Making proofs without Modus Ponens: An introduction to the combinatorics and complexity of cut elimination

This paper is intended to provide an introduction to cut elimination which is accessible to a broad mathematical audience. Gentzen's cut elimination theorem is not as well known as it deserves to be, and it is tied to a lot of interesting mathematical structure. In particular we try to indicate some dynamical and combinatorial aspects of cut elimination, as well as its connections to complexity theory. We discuss two concrete examples where one can see the structure of short proofs with cuts, one concerning feasible numbers and the other concerning "bounded mean oscillation" from real analysis

Scalable computation of intracellular metabolite concentrations

Current mathematical frameworks for predicting the flux state and macromolecular composition of the cell do not rely on thermodynamic constraints to determine the spontaneous direction of reactions. These predictions may be biologically infeasible as a result. Imposing thermodynamic constraints requires accurate estimations of intracellular metabolite concentrations. These concentrations are constrained within physiologically possible ranges to enable an organism to grow in extreme conditions and adapt to its environment. Here, we introduce tractable computational techniques to characterize intracellular metabolite concentrations within a constraint-based modeling framework. This model provides a feasible concentration set, which can generally be nonconvex and disconnected. We examine three approaches based on polynomial optimization, random sampling, and global optimization. We leverage the sparsity and algebraic structure of the underlying biophysical models to enhance the computational efficiency of these techniques. We then compare their performance in two case studies, showing that the global-optimization formulation exhibits more desirable scaling properties than the random-sampling and polynomial-optimization formulation, and, thus, is a promising candidate for handling large-scale metabolic networks

Prime Ideals of Fixed Rings

Throughout this thesis, S is a ring, G is a finite group of automorphisms of S and R is the fixed ring SG. We are concerned here with the correlation between properties of R and properties of S. In Chapter 2, we discuss certain finiteness conditions for the ring R. D.S. Passman has asked, "Is the fixed ring of kH, where k is a field and H is a polycyclic-by-finite group, Noetherian for any finite group G ?" We produce infinitely many examples for which the answer to this question is "yes". The most important results of this thesis are contained in Chapter 3. We develop the Morita prime correspondence of Chapter 1, §2, to produce results relating SpectR := (p ∈ SpecR: tr(S) ∉ p) to SpecfS := (P ∈ SpecS: Eg∈G g ∉ (P0*G)) where Po*G is an ideal in the skew group ring S*G. S. In Chapter 4, we restrict our attention to the case where S is a group algebra. We conclude this thesis in Chapter 5 with some results on localisation in the ring R. Many of these are inspired by the methods of Warfield in [W1]. We find that, with the necessary hypotheses, SpectR has the strong second layer condition. (Abstract shortened by ProQuest.)

Ahlfors circle maps and total reality: from Riemann to Rohlin

This is a prejudiced survey on the Ahlfors (extremal) function and the weaker {\it circle maps} (Garabedian-Schiffer's translation of "Kreisabbildung"), i.e. those (branched) maps effecting the conformal representation upon the disc of a {\it compact bordered Riemann surface}. The theory in question has some well-known intersection with real algebraic geometry, especially Klein's ortho-symmetric curves via the paradigm of {\it total reality}. This leads to a gallery of pictures quite pleasant to visit of which we have attempted to trace the simplest representatives. This drifted us toward some electrodynamic motions along real circuits of dividing curves perhaps reminiscent of Kepler's planetary motions along ellipses. The ultimate origin of circle maps is of course to be traced back to Riemann's Thesis 1851 as well as his 1857 Nachlass. Apart from an abrupt claim by Teichm\"uller 1941 that everything is to be found in Klein (what we failed to assess on printed evidence), the pivotal contribution belongs to Ahlfors 1950 supplying an existence-proof of circle maps, as well as an analysis of an allied function-theoretic extremal problem. Works by Yamada 1978--2001, Gouma 1998 and Coppens 2011 suggest sharper degree controls than available in Ahlfors' era. Accordingly, our partisan belief is that much remains to be clarified regarding the foundation and optimal control of Ahlfors circle maps. The game of sharp estimation may look narrow-minded "Absch\"atzungsmathematik" alike, yet the philosophical outcome is as usual to contemplate how conformal and algebraic geometry are fighting together for the soul of Riemann surfaces. A second part explores the connection with Hilbert's 16th as envisioned by Rohlin 1978.Comment: 675 pages, 199 figures; extended version of the former text (v.1) by including now Rohlin's theory (v.2

UTRGV Undergraduate Catalog 2015-2017

https://scholarworks.utrgv.edu/utrgvcatalogs/1000/thumbnail.jp

Wellesley College Courses [2008-2009]

https://repository.wellesley.edu/catalogs/1106/thumbnail.jp