Thesis (Ph. D.)--University of Chicago, Department of Mathematics, August 2003. / Includes bibliographical references. Also available on the Internet.

Thesis (M.Sc.)--Memorial University of Newfoundland, 2002. / Bibliography: leaves 42-43.

Thesis (M. Phil.)--University of Hong Kong, 2002. / Includes bibliographical references (leaf 98).

Thesis (M. Phil.)--University of Hong Kong, 2010. / Includes bibliographical references (p. 86-88). Also available in print.

Thesis (M.S.)--West Virginia University, 2003. / Title from document title page. Document formatted into pages; contains vi, 41 p. : ill. (some col.). Vita. Includes abstract. Includes bibliographical references (p. 40-41).

Thesis (M.S.)--Worcester Polytechnic Institute. / Keywords: Newton; globalized; inexact Newton. Includes bibliographical references (p. 61).

Invariants of involutions on central simple algebras have been extensively studied. Many important results have been collected and extended by Knus, Merkurjev, Rost and Tignol in "The Book of Involutions" [BI]. Among those invariants are, for example, the (even) Clifford algebra for involutions of the first kind and the discriminant algebra for involutions of the second kind on an algebra of even degree. In his preprint "Triality, Cocycles, Crossed Products, Involutions, Clifford Algebras and Invariants" [S05], Saltman shows that the definition of the Clifford algebra can be generalized to Azumaya algebras and introduces a special cohomology, the so-called G-H cohomology, to describe its structure. In this dissertation, we prove analogous results about the discriminant algebra D(A; [tau]), which is the algebra of invariants under a special automorphism of order two of the [lambda]-power of an algebra A of even degree n = 2m with involution of the second kind, [tau]. In particular, we generalize its construction to the Azumaya case. We identify the exterior power algebra as defined in "Exterior Powers of Fields and Subfields" [S83] as a splitting subalgebra of the m-th [lambda]-power algebra and prove that a certain invariant subalgebra is a splitting subalgebra of the discriminant algebra. Assuming well-situatedness we show how this splitting subalgebra can be described as the fixed field of an S[subscript n] x C₂- Galois extension and that the corresponding subgroup is [Sigma] = S[subscript m] x S[subscript m] [mathematic symbol] C2. We give an explicit description of the corestriction map and define a lattice E that encodes the corestriction as being trivial. Lattice methods and cohomological tools are applied in order to define the group H²(G;E) which contains the cocycle that will describe the discriminant algebra as a crossed product. We compute this group to have order four and conjecture that it is the Klein 4-group and that the mixed element is the desired cocycle. / text

This thesis introduces the infinitesimal symmetries of Dixmier-Douady gerbes over smooth manifolds. The collection of these symmetries are the counterpart for gerbes of the Lie algebra of circle invariant vector fields on principal circle bundles, and are intimately related to connective structures and curvings. We prove that these symmetries possess a Lie 2-algebra structure, and relate them to equivariant gerbes via a "differentiation functor". We also explain the relationship between the infinitesimal symmetries of gerbes and other mathematical structures including Courant algebroids and the String Lie 2-algebra. / text

It is known that the binary Goldbach problem is one of the open problems on linear equations in primes, and it has the Goldbach-Linnik problem, that is, representation of an even integer in the form of two odd primes and powers of two, as its approximate problem. The theme of my research is on linear equations in primes and powers of two. Precisely, there are two cases: one pair of linear equations in primes and powers of two, and one class of pairs of linear equations in primes and powers of two, in this thesis. In 2002, D.R. Heath-Brown and P.C. Puchta obtained that every sufficiently large even integer is the sum of two odd primes and k powers of two. Here k = 13, or = 7 under the generalized Riemann hypothesis. In 2010, B. Green and T. Tao obtained that every pair of linear equations in four prime variables with coefficients matrix A = (a_ij)s×t with s ≤ t, satisfying nondegenerate condition, that is, A has full rank and the only elements of the row-space of A over Q with two or fewer nonzero entries is the zero vector, is solvable. The restriction on the coefficient matrix means that they excluded the case of the binary Goldbach problem. Motivated by the above results, it is obtained that for every pair of sufficiently large positive even integers B1, B2, the simultaneous equation {█({B1 = p1 + p2 + 2v1 + 2v2 + · · · + 2vk ,@B2 = p3 + p4 + 2v1 + 2v2 + · · · + 2vk ,)┤ (1) is solvable, where p1, · · · , p4 are odd primes, each vi is a positive integer, and the positive integer k ≥ 63 or ≥ 31 under the generalized Riemann hypothesis. Note that, in 1989, M.C. Liu and K.M. Tsang have obtained that subject to some natural conditions on the coefficients, every pair of linear equations in five prime variables is solvable. Therefore one class of pairs of linear equations in four prime variables with special coefficient matrix and powers of two is considered. Indeed, it is deduced that every pair of integers B1 and B2 satisfying B1 ≡ 0 (mod 2), 3BB1 > e^(eB^48 ), B2 ≡ ∑_1^4▒= 1^(a_i ) (mod 2) and |B2| < BB1, where B = max1≤j≤4(2, |aj|), can be represented as {█(B1 = 〖p1〗_1 + p2 + 2^(v_1 ) + 2^(v_2 )+ · · · + 2^(v_k )@B2 = a1p1 + a2p2 + a3p3 + a4p4 + 2^(v_1 )+ 2^(v_2 )+ · · · + 2^(v_k ) )┤ (2) with k being a positive integer. Here p1, · · · p4 are odd primes, each 〖v 〗_iis a positive integer and the integral coefficients ai (i = 1, 2, 3, 4) satisfy {█((〖a 〗_1- 〖a 〗_2, 〖a 〗_3, 〖a 〗_4) = 1,@〖a 〗_1 〖a 〗_2< 0, 〖a 〗_3 〖a 〗_4<0,)┤ Moreover it is calculated that the positive integer k ≥ g(〖a 〗_1- 〖a 〗_2, 〖a 〗_3, 〖a 〗_4) where g(〖a 〗_21- 〖a 〗_22, 〖a 〗_23, 〖a 〗_24) = [(log⁡〖G(〖a 〗_21, …, 〖a 〗_24 〗)-log⁡〖F (〖a 〗_21, …, 〖a 〗_24)〗)/log0.975805-84.0285], (3) G(〖a 〗_21, 〖a 〗_22, 〖a 〗_23, 〖a 〗_24) = (min(1/(|a_24 |), 1/(|a_23 |)) - (〖|a〗_(21 )- a_22 |)/(|〖a_23 a〗_24 |) 〖(3B)〗^(-1) ×〖(3B)〗^(-1) (1-0.000001)- 〖(3B)〗^(-1-4), with B = max1≤j≤4(2, |a2j|), and F(a_21, …, a_24) = √(f(a_21)f〖(a〗_22 )) with f(a_2i) = {█(4414.15h (a_21-1)+5.088331 if a_21≠1@59.8411 if a_21=1,)┤ for i = 1, 2, and h(n) =∏_(p|n,p>2)▒(p-1)/(p-2). This result, if without the powers of two, can make up some of the cases excluded in Green and Tao’s paper. / published_or_final_version / Mathematics / Doctoral / Doctor of Philosophy

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