scholarly journals Modifying AMG Coarse Spaces with Weak Approximation Property to Exhibit Approximation in Energy Norm

2019 ◽  
Vol 40 (3) ◽  
pp. 1131-1152
Author(s):  
Xiaozhe Hu ◽  
Panayot S. Vassilevski
Keyword(s):  
Approximation Property ◽  
Energy Norm ◽  
Weak Approximation

Right and Left Weak Approximation Properties in Banach Spaces

2009 ◽  
Vol 52 (1) ◽  
pp. 28-38 ◽  
Author(s):  
Changsun Choi ◽  
Ju Myung Kim ◽  
Keun Young Lee
Keyword(s):  
Banach Spaces ◽  
Approximation Property ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Weak Approximation ◽  
Weak Version ◽  
Approximation Properties ◽  
Necessary And Sufficient

AbstractNew necessary and sufficient conditions are established for Banach spaces to have the approximation property; these conditions are easier to check than the known ones. A shorter proof of a result of Grothendieck is presented, and some properties of a weak version of the approximation property are addressed.

scholarly journals Amenable dynamical systems over locally compact groups

2021 ◽  
pp. 1-41
Author(s):  
ALEX BEARDEN ◽  
JASON CRANN
Keyword(s):  
Dynamical Systems ◽  
Approximation Property ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Weak Approximation ◽  
Locally Compact ◽  
Completely Positive ◽  
Cambridge Philos ◽  
Schur Multipliers ◽  
Approximation Of The Identity

Abstract We establish several new characterizations of amenable $W^*$ - and $C^*$ -dynamical systems over arbitrary locally compact groups. In the $W^*$ -setting we show that amenability is equivalent to (1) a Reiter property and (2) the existence of a certain net of completely positive Herz–Schur multipliers of $(M,G,\alpha )$ converging point weak* to the identity of $G\bar {\ltimes }M$ . In the $C^*$ -setting, we prove that amenability of $(A,G,\alpha )$ is equivalent to an analogous Herz–Schur multiplier approximation of the identity of the reduced crossed product $G\ltimes A$ , as well as a particular case of the positive weak approximation property of Bédos and Conti [On discrete twisted $C^*$ -dynamical systems, Hilbert $C^*$ -modules and regularity. Münster J. Math.5 (2012), 183–208] (generalized to the locally compact setting). When $Z(A^{**})=Z(A)^{**}$ , it follows that amenability is equivalent to the1-positive approximation property of Exel and Ng [Approximation property of $C^*$ -algebraic bundles. Math. Proc. Cambridge Philos. Soc.132(3) (2002), 509–522]. In particular, when $A=C_0(X)$ is commutative, amenability of $(C_0(X),G,\alpha )$ coincides with topological amenability of the G-space $(G,X)$ .

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.

The tight approximation property

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Olivier Benoist ◽  
Olivier Wittenberg
Keyword(s):  
Function Field ◽  
Approximation Property ◽  
Algebraic Curves ◽  
Homogeneous Spaces ◽  
Algebraic Groups ◽  
Linear Algebraic Group ◽  
Weak Approximation ◽  
Property A ◽  
Linear Algebraic Groups ◽  
Real Curve

Abstract This article introduces and studies the tight approximation property, a property of algebraic varieties defined over the function field of a complex or real curve that refines the weak approximation property (and the known cohomological obstructions to it) by incorporating an approximation condition in the Euclidean topology. We prove that the tight approximation property is a stable birational invariant, is compatible with fibrations, and satisfies descent under torsors of linear algebraic groups. Its validity for a number of rationally connected varieties follows. Some concrete consequences are: smooth loops in the real locus of a smooth compactification of a real linear algebraic group, or in a smooth cubic hypersurface of dimension ≥ 2 {\geq 2} , can be approximated by rational algebraic curves; homogeneous spaces of linear algebraic groups over the function field of a real curve satisfy weak approximation.

NORMAL FORMS FOR FUZZY LOGIC — AN APPLICATION OF KOLMOGOROV’S THEOREM

1996 ◽  
Vol 04 (04) ◽  
pp. 331-349 ◽  
Author(s):  
VLADIK KREINOVICH ◽  
HUNG T. NGUYEN ◽  
DAVID A. SPRECHER
Keyword(s):  
Neural Networks ◽  
Fuzzy Logic ◽  
Normal Form ◽  
Normal Forms ◽  
Approximation Property ◽  
Universal Approximation ◽  
Approximation Result ◽  
Logical Operations ◽  
Kolmogorov's Theorem

This paper addresses mathematical aspects of fuzzy logic. The main results obtained in this paper are: 1. the introduction of a concept of normal form in fuzzy logic using hedges; 2. using Kolmogorov’s theorem, we prove that all logical operations in fuzzy logic have normal forms; 3. for min-max operators, we obtain an approximation result similar to the universal approximation property of neural networks.

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