On uniformly bounded sequences in Orlicz spaces

A useful result for uniformly bounded sequences of functions in Orlicz spaces is generalised by means of a recent extension of Komlós' theorem. The same generalisation can also be proven differently, by means of Young measures.

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A note on H-convergence

Researches in Mathematics ◽

10.15421/240923 ◽

2021 ◽

Vol 17 ◽

pp. 150

Author(s):

O.P. Kogut ◽

P.I. Kogut ◽

T.N. Rudyanova

Keyword(s):

Convergence Property ◽

Sufficient Conditions ◽

Weak Limit ◽

Uniformly Bounded ◽

Bounded Sequences

In this paper we study the H-convergence property for the uniformly bounded sequences of square matrices $\left\{ A_{\varepsilon} \in L^{\infty} (D; \mathbb{R}^{n \times n}) \right\}_{\varepsilon > 0}$. We derive the sufficient conditions, which guarantee the coincidence of $H$-limit with the weak-* limit of such sequences in $L^{\infty} (D; \mathbb{R}^{n \times n})$.

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The natural best approximant in Orlicz spaces of Young measures

Nonlinear Analysis ◽

10.1016/j.na.2004.02.002 ◽

2004 ◽

Vol 57 (1) ◽

pp. 99-110

Author(s):

Cristian Constantin Popa

Keyword(s):

Orlicz Spaces ◽

Young Measures

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Stop rule inequalities for uniformly bounded sequences of random variables

Transactions of the American Mathematical Society ◽

10.1090/s0002-9947-1983-0697070-7 ◽

1983 ◽

Vol 278 (1) ◽

pp. 197-197 ◽

Cited By ~ 22

Author(s):

Theodore P. Hill ◽

Robert P. Kertz

Keyword(s):

Random Variables ◽

Stop Rule ◽

Uniformly Bounded ◽

Bounded Sequences

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Comparison of optimal value and constrained maxima expectations for independent random variables

Advances in Applied Probability ◽

10.1017/s0001867800015780 ◽

1986 ◽

Vol 18 (02) ◽

pp. 311-340

Author(s):

Robert P. Kertz

Keyword(s):

Random Variables ◽

Poisson Approximation ◽

Independent Random Variables ◽

Problem Analysis ◽

Universal Inequalities ◽

Optimal Value ◽

Approximation Type ◽

Stop Rule ◽

Uniformly Bounded ◽

Bounded Sequences

For all uniformly bounded sequences of independent random variablesX1, X2,···, a complete comparison is made between the optimal valueV(X1, X2, ···) = sup {EXt:tis an (a.e.) finite stop rule forX1,X2, ···} and, whereMi(X1,X2, ···) is theith largest order statistic forX1, X2, ··· In particular, fork>1, the set of ordered pairs {(x,y):x=V(X1, X2,···) andfor some independent random variablesX1, X2, ··· taking values in [0, 1]} is precisely the set, whereBk(0) = 0,Bk(1) = 1, and forThe result yields sharp, universal inequalities for independent random variables comparing two choice mechanisms, the mortal&s value of the gameV(X1, X2,···) and the prophet&s constrained maxima expectation of the game. Techniques of proof include probability- and convexity-based reductions; calculus-based, multivariate, extremal problem analysis; and limit theorems of Poisson-approximation type. Precise results are also given for finite sequences of independent random variables.

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A-quasi-convexity with variable coefficients

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500003711 ◽

2004 ◽

Vol 134 (6) ◽

pp. 1219-1237 ◽

Cited By ~ 4

Author(s):

Pedro M. Santos

Keyword(s):

Variable Coefficients ◽

Young Measures ◽

Regularity Conditions ◽

Lower Semi ◽

Constant Coefficients ◽

Additional Regularity ◽

Image Position ◽

Quasi Convexity ◽

Bounded Sequences

It is shown that, for integrals of the type with Ω RN open, bounded and f: Ω × Rm × Rd → [0, + ∞) Carathéodory satisfying a growth condition 0 ≤ f(x, u, υ) ≤ C(1 + |υ|p), for some p ∈ (1, + ∞), a sufficient condition for lower semi-continuity along sequences un → u in measure, υn → υ in Lp, Aυn → 0 in W−1, p is the Ax-quasi-convexity of f(x, u, ·). Here, A is a variable coefficients operator of the form with A(i) ∈ C∞ (Ω; Ml × d) ∩ W1, ∞, i = 1, …, N, satisfying the condition and Ax denotes the constant coefficients operator one obtains by freezing x. Under additional regularity conditions on f, it is proved that the condition above is also necessary. A characterization of the Young measures generated by bounded sequences {υn} in Lp satisfying the condition Aυn → 0 in W−1,p, is obtained.

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ON NON-NEWTONIAN FLUIDS WITH A PROPERTY OF RAPID THICKENING UNDER DIFFERENT STIMULUS

Mathematical Models and Methods in Applied Sciences ◽

10.1142/s0218202508002954 ◽

2008 ◽

Vol 18 (07) ◽

pp. 1073-1092 ◽

Cited By ~ 35

Author(s):

PIOTR GWIAZDA ◽

AGNIESZKA ŚWIERCZEWSKA-GWIAZDA

Keyword(s):

Orlicz Spaces ◽

Shear Thickening ◽

Growth Conditions ◽

Young Measures ◽

Cauchy Stress ◽

Cauchy Stress Tensor ◽

Energy Equality ◽

Existence Of Weak Solutions ◽

Nonstandard Growth ◽

Nonstandard Growth Conditions

The paper concerns the model of a flow of non-Newtonian fluid with nonstandard growth conditions of the Cauchy stress tensor. Contrary to standard power-law type rheology, we propose the formulation with the help of the spatially-dependent convex function. This framework includes e.g. rapidly shear thickening and magnetorheological fluids. We provide the existence of weak solutions. The nonstandard growth conditions yield the analytical formulation of the problem in generalized Orlicz spaces. Basing on the energy equality, we exploit the tools of Young measures.

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Comparison of optimal value and constrained maxima expectations for independent random variables

Advances in Applied Probability ◽

10.2307/1427302 ◽

1986 ◽

Vol 18 (2) ◽

pp. 311-340 ◽

Cited By ~ 5

Author(s):

Robert P. Kertz

Keyword(s):

Random Variables ◽

Poisson Approximation ◽

Independent Random Variables ◽

Problem Analysis ◽

Universal Inequalities ◽

Optimal Value ◽

Approximation Type ◽

Stop Rule ◽

Uniformly Bounded ◽

Bounded Sequences

For all uniformly bounded sequences of independent random variables X1, X2, ···, a complete comparison is made between the optimal value V(X1, X2, ···) = sup {EXt:t is an (a.e.) finite stop rule for X1,X2, ···} and , where Mi(X1,X2, ···) is the ith largest order statistic for X1, X2, ··· In particular, for k> 1, the set of ordered pairs {(x, y):x = V(X1, X2, ···) and for some independent random variables X1, X2, ··· taking values in [0, 1]} is precisely the set , where Bk(0) = 0, Bk(1) = 1, and for The result yields sharp, universal inequalities for independent random variables comparing two choice mechanisms, the mortal&s value of the game V(X1, X2, ···) and the prophet&s constrained maxima expectation of the game . Techniques of proof include probability- and convexity-based reductions; calculus-based, multivariate, extremal problem analysis; and limit theorems of Poisson-approximation type. Precise results are also given for finite sequences of independent random variables.

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Applications of Through-Focus Dark Field Image Shifts to Phase Identification

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100114517 ◽

1975 ◽

Vol 33 ◽

pp. 178-179

Author(s):

Prakash Rao

Keyword(s):

Dark Field Image ◽

Thin Foil ◽

Dark Field ◽

The Other ◽

Stereo Image ◽

Phase Identification ◽

The Past ◽

Field Electron Microscopy ◽

Dark Field Electron ◽

Recent Extension

Image shifts in out-of-focus dark field images have been used in the past to determine, for example, epitaxial relationships in thin films. A recent extension of the use of dark field image shifts has been to out-of-focus images in conjunction with stereoviewing to produce an artificial stereo image effect. The technique, called through-focus dark field electron microscopy or 2-1/2D microscopy, basically involves obtaining two beam-tilted dark field images such that one is slightly over-focus and the other slightly under-focus, followed by examination of the two images through a conventional stereoviewer. The elevation differences so produced are usually unrelated to object positions in the thin foil and no specimen tilting is required.In order to produce this artificial stereo effect for the purpose of phase separation and identification, it is first necessary to select a region of the diffraction pattern containing more than just one discrete spot, with the objective aperture.

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