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This investigation studies the nonemptiness problems of Wang cubes with two colors. Wang cubes are unit cubes with colored faces, which are generalized from Wang tiles. For a set $B$ of Wang cubs, $\Sigma(B)$ is the set of all global patterns on $\mathbb{Z}^3$ that can be constructed by the cubes in $B$. The nonemptiness problem is to determine whether $\Sigma(B) \neq \emptyset$ or not. Denote by $\mathcal{P}(B)$ the set of all periodic patterns on $\mathbb{Z}^3$ that can be constructed by the cubes in $B$. For Wang cubes, the corresponding Wang's conjecture is that if $\Sigma(B) \neq \emptyset$, then $\mathcal{P}(B) \neq \emptyset$. ¶ We introduce the transition matrices and trace operators to determine whether $\Sigma(B) \neq \emptyset$ and $\mathcal{P}(B) \neq \emptyset$ or not, respectively. A basic set $B$ is called a minimal cycle generator if $\mathcal{P}(B) \neq \emptyset$ but $\mathcal{P}(B') = \emptyset$ for all $B' \subsetneqq B$. By computer computation, there exist $86$ equivalence classes of minimal cycle generators with two colors. By verifying that the basic sets $B$ that contains no minimal cycle generators satisfy $\Sigma(B) = \emptyset$, we prove that the Wang's conjecture for Wang cubes with two colors is true.

The commonly used topological entropy $h_{top}(\mathcal{U})$ of the multi-dimensional shift space $\mathcal{U}$ is the rectangular spatial entropy $h_{r}(\mathcal{U})$ which is the limit of growth rate of admissible local patterns on finite rectangular sublattices which expands to whole space $\mathbb{Z}^{d}$, $d\geq 2$. This work studies spatial entropy $h_{\Omega}(\mathcal{U})$ of shift space $\mathcal{U}$ on general expanding system $\Omega=\{\Omega(n)\}_{n=1}^{\infty}$ where $\Omega(n)$ is increasing finite sublattices and expands to $\mathbb{Z}^{d}$. $\Omega$ is called genuinely $d$-dimensional if $\Omega(n)$ contains no lower-dimensional part whose size is comparable to that of its $d$-dimensional part. We show that $h_{r}(\mathcal{U})$ is the supremum of $h_{\Omega}(\mathcal{U})$ for all genuinely $d$-dimensional $\Omega$. Furthermore, when $\Omega$ is genuinely $d$-dimensional and satisfies certain conditions, then $h_{\Omega}(\mathcal{U})=h_{r}(\mathcal{U})$. On the contrary, when $\Omega(n)$ contains a lower-dimensional part which is comparable to its $d$-dimensional part, then $h_{r}(\mathcal{U}) < h_{\Omega}(\mathcal{U})$ for some $\mathcal{U}$. Therefore, $h_{r}(\mathcal{U})$ is appropriate to be the $d$-dimensional spatial entropy.

We study one-dimensional motions of polytropic gas governed by the compressible Euler equations. The problem on the half space under a constant gravity gives an equilibrium which has free boundary touching the vacuum and the linearized approximation at this equilibrium gives time periodic solutions. But it is difficult to justify the existence of long-time true solutions for which this time periodic solution is the first approximation. The situation is in contrast to the problem of free motions without gravity. The reason is that the usual iteration method for quasilinear hyperbolic problem cannot be used because of the loss of regularities which causes from the touch with the vacuum. Due to this reason, we try to find a family of solutions expanded by a small parameter and apply the Nash–Moser Theorem to justify this expansion. Note that the application of Nash–Moser Theorem is necessary for the sake of conquest of the trouble with loss of regularities, and the justification of the applicability requires a very delicate analysis of the problem.

This study investigates the complexity of the global set of output patterns for one-dimensional multi-layer cellular neural networks with input. Applying labeling to the output space produces a sofic shift space. Two invariants, namely spatial entropy and dynamical zeta function, can be exactly computed by studying the induced sofic shift space. This study gives sofic shift a realization through a realistic model. Furthermore, a new phenomenon, the broken of symmetry of entropy, is discovered in multi-layer cellular neural networks with input.

In this paper we develop a general approach for investigating pattern generation problems in multi-dimensional lattice models. Let $\mathcal S$ be a set of $p$ symbols or colors, $\mathbf Z_N$ a fixed finite rectangular sublattice of $\mathbf Z^d$, $d\geq 1$ and $N$ a $d$-tuple of positive integers. Functions $U:\mathbf Z^d\rightarrow \mathcal S$ and $U_N:\mathbf Z_N\rightarrow \mathcal S$ are called a global pattern and a local pattern on $\mathbf Z_N$, respectively. We introduce an ordering matrix $\mathbf X_N$ for $\Sigma_N$, the set of all local patterns on $\mathbf Z_N$. For a larger finite lattice , , we derive a recursion formula to obtain the ordering matrix of from $\mathbf X_N$. For a given basic admissible local patterns set $\mathcal B\subset \Sigma_N$, the transition matrix $\mathbf T_N(\mathcal B)$ is defined. For each , denoted by the set of all local patterns which can be generated from $\mathcal B$, the cardinal number of is the sum of entries of the transition matrix which can be obtained from $\mathbf T_N(\mathcal B)$ recursively. The spatial entropy $h(\mathcal B)$ can be obtained by computing the maximum eigenvalues of a sequence of transition matrices $\mathbf T_n(\mathcal B)$. The results can be applied to study the set of global stationary solutions in various Lattice Dynamical Systems and Cellular Neural Networks.

This study investigates bifurcations and chaos in two-cell Cellular Neural Networks (CNN) with periodic inputs. Without the inputs, the time periodic solutions are obtained for template A=[r,p,s] with p>1, r>p-1 and -s>p-1. The number of periodic solutions can be proven to be no more than two in exterior regions. The input is b sin 2πt/T with period T>0 and amplitude b>0. The typical trajectories Γ(b,T,A) and their ω-limit set ω(b,T,A) vary with b, T and A are also considered. The asymptotic limit cycles Λ∞(T,A) with period T of Γ(b,T,A) are obtained as b→∞. When [Formula: see text] (given in (67)), Λ∞and -Λ∞can be separated. The onset of chaos can be induced by crises of ω(b,T,A) and -ω(b,T,A) for suitable T and b. The ratio [Formula: see text], of largest amplitude a1(b) except for T-mode and amplitude of the T-mode of the Fast Fourier Transform (FFT) of Γ(b,T,A), can be used to compare the strength of sustained periodic cycle Λ0(A) and the inputs. When [Formula: see text], Λ0(A) dominates and the attractor ω(b,T,A) is either a quasi-periodic or a periodic. Moreover, the range b of the window of periodic cycles constitutes a devil's staircase. When [Formula: see text], finitely many chaotic regions and window regions exist and interweave with each other. In each window, the basic periodic cycle can be identified. A sequence of period-doubling is observed to the left of the basic periodic cycle and a quasi-periodic region is observed to the right of it. For large b, the input dominates, ω(b,T,A) becomes simpler, from quasi-periodic to periodic as b increases.

This work elucidates the measure-theoretical entropy and dimensions of a unimodal map with a horizontal gap. The measure-theoretical entropy and dimensions of the F t (which is defined later) are shown to form a devil’s staircase structure with respect to the gap size t. Pesin’s formula for gap maps is also considered.

This work investigates the spherical symmetric solutions of more realistic equation of states. We generalize the method of Hsu et al. (Methods Appl. Anal. 8 (2001) 159) to show the existence of spherical symmetric weak solution of the relativistic Euler equation with initial data containing the vacuum state.

In this paper, we prove the Lagrangian stability of the quasi-periodic system d 2 x / dt 2 + G x ( x , t )=0, where G is quasi-periodic in both x and t , respectively.

This study demonstrates the devil's staircase structure of topological entropy functions for one-dimensional symmetric unimodal maps with a gap inside. The results are obtained by using kneading theory and are helpful in investigating the communication of chaos.

This study describes the spatial disorder of one-dimensional Cellular Neural Networks (CNN) with a biased term by applying the iteration map method. Under certain parameters, the map is one-dimensional and the spatial entropy of stable stationary solutions can be obtained explicitly as a staircase function.

This paper shows the spatial disorder of one-dimensional Cellular Neural Networks (CNN) using the iteration map method. Under certain parameters, the map is one-dimensional and the spatial entropy of stable stationary solutions can be obtained explicitly as a devil-staircase like function.

This work investigates binary pattern formations of two-dimensional standard cellular neural networks (CNN) as well as the complexity of the binary patterns. The complexity is measured by the exponential growth rate in which the patterns grow as the size of the lattice increases, i.e. spatial entropy. We propose an algorithm to generate the patterns in the finite lattice for general two-dimensional CNN. For the simplest two-dimensional template, the parameter space is split up into finitely many regions which give rise to different binary patterns. Qualitatively, the global patterns are classified for each region. Quantitatively, the upper bound of the spatial entropy is estimated by computing the number of patterns in the finite lattice, and the lower bound is given by observing a maximal set of patterns of a suitable size which can be adjacent to each other.

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.

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.

where 5 = 5a;b = {x ∈ Rn; 0iai |x|ib} is an annular domain in Rn; n ≥ 3. It is well known that the uniqueness problems of solutions of (1.1) and (1.2) are both fundamental and often di8cult. During the last decade, there has been tremendous progress in studying these problems when 5 is a ball in Rn or entire Rn n ≥ 3, see, e.g., [1,2,5,10,11]. However, very little was known concerning the annular domain until the work of Ni and Nussbaum [12]. Until very recently, Co

This work consider a more realistic equation of states. We generalize the method of DiPerna and G.-Q. Chen et al to show the existence of weak solution of the relativistic Euler equation with initial data containing the vacuum state

In this paper, two recursive formulas for computing the spatial entropy of two-dimensional subshifts of finite type are given. The exact entropy of a nontrivial example arising in cellular neural networks is obtained by using such formulas. We also establish some general theory concerning the spatial entropy of two-dimensional subshifts of finite type. In particular, we show that if either of the transition matrices is rank-one, then the associated exact entropy can be explicitly obtained. The generalization of our results to higher dimension can be similarly obtained. Furthermore, these formulas can be used numerically for estimating the spatial entropy.

This work proves the existence and multiplicity results of monotonic traveling wave solutions for some lattice differential equations by using the monotone iteration method. Our results include the model of cellular neural networks (CNN). In addition to the monotonic traveling wave solutions, non-monotonic and oscillating traveling wave solutions in the delay type of CNN are also obtained.

We consider a cellular neural network (CNN) with a bias term z in the integer lattice Z 2 on the plane R 2 . We impose a symmetric coupling between nearest neighbors, and also between next-nearest neighbors. Two parameters, a and ", are used to describe the weights between such interacting cells. We study patterns that can exist as stable equilibria. In particular, the relationship between mosaic patterns and the parameter space (z;a;") can be completely characterized. This, in turn, addresses the so-called learning problem in CNNs. The complexities of mosaic patterns are also addressed.

In this paper, we study the structure of traveling wave solutions of Cellular Neural Networks of the advanced type. We show the existence of monotone traveling wave, oscillating wave and eventually periodic wave solutions by using shooting method and comparison principle. In addition, we obtain the existence of periodic wave train solutions.

We consider cellular neural networks with symmetric space-variant feedback template. The complete stability is proved via detailed analysis on the energy function. The proof is presented for the two-dimensional case with Dirichlet boundary condition. It can be extended to other dimensions with minor adjustments. Modifications to the cases of Neumann and periodic boundary conditions are also mentioned.

For self-gravitating, spherically symmetric and isentropic gaseous star, there is a family of particular solutions when the adiabatic index γ = 4/3. We found that there is a critical total mass M0 associated with these particular solutions. If the total massM of star less than M0, then the star expands infinitely and its density ultimately tends to approach zero. WhenM ≥ M0 and the initial velocity is slower than escape velocity, then the gas is trapped and collapses toward the star’s center in a finite period of time.

We study the linearized stability of stationary solutions of gaseous stars which are in spherically symmetric and isentropic motion. If viscosity is ignored, we have following three types of problems: (EC), Euler equation with a solid core; (EP), Euler{Poisson equation without a solid core; (EPC), Euler{Poisson equation with a solid core. In Lagrangian formulation, we prove that any solution of (EC) is neutrally stable. Any solution of (EP) and (EPC) is also neutrally stable when the adiabatic index 2 ( 4 ; 2) and unstable for (EP) when 2 (1; 4 ). Moreover, for (EPC) and 2 (1, 2), any solution with small total mass is also neutrally stable. When viscosity is present ( >0), the velocity disturbance on the outer surface of gas is important. For >0, we prove that the neutrally stable solution (when = 0) is now stable with respect to positive-type disturbances, which include Dirichlet and Neumann boundary conditions. The solution can be unstable with respect to disturbances of some other types. The problems were studied through spectral analysis of the linearized operators with singularities at the endpoints of intervals.

We study the asymptotic behavior of positive solutions of the semilinear elliptic equation Δu + ƒ(u) = 0 in Ωa, u = 0 on ∂Ωa, where Ωa = {x ∈ of RN: a < |x| < a + 1} are expanding annuli as a → ∞, and ƒ is positive and superlinear at both 0 and ∞. We first show that there are a priori bounds for some positive solutions ua(x) as a → ∞. Then, if we fix any direction, after a suitable translation of ua the limiting solutions are non-negative solutions on the infinite strip. We can obtain more detailed descriptions of these limits if ua is radially symmetric, least-energy, or least-energy with a particular symmetry.

This investigation completely classifies the spatial chaos problem in plane edge coloring (Wang tiles) with two symbols. For a set of Wang tiles $\mathcal{B}$, spatial chaos occurs when the spatial entropy $h(\mathcal{B})$ is positive. $\mathcal{B}$ is called a minimal cycle generator if $\mathcal{P}(\mathcal{B})\neq\emptyset$ and $\mathcal{P}(\mathcal{B}')=\emptyset$ whenever $\mathcal{B}'\subsetneqq \mathcal{B}$, where $\mathcal{P}(\mathcal{B})$ is the set of all periodic patterns on $\mathbb{Z}^{2}$ generated by $\mathcal{B}$. Given a set of Wang tiles $\mathcal{B}$, write $\mathcal{B}=C_{1}\cup C_{2} \cup\cdots \cup C_{k} \cup N$, where $C_{j}$, $1\leq j\leq k$, are minimal cycle generators and $\mathcal{B}$ contains no minimal cycle generator except those contained in $C_{1}\cup C_{2} \cup\cdots \cup C_{k}$. Then, the positivity of spatial entropy $h(\mathcal{B})$ is completely determined by $C_{1}\cup C_{2} \cup\cdots \cup C_{k}$. Furthermore, there are 39 equivalent classes of marginal positive-entropy (MPE) sets of Wang tiles and 18 equivalent classes of saturated zero-entropy (SZE) sets of Wang tiles. For a set of Wang tiles $\mathcal{B}$, $h(\mathcal{B})$ is positive if and only if $\mathcal{B}$ contains an MPE set, and $h(\mathcal{B})$ is zero if and only if $\mathcal{B}$ is a subset of an SZE set.

This study investigates a multiplicative integer system using a method that was developed for studying pattern generation problems. The entropy and the Minkowski dimensions of general multiplicative systems can thus be computed. A multi-dimensional decoupled system is investigated in three main steps. (I) Identify the admissible lattices of the system; (II) compute the density of copies of admissible lattices of the same length, and (III) compute the number of admissible patterns on the admissible lattices. A coupled system can be decoupled by removing the multiplicative relation set and then performing procedures similar to those applied to a decoupled system . The admissible lattices are chosen to be the\ maximum graphs of different degrees which are mutually independent. The entropy can be obtained after the remaining error term is shown to approach zero as the degree of the admissible lattice tends to infinity.

The degree of mixing is a fundamental property of a dynamical system. General multi-dimensional shifts cannot be systematically determined. This work introduces constructive and systematic methods for verifying the degree of mixing, from topological mixing to strong specification (or strong irreducibility) for two-dimensional shifts of finite type. First, transition matrices on infinite strips of width $n$ are introduced for all $n\geq 2$. To determine the primitivity of the transition matrices, connecting operators are introduced to reduce the order of high-order transition matrices to yield lower-order transition matrices. Two sufficient conditions for primitivity are provided; they are invariant diagonal cycles and primitive commutative cycles of connecting operators. After primitivity is established, the corner-extendability and crisscross-extendability are used to demonstrate topological mixing. In addition, the hole-filling condition yields the strong specification. All mentioned conditions can be verified to apply in a finite number of steps.