We consider a nonsymmetric algebraic matrix Riccati equation arising from transport theory. The nonnegative solutions of the equation can be explicitly constructed via the inversion formula of a Cauchy matrix. An error analysis and numerical results are given. We also show a comparison theorem of the nonnegative solutions.B −à , whereB,C are symmetric andà is arbitrary. On the other hand, nonsymmetric Riccati equations (see, e.g., [7,18]) are less well understood than their symmetric counterparts. Note that H, given in (10), is a Hamiltonian matrix only when c = 1 and α = 0, which is why we call it a Hamiltonian-like matrix. Moreover, we are seeking a nonnegative solution of (6), as opposed to the positive semidefinite solutions found in linear-quadratic control problems or the nonsingular solutions found in polynomial factorizations. This paper is organized as follows. In section 2, we analyze the eigenvalue distribution of H and characterize the components of the associated eigenvectors. In section 3, a complete representation and bifurcation diagram of the nonnegative solutions of (6) are established. In particular, we show that (6) has a unique nonnegative solution when c = 1 and α = 0 and two nonnegative solutions otherwise. An error analysis and some numerical experiments for the computation of the nonnegative solutions are given in section 4. In section 5, some comparison results are derived. Specifically, we are able to show that the minimal solution of (6) is increasing in c and decreasing in α. Our concluding section primarily contains some thoughts regarding possible future research related to the results presented here. For completeness and

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 ε, 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; ε) can be completely characterized. This, in turn, addresses the so-called learning problem in CNNs. The complexities of mosaic patterns are also addressed.

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The onset of entanglement was determined from studying the viscoelastic spectra of the blends consisting of two nearly monodisperse polystyrene polymers:  component one having a molecular weight slightly less than the entanglement molecular weight M e (=4ρRT/5G N), whereas that of component two being greater than M e (such a blend is referred to as the blend solution). It is shown that because of the absence of the hydrodynamic interaction the viscoelastic spectrum of the blend solution can be described by applying the Rouse theory to both components in the entanglement-free region. As entanglements among the chains of component two occur, when its weight fraction, W 2, increases above a critical point, the viscoelastic response of component one remains described by the Rouse theory. With the Rouse viscoelastic response of component one as the internal reference, the onset of chain entanglement among the chains of component two can be determined by monitoring the deviation of the viscoelastic response of component two from being described by the Rouse theory as a function of increasing W 2. According to the study, entanglement starts to occur in the close neighborhood of M e‘ (the entanglement molecular weight of the blend solution and M e‘ = M e W 2 -1) for the polymer blend solution or equivalently M e for the monodisperse polymer melt well below M c‘ or M c (M c‘ = M c W 2 -1, where M c is the critical molecular weight of the zero shear viscosity of the polymer melt).

We consider a matrix Riccati equation containing two parameters c and ␣. The quantity c denotes the average total number of particles emerging from a collision, Ž . Ž . which is assumed to be conservative i.e., 0 -c F 1 , and ␣ 0 F ␣ -1 is an ÄŽ . 4 angular shift. Let S s c, ␣ : 0 -c F 1 and 0 F ␣ -1 . Stability analysis for two steady-state solutions X and X are provided. In particular, we prove that