scholarly journals History, Developments and Open Problems on Approximation Properties

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1117
Author(s):  
Ju Myung Kim ◽  
Bentuo Zheng
Keyword(s):  
Recent Progress ◽  
Strong Approximation ◽  
Approximation Property ◽  
Approximation Properties ◽  
Open Problems ◽  
Comprehensive Review ◽  
Classical Approximation

In this paper, we give a comprehensive review of the classical approximation property. Then, we present some important results on modern variants, such as the weak bounded approximation property, the strong approximation property and p-approximation property. Most recent progress on E-approximation property and open problems are discussed at the end.

scholarly journals Computability-theoretic complexity of effective Banach spaces

2022 ◽  
Author(s):  
◽  
Long Qian
Keyword(s):  
Banach Space ◽  
Lower Bound ◽  
Banach Spaces ◽  
Lower Bounds ◽  
Approximation Property ◽  
Upper Bounds ◽  
Space Theory ◽  
Approximation Properties ◽  
Open Problems

<p><b>We investigate the geometry of effective Banach spaces, namely a sequenceof approximation properties that lies in between a Banach space having a basis and the approximation property.</b></p> <p>We establish some upper bounds on suchproperties, as well as proving some arithmetical lower bounds. Unfortunately,the upper bounds obtained in some cases are far away from the lower bound.</p> <p>However, we will show that much tighter bounds will require genuinely newconstructions, and resolve long-standing open problems in Banach space theory.</p> <p>We also investigate the effectivisations of certain classical theorems in Banachspaces.</p>

scholarly journals COARSE SPACES BY ALGEBRAIC MULTIGRID: MULTIGRID CONVERGENCE AND UPSCALING ERROR ESTIMATES

2011 ◽  
Vol 03 (01n02) ◽  
pp. 229-249 ◽  
Author(s):  
PANAYOT S. VASSILEVSKI
Keyword(s):  
Strong Approximation ◽  
Approximation Property ◽  
Algebraic Multigrid ◽  
Necessary Condition ◽  
Weak Approximation ◽  
Approximation Properties ◽  
Element Discretization ◽  
Multigrid Convergence ◽  
Grid Convergence ◽  
Algebraic Multigrid Methods

We give an overview of a number of algebraic multigrid methods targeting finite element discretization problems. The focus is on the properties of the constructed hierarchy of coarse spaces that guarantee (two-grid) convergence. In particular, a necessary condition known as "weak approximation property," and a sufficient one, referred to as "strong approximation property," are discussed. Their role in proving convergence of the TG method (as iterative method) and also on the approximation properties of the algebraic mottigrid (AMG) coarse spaces if used as discretization tool is pointed out. Some preliminary numerical results illustrating the latter aspect are also reported.

scholarly journals Computability-theoretic complexity of effective Banach spaces

2022 ◽  
Author(s):  
◽  
Long Qian
Keyword(s):  
Banach Space ◽  
Lower Bound ◽  
Banach Spaces ◽  
Lower Bounds ◽  
Approximation Property ◽  
Upper Bounds ◽  
Space Theory ◽  
Approximation Properties ◽  
Open Problems

<p><b>We investigate the geometry of effective Banach spaces, namely a sequenceof approximation properties that lies in between a Banach space having a basis and the approximation property.</b></p> <p>We establish some upper bounds on suchproperties, as well as proving some arithmetical lower bounds. Unfortunately,the upper bounds obtained in some cases are far away from the lower bound.</p> <p>However, we will show that much tighter bounds will require genuinely newconstructions, and resolve long-standing open problems in Banach space theory.</p> <p>We also investigate the effectivisations of certain classical theorems in Banachspaces.</p>

Recent Progress in Iodine‐Catalysed C−O/C−N Bond Formation of 1,3‐Oxazoles: A Comprehensive Review

2021 ◽  
Vol 6 (4) ◽  
pp. 754-787
Author(s):  
Sanjeev Dhawan ◽  
Vishal Kumar ◽  
Pankaj S. Girase ◽  
Sithabile Mokoena ◽  
Rajshekhar Karpoormath
Keyword(s):  
Recent Progress ◽  
Bond Formation ◽  
Comprehensive Review

scholarly journals Quantum chaos for point scatterers on flat tori

2014 ◽  
Vol 372 (2007) ◽  
pp. 20120509 ◽  
Author(s):  
Henrik Ueberschär
Keyword(s):  
Wave Function ◽  
Strong Coupling ◽  
Quantum Chaos ◽  
Recent Progress ◽  
Three Dimensional ◽  
Point Scatterer ◽  
Open Problems ◽  
Survey Article ◽  
Delta Potential ◽  
Flat Tori

This survey article deals with a delta potential—also known as a point scatterer—on flat two- and three-dimensional tori. We introduce the main conjectures regarding the spectral and wave function statistics of this model in the so-called weak and strong coupling regimes. We report on recent progress as well as a number of open problems in this field.

scholarly journals Recent progress in Si hetero-junction solar cell: A comprehensive review

2018 ◽  
Vol 82 ◽  
pp. 1990-2004 ◽  
Author(s):  
Bablu K. Ghosh ◽  
Chadwin N.J. Weoi ◽  
Aminul Islam ◽  
Swapan K. Ghosh
Keyword(s):  
Solar Cell ◽  
Recent Progress ◽  
Comprehensive Review ◽  
Junction Solar Cell

scholarly journals Delta Scuti Stars: Selected Recent Highlights

2000 ◽  
Vol 176 ◽  
pp. 408-414
Author(s):  
G. Handler
Keyword(s):  
Recent Progress ◽  
Open Problems ◽  
Delta Scuti ◽  
Scuti Stars ◽  
Δ Scuti Stars

AbstractAn overview of recent progress in the research on δ Scuti stars is given. Some intriguing results and open problems are pointed out, and some ideas for future investigations are provided.

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