scholarly journals Pattern generation problems arising in multiplicative integer systems

2017 ◽  
Vol 39 (5) ◽  
pp. 1234-1260
Author(s):  
JUNG-CHAO BAN ◽  
WEN-GUEI HU ◽  
SONG-SUN LIN
Keyword(s):  
Error Term ◽  
Mean Shift ◽  
Coupled System ◽  
Pattern Generation ◽  
Invariant Subset ◽  
Golden Mean ◽  
Multiplicative Systems ◽  
Spatial Entropy ◽  
Full Shift ◽  
Approach Zero

This study investigates a multiplicative integer system, an invariant subset of the full shift under the action of the semigroup of multiplicative integers, by using a method that was developed for studying pattern generation problems. The spatial entropy and the Minkowski dimensions of general multiplicative systems can thus be computed. A coupled system is the intersection of a multiplicative integer system and the golden mean shift, which can be decoupled by removing the multiplicative relation set and then performing procedures similar to those applied to a decoupled system. The spatial entropy can be obtained after the remaining error term is shown to approach zero.

scholarly journals Hausdorff dimension of the multiplicative golden mean shift

2011 ◽  
Vol 349 (11-12) ◽  
pp. 625-628 ◽  
Author(s):  
Richard Kenyon ◽  
Yuval Peres ◽  
Boris Solomyak
Keyword(s):  
Hausdorff Dimension ◽  
Mean Shift ◽  
Golden Mean

ON THE SPATIAL ENTROPY OF TWO-DIMENSIONAL GOLDEN MEAN

2004 ◽  
Vol 14 (01) ◽  
pp. 309-319
Author(s):  
JONQ JUANG ◽  
SHIH-FENG SHIEH
Keyword(s):  
Lower Bound ◽  
Two Dimensional ◽  
Golden Mean ◽  
Spatial Entropy

The aim of this paper is to derive a sharper lower bound for the spatial entropy of two-dimensional golden mean.

scholarly journals A characterization of ω-limit sets in shift spaces

2009 ◽  
Vol 30 (1) ◽  
pp. 21-31 ◽  
Author(s):  
ANDREW BARWELL ◽  
CHRIS GOOD ◽  
ROBIN KNIGHT ◽  
BRIAN E. RAINES
Keyword(s):  
Finite Type ◽  
Limit Set ◽  
Invariant Subset ◽  
Limit Sets ◽  
Tent Map ◽  
Shift Of Finite Type ◽  
Sofic Shift ◽  
Shift Space ◽  
Full Shift

AbstractA set Λ is internally chain transitive if for any x,y∈Λ and ϵ>0 there is an ϵ-pseudo-orbit in Λ between x and y. In this paper we characterize all ω-limit sets in shifts of finite type by showing that, if Λ is a closed, strongly shift-invariant subset of a shift of finite type, X, then there is a point z∈X with ω(z)=Λ if and only if Λ is internally chain transitive. It follows immediately that any closed, strongly shift-invariant, internally chain transitive subset of a shift space over some alphabet ℬ is the ω-limit set of some point in the full shift space over ℬ. We use similar techniques to prove that, for a tent map f, a closed, strongly f-invariant, internally chain transitive subset of the interval is the ω-limit set of a point provided it does not contain the image of the critical point. We give an example of a sofic shift space Z𝒢 (a factor of a shift space of finite type) that is not of finite type that has an internally chain transitive subset that is not the ω-limit set of any point in Z𝒢.

THREE-DIMENSIONAL CELLULAR NEURAL NETWORKS AND PATTERN GENERATION PROBLEMS

2008 ◽  
Vol 18 (04) ◽  
pp. 957-984 ◽  
Author(s):  
JUNG-CHAO BAN ◽  
SONG-SUN LIN ◽  
YIN-HENG LIN
Keyword(s):  
Neural Networks ◽  
Transition Matrix ◽  
Finite Lattice ◽  
Three Dimensional ◽  
Recursive Formula ◽  
Cellular Neural Networks ◽  
Pattern Generation ◽  
Transition Matrices ◽  
Spatial Entropy ◽  
Local Patterns

This work investigates three-dimensional pattern generation problems and their applications to three-dimensional Cellular Neural Networks (3DCNN). An ordering matrix for the set of all local patterns is established to derive a recursive formula for the ordering matrix of a larger finite lattice. For a given admissible set of local patterns, the transition matrix is defined and the recursive formula of high order transition matrix is presented. Then, the spatial entropy is obtained by computing the maximum eigenvalues of a sequence of transition matrices. The connecting operators are used to verify the positivity of the spatial entropy, which is important in determining the complexity of the set of admissible global patterns. The results are useful in studying a set of global stationary solutions in various Lattice Dynamical Systems and Cellular Neural Networks.

AlCu Pattern Generation on 3D StructuredWafer Using Multi Level Exposure Method on Electrodeposited Polymer Material

2003 ◽  
Vol 766 ◽  
Author(s):  
Vineet Sharma ◽  
Arief B. Suriadi ◽  
Frank Berauer ◽  
Laurie S. Mittelstadt
Keyword(s):  
Line Width ◽  
Metal Film ◽  
Focal Depth ◽  
Cost Effective ◽  
Pattern Generation ◽  
Promising Technique ◽  
Critical Dimensions ◽  
Exposure Method ◽  
Multi Level ◽  
3D Surfaces

AbstractNormal photolithography tools have focal depth limitations and are unable to meet the expectations of high resolution photolithography on highly topographic structures. This paper shows a cost effective and promising technique of combining two different approaches to achieve critical dimensions of traces on slope pattern continuity on highly topographic structures. Electrophoretically deposited photoresist is used on 3-D structured wafers. This photoresist coating technique is fairly known in the MEMS industries to achieve uniform and conformal photoresist films on 3D surfaces. Multi step exposures are used to expose electrophoretically deposited photoresist. AlCu (Cu-0.5%), 0.47-0.53 μm thick metal film is deposited on 3D structured silicon substrate to plate photoresist. By combining these two novel methods, metal (AlCu) traces of 75 μm line width and 150 μm pitch (from top flat to down the slope) have been demonstrated on isotropically etched 350 μm deep trenches with 5-10% line width loss.

Shape Optimization of Hydraulic Structures: an Example of an Optimum Design of a Fish Passage

10.29007/2k64 ◽  
2018 ◽  
Author(s):  
Pat Prodanovic ◽  
Cedric Goeury ◽  
Fabrice Zaoui ◽  
Riadh Ata ◽  
Jacques Fontaine ◽  
...  
Keyword(s):  
Numerical Model ◽  
Shape Optimization ◽  
Objective Function ◽  
Optimum Design ◽  
Optimization Problem ◽  
A Priori ◽  
Coupled System ◽  
Fish Passage ◽  
Hydraulic Structures ◽  
Design Variables

This paper presents a practical methodology developed for shape optimization studies of hydraulic structures using environmental numerical modelling codes. The methodology starts by defining the optimization problem and identifying relevant problem constraints. Design variables in shape optimization studies are configuration of structures (such as length or spacing of groins, orientation and layout of breakwaters, etc.) whose optimal orientation is not known a priori. The optimization problem is solved numerically by coupling an optimization algorithm to a numerical model. The coupled system is able to define, test and evaluate a multitude of new shapes, which are internally generated and then simulated using a numerical model. The developed methodology is tested using an example of an optimum design of a fish passage, where the design variables are the length and the position of slots. In this paper an objective function is defined where a target is specified and the numerical optimizer is asked to retrieve the target solution. Such a definition of the objective function is used to validate the developed tool chain. This work uses the numerical model TELEMAC- 2Dfrom the TELEMAC-MASCARET suite of numerical solvers for the solution of shallow water equations, coupled with various numerical optimization algorithms available in the literature.

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