Song-Sun Lin

Stability of gaseous stars in spherically symmetric motions

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.

On the existence of positive radial solutions for nonlinear elliptic equations in annular domains

Abstract We study the existence of positive radially symmetric solutions of Δu + g(¦x¦)f(u) = 0 in annulus with Dirichlet (Dirichlet/Neumann) boundary conditions. We show that the equation has a positive radial solution on any annulus if f and f are positive and f is superlinear at 0 and ∞.

COMPLETE STABILITY FOR STANDARD CELLULAR NEURAL NETWORKS

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.

Traveling waves in cellular neural networks

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.

On non-radially symmetric bifurcation in the annulus

We discuss the radially symmetric solutions and the non-radially symmetric bifurcation of the semilinear elliptic equation Δu + 2δeu = 0 in Ω and u = 0 on ∂Ω, where Ω = {xϵ R2: a2 δ∗(0, a). The upper branch of radial solutions has a non-radially symmetric bifurcation (symmetry breaking) at each δ∗(k, a), k ⩾ 1. As a → 0, the radial solutions will tend to the radial solutions on the disk and δ∗(0, a) → δ∗ = 1, the critical number on the disk.

CELLULAR NEURAL NETWORKS: LOCAL PATTERNS FOR GENERAL TEMPLATES

This work investigates the mosaic local patterns for cellular neural networks with general templates. Our results demonstrate that the set of templates can be divided into many nite regions. In each region, the same family of local patterns can be generated. Conversely, our results further demonstrate that some templates can realize a family of local patterns which can be linearly separated by a hyperplane in the conguration space. This study also proposes algorithms for verifying the linear separability for a given family of local patterns and, when it is separable, for obtaining the associated template.

On the number of positive solutions for nonlinear elliptic equations when a parameter is large

wheref(u) > 0 for u > 0 and A > 0, and Q is a bounded smooth domain in R”. Equation (1.1) arises in nonlinear heat generation, in models of combustion, etc. We refer to the survey paper by Lions [ 121. In [lo], the author proved that the positive solution of (1.1) is unique for A is large when f is bounded and satisfies a “concavity” condition. More precisely, (1.1) has an unique positive solution for large A if f satisfies the following assumptions: (i) f E C’W, CQ)), (ii) f(u) 2 m > 0 for each u I 0 and some m > 0, (iii) lim f(u)/u = 0, U’+m (iv) lim inf f(u) > lim sup f’(u)u. U’+m ll++CC We proved that solutions z+, of (1.1) satisfies z+, L ,lCu, if A is large, where u0 is the solution of

On the structure of multi-layer cellular neural networks

Abstract Let Y ⊆ { − 1 , 1 } Z ∞ × n be the mosaic solution space of an n-layer cellular neural network. We decouple Y into n subspaces, say Y ( 1 ) , Y ( 2 ) , … , Y ( n ) , and give a necessary and sufficient condition for the existence of factor maps between them. In such a case, Y ( i ) is a sofic shift for 1 ⩽ i ⩽ n . This investigation is equivalent to study the existence of factor maps between two sofic shifts. Moreover, we investigate whether Y ( i ) and Y ( j ) are topological conjugate, strongly shift equivalent, shift equivalent, or finitely equivalent via the well-developed theory in symbolic dynamical systems. This clarifies, in a multi-layer cellular neural network, each layerʼs structure. As an extension, we can decouple Y into arbitrary k-subspaces, where 2 ⩽ k ⩽ n , and demonstrates each subspaceʼs structure.

On the spatial entropy and patterns of two-dimensional cellular neural networks

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.

SPATIAL DISORDER OF CNN — WITH ASYMMETRIC OUTPUT FUNCTION

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.