GENERALIZED DECIMATION INVARIANT SEQUENCES IN DIMENSION N

2002 ◽  
Vol 12 (04) ◽  
pp. 709-737 ◽  
Author(s):  
A. BARBÉ ◽  
F. VON HAESELER
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
One Dimensional ◽  
Necessary And Sufficient

We generalize the concept of one-dimensional decimation invariant sequences, i.e. sequences which are invariant under a specific rescaling, to dimension N. After discussing the elementary properties of decimation-invariant sequences, we focus our interest on their periodicity. Necessary and sufficient conditions for the existence of periodic decimation invariant sequences are presented.

scholarly journals A classification of explosions in dimension one

2009 ◽  
Vol 29 (2) ◽  
pp. 715-731 ◽  
Author(s):  
E. SANDER ◽  
J. A. YORKE
Keyword(s):  
Sufficient Conditions ◽  
Global Bifurcation ◽  
Necessary And Sufficient Conditions ◽  
Homoclinic Tangency ◽  
Discontinuous Change ◽  
One Dimensional ◽  
One Dimensional Maps ◽  
Dimension One ◽  
Necessary And Sufficient

AbstractA discontinuous change in the size of an attractor is the most easily observed type of global bifurcation. More generally, anexplosionis a discontinuous change in the set of recurrent points. An explosion often results from heteroclinic and homoclinic tangency bifurcations. We prove that, for one-dimensional maps, explosions are generically the result of either tangency or saddle-node bifurcations. Furthermore, we give necessary and sufficient conditions for generic tangency bifurcations to lead to explosions.

On multi-dimensional annealing problems

1989 ◽  
Vol 105 (1) ◽  
pp. 177-184 ◽  
Author(s):  
Terence Chan
Keyword(s):  
Continuous Function ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
One Dimensional ◽  
Dimensional Diffusion ◽  
Deterministic Function ◽  
Image Position ◽  
Necessary And Sufficient

In [1] Chan and Williams considered a one-dimensional diffusion of the formwhere F is a strictly increasing continuous function with F(0) = 0 and ε is a decreasing deterministic function such that ε(0) is finite and ε(t) ↓ 0 as t↑ ∞, and gave necessary and sufficient conditions for Yt →0 a.s. as t→∞.

On the equivalence of boundedness for multiple Hardy-Littlewood averages and related operators

2018 ◽  
Vol 123 (2) ◽  
pp. 273-296
Author(s):  
Dah-Chin Luor
Keyword(s):  
Weight Function ◽  
Geometric Mean ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Maximal Operators ◽  
Averaging Operators ◽  
One Dimensional ◽  
Almost Everywhere ◽  
Carry Over ◽  
Necessary And Sufficient

Necessary and sufficient conditions for the weight function $u$ are obtained, which provide the boundedness for a class of averaging operators from $L_p^+$ to $L_{q,u}^+$. These operators include the multiple Hardy-Littlewood averages and related maximal operators, geometric mean operators, and geometric maximal operators. We show that, under suitable conditions, the boundedness of these operators are equivalent. Our theorems extend several one-dimensional results to multi-dimensional cases and to operators with multiple kernels. We also show that in the case $p<q$, some one-dimensional results do not carry over to the multi-dimensional cases, and the boundedness of $T$ from $L_p^+$ to $L_{q,u}^+$ holds only if $u=0$ almost everywhere.

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