Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1981. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND SCIENCE. / Bibliography: leaves 79-81. / by Ranee Kathryn Gupta. / Ph.D.

The abelian sandpile model, or chip firing game, is a cellular automaton on finite directed graphs often used to describe the phenomenon of self organized criticality. Here we present a thorough introduction to the theory of sandpiles. Additionally, we define a symmetric sandpile configuration, and show that such configurations form a subgroup of the sandpile group. Given a graph, we explore the existence of a quotient graph whose sandpile group is isomorphic to the symmetric subgroup of the original graph. These explorations are motivated by possible applications to counting the domino tilings of a 2n × 2n grid.

In this thesis we study moduli stacks \calM_\Phi^n, indexed by an integer n>0 and a CM-type (K,\Phi), which parametrize abelian schemes equipped with action by \OK and an \OK-linear principal polarization, such that the representation of \OK on the relative Lie algebra of the abelian scheme consists of n copies of each character in \Phi. We do this by systematically applying Serre's tensor construction, and for that we first establish a general correspondence between polarizations on abelian schemes M\otimes_R A arising from this construction and polarizations on the abelian scheme A, along with positive definite hermitian forms on the module M. Next we describe a tensor product of categories and apply it to the category \Herm_n(\OK) of finite non-degenerate positive-definite \OK-hermitian modules of rank n and the category fibred in groupoids \calM_\Phi^1 of principally polarized CM abelian schemes. Assuming n is prime to the class number of K, we show that Serre's tensor construction provides an identification of this tensor product with a substack of the moduli space \calM_\Phi^n, and that in some cases, such as when the base is finite type over \CC or an algebraically closed field of characteristic zero, this substack is the entire space. We then use this characterization to describe the Galois action on \calM_\Phi^n(\overline{\QQ}), by using the description of the action on \calM_\Phi^1(\overline{\QQ}) supplied by the main theorem of complex multiplication.

In this thesis we study moduli stacks \calM_\Phi^n, indexed by an integer n>0 and a CM-type (K,\Phi), which parametrize abelian schemes equipped with action by \OK and an \OK-linear principal polarization, such that the representation of \OK on the relative Lie algebra of the abelian scheme consists of n copies of each character in \Phi. We do this by systematically applying Serre's tensor construction, and for that we first establish a general correspondence between polarizations on abelian schemes M\otimes_R A arising from this construction and polarizations on the abelian scheme A, along with positive definite hermitian forms on the module M. Next we describe a tensor product of categories and apply it to the category \Herm_n(\OK) of finite non-degenerate positive-definite \OK-hermitian modules of rank n and the category fibred in groupoids \calM_\Phi^1 of principally polarized CM abelian schemes. Assuming n is prime to the class number of K, we show that Serre's tensor construction provides an identification of this tensor product with a substack of the moduli space \calM_\Phi^n, and that in some cases, such as when the base is finite type over \CC or an algebraically closed field of characteristic zero, this substack is the entire space. We then use this characterization to describe the Galois action on \calM_\Phi^n(\overline{\QQ}), by using the description of the action on \calM_\Phi^1(\overline{\QQ}) supplied by the main theorem of complex multiplication.

The purpose of this paper was to find a general formula to count the number of automorphisms of any finite abelian group. These groups were separated into five different types. For each of the first three types, theorems were proven, and formulas were derived based on the theorems. A formula for the last two types of groups was derived from a theorem based on a conjecture which was proven in only one direction. Then it was shown that a count found from any of the first three formulas could also be found using the last formula. The result of these comparisons gave credence to the conjecture. Thus we found that the last formula is a general formula to count the number of automorphisms of finite abelian groups. / Department of Mathematical Sciences

Ribosomal RNA genes (rDNA) exist in yeast both as a single chromosomal array of tandemly repeated units and as extrachromosomal units named 3um plasmids, although the relationship between these two forms is unclear. Inheritance of rDNA was studied using two systems. The first used a naturally occuring rDNA restriction enzyme polymorphism between two strains to distinguish between their rDNA arrays, and the second involved cloning a tRNA suppressor gene into rDNA to label individual rDNA units. An added interest to the study of the inheritance of rDNA in yeast was the possible association between it and the inheritance of the Psi factor, an enigmatic type of nonsense suppressor in yeast which shows extra-chromosomal inheritance. In a cross heterozygous for the rDNA polymorphism and the psi factor, tetrad analysis suggested that the psi factor had segregated 4:0. The majority of the rDNA units segregated in a 2:2 fashion, which suggested that reciprocal recombination in the rDNA of psi⁺ diploids is heavily suppressed as was previously shown for psi^- diploids. A heterologous plasmid containing the tRNA suppressor gene was constructed and transformed into haploid and diploid hosts. A series of transformants was obtained and physical and genetic analysis suggested that they contained tRNA suppressor gene(s) integrated into their rDNA. In a cross heterozygous for rDNA-tRNA gene insert(s), 6% of the tetrads dissected showed a meiotic segregation of the suppressed phenotype which could most probably be accounted for by inter-chromosomal gene conversion. This observation could be interpreted in two ways. Firstly, recombination intermediates between rDNA on homologues may occur in meiosis, but they are mostly resolved as gene conversions without reciprocal cross-over. Alternatively, gene conversion tracts in rDNA are rare but very long so that the tRNA gene insert was always included in the event. 3um rDNA plasmids containing the tRNA gene marker were not detected in any of the transformants analysed. An extensive quantitative analysis of the rate of reversion of the suppressed phenotype amongst these transformants identified a particulary unstable transformant group. It was proposed that the mechanism of reversion was loss of the tRNA gene insert by unequal sisterstrand exchange, and the mechanism was shown to be independent of the recombination/repair genes RAD1, RAD52, and RAD51. A genetic analysis of stability suggested that there may have been at least two loci segregating in the host strains with additive effects on stability.

"Definition 1. A group G is an extension of a group A by a group B if and only if A is a normal subgroup of G and the factor group G/A is isomorphic to B. Definition 2. Two extensions G and H of A by B are called equivalent if and only if there exists an isomorphism between G and H that on A coincides with the identity automorphism and that maps onto each other the cosets of A corresponding to one and the same element of B. Consider the following example: let G be the cyclic group of order 4, that is G = {1, a, a², a³} and let H[subscript G] = {1, a²} be a normal subgroup of G. Now let V be the Klein four-group, that is, V = {1, a, b, c : a²=b²=c²=1} and H[subscript V] = {1, b} a normal subgroup of V. Since H[subscript G] and H[subscript V] are cyclic groups of order 2, set H[subscript V] = H[subscript G] = H. G and V are extensions of H by itself but are not equivalent extensions since no isomorphism exists between G and V. So the question arises: what are the necessary and sufficient conditions that two extensions G and G' of a group A by a group B be equivalent?"--Introduction. / Typescript. / "January 1960." / "Submitted to the Graduate Council of Florida State University in partial fulfillment of the requirements for the degree of Master of Arts." / Advisor: Paul J. McCarthy, Professor Directing Paper. / Includes bibliographical references (leaf 37).

published_or_final_version / abstract / Mathematics / Master / Master of Philosophy

No description available.

Let R be a commutative ring with identity and let G and H be abelian groups with the group algebras RG and RH isomorphic as R-algebras. In this dissertation we investigate the relationships between G and H. Let inv(R) denote the set of rational prime numbers that are units in R and let G(R) (respectively, H(R)) be the direct sum of the p-components of G (respectively, H) with p ∈ inv(R). It is known that if G(R) or H(R) is nontrivial then it is not necessarily true that G and H are isomorphic. However, if R is an integral domain of characteristic 0 or a finitely generated indecomposable ring of characteristic 0 then G/G(R) ≅ H/H(R). This leads us to make the following definition: We say that R satisfies the Isomorphism Theorem if whenever RG ≅ RH as R-algebras for abelian groups G and H then G/G(R) ≅ H/H(R). Thus integral domains of characteristic 0 and finitely generated indecomposable rings of characteristic 0 satisfy the Isomorphism Theorem. Our first major result shows that indecomposable rings of characteristic 0 (no restrictions on generation) satisfy the Isomorphism theorem. It has been conjectured that all rings R satisfy the Isomorphism Theorem. However, we show that the conjecture may fail if nontrivial idempotents are present in R. This leads us to consider necessary and sufficient conditions for rings to satisfy the Isomorphism Theorem. We say that R is an ND-ring if whenever R is written as a finite product of rings then one of the factors, say Rᵢ, satisfies inv(Rᵢ) = inv(R). We show that every ring satisfying the Isomorphism Theorem is an ND-ring. Moreover, if R is an ND-ring and if inv(R) is not the complement of a single prime we show that R must satisfy the Isomorphism Theorem. This result together with some other fragmentary evidence leads us to conjecture that R satisfies the Isomorphism Theorem if and only if R is an ND-ring. Finally we obtain several equivalent formulations of our conjecture. Among them is the result that every ND-ring satisfies the Isomorphism Theorem if and only if every field of prime characteristic satisfies the Isomorphism Theorem.