scholarly journals Okounkov bodies of filtered linear series

2011 ◽  
Vol 147 (4) ◽  
pp. 1205-1229 ◽  
Author(s):  
Sébastien Boucksom ◽  
Huayi Chen
Keyword(s):  
Line Bundle ◽  
Lebesgue Measure ◽  
Asymptotic Distribution ◽  
Approximation Theorem ◽  
Concave Function ◽  
Linear Series ◽  
Type Approximation ◽  
Graded Linear Series ◽  
Arakelov Geometry ◽  
Big Line Bundle

AbstractWe associate to certain filtrations of a graded linear series of a big line bundle a concave function on its Okounkov body, whose law with respect to the Lebesgue measure describes the asymptotic distribution of the jumps of the filtration. As a consequence, we obtain a Fujita-type approximation theorem in this general filtered setting. We then specialize these results to the filtrations by minima in the usual context of Arakelov geometry (and for more general adelically normed graded linear series), thereby obtaining in a simple way a natural construction of an arithmetic Okounkov body, the existence of the arithmetic volume as a limit and an arithmetic Fujita approximation theorem for adelically normed graded linear series. We also obtain an easy proof of the existence of the sectional capacity previously obtained by Lau, Rumely and Varley.

scholarly journals Test configurations and Okounkov bodies

2012 ◽  
Vol 148 (6) ◽  
pp. 1736-1756 ◽  
Author(s):  
David Witt Nyström
Keyword(s):  
Line Bundle ◽  
Recent Work ◽  
Lebesgue Measure ◽  
Asymptotic Distribution ◽  
Concave Function ◽  
Toric Geometry ◽  
Piecewise Polynomial ◽  
Test Configuration ◽  
Special Case

AbstractWe associate to a test configuration for a polarized variety a filtration of the section ring of the line bundle. Using the recent work of Boucksom and Chen we get a concave function on the Okounkov body whose law with respect to Lebesgue measure determines the asymptotic distribution of the weights of the test configuration. We show that this is a generalization of a well-known result in toric geometry. As an application, we prove that the pushforward of the Lebesgue measure on the Okounkov body is equal to a Duistermaat–Heckman measure of a certain deformation of the manifold. Via the Duisteraat–Heckman formula, we get as a corollary that in the special case of an effective ℂ×-action on the manifold lifting to the line bundle, the pushforward of the Lebesgue measure on the Okounkov body is piecewise polynomial.

On αβ-statistical convergence for sequences of fuzzy mappings and Korovkin type approximation theorem

Filomat ◽  
2017 ◽  
Vol 31 (12) ◽  
pp. 3749-3760 ◽  
Author(s):  
Ali Karaisa ◽  
Uğur Kadak
Keyword(s):  
Statistical Convergence ◽  
Approximation Theorem ◽  
Linear Operators ◽  
Positive Linear Operators ◽  
Bernstein Operator ◽  
Korovkin Type Approximation Theorem ◽  
Type Approximation ◽  
Inclusion Relations ◽  
Prior Investigation

Upon prior investigation on statistical convergence of fuzzy sequences, we study the notion of pointwise ??-statistical convergence of fuzzy mappings of order ?. Also, we establish the concept of strongly ??-summable sequences of fuzzy mappings and investigate some inclusion relations. Further, we get an analogue of Korovkin-type approximation theorem for fuzzy positive linear operators with respect to ??-statistical convergence. Lastly, we apply fuzzy Bernstein operator to construct an example in support of our result.

scholarly journals A Spline Group – Korovkin Approximation Theorem

2015 ◽  
Vol 7 (2) ◽  
pp. 110
Author(s):  
Malik Saad Al-Muhja
Keyword(s):  
Linear Operator ◽  
Approximation Theorem ◽  
Positive Linear Operator ◽  
Homogeneous Groups ◽  
Korovkin Type Approximation Theorem ◽  
Type Approximation ◽  
Korovkin Approximation ◽  
Korovkin Approximation Theorem

In this paper, using homogeneous groups, we prove a Korovkin type approximation theorem for a spline groupby using the notion of a generalization of positive linear operator.

scholarly journals Statistical Deferred Nörlund Summability and Korovkin-Type Approximation Theorem

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 636 ◽  
Author(s):  
Hari Mohan Srivastava ◽  
Bidu Bhusan Jena ◽  
Susanta Kumar Paikray
Keyword(s):  
Banach Space ◽  
Statistical Convergence ◽  
Approximation Theorem ◽  
Real Sequence ◽  
Test Functions ◽  
Real Numbers ◽  
Korovkin Type Approximation Theorem ◽  
Type Approximation ◽  
Algebraic Test

The concept of the deferred Nörlund equi-statistical convergence was introduced and studied by Srivastava et al. [Rev. Real Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. (RACSAM) 112 (2018), 1487–1501]. In the present paper, we have studied the notion of the deferred Nörlund statistical convergence and the statistical deferred Nörlund summability for sequences of real numbers defined over a Banach space. We have also established a theorem presenting a connection between these two interesting notions. Moreover, based upon our proposed methods, we have proved a new Korovkin-type approximation theorem with algebraic test functions for a sequence of real numbers on a Banach space and demonstrated that our theorem effectively extends and improves most of the earlier existing results (in classical and statistical versions). Finally, we have presented an example involving the generalized Meyer–König and Zeller operators of a real sequence demonstrating that our theorem is a stronger approach than its classical and statistical versions.

scholarly journals Construction of Stancu-Type Bernstein Operators Based on Bézier Bases with Shape Parameter λ

Symmetry ◽  
2019 ◽  
Vol 11 (3) ◽  
pp. 316 ◽  
Author(s):  
Hari Srivastava ◽  
Faruk Özger ◽  
S. Mohiuddine
Keyword(s):  
Uniform Convergence ◽  
Shape Parameter ◽  
Approximation Theorem ◽  
Modulus Of Smoothness ◽  
Bernstein Operators ◽  
Global Approximation ◽  
Approximation Result ◽  
Approximation Theorems ◽  
Type Approximation ◽  
Direct Approximation

We construct Stancu-type Bernstein operators based on Bézier bases with shape parameter λ ∈ [ - 1 , 1 ] and calculate their moments. The uniform convergence of the operator and global approximation result by means of Ditzian-Totik modulus of smoothness are established. Also, we establish the direct approximation theorem with the help of second order modulus of smoothness, calculate the rate of convergence via Lipschitz-type function, and discuss the Voronovskaja-type approximation theorems. Finally, in the last section, we construct the bivariate case of Stancu-type λ -Bernstein operators and study their approximation behaviors.

A Korovkin type approximation theorem in statistical sense

2006 ◽  
Vol 43 (3) ◽  
pp. 285-294 ◽  
Author(s):  
Esra Erkuş ◽  
Oktay Duman
Keyword(s):  
Compact Subset ◽  
Statistical Convergence ◽  
Approximation Theorem ◽  
Dimensional Space ◽  
Linear Operators ◽  
Approximation Result ◽  
The Real ◽  
Korovkin Type Approximation Theorem ◽  
Type Approximation ◽  
Statistical Sense

In this study, using the concept of A-statistical convergence we investigate a Korovkin type approximation result for a sequence of positive linear operators defined on the space of all continuous real valued functions on any compact subset of the real m-dimensional space.

On the limit of spectral measures associated to a test configuration of a polarized Kähler manifold

2016 ◽  
Vol 2016 (713) ◽  
Author(s):  
Tomoyuki Hisamoto
Keyword(s):  
Integral Formula ◽  
Kähler Manifold ◽  
Linear Series ◽  
Spectral Measures ◽  
Test Configuration ◽  
Kahler Manifold ◽  
The Family ◽  
Graded Linear Series

AbstractWe apply our integral formula of volumes to the family of graded linear series constructed from any test configuration. This solves the conjecture raised by Witt Nyström to the effect that the sequence of spectral measures for the induced ℂ

scholarly journals On linear series and a conjecture of D. C. Butler

2015 ◽  
Vol 26 (02) ◽  
pp. 1550007 ◽  
Author(s):  
U. N. Bhosle ◽  
L. Brambila-Paz ◽  
P. E. Newstead
Keyword(s):  
Line Bundle ◽  
Linear Subspace ◽  
Linear Series ◽  
Projective Curve ◽  
Coherent Systems ◽  
Natural Context ◽  
Wall Crossing ◽  
The Stability

Let C be a smooth irreducible projective curve of genus g and L a line bundle of degree d generated by a linear subspace V of H0(L) of dimension n + 1. We prove a conjecture of D. C. Butler on the semistability of the kernel of the evaluation map V ⊗ 𝒪C → L and obtain new results on the stability of this kernel. The natural context for this problem is the theory of coherent systems on curves and our techniques involve wall crossing formulae in this theory.

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