scholarly journals Assignments to sheaves of pseudometric spaces

2020 ◽  
Vol 2 ◽  
pp. 2
Author(s):  
Michael Robinson
Keyword(s):  
Noisy Data ◽  
Base Space ◽  
Local Section ◽  
Structure Preserving ◽  
Continuous Map ◽  
Open Set

An assignment to a sheaf is the choice of a local section from each open set in the sheaf's base space, without regard to how these local sections are related to one another. This article explains that the consistency radius --- which quantifies the agreement between overlapping local sections in the assignment --- is a continuous map. When thresholded, the consistency radius produces the consistency filtration, which is a filtration of open covers. This article shows that the consistency filtration is a functor that transforms the structure of the sheaf and assignment into a nested set of covers in a structure-preserving way. Furthermore, this article shows that consistency filtration is robust to perturbations, establishing its validity for arbitrarily thresholded, noisy data.

Locally Convex Hypersurfaces

1973 ◽  
Vol 25 (3) ◽  
pp. 531-538 ◽  
Author(s):  
L. B. Jonker ◽  
R. D. Norman
Keyword(s):  
Convex Body ◽  
Open Subset ◽  
Convex Hypersurface ◽  
Topological Manifold ◽  
Continuous Map ◽  
Open Set ◽  
Locally Convex ◽  
Convex Hypersurfaces

Let M be an n-dimensional connected topological manifold. Let ξ : M → Rn+1 be a continuous map with the following property: to each x ∈ M there is an open set x ∈ Ux ⊂ M, and a convex body Kx ⊂ Rn+1 such that ξ(UX) is an open subset of ∂Kx and such that is a homeomorphism onto its image. We shall call such a mapping ξ a locally convex immersion and, along with Van Heijenoort [8] we shall call ξ(M) a locally convex hypersurface of Rn+1.

On α*-I-Open Set and α*-I-Continuous Map

2018 ◽  
Vol 56 (2) ◽  
pp. 101-103
Author(s):  
J. K. Ma itra ◽  
◽  
Rajesh Kumar Tiwari
Keyword(s):  
Continuous Map ◽  
Open Set

scholarly journals Robust Structure Preserving Nonnegative Matrix Factorization for Dimensionality Reduction

2016 ◽  
Vol 2016 ◽  
pp. 1-14
Author(s):  
Bingfeng Li ◽  
Yandong Tang ◽  
Zhi Han
Keyword(s):  
Dimensionality Reduction ◽  
Matrix Factorization ◽  
Nonnegative Matrix Factorization ◽  
Noisy Data ◽  
Nonnegative Matrix ◽  
Structure Preserving ◽  
Structure Preservation ◽  
Significant Performance ◽  
Linear Dimensionality Reduction ◽  
Low Dimensional

As a linear dimensionality reduction method, nonnegative matrix factorization (NMF) has been widely used in many fields, such as machine learning and data mining. However, there are still two major drawbacks for NMF: (a) NMF can only perform semantic factorization in Euclidean space, and it fails to discover the intrinsic geometrical structure of high-dimensional data distribution. (b) NMF suffers from noisy data, which are commonly encountered in real-world applications. To address these issues, in this paper, we present a new robust structure preserving nonnegative matrix factorization (RSPNMF) framework. In RSPNMF, a local affinity graph and a distant repulsion graph are constructed to encode the geometrical information, and noisy data influence is alleviated by characterizing the data reconstruction term of NMF withl2,1-norm instead ofl2-norm. With incorporation of the local and distant structure preservation regularization term into the robust NMF framework, our algorithm can discover a low-dimensional embedding subspace with the nature of structure preservation. RSPNMF is formulated as an optimization problem and solved by an effective iterative multiplicative update algorithm. Experimental results on some facial image datasets clustering show significant performance improvement of RSPNMF in comparison with the state-of-the-art algorithms.

scholarly journals Le lemme fondamental pondéré. I. Constructions géométriques

2010 ◽  
Vol 146 (6) ◽  
pp. 1416-1506 ◽  
Author(s):  
Pierre-Henri Chaudouard ◽  
Gérard Laumon
Keyword(s):  
Finite Field ◽  
Affine Space ◽  
Base Space ◽  
Characteristic Polynomials ◽  
Selberg Trace Formula ◽  
Orbital Integrals ◽  
Fundamental Lemma ◽  
Open Set ◽  
Simple Set ◽  
Hitchin Fibration

AbstractThis work is the geometric part of our proof of the weighted fundamental lemma, which is an extension of Ngô Bao Châu’s proof of the Langlands–Shelstad fundamental lemma. Ngô’s approach is based on a study of the elliptic part of the Hichin fibration. The total space of this fibration is the algebraic stack of Hitchin bundles and its base space is the affine space of ‘characteristic polynomials’. Over the elliptic set, the Hitchin fibration is proper and the number of points of its fibers over a finite field can be expressed in terms of orbital integrals. In this paper, we study the Hitchin fibration over an open set larger than the elliptic set, namely the ‘generically regular semi-simple set’. The fibers are in general neither of finite type nor separated. By analogy with Arthur’s truncation, we introduce the substack of ξ-stable Hitchin bundles. We show that it is a Deligne–Mumford stack, smooth over the base field and proper over the base space of ‘characteristic polynomials’. Moreover, the number of points of the ξ-stable fibers over a finite field can be expressed as a sum of weighted orbital integrals, which appear in the Arthur–Selberg trace formula.

ON THE SPECTRALIFICATION OF A HEMISPECTRAL SPACE

2011 ◽  
Vol 10 (04) ◽  
pp. 687-699
Author(s):  
OTHMAN ECHI ◽  
MOHAMED OUELD ABDALLAHI
Keyword(s):  
Topological Space ◽  
Open Subset ◽  
Full Subcategory ◽  
Topological Spaces ◽  
Open Problems ◽  
Complete Answer ◽  
Reflective Subcategory ◽  
Continuous Map ◽  
Open Set ◽  
Spectral Spaces

An open subset U of a topological space X is called intersection compact open, or ICO, if for every compact open set Q of X, U ∩ Q is compact. A continuous map f of topological spaces will be called spectral if f-1 carries ICO sets to ICO sets. Call a topological space Xhemispectral, if the intersection of two ICO sets of X is an ICO. Let HSPEC be the category whose objects are hemispectral spaces and arrows spectral maps. Let SPEC be the full subcategory of HSPEC whose objects are spectral spaces. The main result of this paper proves that SPEC is a reflective subcategory of HSPEC. This gives a complete answer to Problem BST1 of "O. Echi, H. Marzougui and E. Salhi, Problems from the Bizerte–Sfax–Tunis seminar, in Open Problems in Topology II, ed. E. Pearl (Elsevier, 2007), pp. 669–674."

scholarly journals DISCUSSION ON QUOTIENT BI-SPACE AND ON PAIRWISE REGULAR AND NORMAL SPACES IN BITOPOLOGICAL SPACES

2019 ◽  
pp. 13-15
Author(s):  
M. Arunmaran ◽  
K. Kannan
Keyword(s):  
Compact Subset ◽  
Hausdorff Space ◽  
Bitopological Space ◽  
Closed Set ◽  
Continuous Map ◽  
Open Set ◽  
Bitopological Spaces

In this paper, we introduce the concept “Quotient bi-space” in bitopological spaces. In addition, we investigate the results related with quotient bi-space. Moreover, we have discussed the results related with pairwise regular and normal spaces in bitopological space. For a non-empty set X, we can define two topologies (these may be same or distinct topologies) τ1 and τ2 on X. Then, the triple (X, τ1 , τ2 ) is known as bitopological space. Let (X, τ1 , τ2 ) be bitopological space, (Y, σ1 , σ2 ) be trivial bitopological space and f : (X, τ1 , τ2 ) → (Y, σ1 , σ2 ) be onto map. Then f is τ1 τ2 −continuous map. If η = {G (σ − open set in Y ) : f ^{−1} (G) is τ1 τ2 − open in X} then η is a topology on Y . Moreover, if (Y, σ, σ) be a quotient bi-space of (X, τ1 , τ2) under f : (X, τ1 , τ2 ) → (Y, σ, σ) and g : (Y, σ, σ) → (Z, η1 , η2 ) be a map, then, gis σ − continuous if and only if g ◦ f : (X, τ1 , τ2 ) → (Z, η1 , η2 ) is τ1 τ2 −continuous. Let (X, τ1 , τ2) be bitopological space and A be τ1 τ2 − compact subset of pairwise Hausdorff space X. Then, A is τ1 τ2 − closed set. Finally, we have discussed the following : Let (X, τ1 , τ2 ) be bitopological space and τ1 τ2 −compact pairwise Hausdorff space. Then, the space (X, τ1 , τ2 ) is pairwise normal.

Noisy Data: Incorporating Variability into Your Analysis

2014 ◽  
Vol 2 (1) ◽  
pp. 1
Author(s):  
Richard Schwartz
Keyword(s):  
Noisy Data

The smooth case

2017 ◽  
Author(s):  
Ehud Hrushovski ◽  
François Loeser
Keyword(s):  
Homotopy Type ◽  
Unit Vector ◽  
Smooth Case ◽  
Open Set ◽  
Birational Invariant

This chapter examines the simplifications occurring in the proof of the main theorem in the smooth case. It begins by stating the theorem about the existence of an F-definable homotopy h : I × unit vector X → unit vector X and the properties for h. It then presents the proof, which depends on two lemmas. The first recaps the proof of Theorem 11.1.1, but on a Zariski dense open set V₀ only. The second uses smoothness to enable a stronger form of inflation, serving to move into V₀. The chapter also considers the birational character of the definable homotopy type in Remark 12.2.4 concerning a birational invariant.

ON A NEW CLASS OF SEMI GENERALIZED CLOSED SETS IN STRONG GENERALIZED TOPOLOGICAL SPACES

2020 ◽  
Vol 9 (11) ◽  
pp. 9353-9360
Author(s):  
G. Selvi ◽  
I. Rajasekaran
Keyword(s):  
Topological Spaces ◽  
Closed Set ◽  
Basic Properties ◽  
Closed Sets ◽  
New Class ◽  
Open Set ◽  
Continuous Maps

This paper deals with the concepts of semi generalized closed sets in strong generalized topological spaces such as $sg^{\star \star}_\mu$-closed set, $sg^{\star \star}_\mu$-open set, $g^{\star \star}_\mu$-closed set, $g^{\star \star}_\mu$-open set and studied some of its basic properties included with $sg^{\star \star}_\mu$-continuous maps, $sg^{\star \star}_\mu$-irresolute maps and $T_\frac{1}{2}$-space in strong generalized topological spaces.

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