Physical Sciences and Mathematics Commons^™

A Principled Approach To Policy Composition For Runtime Enforcement Mechanisms, Zachary Negual Carter

USF Tampa Graduate Theses and Dissertations

Runtime enforcement mechanisms are an important and well-employed method for ensuring an execution only exhibits acceptable behavior, as dictated by a security policy. Wherever interaction occurs between two or more parties that do not completely trust each other, it is most often the case that a runtime enforcement mechanism is between them in some form, monitoring the exchange. Considering the ubiquity of such scenarios in the computing world, there has been an increased effort to build formal models of runtime monitors that closely capture their capabilities so that their effectiveness can be analysed more precisely. While models have grown more …

Lattice-Ordered Algebras That Are Subdirect Products Of Valuation Domains, Melvin Henriksen, Suzanne Larson, Jorge Martinez, R. G. Woods

Suzanne Larson

An f-ring (i.e., a lattice-ordered ring that is a subdirect product of totally ordered rings) A is called an SV-ring if A/P is a valuation domain for every prime ideal P of A. If M is a maximal ℓ-ideal of A , then the rank of A at M is the number of minimal prime ideals of A contained in M, rank of A is the sup of the ranks of A at each of its maximal ℓ-ideals. If the latter is a positive integer, then A is said to have finite rank, and if A = C(X) is the …

Supermodular Lattices, Florentin Smarandache, Iqbal Unnisa, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

In lattice theory the two well known equational class of lattices are the distributive lattices and the modular lattices. All distributive lattices are modular however a modular lattice in general is not distributive.

In this book, new classes of lattices called supermodular lattices and semi-supermodular lattices are introduced and characterized as follows: A subdirectly irreducible supermodular lattice is isomorphic to the two element chain lattice C2 or the five element modular lattice M3. A lattice L is supermodular if and only if L is a subdirect union of a two element chain C2 and the five element modular lattice M3.

Ore's Theorem, Jarom Viehweg

Theses Digitization Project

The purpose of this project was to study the classical result in this direction discovered by O. Ore in 1938, as well as related theorems and corollaries. Ore's Theorem and its corollaries provide us with several results relating distributive lattices with cyclic groups.

Band Structures Of Layered Carbon/Boron Nitride Materials With Commensurate Lattices, Christopher C. Wells

Legacy Theses & Dissertations (2009 - 2024)

The electronic structures of systems consisting of hexagonal boron nitride layers and graphite sheets have been investigated in detail using density functional theory methods with two exchange correlation functions (local density approximation and generalized gradient approximation). The experimental data of graphene, graphite, monolayer hexagonal BN, and hexagonal BN were reproduced well with computational models. The commensurate models used in the investigation were generated by taking the averages of the lattice constants for graphite and h-BN.

Lattice Vertex Algebras And Combinatorial Bases, Michael Eugene Leslie Penn

Legacy Theses & Dissertations (2009 - 2024)

We explore the structure of a certain ``principal'' subalgebra, $W_L(\mathcal{B})$, of a lattice vertex (super)-algebra, $V_L$, where $L$ is a non-degenerate integral lattice, and $\mathcal{B}$ is a $\mathbb{Z}$-basis of $L$. Under a certain positivity condition on $\mathcal{B}$ we find a presentation of $W_L(\mathcal{B})$ and of $W_L(\mathcal{B})$-modules. In a more general case we also find their combinatorial bases. For both cases we calculate the (multi)-graded dimensions of modules expressed as fermionic $q$-series . This work generalizes some of the results from \cite{CalLM}, which involved a root lattice of type $A-D-E$, and where $\mathcal{B}$ was the set of simple roots.

On Distribution Of Well-Rounded Sublattices Of Z², Lenny Fukshansky

CMC Faculty Publications and Research

Lecture given at Institut de Mathématiques in Bordeaux, France, June 2008.

Sphere Packing, Lattices, And Epstein Zeta Function, Lenny Fukshansky

CMC Faculty Publications and Research

The sphere packing problem in dimension N asks for an arrangement of non-overlapping spheres of equal radius which occupies the largest possible proportion of the corresponding Euclidean space. This problem has a long and fascinating history. In 1611 Johannes Kepler conjectured that the best possible packing in dimension 3 is obtained by a face centered cubic and hexagonal arrangements of spheres. A proof of this legendary conjecture has finally been published in 2005 by Thomas Hales. The analogous problem in dimension 2 has been solved by Laszlo Fejes Toth in 1940, and this really is the extent of our current …

On Distribution Of Integral Well-Rounded Lattices In Dimension Two, Lenny Fukshansky

CMC Faculty Publications and Research

Lecture given at the Illinois Number Theory Fest, May 2007.

Stone's Representation Theorem, Ion Radu

Theses Digitization Project

The thesis analyzes some aspects of the theory of distributive lattices, particularly two representation theorems: Birkhoff's representation theorem for finite distributive lattices and Stone's representation theorem for infinite distributive lattices.

Frobenius Problem And The Covering Radius Of A Lattice, Lenny Fukshansky, Sinai Robins

CMC Faculty Publications and Research

Abstract. Let N ≥ 2 and let 1 < a(1) < ... < a(N) be relatively prime integers. The Frobenius number of this N-tuple is defined to be the largest positive integer that cannot be expressed as Sigma(N)(i=1) a(i) x(i) where x(1),..., x(N) are non-negative integers. The condition that gcd(a(1),..., a(N)) = 1 implies that such a number exists. The general problem of determining the Frobenius number given N and a(1),..., a(N) is NP-hard, but there have been a number of different bounds on the Frobenius number produced by various authors. We use techniques from the geometry of numbers to produce a new bound, relating the Frobenius number to the covering radius of the null-lattice of this N-tuple. Our bound is particularly interesting in the case when this lattice has equal successive minima, which, as we prove, happens infinitely often.

Siegel’S Lemma With Additional Conditions, Lenny Fukshansky

CMC Faculty Publications and Research

Let K be a number field, and let W be a subspace of K-N, N >= 1. Let V-1,..., V-M be subspaces of KN of dimension less than dimension of W. We prove the existence of a point of small height in W\boolean OR(M)(i=1) V-i, providing an explicit upper bound on the height of such a point in terms of heights of W and V-1,..., V-M. Our main tool is a counting estimate we prove for the number of points of a subspace of K-N inside of an adelic cube. As corollaries to our main result we derive an explicit …

Integral Points Of Small Height Outside Of A Hypersurface, Lenny Fukshansky

CMC Faculty Publications and Research

Let F be a non-zero polynomial with integer coefficients in N variables of degree M. We prove the existence of an integral point of small height at which F does not vanish. Our basic bound depends on N and M only. We separately investigate the case when F is decomposable into a product of linear forms, and provide a more sophisticated bound. We also relate this problem to a certain extension of Siegel’s Lemma as well as to Faltings’ version of it. Finally we exhibit an application of our results to a discrete version of the Tarski plank problem.

Counting Lattice Points In Admissible Adelic Sets, Lenny Fukshansky

CMC Faculty Publications and Research

Lecture given at the Midwest Number Theory Conference for Graduate Students and Recent PhDs II, February 2005.

Problems From The Cottonwood Room, Matthias Beck, Beifang Chen, Lenny Fukshansky, Christian Haase, Allen Knutson, Bruce Reznick, Sinai Robins, Achill Schürmann

CMC Faculty Publications and Research

This collection was compiled by Christian Haase and Bruce Reznick from problems presented at the problem sessions, and submissions solicited from the participants of the AMS/IMS/SIAM summer Research Conference on Integer points in polyhedra. Lattice points in homogeneously expanding compact domains. Presented by Lenny Fukshansky (Texas A&M University).

A Note On Lattice Chains And Delannoy Numbers, John S. Caughman Iv, Clifford R. Haithcock, J. J. P. Veerman

Mathematics and Statistics Faculty Publications and Presentations

Fix nonnegative integers n₁,…,n_d and let L denote the lattice of integer points (a₁,…,a_d)∈Z^d satisfying 0⩽a_i⩽n_i for 1⩽i⩽d. Let L be partially ordered by the usual dominance ordering. In this paper we offer combinatorial derivations of a number of results concerning chains in L. In particular, the results obtained are established without recourse to generating functions or recurrence relations. We begin with an elementary derivation of the number of chains in L of a given size, from which one can deduce the classical expression for the total number …

Higher Order Isotropic Velocity Grids In Lattice Methods, Pavol Pavlo, George Vahala, Linda L. Vahala

Electrical & Computer Engineering Faculty Publications

Kinetic lattice methods are a very attractive representation of nonlinear macroscopic systems because of their inherent parallelizability on multiple processors and their avoidance of the nonlinear convective terms. By uncoupling the velocity lattice from the spatial grid, one can employ higher order (non-space-filling) isotropic lattices-lattices which greatly enhance the stable parameter regions, particularly in thermal problems. In particular, the superiority of the octagonal lattice over previous models used in 2D (hexagonal or square) and 3D (projected face-centered hypercube) is shown.

Lattice-Ordered Algebras That Are Subdirect Products Of Valuation Domains, Melvin Henriksen, Suzanne Larson, Jorge Martinez, R. G. Woods

All HMC Faculty Publications and Research

An f-ring (i.e., a lattice-ordered ring that is a subdirect product of totally ordered rings) A is called an SV-ring if A/P is a valuation domain for every prime ideal P of A. If M is a maximal ℓ-ideal of A , then the rank of A at M is the number of minimal prime ideals of A contained in M, rank of A is the sup of the ranks of A at each of its maximal ℓ-ideals. If the latter is a positive integer, then A is said to have finite rank, and if A …

Coexistence And Spinodal Curves In Directionally Bonded Liquids Using The Four-Cluster Approximation, Erik Bodegom, Paul H. Meijer

Physics Faculty Publications and Presentations

We derive the phase diagrams and spinodals of binary liquid systems with anisotropic interactions, such as hydrogen-bonded molecules. The work is based on the four-particle cluster variation method, using a different potential for different contact points. It is shown that the introduction of a cluster larger than previously used by Barker and Fock, leads to a considerable improvement in the shape of the phase diagram and avoids some of the difficulties encountered in their calculation. Phase diagrams are displayed for various choices of the parameters: the number of contact points, the interaction potential, and the order of the approximation.

Some Sufficient Conditions For The Jacobson Radical Of A Commutative Ring With Identity To Contain A Prime Ideal, Melvin Henriksen

All HMC Faculty Publications and Research

Throughout, the word "ring" will abbreviate the phrase "commutative ring with identity element 1" unless the contrary is stated explicitly. An ideal I of a ring R is called pseudoprime if ab = 0 implies a or b is in I. This term was introduced by C. Kohls and L. Gillman who observed that if I contains a prime ideal, then I is pseudoprime, but, in general, the converse need not hold. In [9 p. 233], M. Larsen, W. Lewis, and R. Shores ask if whenever the Jacobson radical J(R) of an arithmetical ring is pseudoprime, it follows that J(R) …

On The Structure Of A Class Of Archimedean Lattice-Ordered Algebras, Melvin Henriksen, D. G. Johnson

All HMC Faculty Publications and Research

By a Φ-algebra A, we mean an Archimedean lattice-ordered algebra over the real field R which has an identity element 1 that is a weak order unit. The Φ-algebras constitute the class of the title. It is shown that every ф-algebra is isomorphic to an algebra of continuous functions on a compact space X into the two-point compactification of the real line R, each of which is real-valued on an (open) everywhere dense subset of X. Under more restrictive assumptions on A, ropresentations of this sort have long been known. An (incomplete) history of them …

Lattice-Ordered Rings And Function Rings, Melvin Henriksen, John R. Isbell

All HMC Faculty Publications and Research

This paper treats the structure of those lattice-ordered rings which are subdirect sums of totally ordered rings -- the f-rings of Birkhoff and Pierce [4]. Broadly, it splits into two parts, concerned respectively with identical equations and with ideal structure; but there is an important overlap at the beginning.