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Thesis (Ph. D.)--University of Texas at Austin, 2000. / Vita. Includes bibliographical references (leaves 156-164). Available also in a digital version from Dissertation Abstracts.

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The main objects of study of the thesis are two classes of special vector configurations appeared in the geometry and the theory of integrable systems. In the first part we consider a special class of vector configurations known as the V-systems, which appeared in the theory of the generalised Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations. Several families of V-systems are known, but their classification is an open problem. We derive the relations describing the infinitesimal deformations of V-systems and use them to study the classification problem for V-systems in dimension 3. In particular, we prove that the isolated cases in Feigin-Veselov list admit only trivial deformations. We present the catalogue of all known 3D V-systems including graphical representations of the corresponding matroids and values of v-functions. In the second part we study the vector configurations, which form vertex sets for a new class of polyhedra called affine B-regular. They are defined by a 3-dimensional analogue of the Buffon procedure proposed by Veselov and Ward. The main result is the proof of existence of star-shaped affine B-regular polyhedron with prescribed combinatorial structure, under partial symmetry and simpliciality assumptions. The proof is based on deep results from spectral graph theory due to Colin de Verdière and Lovász.

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The purpose of this thesis is to investigate how "spect ralness" properties of a torsion theory T on R - Mod are reflected by properties of the ring R and its ring of quotients R,.. The development of "spectral" torsion theory owes much to Zelmanowitz [50] and Gomez-Pardo [23] . Gomez-Pardo proved that there exists a bijective correspondence between the set of spectral torsion theories on R - Modand rings of quotients of R that are Von Neumann regular and left self-injective. Chapter 1 is concerning with the notation used in the thesis and a summary of main results which are needed for understanding the sequel. Chapter 2 is concerned with the construction of a maximal ring of quotients of an arbitrary ring R by using the notion of denseness and relative injective hull. In Chapter 3, we survey the three equivalent ways of formulating Torsion Theory: by means of preradical functors on the category R- Mod, pairs of torsion / torsion-free classes and topologizing filters on rings. We shall show that Golan's approach to Torsion Theory via equivalence classes of injectives; and Dickson's one (as presented by Stenstrom) are equivalent. With a torsion theory T defined on R-Mod we associate R,. a ring of quotients of R. The full subcategory (R, T) - Mod of R- Mod whose objects are the T-torsion-free r-injective left R-modules is a Grothendieck category called the quotient category of R - Mod with respect to T. A left R,.-module that is r-torsion-free T-injective as a left R-module is injective if and only if it is injective as a left R-module (Proposition 3.6.4). Because of its use in the sequel , particular attention is paid to the lattice isomorphism that exists between the lattice of .r-pure submodules of a left Rmodule M and the lattice of subobjects of the quotient module M; in the category (R , T) - Mod. Chapter 4 introduces the definition of a spectral torsion theory: a Vll torsion theory r on R - Mod is said to be spectral if the Grothendieck category (R, r) - Mod is spectral. Using the notion of relative essential submodule, one can construct a spectral torsion theory from an arbitrary torsion theory on R - Mod. We shall show how an investigation of a general spectral torsion theory on R - Mod reduces to the Goldie torsion theory on R/tT (R) - Mod. Moreover, we shall exhibit necessary and sufficient conditions for R; to be a regular left self-injective ring (Theorem 4.2.10). In Chapter 5, after constructing the torsion functor Soce(-) which is associated with the pseudocomplement r.l of r in R - tors, we show how semiartinian rings can be characterized by means of spectral torsion theories: if a spectral torsion theory r on R - Mod is generated by the class of r-torsion simple left R-modules or, equivalently, cogenerated by the class of r-torsion-free simple left R-modules, then R is a left semiartinian ring (Proposition 5.3.2). Chapter 6 gives Zelmanowitz' important result [50]: R; is a semisimple artinian ring if and only if the torsion theory r is spectral and the associated left Gabriel topology has a basis of finitely generated left ideals. We also exhibit results due to M.J. Arroyo and J. Rios ([4] and [5]) which illustrate how spectral torsion theories can be used to describe when R; is (1) prime regular and left self-injective, (2) a left full linear ring, and (3) a direct product of left full linear rings. We also study the relationship between the flatness of the ring of quotients R; and the r- coherence of the ring R when r is a spectral torsion theory. It is proved that if r is a spectral torsion theory on R - Mod then the following conditions are equivalent: (1) R is left r-coherent; (2) (Rr)R is flat; (3) every right Rr-module is flat as a right R-module (Proposition 6.3.9). This result is an extension of Cateforis' results. / Thesis (M.Sc) - University of Natal, Pietermaritzburg, 2003

Thesis (M.S.)--Rochester Institute of Technology, 1990. / Includes bibliographical references.

Spectral moment formulae of various shapes have proven to be very successful in studying the statistics of central 𝐿-values. In this article, we establish, in a completely explicit fashion, such formulae for the family of 𝐺𝐿(3) × 𝐺𝐿(2) Rankin-Selberg 𝐿-functions using the period integral method. The Kuznetsov and the Voronoi formulae are not needed in our argument. We also prove the essential analytic properties and explicit formulae for the integral transform of our moment formulae. It is hoped that our method will provide insights into moments of 𝐿-functions for higher-rank groups.