Thesis (M. Sc.)--University of Canterbury, 2008. / Typescript (photocopy). Includes bibliographical references (p. [58]). Also available via the World Wide Web.

Thesis (M. Sc.)--Dept. of Mathematics and Statistics, Concordia University, 1996. / "May 1996." Includes bibliographical references. Available also on the Internet.

We discuss Lojasiewicz's beautiful proof of the division theorem for smooth functions. The standard proofs are based on the Weierstrass preparation theorem for analytic functions and use techniques from the theory of partial differential equations. Lojasiewicz's approach is more geometric and syn¬thetic. In the appendices appear new proofs of results which are required for the theorem. / Dissertation (MSc (Mathematics))--University of Pretoria, 2006. / Mathematics and Applied Mathematics / unrestricted

Modern data science applications demand solving large-scale optimization problems. The prevalent approaches are first-order methods, valued for their scalability. These methods are implemented to tackle highly irregular problems where assumptions of convexity and smoothness are untenable. Seeking to deepen the understanding of these methods, we study first-order methods with constant step size for minimizing locally Lipschitz tame functions. To do so, we propose notions of discrete Lyapunov stability for optimization methods. Concerning common first-order methods, we provide necessary and sufficient conditions for stability. We also show that certain local minima can be unstable, without additional noise in the method. Our analysis relies on the connection between the iterates of the first-order methods and continuous-time dynamics.

In this thesis, we prove a Kudla-Rapoport conjecture for 𝓨-cycles on exotic smooth unitary Rapoport-Zink spaces of odd arithmetic dimension, i.e. the arithmetic intersection numbers for 𝓨-cycles equals the derivatives of local representation density. We also compare 𝓩-cycles and 𝓨-cycles on these RZ spaces. The method is to relate both geometric and analytic sides to the even dimensional case and reduce the conjecture to the results in \cite{LL22}.