scholarly journals Polynomially convex embeddings of odd-dimensional closed manifolds

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Purvi Gupta ◽  
Rasul Shafikov
Keyword(s):  
Lower Bound ◽  
Continuous Functions ◽  
Topological Approach ◽  
Analytic Methods ◽  
Complex Dimension ◽  
Polynomially Convex ◽  
Orientable Manifold

Abstract It is shown that any smooth closed orientable manifold of dimension 2 ⁢ k + 1 {2k+1} , k ≥ 2 {k\geq 2} , admits a smooth polynomially convex embedding into ℂ 3 ⁢ k {\mathbb{C}^{3k}} . This improves by 1 the previously known lower bound of 3 ⁢ k + 1 {3k+1} on the possible ambient complex dimension for such embeddings (which is sharp when k = 1 {k=1} ). It is further shown that the embeddings produced have the property that all continuous functions on the image can be uniformly approximated by holomorphic polynomials. Lastly, the same technique is modified to construct embeddings whose images have nontrivial hulls containing no nontrivial analytic disks. The distinguishing feature of this dimensional setting is the appearance of nonisolated CR-singularities, which cannot be tackled using only local analytic methods (as done in earlier results of this kind), and a topological approach is required.

BEST BOUNDS ON DENSITY AND DEPTH FROM GRAVITY DATA

Geophysics ◽  
1974 ◽  
Vol 39 (5) ◽  
pp. 644-649 ◽  
Author(s):  
Robert L. Parker
Keyword(s):  
Lower Bound ◽  
General Theory ◽  
Upper Bound ◽  
Incomplete Data ◽  
Extreme Values ◽  
Gravity Data ◽  
Maximum Depth ◽  
Analytic Methods ◽  
Depth Of Burial

Gravity data cannot usually be inverted to yield unique structures from incomplete data; however, there is a smallest density compatible with the data or, if the density is known, a deepest depth of burial. A general theory is derived which gives the greatest lower bound on density or the least upper bound on depth. These bounds are discovered by consideration of a class of “ideal” bodies which achieve the extreme values of depth or density. The theory is illustrated with several examples which are solved by analytic methods. New maximum depth rules derived by this theory are, unlike some earlier rules of this type, optimal for the data they treat.

Carleman Approximation by Entire Functions on the Union of Two Totally Real Subspaces of Cn

1994 ◽  
Vol 37 (4) ◽  
pp. 522-526
Author(s):  
Per E. Manne
Keyword(s):  
Entire Functions ◽  
Continuous Functions ◽  
Real Dimension ◽  
Carleman Approximation ◽  
Polynomially Convex ◽  
Cauchy Riemann ◽  
Totally Real

AbstractLet L1, L2 ⊂ Cn be two totally real subspaces of real dimension n, and such that L1 ∩ L2 = {0}. We show that continuous functions on L1 ∪L2 allow Carleman approximation by entire functions if and only if L1 ∪L2 is polynomially convex. If the latter condition is satisfied, then a function f:L1 ∪L2 —> C such that f|LiCk(Li), i = 1,2, allows Carleman approximation of order k by entire functions if and only if f satisfies the Cauchy-Riemann equations up to order k at the origin.

scholarly journals ADVANCES IN MULTIDIMENSIONAL SIZE THEORY

2011 ◽  
Vol 29 (1) ◽  
pp. 19 ◽  
Author(s):  
Andrea Cerri ◽  
Patrizio Frosini
Keyword(s):  
Computer Vision ◽  
Pattern Recognition ◽  
Continuous Functions ◽  
Research Topic ◽  
Topological Spaces ◽  
Topological Approach ◽  
Shape Comparison ◽  
Size Function ◽  
Multidimensional Information ◽  
Vector Valued

Size Theory was proposed in the early 90's as a geometrical/topological approach to the problem of Shape Comparison, a very lively research topic in the fields of Computer Vision and Pattern Recognition. The basic idea is to discriminate shapes by comparing shape properties that are provided by continuous functions valued in R, called measuring functions and defined on topological spaces associated to the objects to be studied. In this way, shapes can be compared by using a descriptor named size function, whose role is to capture the features described by measuring functions and represent them in a quantitative way. However, a common scenario in applications is to deal with multidimensional information. This observation has led to considering vector-valued measuring functions, and consequently the multidimensional extension of size functions, namely the k-dimensional size functions. In this work we survey some recent results about size functions in this multidimensional setting, with particular reference to the localization of their discontinuities.

A Compactness Theorem For Affine Equivalence-Classes of Convex Regions

1951 ◽  
Vol 3 ◽  
pp. 54-61 ◽  
Author(s):  
A. M. Macbeath
Keyword(s):  
Lower Bound ◽  
Continuous Functions ◽  
Equivalence Classes ◽  
Compactness Theorem ◽  
Upper And Lower Bounds ◽  
Geometry Of Numbers ◽  
Selection Theorem ◽  
Critical Determinant ◽  
Affine Equivalence ◽  
Affine Invariants

In some parts of the Geometry of Numbers it is convenient to know that certain affine invariants associated with convex regions attain their upper and lower bounds. A classical example is the quotient of the critical determinant by the content (if the region is symmetrical) for which Minkowski determined the exact lower bound 2–n. The object of this paper is to prove that for continuous functions of bounded regions the bounds are attained. The result is, of course, deduced from the selection theorem of Blaschke, and itself is a compactness theorem about the space of affine equivalence-classes.

A Topological Approach in the Extended Fraenkel-Mostowski Model of Set Theory

2014 ◽  
Vol 60 (2) ◽  
pp. 261-277
Author(s):  
Andrei Alexandru ◽  
Gabriel Ciobanu
Keyword(s):  
Set Theory ◽  
Continuous Functions ◽  
New Order ◽  
Scott Topology ◽  
Topological Properties ◽  
Arbitrary Group ◽  
Topological Approach ◽  
Lattices Of Subgroups ◽  
Group A ◽  
Algebraic Domains

Abstract Lattices of subgroups are presented as algebraic domains. Given an arbitrary group, we define the Scott topology over the subgroups lattice of that group. A basis for this topology is expressed in terms of finitely generated subgroups. Several properties of the continuous functions with respect the Scott topology are obtained; they provide new order properties of groups. Finally there are expressed several properties of the group of permutations of atoms in a permutative model of set theory. We provide new properties of the extended interchange function by presenting some topological properties of its domain. Several order and topological properties of the sets in the Fraenkel-Mostowski model remains also valid in the Extended Fraenkel-Mostowski model, even one axiom in the axiomatic description of the Extended Fraenkel-Mostowski model is weaker than its homologue in the axiomatic description of the Fraenkel-Mostowski model.

Polynomial Hulls of Sets Invariant Under an Action of the Special Unitary Group

1988 ◽  
Vol 40 (5) ◽  
pp. 1256-1271
Author(s):  
John T. Anderson
Keyword(s):  
Continuous Functions ◽  
Difficult Problem ◽  
Ideal Space ◽  
Analytic Structure ◽  
Compact Sets ◽  
Polynomially Convex ◽  
Polynomial Hulls ◽  
Polynomial Hull ◽  
Image Position

If K is a compact subset of Cn, will denote the polynomial hull of K: arises in the study of uniform algebras as the maximal ideal space of the algebra P(K) of uniform limits on K of polynomials (see [3]). The condition (K is polynomially convex) is a necessary one for uniform approximation on K of continuous functions by polynomials (P(K) = C(K)). If K is not polynomially convex, the question of existence of analytic structure in is of particular interest. For n = 1, is the union of K and the bounded components of C\K. The determination of in dimensions greater than one is a more difficult problem. Among the special classes of compact sets K whose polynomial hulls have been determined are those invariant under certain group actions on Cn.

Approximation in statistical sense to B -continuous functions by positive linear operators

2010 ◽  
Vol 47 (3) ◽  
pp. 289-298 ◽  
Author(s):  
Fadime Dirik ◽  
Oktay Duman ◽  
Kamil Demirci
Keyword(s):  
Compact Subset ◽  
Real Line ◽  
Statistical Convergence ◽  
Approximation Theorem ◽  
Continuous Functions ◽  
Linear Operators ◽  
Positive Linear Operators ◽  
Statistical Approximation ◽  
Classical Version ◽  
Statistical Sense

In the present work, using the concept of A -statistical convergence for double real sequences, we obtain a statistical approximation theorem for sequences of positive linear operators defined on the space of all real valued B -continuous functions on a compact subset of the real line. Furthermore, we display an application which shows that our new result is stronger than its classical version.

Analytical Study on Big Data

2018 ◽  
Vol 8 (5) ◽  
pp. 75
Author(s):  
Vivek Raich ◽  
Pankaj Maurya
Keyword(s):  
Information Technology ◽  
Big Data ◽  
Decision Maker ◽  
Analytical Study ◽  
Large Data ◽  
Decision Makers ◽  
Continuous Increase ◽  
Analytic Methods ◽  
Data Store ◽  
Business Engineering

in the time of the Information Technology, the big data store is going on. Due to which, Huge amounts of data are available for decision makers, and this has resulted in the progress of information technology and its wide growth in many areas of business, engineering, medical, and scientific studies. Big data means that the size which is bigger in size, but there are several types, which are not easy to handle, technology is required to handle it. Due to continuous increase in the data in this way, it is important to study and manage these datasets by adjusting the requirements so that the necessary information can be obtained.The aim of this paper is to analyze some of the analytic methods and tools. Which can be applied to large data. In addition, the application of Big Data has been analyzed, using the Decision Maker working on big data and using enlightened information for different applications.

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