acase@tulane.edu

acase@tulane.edu

acase@tulane.edu

Philip Rosenau introduced the equation $u\sb{t} + (u + u\sp2)\sb{x} + u\sb{xxxxt} = 0,$ which models approximately the dynamics of certain large discrete systems. We study the Rosenau equation with initial and boundary conditions. First we establish global existence and uniqueness of solutions to the mixed problem for the generalized one dimensional Rosenau equation. Secondly we establish global existence and uniqueness in higher dimensional spaces. Then we study qualitative properties of the solutions, considering the initial value problem for the generalized one dimensional Rosenau equation. Of particular concern are pointwise decay estimates / acase@tulane.edu

acase@tulane.edu

As evolutionary psychologists argue, the thinking mind as a product of evolution contains domain-specific modules that are designed to execute specific adaptive functions in the ancestral environments. However, cultural diversity and novelty indicate cognitive domains that do not match those specific adaptive functions. This dissertation tries to solve this domain-inspecificity problem by investigating how a mind that is designed to be modular can lose domain specificity The notion of demodularization is raised to characterize the phenomenon of losing domain specificity of a module. The key to explaining how demodularization of a modular mind is possible is to explore a particular effect of the metarepresentational capability. I argue that, as a capability of representing representing relations, the metarepresentational capability can be divided into two kinds: the metarepresentational (R) capability and the metarepresentational (I) capability. Metarepresentational (R) capability explains demodularization by bringing about a new semantics (mental attitudes) and function of representations, and therefore, reformats representations into a common format. The common format allows domain-inspecific computations, enables new cognitive functions. A result is that representations become manipulable, which in turn explains cultural explosion. Metarepresentational (I) capability explains how a modular mind can reach the level of metarepresentation (R). As part of the theory-of-mind module, the metarepresentational (I) capability mediates between a modular mind and demodularization. Speculations are made to support this claim / acase@tulane.edu

acase@tulane.edu

acase@tulane.edu

This dissertation consists of two chapters. Chapter I concerns the equation (1) du/dt = Au + f(t,u), u(0) = u(,0), where, for X a Banach space, u(,0) (ELEM) X, f : {0,(tau)} x X (--->) X is jointly continuous and uniformly approximable by functions which are Lipschitz continuous in X, and A generates a C(,0)-semigroup, {S(t) : t (GREATERTHEQ) 0}, on X such that for t > 0, S(t) is compact In the case where f is bounded, it is shown that mild solutions of (1) exist on {0,(tau)} and that the set of all mild solutions of (1), i.e. {u (ELEM) C({0,(tau)},X) : u is a mild solution of (1)} is a compact set and is homeomorphic to the intersection of a decreasing sequence of absolute retracts In the general case, it is shown that mild solutions of (1) exist on a fixed interval {0,(eta)} (L-HOOK EQ) {0,(tau)}, and the set of all mild solutions on {0,(eta)} is compact and is homeomorphic to the intersection of a decreasing sequence of absolute retracts Chapter II concerns an ergodic theorem due to N. Wiener. He proved that if u is a finite Borel measure, u = u(,c) + u(,d) the Lebesgue decomposition of u into continuous and discrete parts and if u(t) = 1/(2(pi))(' 1/2) (INT)(,(//R)) e('itx) u(dx) is the Fourier transform of u, then (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) as T (--->) (INFIN) There is a natural reformulation of this result. Let iA generate a C(,0)-unitary group, {U(t) : t (ELEM) (//R)}, on a Hilbert space H, where A is a self-adjoint operator with the spectral family {E(,A)((gamma)) : (gamma) (ELEM) (//R)}. Let (sigma)(,p)(A) be the point spectrum of A and f, g (ELEM) H. Then (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) as T (--->) (INFIN) The main generalization of this is as follows. Let A generate a C(,0)-contraction semigroup, {S(t) : t (GREATERTHEQ) 0}, on a Hilbert space H. Let P(,(gamma)) be the projection onto ker(A-(gamma)I), and f, g (ELEM) H. Then (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) as T (--->) (INFIN) Formula (2) was known to N. Wiener {21}. Chapter II contains simple proofs of (2) and (3). Moreover, necessary and sufficient conditions are given for the convergence in (2) and (3) to occur at a rate 0((alpha)) where (alpha) : (//R)('+) (--->) (//R)('+) and (alpha)(t) (--->) 0 as t (--->) (INFIN) Results are obtained with the concept of almost convergence replacing the slightly weaker Cesaro convergence of (2) and (3). An extension of these results to the case of a semigroup which is similar to a contraction semigroup is also given / acase@tulane.edu

acase@tulane.edu