Stig I. Andersson - Böcker

most polynomial growth on every half-space Re (z) ::::: c. Moreover, Op(t) depends holomorphically on t for Re t > O. General references for much of the material on the derivation of spectral functions, asymptotic expansions and analytic properties of spectral functions are [A-P-S] and [Sh], especially Chapter 2. To study the spectral functions and their relation to the geometry and topology of X, one could, for example, take the natural associated parabolic problem as a starting point. That is, consider the 'heat equation': (%t + p) u(x, t) = 0 { u(x,O) = Uo(x), tP which is solved by means of the (heat) semi group V(t) = e-; namely, u(*, t) = V(t)uoU* Assuming that V(t) is of trace class (which is guaranteed, for instance, if P has a positive principal symbol), it has a Schwartz kernel K E COO(X x X x Rt,E* (R)E), locally given by 00 K(x,y; t) = L>-IAk(~k (R) 'Pk)(X,y), k=O for a complete set of orthonormal eigensections 'Pk E COO(E). Taking the trace, we then obtain: 00 tA Op(t) = trace(V(t)) = 2::>- k. k=O Now, using, e. g., the Dunford calculus formula (where C is a suitable curve around a(P)) as a starting point and the standard for- malism of pseudodifferential operators, one easily derives asymptotic expansions for the spectral functions, in this case for Op.

This volume constitutes the documentation of the advanced course on Analysis of Dynamical and Cognitive Systems, held during the Summer University of Southern Stockholm in Stockholm, Sweden in August 1993.The volume contains eight carefully revised full versions of the invited three-to-four hour presentations as well as two abstracts. As a consequence of the interdisciplinary topic, several aspects of dynamical and cognitive systems are addressed: there are three papers on computability and undecidability, five tutorials on diverse aspects of universal cellular neural networks, and two presentations on dynamical systems and complexity.

most polynomial growth on every half-space Re (z) ::::: c. Moreover, Op(t) depends holomorphically on t for Re t > O. General references for much of the material on the derivation of spectral functions, asymptotic expansions and analytic properties of spectral functions are [A-P-S] and [Sh], especially Chapter 2. To study the spectral functions and their relation to the geometry and topology of X, one could, for example, take the natural associated parabolic problem as a starting point. That is, consider the 'heat equation': (%t + p) u(x, t) = 0 { u(x,O) = Uo(x), tP which is solved by means of the (heat) semi group V(t) = e-; namely, u(*, t) = V(t)uoU* Assuming that V(t) is of trace class (which is guaranteed, for instance, if P has a positive principal symbol), it has a Schwartz kernel K E COO(X x X x Rt,E* (R)E), locally given by 00 K(x,y; t) = L>-IAk(~k (R) 'Pk)(X,y), k=O for a complete set of orthonormal eigensections 'Pk E COO(E). Taking the trace, we then obtain: 00 tA Op(t) = trace(V(t)) = 2::>- k. k=O Now, using, e. g., the Dunford calculus formula (where C is a suitable curve around a(P)) as a starting point and the standard for- malism of pseudodifferential operators, one easily derives asymptotic expansions for the spectral functions, in this case for Op.

For the mathematical modeling of complex system behavior, dynamical systems play an increasing role. The flexibility and very rich phenomenology exhibited by such systems make them indispensible in this context. Control theory for dynamical systems is also a highly active field of research where a number of important results have been achieved recently.This combined course and workshop deals with recent results regarding dynamical systems and control theory, primarily in differential geometric terms as well as the applications of these fields to biological systems, with an emphasis on various aspects of the immune system and on neural networks.