Locally recurrent functions, density topologies and algebraic genericity

We study the algebraic and topological genericity of certain subsets of locally recurrent functions, obtaining (among other results) algebrability and spaceability within these classes.

Locally recurrent functions, density topologies and algebraic genericity

We study the algebraic and topological genericity of certain subsets of locally recurrent functions, obtaining (among other results) algebrability and spaceability within these classes.

A monoidal algebraic model for rational SO(2)-spectra

The category of rational SO(2)--equivariant spectra admits an algebraic model. That is, there is an abelian category A(SO(2)) whose derived category is equivalent to the homotopy category of rational$SO(2)--equivariant spectra. An important question is: does this algebraic model capture the smash product of spectra? The category A(SO(2)) is known as Greenlees' standard model, it is an abelian category that has no projective objects and is constructed from modules over a non--Noetherian ring. As a consequence, the standard techniques for constructing a monoidal model structure cannot be applied. In this paper a monoidal model structure on A(SO(2)) is constructed and the derived tensor product on the homotopy category is shown to be compatible with the smash product of spectra. The method used is related to techniques developed by the author in earlier joint work with Roitzheim. That work constructed a monoidal model structure on Franke's exotic model for the K_(p)--local stable homotopy category. A monoidal Quillen equivalence to a simpler monoidal model category that has explicit generating sets is also given. Having monoidal model structures on the two categories removes a serious obstruction to constructing a series of monoidal Quillen equivalences between the algebraic model and rational SO(2)--equivariant spectra.

Problems in Algebraic Vision

Thesis (Ph.D.)--University of Washington, 2016-08

Ein Ansatz zur methoden- und werkzeuggestützten Anforderungs- und Systemspezifikation auf der Grundlage objektorientierter Modellierungs- und Simulationstechniken

Vorgestellt wird ein Ansatz zur objektorientierten Modellierung, Simulation und Animation von Informationssystemen. Es wird ein Vorgehensmodell dargestellt, mit dem unter Verwendung des beschriebenen Ansatzes Anforderungs- oder Systemspezifikationen von Rechnergestützten Informationssystemen erstellt werden können. Der Ansatz basiert auf einem Metamodell zur Beschreibung Rechnergestützter Informationssysteme und verfügt über eine rechnergestützte Modellierungsumgebung. Anhand eines Projektes zur Entwicklung einer Anforderungsspezifikation für ein rechnergestütztes Pflegedokumentations- und -kommunikationssystems wird der Einsatz der Methode beispielhaft illustriert.

Behavioral equivalence of hidden k-logics: an abstract algebraic approach

This work advances a research agenda which has as its main aim the application of Abstract Algebraic Logic (AAL) methods and tools to the specification and verification of software systems. It uses a generalization of the notion of an abstract deductive system to handle multi-sorted deductive systems which differentiate visible and hidden sorts. Two main results of the paper are obtained by generalizing properties of the Leibniz congruence — the central notion in AAL. In this paper we discuss a question we posed in [1] about the relationship between the behavioral equivalences of equivalent hidden logics. We also present a necessary and sufficient intrinsic condition for two hidden logics to be equivalent.

Causal structure of the entanglement renormalization ansatz

We show that the multiscale entanglement renormalization ansatz (MERA) can be reformulated in terms of a causality constraint on discrete quantum dynamics. This causal structure is that of de Sitter space with a flat space-like boundary, where the volume of a spacetime region corresponds to the number of variational parameters it contains. This result clarifies the nature of the ansatz, and suggests a generalization to quantum field theory. It also constitutes an independent justification of the connection between MERA and hyperbolic geometry which was proposed as a concrete implementation of the AdS-CFT correspondence.

Strong shift equivalence, algebraic K-theory, and isolating zero-dimensional dynamics on manifolds

We study the relations of shift equivalence and strong shift equivalence for matrices over a ring $\mathcal{R}$, and establish a connection between these relations and algebraic K-theory. We utilize this connection to obtain results in two areas where the shift and strong shift equivalence relations play an important role: the study of finite group extensions of shifts of finite type, and the Generalized Spectral Conjectures of Boyle and Handelman for nonnegative matrices over subrings of the real numbers. We show the refinement of the shift equivalence class of a matrix $A$ over a ring $\mathcal{R}$ by strong shift equivalence classes over the ring is classified by a quotient $NK_{1}(\mathcal{R}) / E(A,\mathcal{R})$ of the algebraic K-group $NK_{1}(\calR)$. We use the K-theory of non-commutative localizations to show that in certain cases the subgroup $E(A,\mathcal{R})$ must vanish, including the case $A$ is invertible over $\mathcal{R}$. We use the K-theory connection to clarify the structure of algebraic invariants for finite group extensions of shifts of finite type. In particular, we give a strong negative answer to a question of Parry, who asked whether the dynamical zeta function determines up to finitely many topological conjugacy classes the extensions by $G$ of a fixed mixing shift of finite type. We apply the K-theory connection to prove the equivalence of a strong and weak form of the Generalized Spectral Conjecture of Boyle and Handelman for primitive matrices over subrings of $\mathbb{R}$. We construct explicit matrices whose class in the algebraic K-group $NK_{1}(\mathcal{R})$ is non-zero for certain rings $\mathcal{R}$ motivated by applications. We study the possible dynamics of the restriction of a homeomorphism of a compact manifold to an isolated zero-dimensional set. We prove that for $n \ge 3$ every compact zero-dimensional system can arise as an isolated invariant set for a homeomorphism of a compact $n$-manifold. In dimension two, we provide obstructions and examples.

The Real-Quaternionic Indicator of Irreducible Self-Conjugate Representations of Real Reductive Algebraic Groups and A Comment on the Local Langlands Correspondence of GL(2, F )

The real-quaternionic indicator, also called the $\delta$ indicator, indicates if a self-conjugate representation is of real or quaternionic type. It is closely related to the Frobenius-Schur indicator, which we call the $\varepsilon$ indicator. The Frobenius-Schur indicator $\varepsilon(\pi)$ is known to be given by a particular value of the central character. We would like a similar result for the $\delta$ indicator. When $G$ is compact, $\delta(\pi)$ and $\varepsilon(\pi)$ coincide. In general, they are not necessarily the same. In this thesis, we will give a relation between the two indicators when $G$ is a real reductive algebraic group. This relation also leads to a formula for $\delta(\pi)$ in terms of the central character. For the second part, we consider the construction of the local Langlands correspondence of $GL(2,F)$ when $F$ is a non-Archimedean local field with odd residual characteristics. By re-examining the construction, we provide new proofs to some important properties of the correspondence. Namely, the construction is independent of the choice of additive character in the theta correspondence.

Algebraic geometric codes over rings

The techniques of algebraic geometry have been widely and successfully applied to the study of linear codes over finite fields since the early 1980's. Recently, there has been an increased interest in the study of linear codes over finite rings. In this thesis, we combine these two approaches to coding theory by introducing and studying algebraic geometric codes over rings.

Doing difference – Die Reflexion auf Unterscheidungen als Ansatz Politischer Erwachsenenbildung

Ein Verständnis von Demokratie als „stets im Kommen oder im Werden“ schließt fort- und immerwährende Verhandlungsprozesse mit ein. Das Ausbleiben von Konflikten oder die Versuche des Einebnens und Nivellierens von Widersprüchen oder gar deren Negation sind dann Indizien der Gefährdung von Demokratisierung und gesamtgesellschaftlicher Entwicklungsmöglichkeiten. Der vorliegende Beitrag betont die Bedeutung von Widerständigkeit für Demokratisierung. Der vorgestellte Ansatz der Reflexion auf Unterscheidungen - un/doing difference - verabschiedet die stillschweigende Vorstellung, dass es vorab feststehende und gegebene Individuen oder Gruppen mit bestimmten Eigenschaften gibt, an die unterschiedliche Angebote und Interventionen gerichtet werden können. Es ist eine differenzsensible Herangehensweise, die latente Unterschiede aufgreift und ihnen im Handeln - im Sinne von Ungleichheiten - Bedeutung verleiht. Den Abschluss des Beitrages bilden vier Strategien im Kontext politischer Bildungsarbeit, um Pluralität und Kontingenz sichtbar zu machen: Pluralisierung und Konkurrenz ermöglichen; Löschung durch Nichtbeachten; ironische Entlarvung von Selbstverständlichkeitsannahmen; eingeführte Kategorien stehen lassen und zugleich Alternativen fördern. (DIPF/Orig.)

Der Anti-Bias-Ansatz als Methode politischer Erwachsenenbildung

Vorurteile, Macht und Diskriminierung sind die zentralen Themen des Anti-Bias-Ansatzes, der Diskriminierung auf zwischenmenschlicher, struktureller und gesellschaftlich-kultureller Ebene berücksichtigt. Die eigene Verwobenheit in Machtverhältnisse und damit verbundene Erfahrungen von Diskriminierung und Privilegierung sind dabei der Ausgangspunkt des Lernens. Vision ist eine vorurteilsbewusste, diskriminierungskritische und machtsensible Gesellschaft. Der vorliegende Beitrag stellt den Anti-Bias-Ansatz als Methode politischer Erwachsenenbildung vor. Hierfür werden die (Macht-)Mechanismen der Entstehung, Verinnerlichung und Reproduktion von Vorurteilen und Diskriminierung beleuchtet und wird die Intersektionalität (Überkreuzung) verschiedener Differenzlinien ausgemacht und aufgedeckt. Den Abschluss bildet ein Beispiel konkreter Umsetzung. (DIPF/Orig.)

Bethe graphs attached to the vertices of a connected graph: a spectral approach

A weighted Bethe graph $B$ is obtained from a weighted generalized Bethe tree by identifying each set of children with the vertices of a graph belonging to a family $F$ of graphs. The operation of identifying the root vertex of each of $r$ weighted Bethe graphs to the vertices of a connected graph $\mathcal{R}$ of order $r$ is introduced as the $\mathcal{R}$-concatenation of a family of $r$ weighted Bethe graphs. It is shown that the Laplacian eigenvalues (when $F$ has arbitrary graphs) as well as the signless Laplacian and adjacency eigenvalues (when the graphs in $F$ are all regular) of the $\mathcal{R}$-concatenation of a family of weighted Bethe graphs can be computed (in a unified way) using the stable and low computational cost methods available for the determination of the eigenvalues of symmetric tridiagonal matrices. Unlike the previous results already obtained on this topic, the more general context of families of distinct weighted Bethe graphs is herein considered.

A divide-and-conquer bound for aggregate's quality and algebraic connectivity

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