scholarly journals L0-Linear Modulus of a Random Linear Operator

2014 ◽  
Vol 2014 ◽  
pp. 1-4
Author(s):  
Ming Liu ◽  
Xia Zhang
Keyword(s):  
Linear Operator ◽  
Classical Case ◽  
Random Normed Module ◽  
Normed Module

We prove that there exists a uniqueL0-linear modulus for an a.s. bounded random linear operator on a specifical random normed module, which generalizes the classical case.

THE RANDOM SUBREFLEXIVITY OF COMPLETE RANDOM NORMED MODULES

2012 ◽  
Vol 23 (03) ◽  
pp. 1250047 ◽  
Author(s):  
SHIEN ZHAO ◽  
TIEXIN GUO
Keyword(s):  
Random Normed Module ◽  
Convex Topology ◽  
Normed Module ◽  
Random Normed Modules

Combining respective advantages of the (ε, λ)-topology and the locally L0-convex topology we first prove that every complete random normed module is random subreflexive under the (ε, λ)-topology. Further, we prove that every complete random normed module with the countable concatenation property is also random subreflexive under the locally L0-convex topology, at the same time we also provide a counterexample which shows that it is necessary to require the random normed module to have the countable concatenation property.

scholarly journals A Minimax Theorem forL-0-Valued Functions on Random Normed Modules

2013 ◽  
Vol 2013 ◽  
pp. 1-10
Author(s):  
Shien Zhao ◽  
Yuan Zhao
Keyword(s):  
Lower Semicontinuous ◽  
Saddle Points ◽  
Minimax Theorem ◽  
Lower Semicontinuous Function ◽  
Semicontinuous Function ◽  
Minimax Theorems ◽  
Random Normed Module ◽  
Normed Module ◽  
Random Normed Modules ◽  
Definition Of

We generalize the well-known minimax theorems toL¯0-valued functions on random normed modules. We first give some basic properties of anL0-valued lower semicontinuous function on a random normed module under the two kinds of topologies, namely, the (ε,λ)-topology and the locallyL0-convex topology. Then, we introduce the definition of random saddle points. Conditions for anL0-valued function to have a random saddle point are given. The most greatest difference between our results and the classical minimax theorems is that we have to overcome the difficulty resulted from the lack of the condition of compactness. Finally, we, using relations between the two kinds of topologies, establish the minimax theorem ofL¯0-valued functions in the framework of random normed modules and random conjugate spaces.

Criterion of cyclically compactness infinite-dimensional normed module

2018 ◽  
Vol 2018 (3) ◽  
pp. 51-62
Author(s):  
V.I. Chilin ◽  
J.A. Karimov
Keyword(s):  
Infinite Dimensional ◽  
Normed Module

A lambda calculus with naive substitution

1979 ◽  
Vol 28 (3) ◽  
pp. 269-282 ◽  
Author(s):  
John Staples
Keyword(s):  
Lambda Calculus ◽  
Classical Case ◽  
Syntactic Theory ◽  
Alternative Approach ◽  
New Formulation

AbstractAn alternative approach is proposed to the basic definitions of the lassical lambda calculus. A proof is sketched of the equivalence of the approach with the classical case. The new formulation simplifies some aspects of the syntactic theory of the lambda calculus. In particular it provides a justification for omitting in syntactic theory discussion of changes of bound variable.

A note on normal operators

1963 ◽  
Vol 59 (4) ◽  
pp. 727-729 ◽  
Author(s):  
S. J. Bernau ◽  
F. Smithies
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Adjoint Operator ◽  
Spectral Theorem ◽  
Unitary Operators ◽  
Unitary Space ◽  
Bounded Linear ◽  
Finite Dimensional ◽  
Normal Operators ◽  
Finite Dimensional Case

We recall that a bounded linear operator T in a Hilbert space or finite-dimensional unitary space is said to be normal if T commutes with its adjoint operator T*, i.e. TT* = T*T. Most of the proofs given in the literature for the spectral theorem for normal operators, even in the finite-dimensional case, appeal to the corresponding results for Hermitian or unitary operators.

scholarly journals Convergence Rates of First- and Higher-Order Dynamics for Solving Linear Ill-Posed Problems

2021 ◽  
Author(s):  
Radu Boţ ◽  
Guozhi Dong ◽  
Peter Elbau ◽  
Otmar Scherzer
Keyword(s):  
Linear Operator ◽  
Operator Equation ◽  
Convex Analysis ◽  
Convergence Rates ◽  
Linear Operator Equation ◽  
Optimal Convergence Rates ◽  
Ill Posed ◽  
Thresholding Algorithm ◽  
The Given ◽  
Theoretical Results

AbstractRecently, there has been a great interest in analysing dynamical flows, where the stationary limit is the minimiser of a convex energy. Particular flows of great interest have been continuous limits of Nesterov’s algorithm and the fast iterative shrinkage-thresholding algorithm, respectively. In this paper, we approach the solutions of linear ill-posed problems by dynamical flows. Because the squared norm of the residual of a linear operator equation is a convex functional, the theoretical results from convex analysis for energy minimising flows are applicable. However, in the restricted situation of this paper they can often be significantly improved. Moreover, since we show that the proposed flows for minimising the norm of the residual of a linear operator equation are optimal regularisation methods and that they provide optimal convergence rates for the regularised solutions, the given rates can be considered the benchmarks for further studies in convex analysis.

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