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Articles 1 - 17 of 17
Full-Text Articles in Physical Sciences and Mathematics
Higher Diffeology Theory, Emilio Minichiello
Higher Diffeology Theory, Emilio Minichiello
Dissertations, Theses, and Capstone Projects
Finite dimensional smooth manifolds have been studied for hundreds of years, and a massive theory has been built around them. However, modern mathematicians and physicists are commonly dealing with objects outside the purview of classical differential geometry, such as orbifolds and loop spaces. Diffeology is a new framework for dealing with such generalized smooth spaces. This theory (whose development started in earnest in the 1980s) has started to catch on amongst the wider mathematical community, thanks to its simplicity and power, but it is not the only approach to dealing with generalized smooth spaces. Higher topos theory is another such …
An Explicit Construction Of Sheaves In Context, Tyler A. Bryson
An Explicit Construction Of Sheaves In Context, Tyler A. Bryson
Dissertations, Theses, and Capstone Projects
This document details the body of theory necessary to explicitly construct sheaves of sets on a site together with the development of supporting material necessary to connect sheaf theory with the wider mathematical contexts in which it is applied. Of particular interest is a novel presentation of the plus construction suitable for direct application to a site without first passing to the generated grothendieck topology.
Sl(2,Z) Representations And 2-Semiregular Modular Categories, Samuel Nathan Wilson
Sl(2,Z) Representations And 2-Semiregular Modular Categories, Samuel Nathan Wilson
LSU Doctoral Dissertations
We address the open question of which representations of the modular group SL(2,Z) can be realized by a modular category. In order to investigate this problem, we introduce the concept of a symmetrizable representation of SL(2,Z) and show that this property is necessary for the representation to be realized. We then prove that all congruence representations of SL(2,Z) are symmetrizable. The proof involves constructing a symmetric basis, which greatly aids in further calculation. We apply this result to the reconstruction of modular category data from representations, as well as to the classification of semiregular categories, which are defined via an …
Bicategorical Traces And Cotraces, Justin Barhite
Bicategorical Traces And Cotraces, Justin Barhite
Theses and Dissertations--Mathematics
Familiar constructions like the trace of a matrix and the Euler characteristic of a closed smooth manifold are generalized by a notion of trace of an endomorphism of a dualizable object in a bicategory equipped with a piece of additional structure called a shadow functor. Another example of this bicategorical trace, in the form of maps between Hochschild homology of bimodules, appears in a 1987 paper by Joseph Lipman, alongside a more mysterious ”cotrace” map involving Hochschild cohomology. Putting this cotrace on the same category-theoretic footing as the trace has led us to propose a ”bicategorical cotrace” in a closed …
The Algebra Of Type Unification, Verity James Scheel
The Algebra Of Type Unification, Verity James Scheel
Senior Projects Spring 2022
Type unification takes type inference a step further by allowing non-local flow of information. By exposing the algebraic structure of type unification, we obtain even more flexibility as well as clarity in the implementation. In particular, the main contribution is an explicit description of the arithmetic of universe levels and consistency of constraints of universe levels, with hints at how row types and general unification/subsumption can fit into the same framework of constraints. The compositional nature of the algebras involved ensure correctness and reduce arbitrariness: properties such as associativity mean that implementation details of type inference do not leak in …
Unique Lifting To A Functor, Mark Myers
Unique Lifting To A Functor, Mark Myers
West Chester University Master’s Theses
We develop a functorial approach to quotient constructions, defining morphisms quotient relative to a functor and the dual concept of unique liftings relative to a functor. Various classes of epimorphism are given detailed analysis and their relationship to quotient morphisms characterized. The behavior of unique lifting morphisms with respect to products, equalizers, and general limits in a category are studied. Applications to generalized covering space theory, coreflective subcategories of topological spaces, topological groups and rings, and Galois theory are explored. Finally, we give conditions for the product of two quotient morphisms to be quotient in a braided monoidal closed category.
Categorical Aspects Of Graphs, Jacob D. Ender
Categorical Aspects Of Graphs, Jacob D. Ender
Undergraduate Student Research Internships Conference
In this article, we introduce a categorical characterization of directed and undirected graphs, and explore subcategories of reflexive and simple graphs. We show that there are a number of adjunctions between such subcategories, exploring varying combinations of graph types.
A Study On Approximations Of Totally Acyclic Complexes, Tyler Dean Anway
A Study On Approximations Of Totally Acyclic Complexes, Tyler Dean Anway
Mathematics Dissertations
Let $R$ be a commutative local ring to which we associate the subcategory $\Ktac(R)$ of the homotopy category of $R$-complexes, consisting of totally acyclic complexes. Further suppose there exists a surjection of Gorenstein local rings $Q \xrightarrowdbl{\varphi} R$ such that $R$ can be viewed as a $Q$-module with finite projective dimension. Under these assumptions, Bergh, Jorgensen, and Moore define the notion of approximations of totally acyclic complexes. In this dissertation we make extensive use of these approximations and define several novel applications. In particular, we extend Auslander-Reiten theory from the category of $R$-modules over a Henselian Gorenstein ring and show …
At The Interface Of Algebra And Statistics, Tai-Danae Bradley
At The Interface Of Algebra And Statistics, Tai-Danae Bradley
Dissertations, Theses, and Capstone Projects
This thesis takes inspiration from quantum physics to investigate mathematical structure that lies at the interface of algebra and statistics. The starting point is a passage from classical probability theory to quantum probability theory. The quantum version of a probability distribution is a density operator, the quantum version of marginalizing is an operation called the partial trace, and the quantum version of a marginal probability distribution is a reduced density operator. Every joint probability distribution on a finite set can be modeled as a rank one density operator. By applying the partial trace, we obtain reduced density operators whose diagonals …
A Coherent Proof Of Mac Lane's Coherence Theorem, Luke Trujillo
A Coherent Proof Of Mac Lane's Coherence Theorem, Luke Trujillo
HMC Senior Theses
Mac Lane’s Coherence Theorem is a subtle, foundational characterization of monoidal categories, a categorical concept which is now an important and popular tool in areas of pure mathematics and theoretical physics. Mac Lane’s original proof, while extremely clever, is written somewhat confusingly. Many years later, there still does not exist a fully complete and clearly written version of Mac Lane’s proof anywhere, which is unfortunate as Mac Lane’s proof provides very deep insight into the nature of monoidal categories. In this thesis, we provide brief introductions to category theory and monoidal categories, and we offer a precise, clear development of …
Relationships Between Category Theory And Functional Programming With An Application, Alper Odabaş, Eli̇s Soylu
Relationships Between Category Theory And Functional Programming With An Application, Alper Odabaş, Eli̇s Soylu
Turkish Journal of Mathematics
The most recent studies in mathematics are concerned with objects, morphisms, and the relationship between morphisms. Prominent examples can be listed as functions, vector spaces with linear transformations, and groups with homomorphisms. Category theory proposes and constitutes new structures by examining objects, morphisms, and compositions. Source and target of a morphism in category theory corresponds to input and output in programming language. Thus, a connection can be obtained between category theory and functional programming languages. From this point, this paper constructs a small category implementation in a functional programming language called Haskell.
Logic -> Proof -> Rest, Maxwell Taylor
Logic -> Proof -> Rest, Maxwell Taylor
Senior Independent Study Theses
REST is a common architecture for networked applications. Applications that adhere to the REST constraints enjoy significant scaling advantages over other architectures. But REST is not a panacea for the task of building correct software. Algebraic models of computation, particularly CSP, prove useful to describe the composition of applications using REST. CSP enables us to describe and verify the behavior of RESTful systems. The descriptions of each component can be used independently to verify that a system behaves as expected. This thesis demonstrates and develops CSP methodology to verify the behavior of RESTful applications.
Connecting Models Of Configuration Spaces: From Double Loops To Strings, Jason M. Lucas
Connecting Models Of Configuration Spaces: From Double Loops To Strings, Jason M. Lucas
Open Access Dissertations
Foundational to the subject of operad theory is the notion of an En operad, that is, an operad that is quasi-isomorphic to the operad of little n-cubes Cn. They are central to the study of iterated loop spaces, and the specific case of n = 2 is key in the solution of the Deligne Conjecture. In this paper we examine the connection between two E 2 operads, namely the little 2-cubes operad C 2 itself and the operad of spineless cacti. To this end, we construct a new suboperad of C2, which we name the operad of tethered …
Mathematical Frameworks For Consciousness, Menas C. Kafatos, Ashok Narasimhan
Mathematical Frameworks For Consciousness, Menas C. Kafatos, Ashok Narasimhan
Mathematics, Physics, and Computer Science Faculty Articles and Research
If Awareness is fundamental in the universe, mathematical frameworks are better suited to reveal its fundamental aspects than physical models. Awareness operates through three fundamental laws which apply at all levels of reality and is characterized by three universal powers. We explore and summarize in general terms mathematical formalisms that may take us as close as possible to conscious awareness, beginning with the primary relationships between the observer with the observed, using a Hilbert space approach. We also examine insights from category theory, and the calculus of indications or laws of forms. Mathematical frameworks as fundamental languages of our interaction …
Coalgebraic Semantics Of Reflexive Economics (Dagstuhl Seminar 15042), Samson Abramsky, Alexander Kurz, Pierre Lescanne, Viktor Winschel
Coalgebraic Semantics Of Reflexive Economics (Dagstuhl Seminar 15042), Samson Abramsky, Alexander Kurz, Pierre Lescanne, Viktor Winschel
Engineering Faculty Articles and Research
This report documents the program and the outcomes of Dagstuhl Seminar 15042 “Coalgebraic Semantics of Reflexive Economics”.
Diamond Semiotic Short Studies, Rudolf Kaehr
Diamond Semiotic Short Studies, Rudolf Kaehr
Rudolf Kaehr
A collection of papers on semiotics, polycontexturality and diamond theory
Double Cross Playing Diamonds, Rudolf Kaehr
Double Cross Playing Diamonds, Rudolf Kaehr
Rudolf Kaehr
Understanding interactivity in/between bigraphs and diamonds Grammatologically, the Western notational system is not offering space in itself to place sameness and otherness necessary to realize interaction/ality. Alphabetism is not prepared to challenge the dynamics of interaction directly. The Chinese writing system in its scriptural structuration, is able to place complex differences into itself, necessary for the development and design of formal systems and programming languages of interaction. The challenge of interactionality to Western thinking, modeling and design interactivity has to be confronted with the decline of the scientific power of alpha-numeric notational systems as media of living in a complex …