Matching Items (11)

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GROUPOIDS AND C*-ALGEBRAS FOR CATEGORIES OF PATHS

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In this paper we describe a new method of defining C[superscript *]-algebras from oriented combinatorial data, thereby generalizing the construction of algebras from directed graphs, higher-rank graphs, and ordered groups.

In this paper we describe a new method of defining C[superscript *]-algebras from oriented combinatorial data, thereby generalizing the construction of algebras from directed graphs, higher-rank graphs, and ordered groups. We show that only the most elementary notions of concatenation and cancellation of paths are required to define versions of Cuntz-Krieger and Toeplitz-Cuntz-Krieger algebras, and the presentation by generators and relations follows naturally. We give sufficient conditions for the existence of an AF core, hence of the nuclearity of the C[superscript *]-algebras, and for aperiodicity, which is used to prove the standard uniqueness theorems.

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  • 2014-11-01

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Representing Certain Continued Fraction AF Algebras as C*-algebras of Categories of Paths and non-AF Groupoids

Description

C*-algebras of categories of paths were introduced by Spielberg in 2014 and generalize C*-algebras of higher rank graphs. An approximately finite dimensional (AF) C*-algebra is one which is isomorphic to

C*-algebras of categories of paths were introduced by Spielberg in 2014 and generalize C*-algebras of higher rank graphs. An approximately finite dimensional (AF) C*-algebra is one which is isomorphic to an inductive limit of finite dimensional C*-algebras. In 2012, D.G. Evans and A. Sims proposed an analogue of a cycle for higher rank graphs and show that the lack of such an object is necessary for the associated C*-algebra to be AF. Here, I give a class of examples of categories of paths whose associated C*-algebras are Morita equivalent to a large number of periodic continued fraction AF algebras, first described by Effros and Shen in 1980. I then provide two examples which show that the analogue of cycles proposed by Evans and Sims is neither a necessary nor a sufficient condition for the C*-algebra of a category of paths to be AF.

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Date Created
  • 2020

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Toward the enumeration of maximal chains in the Tamari lattices

Description

The Tamari lattices have been intensely studied since they first appeared in Dov Tamari’s thesis around 1952. He defined the n-th Tamari lattice T(n) on bracketings of a set of

The Tamari lattices have been intensely studied since they first appeared in Dov Tamari’s thesis around 1952. He defined the n-th Tamari lattice T(n) on bracketings of a set of n+1 objects, with a cover relation based on the associativity rule in one direction. Despite their interesting aspects and the attention they have received, a formula for the number of maximal chains in the Tamari lattices is still unknown. The purpose of this thesis is to convey my results on progress toward the solution of this problem and to discuss future work.

A few years ago, Bergeron and Préville-Ratelle generalized the Tamari lattices to the m-Tamari lattices. The original Tamari lattices T(n) are the case m=1. I establish a bijection between maximum length chains in the m-Tamari lattices and standard m-shifted Young tableaux. Using Thrall’s formula, I thus derive the formula for the number of maximum length chains in T(n).

For each i greater or equal to -1 and for all n greater or equal to 1, I define C(i,n) to be the set of maximal chains of length n+i in T(n). I establish several properties of maximal chains (treated as tableaux) and identify a particularly special property: each maximal chain may or may not possess a plus-full-set. I show, surprisingly, that for all n greater or equal to 2i+4, each member of C(i,n) contains a plus-full-set. Utilizing this fact and a collection of maps, I obtain a recursion for the number of elements in C(i,n) and an explicit formula based on predetermined initial values. The formula is a polynomial in n of degree 3i+3. For example, the number of maximal chains of length n in T(n) is n choose 3.

I discuss current work and future plans involving certain equivalence classes of maximal chains in the Tamari lattices. If a maximal chain may be obtained from another by swapping a pair of consecutive edges with another pair in the Hasse diagram, the two maximal chains are said to differ by a square move. Two maximal chains are said to be in the same equivalence class if one may be obtained from the other by making a set of square moves.

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Date Created
  • 2016

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Operator-valued frames associated with measure spaces

Description

Since Duffin and Schaeffer's introduction of frames in 1952, the concept of a frame has received much attention in the mathematical community and has inspired several generalizations. The focus of

Since Duffin and Schaeffer's introduction of frames in 1952, the concept of a frame has received much attention in the mathematical community and has inspired several generalizations. The focus of this thesis is on the concept of an operator-valued frame (OVF) and a more general concept called herein an operator-valued frame associated with a measure space (MS-OVF), which is sometimes called a continuous g-frame. The first of two main topics explored in this thesis is the relationship between MS-OVFs and objects prominent in quantum information theory called positive operator-valued measures (POVMs). It has been observed that every MS-OVF gives rise to a POVM with invertible total variation in a natural way. The first main result of this thesis is a characterization of which POVMs arise in this way, a result obtained by extending certain existing Radon-Nikodym theorems for POVMs. The second main topic investigated in this thesis is the role of the theory of unitary representations of a Lie group G in the construction of OVFs for the L^2-space of a relatively compact subset of G. For G=R, Duffin and Schaeffer have given general conditions that ensure a sequence of (one-dimensional) representations of G, restricted to (-1/2,1/2), forms a frame for L^{2}(-1/2,1/2), and similar conditions exist for G=R^n. The second main result of this thesis expresses conditions related to Duffin and Schaeffer's for two more particular Lie groups: the Euclidean motion group on R^2 and the (2n+1)-dimensional Heisenberg group. This proceeds in two steps. First, for a Lie group admitting a uniform lattice and an appropriate relatively compact subset E of G, the Selberg Trace Formula is used to obtain a Parseval OVF for L^{2}(E) that is expressed in terms of irreducible representations of G. Second, for the two particular Lie groups an appropriate set E is found, and it is shown that for each of these groups, with suitably parametrized unitary duals, the Parseval OVF remains an OVF when perturbations are made to the parameters of the included representations.

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Date Created
  • 2014

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Symplectic topology and geometric quantum mechanics

Description

The theory of geometric quantum mechanics describes a quantum system as a Hamiltonian dynamical system, with a projective Hilbert space regarded as the phase space. This thesis extends the theory

The theory of geometric quantum mechanics describes a quantum system as a Hamiltonian dynamical system, with a projective Hilbert space regarded as the phase space. This thesis extends the theory by including some aspects of the symplectic topology of the quantum phase space. It is shown that the quantum mechanical uncertainty principle is a special case of an inequality from J-holomorphic map theory, that is, J-holomorphic curves minimize the difference between the quantum covariance matrix determinant and a symplectic area. An immediate consequence is that a minimal determinant is a topological invariant, within a fixed homology class of the curve. Various choices of quantum operators are studied with reference to the implications of the J-holomorphic condition. The mean curvature vector field and Maslov class are calculated for a lagrangian torus of an integrable quantum system. The mean curvature one-form is simply related to the canonical connection which determines the geometric phases and polarization linear response. Adiabatic deformations of a quantum system are analyzed in terms of vector bundle classifying maps and related to the mean curvature flow of quantum states. The dielectric response function for a periodic solid is calculated to be the curvature of a connection on a vector bundle.

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Created

Date Created
  • 2011

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One- and two-variable p-adic measures in Iwasawa theory

Description

In 1984, Sinnott used $p$-adic measures on $\mathbb{Z}_p$ to give a new proof of the Ferrero-Washington Theorem for abelian number fields by realizing $p$-adic $L$-functions as (essentially) the $Gamma$-transform of

In 1984, Sinnott used $p$-adic measures on $\mathbb{Z}_p$ to give a new proof of the Ferrero-Washington Theorem for abelian number fields by realizing $p$-adic $L$-functions as (essentially) the $Gamma$-transform of certain $p$-adic rational function measures. Shortly afterward, Gillard and Schneps independently adapted Sinnott's techniques to the case of $p$-adic $L$-functions associated to elliptic curves with complex multiplication (CM) by realizing these $p$-adic $L$-functions as $Gamma$-transforms of certain $p$-adic rational function measures. The results in the CM case give the vanishing of the Iwasawa $mu$-invariant for certain $mathbb{Z}_p$-extensions of imaginary quadratic fields constructed from torsion points of CM elliptic curves.

In this thesis, I develop the theory of $p$-adic measures on $mathbb{Z}_p^d$, with particular interest given to the case of $d>1$. Although I introduce these measures within the context of $p$-adic integration, this study includes a strong emphasis on the interpretation of $p$-adic measures as $p$-adic power series. With this dual perspective, I describe $p$-adic analytic operations as maps on power series; the most important of these operations is the multivariate $Gamma$-transform on $p$-adic measures.

This thesis gives new significance to product measures, and in particular to the use of product measures to construct measures on $mathbb{Z}_p^2$ from measures on $mathbb{Z}_p$. I introduce a subring of pseudo-polynomial measures on $mathbb{Z}_p^2$ which is closed under the standard operations on measures, including the $Gamma$-transform. I obtain results on the Iwasawa-invariants of such pseudo-polynomial measures, and use these results to deduce certain continuity results for the $Gamma$-transform. As an application, I establish the vanishing of the Iwasawa $mu$-invariant of Yager's two-variable $p$-adic $L$-function from measure theoretic considerations.

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Date Created
  • 2015

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Persistence of discrete dynamical systems in infinite dimensional state spaces

Description

Persistence theory provides a mathematically rigorous answer to the question of population survival by establishing an initial-condition- independent positive lower bound for the long-term value of the population size. This

Persistence theory provides a mathematically rigorous answer to the question of population survival by establishing an initial-condition- independent positive lower bound for the long-term value of the population size. This study focuses on the persistence of discrete semiflows in infinite-dimensional state spaces that model the year-to-year dynamics of structured populations. The map which encapsulates the population development from one year to the next is approximated at the origin (the extinction state) by a linear or homogeneous map. The (cone) spectral radius of this approximating map is the threshold between extinction and persistence. General persistence results are applied to three particular models: a size-structured plant population model, a diffusion model (with both Neumann and Dirichlet boundary conditions) for a dispersing population of males and females that only mate and reproduce once during a very short season, and a rank-structured model for a population of males and females.

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Date Created
  • 2014

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Functorial results for C*-algebras of higher-rank graphs

Description

Higher-rank graphs, or k-graphs, are higher-dimensional analogues of directed graphs, and as with ordinary directed graphs, there are various C*-algebraic objects that can be associated with them. This thesis adopts

Higher-rank graphs, or k-graphs, are higher-dimensional analogues of directed graphs, and as with ordinary directed graphs, there are various C*-algebraic objects that can be associated with them. This thesis adopts a functorial approach to study the relationship between k-graphs and their associated C*-algebras. In particular, two functors are given between appropriate categories of higher-rank graphs and the category of C*-algebras, one for Toeplitz algebras and one for Cuntz-Krieger algebras. Additionally, the Cayley graphs of finitely generated groups are used to define a class of k-graphs, and a functor is then given from a category of finitely generated groups to the category of C*-algebras. Finally, functoriality is investigated for product systems of C*-correspondences associated to k-graphs. Additional results concerning the structural consequences of functoriality, properties of the functors, and combinatorial aspects of k-graphs are also included throughout.

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Date Created
  • 2016

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Toward enumerating the chains of maximum length of Cambrian and m-eralized Cambrian lattices

Description

The Cambrian lattice corresponding to a Coxeter element c of An, denoted Camb(c),

is the subposet of An induced by the c-sortable elements, and the m-eralized Cambrian

lattice corresponding to c, denoted

The Cambrian lattice corresponding to a Coxeter element c of An, denoted Camb(c),

is the subposet of An induced by the c-sortable elements, and the m-eralized Cambrian

lattice corresponding to c, denoted Cambm(c), is dened as a subposet of the

braid group accompanied with the right weak ordering induced by the c-sortable elements

under certain conditions. Both of these families generalize the well-studied

Tamari lattice Tn rst introduced by D. Tamari in 1962. S. Fishel and L. Nelson

enumerated the chains of maximum length of Tamari lattices.

In this dissertation, I study the chains of maximum length of the Cambrian and

m-eralized Cambrian lattices, precisely, I enumerate these chains in terms of other

objects, and then nd formulas for the number of these chains for all m-eralized

Cambrian lattices of A1, A2, A3, and A4. Furthermore, I give an alternative proof

for the number of chains of maximum length of the Tamari lattice Tn, and provide

conjectures and corollaries for the number of these chains for all m-eralized Cambrian

lattices of A5.

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Date Created
  • 2017

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Some Turán-type problems in extremal graph theory

Description

Since the seminal work of Tur ́an, the forbidden subgraph problem has been among the central questions in extremal graph theory. Let ex(n;F) be the smallest number m such that

Since the seminal work of Tur ́an, the forbidden subgraph problem has been among the central questions in extremal graph theory. Let ex(n;F) be the smallest number m such that any graph on n vertices with m edges contains F as a subgraph. Then the forbidden subgraph problem asks to find ex(n; F ) for various graphs F . The question can be further generalized by asking for the extreme values of other graph parameters like minimum degree, maximum degree, or connectivity. We call this type of question a Tura ́n-type problem. In this thesis, we will study Tura ́n-type problems and their variants for graphs and hypergraphs.

Chapter 2 contains a Tura ́n-type problem for cycles in dense graphs. The main result in this chapter gives a tight bound for the minimum degree of a graph which guarantees existence of disjoint cycles in the case of dense graphs. This, in particular, answers in the affirmative a question of Faudree, Gould, Jacobson and Magnant in the case of dense graphs.

In Chapter 3, similar problems for trees are investigated. Recently, Faudree, Gould, Jacobson and West studied the minimum degree conditions for the existence of certain spanning caterpillars. They proved certain bounds that guarantee existence of spanning caterpillars. The main result in Chapter 3 significantly improves their result and answers one of their questions by proving a tight minimum degree bound for the existence of such structures.

Chapter 4 includes another Tur ́an-type problem for loose paths of length three in a 3-graph. As a corollary, an upper bound for the multi-color Ramsey number for the loose path of length three in a 3-graph is achieved.

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Created

Date Created
  • 2018