EXACT NUMBER OF MOSAIC PATTERNS IN CELLULAR NEURAL NETWORKS

2001 ◽  
Vol 11 (06) ◽  
pp. 1645-1653 ◽  
Author(s):  
JUNG-CHAO BAN ◽  
SONG-SUN LIN ◽  
CHIH-WEN SHIH
Keyword(s):  
Neural Networks ◽  
Boundary Conditions ◽  
Cellular Neural Networks ◽  
Transition Matrices ◽  
Lattice Systems ◽  
Range Interaction ◽  
One Dimensional ◽  
Various Boundary Conditions ◽  
Exact Number ◽  
The One

This work investigates mosaic patterns for the one-dimensional cellular neural networks with various boundary conditions. These patterns can be formed by combining the basic patterns. The parameter space is partitioned so that the existence of basic patterns can be determined for each parameter region. The mosaic patterns can then be completely characterized through formulating suitable transition matrices and boundary-pattern matrices. These matrices generate the patterns for the interior cells from the basic patterns and indicate the feasible patterns for the boundary cells. As an illustration, we elaborate on the cellular neural networks with a general 1 × 3 template. The exact number of mosaic patterns will be computed for the system with the Dirichlet, Neumann and periodic boundary conditions respectively. The idea in this study can be extended to other one-dimensional lattice systems with finite-range interaction.

ABUNDANCE OF MOSAIC PATTERNS FOR CNN WITH SPATIALLY VARIANT TEMPLATES

2002 ◽  
Vol 12 (06) ◽  
pp. 1321-1332 ◽  
Author(s):  
CHENG-HSIUNG HSU ◽  
TING-HUI YANG
Keyword(s):  
Neural Network ◽  
Boundary Conditions ◽  
Cellular Neural Network ◽  
One Dimensional ◽  
Various Boundary Conditions ◽  
Exact Number ◽  
Finite Lattices ◽  
Infinite Lattices ◽  
Spatially Variant

This work investigates the complexity of one-dimensional cellular neural network mosaic patterns with spatially variant templates on finite and infinite lattices. Various boundary conditions are considered for finite lattices and the exact number of mosaic patterns is computed precisely. The entropy of mosaic patterns with periodic templates can also be calculated for infinite lattices. Furthermore, we show the abundance of mosaic patterns with respect to template periods and, which differ greatly from cases with spatially invariant templates.

SPATIAL DISORDER OF CNN — WITH ASYMMETRIC OUTPUT FUNCTION

2001 ◽  
Vol 11 (08) ◽  
pp. 2085-2095 ◽  
Author(s):  
JUNG-CHAO BAN ◽  
KAI-PING CHIEN ◽  
SONG-SUN LIN ◽  
CHENG-HSIUNG HSU
Keyword(s):  
Neural Networks ◽  
Steady State ◽  
Three Dimensional ◽  
Cellular Neural Networks ◽  
Output Function ◽  
One Dimensional ◽  
Spatial Entropy ◽  
The One ◽  
Steady State Solutions

This investigation will describe the spatial disorder of one-dimensional Cellular Neural Networks (CNN). The steady state solutions of the one-dimensional CNN can be replaced as an iteration map which is one dimensional under certain parameters. Then, the maps are chaotic and the spatial entropy of the steady state solutions is a three-dimensional devil-staircase like function.

Investigation of the Finite Size Properties of the Ising Model Under Various Boundary Conditions

2020 ◽  
Vol 75 (2) ◽  
pp. 175-182
Author(s):  
Magdy E. Amin ◽  
Mohamed Moubark ◽  
Yasmin Amin
Keyword(s):  
Magnetic Field ◽  
Ising Model ◽  
Boundary Conditions ◽  
Finite Size ◽  
Scaling Functions ◽  
Finite Size Scaling ◽  
One Dimensional ◽  
Various Boundary Conditions ◽  
The One ◽  
Bulk Case

AbstractThe one-dimensional Ising model with various boundary conditions is considered. Exact expressions for the thermodynamic and magnetic properties of the model using different kinds of boundary conditions [Dirichlet (D), Neumann (N), and a combination of Neumann–Dirichlet (ND)] are presented in the absence (presence) of a magnetic field. The finite-size scaling functions for internal energy, heat capacity, entropy, magnetisation, and magnetic susceptibility are derived and analysed as function of the temperature and the field. We show that the properties of the one-dimensional Ising model is affected by the finite size of the system and the imposed boundary conditions. The thermodynamic limit in which the finite-size functions approach the bulk case is also discussed.

scholarly journals Traveling Waves for Delayed Cellular Neural Networks with Nonmonotonic Output Functions

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Zhi-Xian Yu ◽  
Rong Yuan ◽  
Cheng-Hsiung Hsu ◽  
Ming-Shu Peng
Keyword(s):  
Neural Networks ◽  
Traveling Waves ◽  
Signal Transmission ◽  
Distributed Delay ◽  
Traveling Wave Solutions ◽  
Cellular Neural Networks ◽  
Switching Speed ◽  
One Dimensional ◽  
The One ◽  
Nondecreasing Functions

This work investigates traveling waves for a class of delayed cellular neural networks with nonmonotonic output functions on the one-dimensional integer latticeZ. The dynamics of each given cell depends on itself and its nearestmleft orlright neighborhood cells with distributed delay due to, for example, finite switching speed and finite velocity of signal transmission. Our technique is to construct two appropriate nondecreasing functions to squeeze the nonmonotonic output functions. Then we construct a suitable wave profiles set and derive the existence of traveling wave solutions by using Schauder's fixed point theorem.

Boundary Conditions for the DKP Particle in the One-Dimensional Box

2016 ◽  
Vol 71 (5) ◽  
pp. 405-412
Author(s):  
Boutheina Boutabia-Chéraitia ◽  
Abdenacer Makhlouf
Keyword(s):  
Boundary Conditions ◽  
One Dimensional ◽  
Various Boundary Conditions ◽  
The One

AbstractIn this article, we study a relativistic DKP particle in a one-dimensional box. We prove that it is impossible that the wavefunction vanishes completely at the box walls and provides various boundary conditions for this problem.

CELLULAR NEURAL NETWORKS: MOSAIC PATTERNS, BIFURCATION AND COMPLEXITY

2006 ◽  
Vol 16 (01) ◽  
pp. 47-57 ◽  
Author(s):  
JONQ JUANG ◽  
CHIN-LUNG LI ◽  
MING-HUANG LIU
Keyword(s):  
Neural Network ◽  
Neural Networks ◽  
Source Term ◽  
Cellular Neural Network ◽  
Cellular Neural Networks ◽  
Output Function ◽  
Transition Matrices ◽  
One Dimensional ◽  
Bifurcation Phenomenon

We study a one-dimensional Cellular Neural Network with an output function which is nonflat at infinity. Spatial chaotic regions are completely characterized. Moreover, each of their exact corresponding entropy is obtained via the method of transition matrices. We also study the bifurcation phenomenon of mosaic patterns with bifurcation parameters z and β. Here z is a source (or bias) term and β is the interaction weight between the neighboring cells. In particular, we find that by injecting the source term, i.e. z ≠ 0, a lot of new chaotic patterns emerge with a smaller interaction weight β. However, as β increases to a certain range, most of previously observed chaotic patterns disappear, while other new chaotic patterns emerge.

Initial-boundary value problems for the one-dimensional linear advection–dispersion equation with decay

2020 ◽  
Vol 75 (8) ◽  
pp. 713-725 ◽  
Author(s):  
Guenbo Hwang
Keyword(s):  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problems ◽  
One Dimensional ◽  
Advection Dispersion ◽  
Asymptotically Periodic ◽  
The One ◽  
Initial Boundary Value

AbstractInitial-boundary value problems for the one-dimensional linear advection–dispersion equation with decay (LAD) are studied by utilizing a unified method, known as the Fokas method. The method takes advantage of the spectral analysis of both parts of Lax pair and the global algebraic relation coupling all initial and boundary values. We present the explicit analytical solution of the LAD equation posed on the half line and a finite interval with general initial and boundary conditions. In addition, for the case of periodic boundary conditions, we show that the solution of the LAD equation is asymptotically t-periodic for large t if the Dirichlet boundary datum is periodic in t. Furthermore, it can be shown that if the Dirichlet boundary value is asymptotically periodic for large t, then so is the unknown Neumann boundary value, which is uniquely characterized in terms of the given asymptotically periodic Dirichlet boundary datum. The analytical predictions for large t are compared with numerical results showing the excellent agreement.

scholarly journals On a stochastic Burgers equation with Dirichlet boundary conditions

2003 ◽  
Vol 2003 (43) ◽  
pp. 2735-2746 ◽  
Author(s):  
Ekaterina T. Kolkovska
Keyword(s):  
Weak Solution ◽  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Burgers Equation ◽  
Martingale Problem ◽  
Weak Limit ◽  
Dirichlet Boundary Conditions ◽  
One Dimensional ◽  
Stochastic Burgers Equation ◽  
The One

We consider the one-dimensional Burgers equation perturbed by a white noise term with Dirichlet boundary conditions and a non-Lipschitz coefficient. We obtain existence of a weak solution proving tightness for a sequence of polygonal approximations for the equation and solving a martingale problem for the weak limit.

scholarly journals Order parameter profiles in a system with Neumann – Neumann boundary conditions

2018 ◽  
Vol 145 ◽  
pp. 01009 ◽  
Author(s):  
Vassil M. Vassilev ◽  
Daniel M. Dantchev ◽  
Peter A. Djondjorov
Keyword(s):  
Boundary Conditions ◽  
Order Parameter ◽  
Neumann Boundary ◽  
Elliptic Functions ◽  
Thermodynamic System ◽  
Neumann Boundary Conditions ◽  
Ginzburg Landau ◽  
One Dimensional ◽  
The One ◽  
Field Plane

In this article we consider a critical thermodynamic system with the shape of a thin film confined between two parallel planes. It is assumed that the state of the system at a given temperature and external ordering field is described by order-parameter profiles, which minimize the one-dimensional counterpart of the standard ϕ4 Ginzburg–Landau Hamiltonian and meet the so-called Neumann – Neumann boundary conditions. We give analytic representation of the extremals of this variational problem in terms ofWeierstrass elliptic functions. Then, depending on the temperature and ordering field we determine the minimizers and obtain the phase diagram in the temperature-field plane.

CHAOS AND TWO-TORI IN A NEW FAMILY OF 4-CNNS

2007 ◽  
Vol 17 (03) ◽  
pp. 953-963 ◽  
Author(s):  
XIAO-SONG YANG ◽  
YAN HUANG
Keyword(s):  
Neural Networks ◽  
Limit Cycle ◽  
Limit Cycles ◽  
Cellular Neural Networks ◽  
Lyapunov Spectrum ◽  
Regular Array ◽  
Poincaré Maps ◽  
One Dimensional ◽  
Poincare Maps ◽  
New Family

In this paper we demonstrate chaos, two-tori and limit cycles in a new family of Cellular Neural Networks which is a one-dimensional regular array of four cells. The Lyapunov spectrum is calculated in a range of parameters, the bifurcation plots are presented as well. Furthermore, we confirm the nature of limit cycle, chaos and two-tori by studying Poincaré maps.

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