eventually asymptotically nonexpansive maps if X has uniform normal structure or X is uniformly convex in every direction with the Maluta constant D(X) < 1. Also, we study the asymptotic behavior of the sequence {T<sup>n</sup>x} for a pointwise eventually asymptotically nonexpansive map T defined on a nonempty weakly compact convex subset K of a Banach space X whenever X satisfies the uniform Opial condition or X has a weakly continuous duality map.</p>

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26

Feichtinger, Hans Georg. "Homogeneous Banach Spaces as Banach Convolution Modules over M(G)." Mathematics 10, no. 3 (January 25, 2022): 364. http://dx.doi.org/10.3390/math10030364.

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This paper is supposed to form a keystone towards a new and alternative approach to Fourier analysis over LCA (locally compact Abelian) groups G. In an earlier paper the author has already shown that one can introduce convolution and the Fourier–Stieltjes transform on (M(G),∥·∥M), the space of bounded measures (viewed as a space of linear functionals) in an elementary fashion over Rd. Bounded uniform partitions of unity (BUPUs) are easily constructed in the Euclidean setting (by dilation). Moving on to general LCA groups, it becomes an interesting challenge to find ways to construct arbitrary fine BUPUs, ideally without the use of structure theory, the existence of a Haar measure and even Lebesgue integration. This article provides such a construction and demonstrates how it can be used in order to show that any so-called homogeneous Banach space(B,∥·∥B) on G, such as (Lp(G),∥·∥p), for 1≤p<∞, or the Fourier–Stieltjes algebra FM(G), and in particular any Segal algebra is a Banach convolution module over (M(G),∥·∥M) in a natural way. Via the Haar measure we can then identify L1(G),∥·∥1 with the closure (of the embedded version) of Cc(G), the space of continuous functions with compact support, in (M(G),∥·∥M), and show that these homogeneous Banach spaces are essentialL1(G)-modules. Thus, in particular, the approximate units act properly as one might expect and converge strongly to the identity operator. The approach is in the spirit of Hans Reiter, avoiding the use of structure theory for LCA groups and the usual techniques of vector-valued integration via duality. The ultimate (still distant) goal of this approach is to provide a new and elementary approach towards the (extended) Fourier transform in the setting of the so-called Banach–Gelfand triple(S0,L2,S0′)(G), based on the Segal algebra S0(G). This direction will be pursued in subsequent papers.

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27

JOST, JÜRGEN, та YI-HU YANG. "KÄHLER MANIFOLDS AND FUNDAMENTAL GROUPS OF NEGATIVELY δ-PINCHED MANIFOLDS". International Journal of Mathematics 15, № 02 (березень 2004): 151–67. http://dx.doi.org/10.1142/s0129167x04002247.

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In this note, we will show that the fundamental group of any negatively δ-pinched [Formula: see text] manifold cannot be the fundamental group of a quasi-compact Kähler manifold. As a consequence of our proof, we also show that any nonuniform lattice in F4(-20) cannot be the fundamental group of a quasi-compact Kähler manifold. The corresponding result for uniform lattices was proved by Carlson and Hernández [3]. Finally, we follow Gromov and Thurston [6] to give some examples of negatively δ-pinched manifolds [Formula: see text] of finite volume which, as topological manifolds, admit no hyperbolic metric with finite volume under any smooth structure. This shows that our result for δ-pinched manifolds is a nontrivial generalization of the fact that no nonuniform lattice in SO(n,1)(n≥3) is the fundamental group of a quasi-compact Kähler manifold [21].

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28

CHEN, WEIMIN. "ON A NOTION OF MAPS BETWEEN ORBIFOLDS I: FUNCTION SPACES." Communications in Contemporary Mathematics 08, no. 05 (October 2006): 569–620. http://dx.doi.org/10.1142/s0219199706002246.

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This is the first of a series of papers which is devoted to a comprehensive theory of maps between orbifolds. In this paper, we define the maps in the more general context of orbispaces, and establish several basic results concerning the topological structure of the space of such maps. In particular, we show that the space of such maps of Cr class between smooth orbifolds has a natural Banach orbifold structure if the domain of the map is compact, generalizing the corresponding result in the manifold case. Motivations and applications of the theory come from string theory and the theory of pseudoholomorphic curves in symplectic orbifolds.

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29

Huang, Shaosai. "ε-Regularity and Structure of Four-dimensional Shrinking Ricci Solitons". International Mathematics Research Notices 2020, № 5 (18 квітня 2018): 1511–74. http://dx.doi.org/10.1093/imrn/rny069.

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Abstract A closed four-dimensional manifold cannot possess a non-flat Ricci soliton metric with arbitrarily small $L^{2}$-norm of the curvature. In this paper, we localize this fact in the case of gradient shrinking Ricci solitons by proving an $\varepsilon $-regularity theorem, thus confirming a conjecture of Cheeger–Tian [20]. As applications, we will also derive structural results concerning the degeneration of the metrics on a family of complete non-compact four-dimensional gradient shrinking Ricci solitons without a uniform entropy lower bound. In the appendix, we provide a detailed account of the equivariant good chopping theorem when collapsing with locally bounded curvature happens.

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30

Benkő, Zsigmond, Marcell Stippinger, Roberta Rehus, Attila Bencze, Dániel Fabó, Boglárka Hajnal, Loránd G. Eröss, András Telcs, and Zoltán Somogyvári. "Manifold-adaptive dimension estimation revisited." PeerJ Computer Science 8 (January 6, 2022): e790. http://dx.doi.org/10.7717/peerj-cs.790.

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Data dimensionality informs us about data complexity and sets limit on the structure of successful signal processing pipelines. In this work we revisit and improve the manifold adaptive Farahmand-Szepesvári-Audibert (FSA) dimension estimator, making it one of the best nearest neighbor-based dimension estimators available. We compute the probability density function of local FSA estimates, if the local manifold density is uniform. Based on the probability density function, we propose to use the median of local estimates as a basic global measure of intrinsic dimensionality, and we demonstrate the advantages of this asymptotically unbiased estimator over the previously proposed statistics: the mode and the mean. Additionally, from the probability density function, we derive the maximum likelihood formula for global intrinsic dimensionality, if i.i.d. holds. We tackle edge and finite-sample effects with an exponential correction formula, calibrated on hypercube datasets. We compare the performance of the corrected median-FSA estimator with kNN estimators: maximum likelihood (Levina-Bickel), the 2NN and two implementations of DANCo (R and MATLAB). We show that corrected median-FSA estimator beats the maximum likelihood estimator and it is on equal footing with DANCo for standard synthetic benchmarks according to mean percentage error and error rate metrics. With the median-FSA algorithm, we reveal diverse changes in the neural dynamics while resting state and during epileptic seizures. We identify brain areas with lower-dimensional dynamics that are possible causal sources and candidates for being seizure onset zones.

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31

SOLIMAN, AHMED H., MOHAMMAD IMDAD, and MD AHMADULLAH. "Fixed point theorems for uniformly generalized Kannan type semigroup of self-mappings." Creative Mathematics and Informatics 26, no. 2 (2017): 231–40. http://dx.doi.org/10.37193/cmi.2017.02.12.

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In this paper, we consider a new uniformly generalized Kannan type semigroup of self-mappings deﬁned on a closed convex subset of a real Banach space equipped with uniform normal structure and employ the same to show that such semigroup of self-mappings admits a common ﬁxed point provided the underlying semigroup of self-mappings has a bounded orbit.

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32

Yang, Hong. "Stability and Hopf bifurcation for a logistic SIR model with a stage — Structure." International Journal of Biomathematics 09, no. 01 (November 2, 2015): 1650013. http://dx.doi.org/10.1142/s1793524516500133.

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A SIR model of epidemiological dynamics with stage-structure and a type of nonlinear incidence rate is considered under the assumption that the susceptible individual satisfy the logistic equation. The global attractivity of the model is studied using Lyapunov functions and LaSalle's invariance principle. By the uniform persistence theories, the permanence of the system and the existence of the positive equilibrium are obtained. Moreover, by the normal form theory and the center manifold presented by Hassard, a stability and Hopf bifurcation analysis of the system around positive equilibrium from a local perspective are performed. Numerical simulation is carried out to illustrate our results.

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33

Jiang, Ruichao, Javad Tavakoli, and Yiqiang Zhao. "Weyl Prior and Bayesian Statistics." Entropy 22, no. 4 (April 20, 2020): 467. http://dx.doi.org/10.3390/e22040467.

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Анотація:

When using Bayesian inference, one needs to choose a prior distribution for parameters. The well-known Jeffreys prior is based on the Riemann metric tensor on a statistical manifold. Takeuchi and Amari defined the α -parallel prior, which generalized the Jeffreys prior by exploiting a higher-order geometric object, known as a Chentsov–Amari tensor. In this paper, we propose a new prior based on the Weyl structure on a statistical manifold. It turns out that our prior is a special case of the α -parallel prior with the parameter α equaling − n , where n is the dimension of the underlying statistical manifold and the minus sign is a result of conventions used in the definition of α -connections. This makes the choice for the parameter α more canonical. We calculated the Weyl prior for univariate Gaussian and multivariate Gaussian distribution. The Weyl prior of the univariate Gaussian turns out to be the uniform prior.

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34

Saejung, Satit. "On the modulus ofU-convexity." Abstract and Applied Analysis 2005, no. 1 (2005): 59–66. http://dx.doi.org/10.1155/aaa.2005.59.

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Анотація:

We prove that the moduli ofU-convexity, introduced by Gao (1995), of the ultrapowerX˜of a Banach spaceXand ofXitself coincide wheneverXis super-reflexive. As a consequence, some known results have been proved and improved. More precisely, we prove thatuX(1)>0implies that bothXand the dual spaceX∗ofXhave uniform normal structure and hence the “worth” property in Corollary 7 of Mazcuñán-Navarro (2003) can be discarded.

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35

Li, Cai Xia, Si Min Wang, Shi Feng Xu, and Jian Wen. "The Numerical Investigation of Configuration Improvement on Gas Distributor in a Hydrogenated Reactor." Advanced Materials Research 781-784 (September 2013): 2847–50. http://dx.doi.org/10.4028/www.scientific.net/amr.781-784.2847.

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Gas distributor is of vital significance to the stability of hydrogenation reaction. The distribution performance of original style was studied with CFD numerical simulation method and an improved distributor was proposed in this paper. Compared with the original style, the new structure overcame the center-manifold phenomenon. The velocity distribution of the new distributor at evaluation plane was more uniform and pressure drop was smaller. The distribution performance of the new distributor was better, and the improved structure of the new distributor was obtained through further investigation, which will provide instructional information for industry application.

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36

Pogorzelski, Felix. "Convergence theorems for graph sequences." International Journal of Algebra and Computation 24, no. 08 (December 2014): 1233–51. http://dx.doi.org/10.1142/s0218196714500556.

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In this paper, we deal with a notion of Banach space-valued mappings defined on a set consisting of finite graphs with uniformly bounded vertex degree. These functions will be endowed with certain boundedness and additivity criteria. We examine their normalized long-term behavior along a particular class of graph sequences. Using techniques developed by Elek, we show convergence in the topology of the Banach space if the corresponding graph sequence possesses a hyperfinite structure. These considerations extend and complement the corresponding results for amenable groups. As an application, we verify the uniform approximation of the integrated density of states for bounded, finite range operators on discrete structures. Further, we extend results concerning an abstract version of Fekete's lemma for cancellative, amenable groups and semigroups to the geometric situation of convergent graph sequences.

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37

Liu, Song, Lisheng Yang, Shizhong Yang, Qingping Jiang, and Haowei Wu. "Blind Direction-of-Arrival Estimation with Uniform Circular Array in Presence of Mutual Coupling." International Journal of Antennas and Propagation 2016 (2016): 1–7. http://dx.doi.org/10.1155/2016/8109013.

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A blind direction-of-arrival (DOA) estimation algorithm based on the estimation of signal parameters via rotational invariance techniques (ESPRIT) is proposed for a uniform circular array (UCA) when strong electromagnetic mutual coupling is present. First, an updated UCA model with mutual coupling in a discrete Fourier transform (DFT) beam space is deduced, and the new manifold matrix is equal to the product of a centrosymmetric diagonal matrix and a Vandermonde-structure matrix. Then we carry out blind DOA estimation through a modified ESPRIT method, thus avoiding the need for spatial angular searching. In addition, two mutual coupling parameter estimation methods are presented after the DOAs have been estimated. Simulation results show that the new algorithm is reliable and effective especially for closely spaced signals.

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38

Wei, Xiao Ling, Qing Hui Wang, and Min Qiang Pan. "Numerical Simulation of Velocity Distribution among Microchannels with Bifurcation Structures as Manifolds." Advanced Materials Research 108-111 (May 2010): 1009–12. http://dx.doi.org/10.4028/www.scientific.net/amr.108-111.1009.

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Manifold structure shows important effects on the velocity distribution among parallel microchannels. The bifurcation structure was adopted as the manifolds of microchannel array in a plane and a specific case was illustrated to estimate the effects of structural parameters on the velocity distribution using numerical simulation. Simulation result indicated that the velocity distribution appeared to be double W-shape, and somewhat symmetrical. In addition, the velocity values in each W-shape distribution also appeared symmetrical. It also indicated that two bifurcation channels from the same low-level bifurcation channel had different velocity values due to different singular losses. Larger microchannel length, smaller microchannel width or depth, larger RW or RL, suitable RZ were favor in obtaining relatively uniform velocity distribution among microchannels.

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39

Freilich, Daniel V., and Stefan G. Llewellyn Smith. "The Sadovskii vortex in strain." Journal of Fluid Mechanics 825 (July 21, 2017): 479–501. http://dx.doi.org/10.1017/jfm.2017.401.

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The point vortex is the simplest model of a two-dimensional vortex with non-zero circulation. The limitations introduced by its lack of core structure have been remedied by using desingularizations such as vortex patches and vortex sheets. We investigate steady states of the Sadovskii vortex in strain, a canonical model for a vortex in a general flow. The Sadovskii vortex is a uniform patch of vorticity surrounded by a vortex sheet. We recover previously known limiting cases of the vortex patch and hollow vortex, and examine the bifurcations away from these families. The result is a solution manifold spanned by two parameters. The addition of the vortex sheet to the vortex patch solutions immediately leads to splits in the solution manifold at certain bifurcation points. The more circular elliptical family remains attached to the family with a single pinch-off, and this family extends all the way to the simpler solution branch for the pure vortex sheet solutions. More elongated families below this one also split at bifurcation points, but these families do not exist in the vortex sheet limit.

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40

Suri, Ali. "Isomorphism classes for higher order tangent bundles." Advances in Geometry 17, no. 2 (March 28, 2017): 175–89. http://dx.doi.org/10.1515/advgeom-2017-0001.

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AbstractThe tangent bundle TkM of order k of a smooth Banach manifold M consists of all equivalence classes of curves that agree up to their accelerations of order k. In previous work the author proved that TkM, 1 ≤ k ≤∞, admits a vector bundle structure on M if and only if M is endowed with a linear connection, or equivalently if a connection map on TkM is defined. This bundle structure depends heavily on the choice of the connection. In this paper we ask about the extent to which this vector bundle structure remains isomorphic. To this end we define the k-th order differential Tkg : TkM ⟶ TkN for a given differentiable map g between manifolds M and N. As we shall see, Tkg becomes a vector bundle morphism if the base manifolds are endowed with g-related connections. In particular, replacing a connection with a g-related one, where g : M ⟶ M is a diffeomorphism, one obtains invariant vector bundle structures. Finally, using immersions on Hilbert manifolds, convex combinations of connection maps and manifolds of Cr maps we offer three examples for our theory, showing its interaction with known problems such as the Sasaki lift of metrics.

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41

Shen, Yuliang. "VMO-Teichmüller space on the real line." Annales Fennici Mathematici 47, no. 1 (November 29, 2021): 57–82. http://dx.doi.org/10.54330/afm.112456.

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An increasing homeomorphism $h$ on the real line $\mathbb{R}$ is said to be strongly symmetric if it can be extended to a quasiconformal homeomorphism of the upper half plane $\mathbb{U}$ onto itself whose Beltrami coefficient $\mu$ induces a vanishing Carleson measure $|\mu(z)|^2/y\,dx\,dy$ on $\mathbb{U}$. We will deal with the class of strongly symmetric homeomorphisms on the real line and its Teichmüller space, which we call the VMO-Teichmüller space. In particular, we will show that if $h$ is strongly symmetric on the real line, then it is strongly quasisymmetric such that $\log h'$ is a VMO function. This improves some classical results of Carleson (1967) and Anderson-Becker-Lesley (1988) on the problem about the local absolute continuity of a quasisymmetric homeomorphism in terms of the Beltrami coefficient of a quasiconformal extension. We will also discuss various models of the VMO-Teichmüller space and endow it with a complex Banach manifold structure via the standard Bers embedding.

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42

Demidova, Liliya A., and Artyom V. Gorchakov. "Fuzzy Information Discrimination Measures and Their Application to Low Dimensional Embedding Construction in the UMAP Algorithm." Journal of Imaging 8, no. 4 (April 15, 2022): 113. http://dx.doi.org/10.3390/jimaging8040113.

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Dimensionality reduction techniques are often used by researchers in order to make high dimensional data easier to interpret visually, as data visualization is only possible in low dimensional spaces. Recent research in nonlinear dimensionality reduction introduced many effective algorithms, including t-distributed stochastic neighbor embedding (t-SNE), uniform manifold approximation and projection (UMAP), dimensionality reduction technique based on triplet constraints (TriMAP), and pairwise controlled manifold approximation (PaCMAP), aimed to preserve both the local and global structure of high dimensional data while reducing the dimensionality. The UMAP algorithm has found its application in bioinformatics, genetics, genomics, and has been widely used to improve the accuracy of other machine learning algorithms. In this research, we compare the performance of different fuzzy information discrimination measures used as loss functions in the UMAP algorithm while constructing low dimensional embeddings. In order to achieve this, we derive the gradients of the considered losses analytically and employ the Adam algorithm during the loss function optimization process. From the conducted experimental studies we conclude that the use of either the logarithmic fuzzy cross entropy loss without reduced repulsion or the symmetric logarithmic fuzzy cross entropy loss with sufficiently large neighbor count leads to better global structure preservation of the original multidimensional data when compared to the loss function used in the original UMAP algorithm implementation.

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43

Riemer, M., and M. T. Montgomery. "Simple kinematic models for the environmental interaction of tropical cyclones in vertical wind shear." Atmospheric Chemistry and Physics 11, no. 17 (September 12, 2011): 9395–414. http://dx.doi.org/10.5194/acp-11-9395-2011.

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Abstract. A major impediment to the intensity forecast of tropical cyclones (TCs) is believed to be associated with the interaction of TCs with dry environmental air. However, the conditions under which pronounced TC-environment interaction takes place are not well understood. As a step towards improving our understanding of this problem, we analyze here the flow topology of a TC immersed in an environment of vertical wind shear in an idealized, three-dimensional, convection-permitting numerical experiment. A set of distinct streamlines, the so-called manifolds, can be identified under the assumptions of steady and layer-wise horizontal flow. The manifolds are shown to divide the flow around the TC into distinct regions. The manifold structure in our numerical experiment is more complex than the well-known manifold structure of a non-divergent point vortex in uniform background flow. In particular, one manifold spirals inwards and ends in a limit cycle, a meso-scale dividing streamline encompassing the eyewall above the layer of strong inflow associated with surface friction and below the outflow layer in the upper troposphere. From the perspective of a steady and layer-wise horizontal flow model, the eyewall is well protected from the intrusion of environmental air. In order for the environmental air to intrude into the inner-core convection, time-dependent and/or vertical motions, which are prevalent in the TC inner-core, are necessary. Air with the highest values of moist-entropy resides within the limit cycle. This "moist envelope" is distorted considerably by the imposed vertical wind shear, and the shape of the moist envelope is closely related to the shape of the limit cycle. In a first approximation, the distribution of high- and low-θe air around the TC at low to mid-levels is governed by the stirring of convectively modified air by the steady, horizontal flow. Motivated by the results from the idealized numerical experiment, an analogue model based on a weakly divergent point vortex in background flow is formulated. The simple kinematic model captures the essence of many salient features of the manifold structure in the numerical experiment. A regime diagram representing realistic values of TC intensity and vertical wind shear can be constructed for the point-vortex model. The results indicate distinct scenarios of environmental interaction depending on the ratio of storm intensity and vertical-shear magnitude. Further implications of the new results derived from the manifold analysis for TCs in the real atmosphere are discussed.

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44

Bollon, Jordy, Michela Assale, Andrea Cina, Stefano Marangoni, Matteo Calabrese, Chiara Beatrice Salvemini, Jean Marc Christille, Stefano Gustincich, and Andrea Cavalli. "Investigating How Reproducibility and Geometrical Representation in UMAP Dimensionality Reduction Impact the Stratification of Breast Cancer Tumors." Applied Sciences 12, no. 9 (April 22, 2022): 4247. http://dx.doi.org/10.3390/app12094247.

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Advances in next-generation sequencing have provided high-dimensional RNA-seq datasets, allowing the stratification of some tumor patients based on their transcriptomic profiles. Machine learning methods have been used to reduce and cluster high-dimensional data. Recently, uniform manifold approximation and projection (UMAP) was applied to project genomic datasets in low-dimensional Euclidean latent space. Here, we evaluated how different representations of the UMAP embedding can impact the analysis of breast cancer (BC) stratification. We projected BC RNA-seq data on Euclidean, spherical, and hyperbolic spaces, and stratified BC patients via clustering algorithms. We also proposed a pipeline to yield more reproducible clustering outputs. The results show how the selection of the latent space can affect downstream stratification results and suggest that the exploration of different geometrical representations is recommended to explore data structure and samples’ relationships.

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45

Piccione, Paolo, and Daniel V. Tausk. "Lagrangian and Hamiltonian formalism for constrained variational problems." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 132, no. 6 (December 2002): 1417–37. http://dx.doi.org/10.1017/s0308210500002183.

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We consider solutions of Lagrangian variational problems with linear constraints on the derivative. More precisely, given a smooth distribution D ⊂ TM on M and a time-dependent Lagrangian L defined on D, we consider an action functional L defined on the set ΩPQ(M, D) of horizontal curves in M connecting two fixed submanifolds P, Q ⊂ M. Under suitable assumptions, the set ΩPQ(M, D) has the structure of a smooth Banach manifold and we can thus study the critical points of L. If the Lagrangian L satisfies an appropriate hyper-regularity condition, we associate to it a degenerate Hamiltonian H on TM* using a general notion of Legendre transform for maps on vector bundles. We prove that the solutions of the Hamilton equations of H are precisely the critical points of L. In the particular case where L is given by the quadratic form corresponding to a positive-definite metric on D, we obtain the well-known characterization of the normal geodesics in sub-Riemannian geometry (see [8]). By adding a potential energy term to L, we obtain again the equations of motion for the Vakonomic mechanics with non-holonomic constraints (see [6]).

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46

Lin, Heyun, Chaowei Yuan, Jianhe Du, and Zhongwei Hu. "Estimation of DOA for Noncircular Signals via Vandermonde Constrained Parallel Factor Analysis." International Journal of Antennas and Propagation 2018 (2018): 1–9. http://dx.doi.org/10.1155/2018/4612583.

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We provide a complete study on the direction-of-arrival (DOA) estimation of noncircular (NC) signals for uniform linear array (ULA) via Vandermonde constrained parallel factor (PARAFAC) analysis. By exploiting the noncircular property of the signals, we first construct an extended matrix which contains two times sampling number of the received signal. Then, taking the Vandermonde structure of the array manifold matrix into account, the extended matrix can be turned into a tensor model which admits the Vandermonde constrained PARAFAC decomposition. Based on this tensor model, an efficient linear algebra algorithm is applied to obtain the DOA estimation via utilizing the rotational invariance between two submatrices. Compared with some existing algorithms, the proposed method has a better DOA estimation performance. Meanwhile, the proposed method consistently has a higher estimation accuracy and a much lower computational complexity than the trilinear alternating least square- (TALS-) based PARAFAC algorithm. Finally, numerical examples are conducted to demonstrate the effectiveness of the proposed approach in terms of estimation accuracy and computational complexity.

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47

Schmitz, S., H. Hammer, and A. Thiele. "MULTI-FREQUENCY POLINSAR DATA ARE ADVANTAGEOUS FOR LAND COVER CLASSIFICATION – A VISUAL AND QUANTITATIVE ANALYSIS." ISPRS Annals of the Photogrammetry, Remote Sensing and Spatial Information Sciences V-1-2022 (May 17, 2022): 49–56. http://dx.doi.org/10.5194/isprs-annals-v-1-2022-49-2022.

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Abstract. This paper investigates the enhanced potential of using multi-frequency PolInSAR data for land cover classification. In order to enable a descriptive analysis that goes beyond the mere comparison of classification accuracies, a two-step classification process is applied. First, polarimetric and interferometric features are extracted and projected into a 3-dimensional feature space by using the supervised dimension reduction algorithm Uniform Manifold Approximation and Projection (UMAP). Subsequently, based on the expressive 3-dimensional representation a simple yet sufficient k-nearest neighbors (KNN) classifier is applied to assign a land cover class to each pixel. In this way, besides the simplified classification, the visualization of the underlying data structure is possible and contributes to a better explanation and analysis of classification results. The data analyzed in this way are airborne L- and S-band PolInSAR data acquired by the F-SAR system. The visual analysis of reduced feature spaces as well as the quantitative analysis of classification results reveal the benefits of combining both frequencies with regard to class separability.

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48

Saul, Lawrence K. "A tractable latent variable model for nonlinear dimensionality reduction." Proceedings of the National Academy of Sciences 117, no. 27 (June 22, 2020): 15403–8. http://dx.doi.org/10.1073/pnas.1916012117.

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We propose a latent variable model to discover faithful low-dimensional representations of high-dimensional data. The model computes a low-dimensional embedding that aims to preserve neighborhood relationships encoded by a sparse graph. The model both leverages and extends current leading approaches to this problem. Like t-distributed Stochastic Neighborhood Embedding, the model can produce two- and three-dimensional embeddings for visualization, but it can also learn higher-dimensional embeddings for other uses. Like LargeVis and Uniform Manifold Approximation and Projection, the model produces embeddings by balancing two goals—pulling nearby examples closer together and pushing distant examples further apart. Unlike these approaches, however, the latent variables in our model provide additional structure that can be exploited for learning. We derive an Expectation–Maximization procedure with closed-form updates that monotonically improve the model’s likelihood: In this procedure, embeddings are iteratively adapted by solving sparse, diagonally dominant systems of linear equations that arise from a discrete graph Laplacian. For large problems, we also develop an approximate coarse-graining procedure that avoids the need for negative sampling of nonadjacent nodes in the graph. We demonstrate the model’s effectiveness on datasets of images and text.

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49

Suzuki, Taiji, and Masashi Sugiyama. "Sufficient Dimension Reduction via Squared-Loss Mutual Information Estimation." Neural Computation 25, no. 3 (March 2013): 725–58. http://dx.doi.org/10.1162/neco_a_00407.

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The goal of sufficient dimension reduction in supervised learning is to find the low-dimensional subspace of input features that contains all of the information about the output values that the input features possess. In this letter, we propose a novel sufficient dimension-reduction method using a squared-loss variant of mutual information as a dependency measure. We apply a density-ratio estimator for approximating squared-loss mutual information that is formulated as a minimum contrast estimator on parametric or nonparametric models. Since cross-validation is available for choosing an appropriate model, our method does not require any prespecified structure on the underlying distributions. We elucidate the asymptotic bias of our estimator on parametric models and the asymptotic convergence rate on nonparametric models. The convergence analysis utilizes the uniform tail-bound of a U-process, and the convergence rate is characterized by the bracketing entropy of the model. We then develop a natural gradient algorithm on the Grassmann manifold for sufficient subspace search. The analytic formula of our estimator allows us to compute the gradient efficiently. Numerical experiments show that the proposed method compares favorably with existing dimension-reduction approaches on artificial and benchmark data sets.

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50

Brattka, Vasco, and Guido Gherardi. "Effective Choice and Boundedness Principles in Computable Analysis." Bulletin of Symbolic Logic 17, no. 1 (March 2011): 73–117. http://dx.doi.org/10.2178/bsl/1294186663.

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AbstractIn this paper we study a new approach to classify mathematical theorems according to their computational content. Basically, we are asking the question which theorems can be continuously or computably transferred into each other? For this purpose theorems are considered via their realizers which are operations with certain input and output data. The technical tool to express continuous or computable relations between such operations is Weihrauch reducibility and the partially ordered degree structure induced by it. We have identified certain choice principles such as co-finite choice, discrete choice, interval choice, compact choice and closed choice, which are cornerstones among Weihrauch degrees and it turns out that certain core theorems in analysis can be classified naturally in this structure. In particular, we study theorems such as the Intermediate Value Theorem, the Baire Category Theorem, the Banach Inverse Mapping Theorem, the Closed Graph Theorem and the Uniform Boundedness Theorem. We also explore how existing classifications of the Hahn–Banach Theorem and Weak Kőnig's Lemma fit into this picture. Well-known omniscience principles from constructive mathematics such as LPO and LLPO can also naturally be considered as Weihrauch degrees and they play an important role in our classification. Based on this we compare the results of our classification with existing classifications in constructive and reverse mathematics and we claim that in a certain sense our classification is finer and sheds some new light on the computational content of the respective theorems. Our classification scheme does not require any particular logical framework or axiomatic setting, but it can be carried out in the framework of classical mathematics using tools of topology, computability theory and computable analysis. We develop a number of separation techniques based on a new parallelization principle, on certain invariance properties of Weihrauch reducibility, on the Low Basis Theorem of Jockusch and Soare and based on the Baire Category Theorem. Finally, we present a number of metatheorems that allow to derive upper bounds for the classification of the Weihrauch degree of many theorems and we discuss the Brouwer Fixed Point Theorem as an example.

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