archives@tulane.edu / The primary objects studied in this dissertation are ordinary and symbolic powers of monomial ideals in a polynomial ring over a field. In particular, we are interested in studying their local cohomology and Castlenuovo-Mumford regularity. In Chapter 3, we restrict our study to edge ideals of unicyclic graphs, that is, squarefree monomial ideals generated in degree $2$ corresponding to a graph that has a single cycle. When the cycle is even, the symbolic power was known to coincide with the ordinary power. When the cycle is odd, we are able to describe the symbolic powers explicitly, which allows us to compute invariants of the ideal explicitly. Furthermore, in certain cases, we can calculate the Castelnuovo-Mumford regularity. In Chapter 4, we study ideals that can be written as the sum of monomial ideals in different polynomial rings. In order to study the graded local cohomology of these ideals, we use a formula of Takayama which allows us to translate this problem of computing homology of certain simplicial complexes called \textit{degree complexes}. We build up the construction of the degree complexes of ordinary and symbolic powers of sums, and then we use this to discuss their graded local cohomologies. In Chapter 5, we study ideals that can be written as the fiber product of squarefree monomial ideals in different polynomial rings. Building on the construction from Chapter 4, we are able to determine that the nonempty faces in the degree complex of ordinary and symbolic powers of fiber products come from the faces of the degree complexes of powers of the component ideals. This allows us to compute the homology of these degree complexes explicitly. Furthermore, this allows us to compute the regularity of symbolic powers of fiber products of squarefree monomial ideals in terms of the regularities of the component ideals. / 1 / Jonathan O'Rourke

The class of squarefree monomial ideals is a classical object in commutative algebra, which has a strong connection to combinatorics. Our main goal throughout this dissertation is to study the algebraic properties of squarefree monomial ideals using combinatorial structures and invariants of hypergraphs. We focus on the following algebraic properties and invariants: the persistence property, non-increasing depth property, Castelnuovo-Mumford regularity and projective dimension. It has been believed for a long time that squarefree monomial ideals satisfy the persistence property and non-increasing depth property. In a recent work, Kaiser, Stehlik and Skrekovski provided a family of graphs and showed that the cover ideal of the smallest member of this family gives a counterexample to the persistence and non-increasing depth properties. We show that the cover ideals of all members of their family of graphs indeed fail to have the persistence and non-increasing depth properties. Castelnuovo-Mumford regularity and projective dimension are both important invariants in commutative algebra and algebraic geometry that govern the computational complexity of ideals and modules. Our focus is on finding bounds for the regularity in terms of combinatorial data from associated hypergraphs. We provide two upper bounds for the edge ideal of any vertex decomposable graph in terms of induced matching number and the number of cycles. We then give an upper bound for the edge ideal of a special class of vertex decomposable hypergraphs. Moreover, we generalize a domination parameter from graphs to hypergraphs and use it to give an upper bound for the projective dimension of the edge ideal of any hypergraph. / Mengyao Sun

The use of graphs in algebraic studies is ubiquitous, whether the graphs be finite or infinite, directed or undirected. Green and Zacharia have characterized finite generation of the cohomology rings of monomial algebras, and thereafter G. Davis determined a finite criteria for such generation in the case of cycle algebras. Herein, we describe the construction of a finite directed graph upon which criteria can be established to determine finite generation of the cohomology ring of in-spoked cycle" algebras, a class of algebras that includes cycle algebras. We then show the further usefulness of this constructed graph by studying other monomial algebras, including d-Koszul monomial algebras and a new class of monomial algebras which we term "left/right-symmetric" algebras. / Ph. D.

An abundance of information regarding the structure of a finite group can be obtained by studying its irreducible characters. Of particular interest are monomial characters – those induced from a linear character of some subgroup – since Brauer has shown that any irreducible character of a group can be written as an integral linear combination of monomial characters. Our primary focus is the class of M-groups, those groups all of whose irreducible characters are monomial. A classical theorem of Taketa asserts that an M-group is necessarily solvable, and Dade proved that every solvable group can be embedded as a subgroup of an M-group. After discussing results related to M-groups, we will construct explicit families of solvable groups that cannot be embedded as subnormal subgroups of any M-group. We also discuss groups possessing a unique non-monomial irreducible character, and prove that such a group cannot be simple.

In this thesis we have presented original homomorphic images of permutations and monomial progenitors. In some cases we have used the double coset enumeration tech- nique to construct the images and for all of the homomorphic images that we have discovered, the isomorphism type of each group is given. The homomorphic images discovered include Linear groups, Alternating groups, and two sporadic simple groups J1 and J2X2 where J1 is the smallest Janko group and J2 is the second Janko sporadic group.

In this thesis we study the number of periodic points of congruential monomial dynamical system. By concept of index calculus we are able to calculate the number of solutions for congruential equations. We give formula for the number of r-periodic points over prime power. Then we discuss about calculating the total number of periodic points and cycles of length r for prime power.

In this thesis, we study Hilbert functions of monomial ideals in the polynomial ring and the Kruskal-Katona ring. In particular, we classify Gotzmann edge ideals and, more generally, Gotzmann squarefree monomial ideals. In addition, we discuss Betti numbers of Gotzmann ideals and measure how far certain edge ideals are from Gotzmann. This thesis also contains a thorough account the combinatorial relationship between lex segments and Macaulay representations of their dimensions and codimensions.

In this paper, for a select group of quadratic algebras, we investigate restrictions necessary on the generators of the ideal for the resulting algebra to be Koszul. Techniques include the use of Gröbner bases and development of Koszul resolutions. When the quadratic algebra is Koszul, we provide the associated linear resolution of the field. When not Koszul, we describe the maps of the resolution up to the instance of nonlinearity. / Ph. D.

The main objective of this project is to find the original symmetric presentations of some very important finite groups and to give our constructions of some of these groups. We have found the Mathieu sporadic group M11, HS × D5, where HS is the sporadic group Higman-Sim group, the projective special unitary group U(3; 5) and the projective special linear group L2(149) as homomorphic images of the monomial progenitors 11*4 :m (5 :4), 5*6 :m S5 and 149*2 :m D37. We have also discovered 24 : S3 × C2, 24 : A5, (25 : S4), 25 : S3 × S3, 33 : S4 × C2, S6, 29: PGL(2,7), 22 • (S6 : S6), PGL(2,19), ((A5 : A5 × A5) : D6), 6 • (U4(3): 2), 2 • PGL(2,13), S7, PGL (2,8), PSL(2,19), 2 × PGL(2,81), 25 : (S6 × A5), 26 : S4 × D3, U(4,3), 34 : S4, 32 :D6, 2 • (PGL(2,7) :PSL(2,7), 22 : (S5 : S5) and 23 : (PSL3(4) : 2) as homomorphic images of the permutation progenitors 2*8 : (2 × 4 : 2), 2*16: (2 × 4 :C2 × C2), 2*9: (S3 × S3), 2*9: (S3 × A3), 2*9: (32 × 23) and 2*9: (33 × A3). We have also constructed 24: S3 × C2, 24 : A5, (25: S4), 25 : S3 × S3,: 33: S4 × C2, S6, M11 and U (3,5) by using the technique of double coset enumeration. We have determined the isomorphism types of the most of the images mentioned in this thesis. We demonstrate our work for the following examples: 34 : (32 * 23) × 2, 29 : PGL(2,7), 2•S6, (54 : (D4 × S3)), and 3: •PSL(2,19) ×2.

The aim of this project is to keep the x-ray exposure of the patient as low as reasonably achievable while improving the diagnostic image quality for the radiologist. The means to achieve these goals is to develop and evaluate an ecient adaptive ltering (denoising/image enhancement) method that fully explores true 4D image acquisition modes. The proposed prototype system uses a novel lter set having directional lter responses being monomials. The monomial lter concept is used both for estimation of local structure and for the anisotropic adaptive ltering. Initial tests on clinical 4D CT-heart data with ECG-gated exposure has resulted in a signicant reduction of the noise level and an increased detail compared to 2D and 3D methods. Another promising feature is that the reconstruction induced streak artifacts which generally occur in low dose CT are remarkably reduced in 4D.