Open Access. Powered by Scholars. Published by Universities.®
Physical Sciences and Mathematics Commons™
Open Access. Powered by Scholars. Published by Universities.®
- Keyword
-
- Reconstruction (2)
- 1-factor (1)
- Adjoint functor (1)
- Almost Minimizer (1)
- Annihilators (1)
-
- Artinian (1)
- Asymptotic behavior (1)
- Asymptotically Convex (1)
- Automata (1)
- Ball (1)
- Bass number (1)
- Betti number (1)
- Bisection (1)
- Bump attractor (1)
- C*-algebra (1)
- Commutative Algebra (1)
- Compact group (1)
- Continuous attractors (1)
- Crossed product (1)
- Degree sequence (1)
- Discrete Fractional Calculus (1)
- Dynamical system (1)
- Evolutions (1)
- Ext (1)
- Extremal Trees (1)
- Fat points (1)
- Festoon (1)
- Fixed Point Theory (1)
- Fractional Boundary Value Problem (1)
- Fractional Difference Equation (1)
Articles 1 - 18 of 18
Full-Text Articles in Physical Sciences and Mathematics
Are Symbolic Powers Highly Evolved?, Brian Harbourne, Craig Hunkeke
Are Symbolic Powers Highly Evolved?, Brian Harbourne, Craig Hunkeke
Department of Mathematics: Faculty Publications
Searching for structural reasons behind old results and conjectures of Chudnovksy regarding the least degree of a nonzero form in an ideal of fat points in PN, we make conjectures which explain them, and we prove the conjectures in certain cases, including the case of general points in P2. Our conjectures were also partly motivated by the Eisenbud-Mazur Conjecture on evolutions, which concerns symbolic squares of prime ideals in local rings, but in contrast we consider higher symbolic powers of homogeneous ideals in polynomial rings.
Isomorph-Free Generation Of 2-Connected Graphs With Applications, Derrick Stolee
Isomorph-Free Generation Of 2-Connected Graphs With Applications, Derrick Stolee
CSE Technical Reports
Many interesting graph families contain only 2-connected graphs, which have ear decompositions. We develop a technique to generate families of unlabeled 2-connected graphs using ear augmentations and apply this technique to two problems. In the first application, we search for uniquely Kr-saturated graphs and find the list of uniquely K4-saturated graphs on at most 12 vertices, supporting current conjectures for this problem. In the second application, we verify the Edge Reconstruction Conjecture for all 2-connected graphs on at most 12 vertices. This technique can be easily extended to more problems concerning 2-connected graphs.
Covariant Representations Of C*-Dynamical Systems Involving Compact Groups, Firuz Kamalov
Covariant Representations Of C*-Dynamical Systems Involving Compact Groups, Firuz Kamalov
Department of Mathematics: Dissertations, Theses, and Student Research
Given a C*-dynamical system (A, G, σ) the crossed product C*-algebra A x σG encodes the action of G on A. By the universal property of A x σG there exists a one to one correspondence between the set all covariant representations of the system (A, G, σ) and the set of all *-representations of A x σG. Therefore, the study of representations of A x σG is equivalent to that of covariant representations of (A, G, σ).
We study induced covariant representations of systems involving compact groups. We prove that every irreducible (resp. factor) covariant …
On Morrey Spaces In The Calculus Of Variations, Kyle Fey
On Morrey Spaces In The Calculus Of Variations, Kyle Fey
Department of Mathematics: Dissertations, Theses, and Student Research
We prove some global Morrey regularity results for almost minimizers of functionals of the form u → ∫Ω f(x, u, ∇u)dx. This regularity is valid up to the boundary, provided the boundary data are sufficiently regular. The main assumption on f is that for each x and u, the function f(x, u, ·) behaves asymptotically like the function h(|·|)α(x), where h is an N-function.
Following this, we provide a characterization of the class of Young measures that can be generated by a sequence …
Hilbert-Samuel And Hilbert-Kunz Functions Of Zero-Dimensional Ideals, Lori A. Mcdonnell
Hilbert-Samuel And Hilbert-Kunz Functions Of Zero-Dimensional Ideals, Lori A. Mcdonnell
Department of Mathematics: Dissertations, Theses, and Student Research
The Hilbert-Samuel function measures the length of powers of a zero-dimensional ideal in a local ring. Samuel showed that over a local ring these lengths agree with a polynomial, called the Hilbert-Samuel polynomial, for sufficiently large powers of the ideal. We examine the coefficients of this polynomial in the case the ideal is generated by a system of parameters, focusing much of our attention on the second Hilbert coefficient. We also consider the Hilbert-Kunz function, which measures the length of Frobenius powers of an ideal in a ring of positive characteristic. In particular, we examine a conjecture of Watanabe and …
Homology Of Artinian Modules Over Commutative Noetherian Rings, Micah J. Leamer
Homology Of Artinian Modules Over Commutative Noetherian Rings, Micah J. Leamer
Department of Mathematics: Dissertations, Theses, and Student Research
This work is primarily concerned with the study of artinian modules over commutative noetherian rings.
We start by showing that many of the properties of noetherian modules that make homological methods work seamlessly have analogous properties for artinian modules. We prove many of these properties using Matlis duality and a recent characterization of Matlis reflexive modules. Since Matlis reflexive modules are extensions of noetherian and artinian modules many of the properties that hold for artinian and noetherian modules naturally follow for Matlis reflexive modules and more generally for mini-max modules.
In the last chapter we prove that if the Betti …
Groups And Semigroups Generated By Automata, David Mccune
Groups And Semigroups Generated By Automata, David Mccune
Department of Mathematics: Dissertations, Theses, and Student Research
In this dissertation we classify the metabelian groups arising from a restricted class of invertible synchronous automata over a binary alphabet. We give faithful, self-similar actions of Heisenberg groups and upper triangular matrix groups. We introduce a new class of semigroups given by a restricted class of asynchronous automata. We call these semigroups ``expanding automaton semigroups''. We show that this class strictly contains the class of automaton semigroups, and we show that the class of asynchronous automaton semigroups strictly contains the class of expanding automaton semigroups. We demonstrate that undecidability arises in the actions of expanding automaton semigroups and semigroups …
On A Family Of Generalized Wiener Spaces And Applications, Ian Pierce
On A Family Of Generalized Wiener Spaces And Applications, Ian Pierce
Department of Mathematics: Dissertations, Theses, and Student Research
We investigate the structure and properties of a variety of generalized Wiener spaces. Our main focus is on Wiener-type measures on spaces of continuous functions; our generalizations include an extension to multiple parameters, and a method of adjusting the distribution and covariance structure of the measure on the underlying function space.
In the second chapter, we consider single-parameter function spaces and extend a fundamental integration formula of Paley, Wiener, and Zygmund for an important class of functionals on this space. In the third chapter, we discuss measures on very general function spaces and introduce the specific example of a generalized …
Extremal Trees And Reconstruction, Andrew Ray
Extremal Trees And Reconstruction, Andrew Ray
Department of Mathematics: Dissertations, Theses, and Student Research
Problems in two areas of graph theory will be considered.
First, I will consider extremal problems for trees. In these questions we examine the trees that maximize or minimize various invariants. For instance the number of independent sets, the number of matchings, the number of subtrees, the sum of pairwise distances, the spectral radius, and the number of homomorphisms to a fixed graph. I have two general approaches to these problems. To find the extremal trees in the collection of trees on n vertices with a fixed degree bound I use the certificate method. The certificate is a branch invariant, …
Packings And Realizations Of Degree Sequences With Specified Substructures, Tyler Seacrest
Packings And Realizations Of Degree Sequences With Specified Substructures, Tyler Seacrest
Department of Mathematics: Dissertations, Theses, and Student Research
This dissertation focuses on the intersection of two classical and fundamental areas in graph theory: graph packing and degree sequences. The question of packing degree sequences lies naturally in this intersection, asking when degree sequences have edge-disjoint realizations on the same vertex set. The most significant result in this area is Kundu's k-Factor Theorem, which characterizes when a degree sequence packs with a constant sequence. We prove a series of results in this spirit, and we particularly search for realizations of degree sequences with edge-disjoint 1-factors.
Perhaps the most fundamental result in degree sequence theory is the Erdos-Gallai Theorem, characterizing …
Annihilators Of Local Cohomology Modules, Laura Lynch
Annihilators Of Local Cohomology Modules, Laura Lynch
Department of Mathematics: Dissertations, Theses, and Student Research
In many important theorems in the homological theory of commutative local rings, an essential ingredient in the proof is to consider the annihilators of local cohomology modules. We examine these annihilators at various cohomological degrees, in particular at the cohomological dimension and at the height or the grade of the defining ideal. We also investigate the dimension of these annihilators at various degrees and we refine our results by specializing to particular types of rings, for example, Cohen Macaulay rings, unique factorization domains, and rings of small dimension.
Adviser: Thomas Marley
The Theory Of Discrete Fractional Calculus: Development And Application, Michael T. Holm
The Theory Of Discrete Fractional Calculus: Development And Application, Michael T. Holm
Department of Mathematics: Dissertations, Theses, and Student Research
The author's purpose in this dissertation is to introduce, develop and apply the tools of discrete fractional calculus to the arena of fractional difference equations. To this end, we develop the Fractional Composition Rules and the Fractional Laplace Transform Method to solve a linear, fractional initial value problem in Chapters 2 and 3. We then apply fixed point strategies of Krasnosel'skii and Banach to study a nonlinear, fractional boundary value problem in Chapter 4.
Adviser: Lynn Erbe and Allan Peterson
Formalizing Categorical And Algebraic Constructions In Operator Theory, William Benjamin Grilliette
Formalizing Categorical And Algebraic Constructions In Operator Theory, William Benjamin Grilliette
Department of Mathematics: Dissertations, Theses, and Student Research
In this work, I offer an alternative presentation theory for C*-algebras with applicability to various other normed structures. Specifically, the set of generators is equipped with a nonnegative-valued function which ensures existence of a C*-algebra for the presentation. This modification allows clear definitions of a "relation" for generators of a C*-algebra and utilization of classical algebraic tools, such as Tietze transformations.
How Do Neurons Work Together? Lessons From Auditory Cortex, Kenneth D. Harris, Peter Bartho, Paul Chadderton, Carina Curto, Jaime De La Rocha, Liad Hollender, Vladimir Itskov, Artur Luczak, Stephan Marguet, Alfonso Renart, Shuzo Sakata
How Do Neurons Work Together? Lessons From Auditory Cortex, Kenneth D. Harris, Peter Bartho, Paul Chadderton, Carina Curto, Jaime De La Rocha, Liad Hollender, Vladimir Itskov, Artur Luczak, Stephan Marguet, Alfonso Renart, Shuzo Sakata
Department of Mathematics: Faculty Publications
Recordings of single neurons have yielded great insights into the way acoustic stimuli are represented in auditory cortex. However, any one neuron functions as part of a population whose combined activity underlies cortical information processing. Here we review some results obtained by recording simultaneously from auditory cortical populations and individual morphologically identified neurons, in urethane-anesthetized and unanesthetized passively listening rats. Auditory cortical populations produced structured activity patterns both in response to acoustic stimuli, and spontaneously without sensory input. Population spike time patterns were broadly conserved across multiple sensory stimuli and spontaneous events, exhibiting a generally conserved sequential organization lasting approximately …
An Entropy Proof Of The Kahn-Lovasz Theorem, Jonathan Cutler, A. J. Radcliffe
An Entropy Proof Of The Kahn-Lovasz Theorem, Jonathan Cutler, A. J. Radcliffe
Department of Mathematics: Faculty Publications
Bregman [2], gave a best possible upper bound for the number of perfect matchings in a balanced bipartite graph in terms of its degree sequence. Recently Kahn and Lovasz [8] extended Bregman’s theorem to general graphs. In this paper, we use entropy methods to give a new proof of the Kahn-Lovasz theorem. Our methods build on Radhakrishnan’s [9] use of entropy to prove Bregman’s theorem.
Short-Term Facilitation May Stabilize Parametric Working Memory Trace, Vladimir Itskov, David Hansel, Misha Tsodyks
Short-Term Facilitation May Stabilize Parametric Working Memory Trace, Vladimir Itskov, David Hansel, Misha Tsodyks
Department of Mathematics: Faculty Publications
Networks with continuous set of attractors are considered to be a paradigmatic model for parametric working memory (WM), but require fine tuning of connections and are thus structurally unstable. Here we analyzed the network with ring attractor, where connections are not perfectly tuned and the activity state therefore drifts in the absence of the stabilizing stimulus. We derive an analytical expression for the drift dynamics and conclude that the network cannot function as WM for a period of several seconds, a typical delay time in monkey memory experiments. We propose that short-term synaptic facilitation in recurrent connections significantly improves the …
On The Betti Number Of Differential Modules, Justin Devries
On The Betti Number Of Differential Modules, Justin Devries
Department of Mathematics: Dissertations, Theses, and Student Research
Let R = k[x1, ..., xn] with k a field. A multi-graded differential R-module is a multi-graded R-module D with an endomorphism d such that d2 = 0. This dissertation establishes a lower bound on the rank of such a differential module when the underlying R-module is free. We define the Betti number of a differential module and use it to show that when the homology ker d/im d of D is non-zero and finite dimensional over k then there is an inequality rankR D ≥ 2n. This …
Extremal Problems For Independent Set Enumeration, Jonathan Cutler, A. J. Radcliffe
Extremal Problems For Independent Set Enumeration, Jonathan Cutler, A. J. Radcliffe
Department of Mathematics: Faculty Publications
The study of the number of independent sets in a graph has a rich history. Recently, Kahn proved that disjoint unions of Kr,r’s have the maximum number of independent sets amongst r-regular bipartite graphs. Zhao extended this to all r-regular graphs. If we instead restrict the class of graphs to those on a fixed number of vertices and edges, then the Kruskal-Katona theorem implies that the graph with the maximum number of independent sets is the lex graph, where edges form an initial segment of the lexicographic ordering. In this paper, we study three related questions. Firstly, we …