Lefschetz Properties And Enumerations,
2012
University of Kentucky
Lefschetz Properties And Enumerations, David Cook Ii
Theses and Dissertations--Mathematics
An artinian standard graded algebra has the weak Lefschetz property if the multiplication by a general linear form induces maps of maximal rank between consecutive degree components. It has the strong Lefschetz property if the multiplication by powers of a general linear form also induce maps of maximal rank between the appropriate degree components. These properties are mainly studied for the constraints they place, when present, on the Hilbert series of the algebra. While the majority of research on the Lefschetz properties has focused on characteristic zero, we primarily consider the presence of the properties in positive characteristic. We study …
Classification Of Noncommutative Domain Algebras,
2012
University of Denver
Classification Of Noncommutative Domain Algebras, Alvaro Arias, Frédéric Latrémolièr
Mathematics Preprint Series
Noncommutative domain algebras are noncommutative analogues of the algebras of holomorphic functions on domains of Cn defined by holomorphic polynomials, and they generalize the noncommutative Hardy algebras. We present here a complete classification of these algebras based upon techniques inspired by multivariate complex analysis, and more specifically the classification of domains in hermitian spaces up to biholomorphic equivalence.
Shifts Of Finite Type With Nearly Full Entropy,
2012
Department of Mathematics, University of Denver
Shifts Of Finite Type With Nearly Full Entropy, Ronnie Pavlov
Mathematics Preprint Series
For any fixed alphabet A, the maximum topological entropy of a Z d subshift with alphabet A is obviously log |A|. We study the class of nearest neighbor Z d shifts of finite type which have topological entropy very close to this maximum, and show that they have many useful properties. Specifically, we prove that for any d, there exists δd > 0 such that for any nearest neighbor Z d shift of finite type X with alphabet A for which (log |A|) − h(X) < δd, X has a unique measure of maximal entropy µ. We also show that any such X is a measure-theoretic universal model in the sense of [25], that h(X) is a computable number, that µ is measure-theoretically isomorphic to a Bernoulli measure, and that the support of µ has topologically completely positive entropy. Though there are other sufficient conditions in the literature (see [9], [15], [22]) which guarantee a unique measure of maximal entropy for Z d shifts of finite type, this is (to our knowledge) the first such condition which makes no reference to the specific adjacency rules of individual letters of the alphabet.
Problems And Solutions.,
2012
Columbus State University
Regular Octahedrons In {0, 1, K, N} 3,
2012
Columbus State University
Regular Octahedrons In {0, 1, K, N} 3, Eugen J. Ionascu
Faculty Bibliography
In this paper we describe a procedure for calculating the number of regular octahedra, RO(n), which have vertices with coordinates in the set {0, 1, ..., n}. As a result, we introduce a new sequence in The Online Encyclopedia of Integer Sequences (A178797) and list the first one hundred terms of it. We improve the method appeared in [12] which was used to find the number of regular tetrahedra with coordinates of their vertices in {0, 1, ..., n}. A new fact proved here helps increasing considerably the speed of all programs used before. The procedure is put together in …
Solution Of Problem 955,
2012
Columbus State University
Solution Of Problem 947,
2012
Columbus State University
Weight: Does It Really Matter?,
2012
Columbus State University
Weight: Does It Really Matter?, Jennifer Brown
Faculty Bibliography
Differential weighting promises to improve the validity of a measure. This study examines whether similar results would be found using weighted, unweighted and standardized z scores from the All Stars Core survey. It was concluded that the weighted systems were developed to equate the questions within the scales and to ease the process for customers without access to data analysis programs; however, the standardized scores were the more appropriate method for equating the test items.
Odd Or Even: Uncovering Parity Of Rank In A Family Of Rational Elliptic Curves,
2012
Colby College
Odd Or Even: Uncovering Parity Of Rank In A Family Of Rational Elliptic Curves, Anika Lindemann
Honors Theses
Puzzled by equations in multiple variables for centuries, mathematicians have made relatively few strides in solving these seemingly friendly, but unruly beasts. Currently, there is no systematic method for finding all rational values, that satisfy any equation with degree higher than a quadratic. This is bizarre. Solving these has preoccupied great minds since before the formal notion of an equation existed. Before any sort of mathematical formality, these questions were nested in plucky riddles and folded into folk tales. Because they are so simple to state, these equations are accessible to a very general audience. Yet an astounding amount of …
A Note On Acoustic Propagation In Power-Law Fluids: Compact Kinks, Mild Discontinuities, And A Connection To Finite-Scale Theory,
2012
University of New Orleans
A Note On Acoustic Propagation In Power-Law Fluids: Compact Kinks, Mild Discontinuities, And A Connection To Finite-Scale Theory, Dongming Wei
Mathematics Faculty Publications
Acoustic traveling waves in a class of viscous, power-lawfluids are investigated. Both bi-directional and unidirectional versions of the one-dimensional (1D), weakly-nonlinear equation of motion are derived; traveling wave solutions (TWS)s, special cases of which take the form of compact and algebraic kinks, are determined; and the impact of the bulk viscosity on the structure/nature of the kinks is examined. Most significantly, we point out a connection that exists between the power-law model considered here and the recently introduced theory of finite-scale equations.
Equivariant Degenerations Of Spherical Modules For Groups Of Type A,
2012
Universidade Tecnica de Lisboa
Equivariant Degenerations Of Spherical Modules For Groups Of Type A, Stavros Argyrios Papadakis, Bart Van Steirteghem
Publications and Research
Let G be a complex reductive algebraic group. Fix a Borel subgroup B of G and a maximal torus T in B. Call the monoid of dominant weights L+ and let S be a finitely generated submonoid of L+. V. Alexeev and M. Brion introduced a moduli scheme MS which classifies affine G-varieties X equipped with a T-equivariant isomorphism SpecC[X]U → SpecC[S], where U is the unipotent radical of B. Examples of MS have been obtained by S. Jansou, P. Bravi and S. Cupit-Foutou. In this paper, we prove that MS is isomorphic to an affine space when S is …
Rational Approximation On Compact Nowhere Dense Sets,
2012
University of Kentucky
Rational Approximation On Compact Nowhere Dense Sets, Christopher Mattingly
Theses and Dissertations--Mathematics
For a compact, nowhere dense set X in the complex plane, C, define Rp(X) as the closure of the rational functions with poles off X in Lp(X, dA). It is well known that for 1 ≤ p < 2, Rp(X) = Lp(X) . Although density may not be achieved for p > 2, there exists a set X so that Rp(X) = Lp(X) for p up to a given number greater than 2 but not after. Additionally, when p > 2 we shall establish that the support of the annihiliating and …
Hilbert Polynomials And Strongly Stable Ideals,
2012
University of Kentucky
Hilbert Polynomials And Strongly Stable Ideals, Dennis Moore
Theses and Dissertations--Mathematics
Strongly stable ideals are important in algebraic geometry, commutative algebra, and combinatorics. Prompted, for example, by combinatorial approaches for studying Hilbert schemes and the existence of maximal total Betti numbers among saturated ideals with a given Hilbert polynomial, three algorithms are presented. Each of these algorithms produces all strongly stable ideals with some prescribed property: the saturated strongly stable ideals with a given Hilbert polynomial, the almost lexsegment ideals with a given Hilbert polynomial, and the saturated strongly stable ideals with a given Hilbert function. Bounds for the complexity of our algorithms are included. Also included are some applications for …
Visualizing Chaos,
2012
College of Saint Benedict and Saint John's University
Visualizing Chaos, Andrew Nicklawsky
Mathematics Student Work
An important piece of information when dealing with a polynomial in the complex plane is its roots, the value or values of x for a given function f such that f(x)=0. Iterative root finding methods, such as Newton’s method, are utilized to discover an approximate value when these values cannot be explicitly solved. This process can be graphically represented for complex-valued functions and has been achieved with relative ease on a 2-Dimensional plane. However, this process can also be embodied on a sphere through the method of stereographic projection, which has not been attempted. In this research, I worked with …
Stability And Clustering Of Self-Similar Solutions Of Aggregation Equations,
2012
University of San Francisco
Stability And Clustering Of Self-Similar Solutions Of Aggregation Equations, Hui Sun, David Uminsky, Andrea L. Bertozzi
Mathematics
In this paper we consider the linear stability of a family of exact collapsing similarity solutions to the aggregation equation ρ t = ∇ · (ρ∇K * ρ) in Rd , d ⩾ 2, where K(r) = r γ/γ with γ > 2. It was previously observed [Y. Huang and A. L. Bertozzi, “Self-similar blowup solutions to an aggregation equation in Rn,” J. SIAM Appl. Math.70, 2582–2603 (Year: 2010)]10.1137/090774495 that radially symmetric solutions are attracted to a self-similar collapsing shell profile in infinite time for γ > 2. In this paper we compute the stability of the …
A Generalized Birkhoff-Rott Equation For Two-Dimensional Active Scalar Problems,
2012
University of San Francisco
A Generalized Birkhoff-Rott Equation For Two-Dimensional Active Scalar Problems, Hui Sun, David Uminsky, Andrea L. Bertozzi
Mathematics
In this paper we derive evolution equations for the two-dimensional active scalar problem when the solution is supported on one-dimensional curves. These equations are a generalization of the Birkhoff–Rott equation when vorticity is the active scalar. The formulation is Lagrangian and it is valid for nonlocal kernels K that may include both a gradient and an incompressible term. We develop a numerical method for implementing the model which achieves second order convergence in space and fourth order in time. We verify the model by simulating classic active scalar problems such as the vortex sheet problem (in the case of inviscid, …
Network-Based Criterion For The Success Of Cooperation In An Evolutionary Prisoner's Dilemma,
2012
University of San Francisco
Network-Based Criterion For The Success Of Cooperation In An Evolutionary Prisoner's Dilemma, Stephen Devlin, T Treloar
Mathematics
We consider an evolutionary prisoner's dilemma on a random network. We introduce a simple quantitative network-based parameter and show that it effectively predicts the success of cooperation in simulations on the network. The criterion is shown to be accurate on a variety of networks with degree distributions ranging from regular to Poisson to scale free. The parameter allows for comparisons of random networks regardless of their underlying topology. Finally, we draw analogies between the criterion for the success of cooperation introduced here and existing criteria in other contexts.
Convergence And Energy Landscape For Cheeger Cut Clustering,
2012
University of San Francisco
Convergence And Energy Landscape For Cheeger Cut Clustering, X Bresson, T Laurent, David Uminsky, J Von Brecht
Mathematics
This paper provides both theoretical and algorithmic results for the l 1-relaxation of the Cheeger cut problem. The l2- relaxation, known as spectral clustering, only loosely relates to the Cheeger cut; however, it is convex and leads to a simple optimization problem. The l1-relaxation, in contrast, is non-convex but is provably equivalent to the original problem. The l1-relaxation therefore trades convexity for exactness, yielding improved clustering results at the cost of a more challenging optimization. The first challenge is understanding convergence of algorithms. This paper provides the first complete proof of convergence for …
Categoricity And Topological Graphs,
2012
Marquette University
Categoricity And Topological Graphs, Paul Bankston
Mathematics, Statistics and Computer Science Faculty Research and Publications
Let X be a topological graph, with arcs joined only at end points. If Y is any locally connected metrizable compactum that is co-elementarily equivalent to X, then Y is homeomorphic to X. In particular, X and Y are homeomorphic if some lattice base for one is elementarily equivalent to some lattice base for the other.
Complete Multipartite Graphs And The Relaxed Coloring Game,
2012
Linfield College
Complete Multipartite Graphs And The Relaxed Coloring Game, Charles Dunn
Faculty Publications
Let k be a positive integer, d be a nonnegative integer, and G be a finite graph. Two players, Alice and Bob, play a game on G by coloring the uncolored vertices with colors from a set X of k colors. At all times, the subgraph induced by a color class must have maximum degree at most d. Alice wins the game if all vertices are eventually colored; otherwise, Bob wins. The least k such that Alice has a winning strategy is called the d-relaxed game chromatic number of G, denoted χ gd (G). …
