Shadows and traces in bicategories

Traces in symmetric monoidal categories are well-known and have many applications; for instance, their functoriality directly implies the Lefschetz fixed point theorem. However, for some applications, such as generalizations of the Lefschetz theorem, one needs "noncommutative" traces, such as the Hattori-Stallings trace for modules over noncommutative rings. In this paper we study a generalization of the symmetric monoidal trace which applies to noncommutative situations; its context is a bicategory equipped with an extra structure called a "shadow." In particular, we prove its functoriality and 2-functoriality, which are essential to its applications in fixed-point theory. Throughout we make use of an appropriate "cylindrical" type of string diagram, which we justify formally in an appendix.Comment: 46 pages; v2: reorganized and shortened, added proof for cylindrical string diagrams; v3: final version, to appear in JHR

Hidden in plain sight : the sibling relationship and psychodynamic theory

In psychotherapy, psychodynamic theory informs a large part of clinical practice. Throughout time, the prevailing theories have been largely devoted to exploring the parent-child relationship and its influence on development. A noticeable gap in the existing psychodynamic literature is in respect to the unique complexities of the sibling. The intent of this thesis is to further synthesize, explore, and contribute to the relatively small body of psychodynamic literature that addresses the sibling relationship. By the use of two very different major theoretical orientations, this thesis emphasizes the distinct, powerful, and mutual influence of siblings on development. It demonstrates that conflicted or unresolved early sibling relationships can come to influence adult pathology, personality presentation, and transference in the clinical setting. Thus, by advocating for the separate and continued consideration of the horizontal axis, this thesis reinforces the demand for greater attunement to siblings in both a theoretical and a treatment context

Relationship of the Pillars of Character and At-Risk Behaviors in Middle School Students

The purpose of this study was to analyze data to determine the effectiveness of the Character Counts! education program taught in a mid-sized midwestern public school district. Four independent evaluators compared the Six Pillars of Character to questions from the Youth Risk Behavior Survey (YRBS). Four research questions asked whether differences of the YRBS scores of risk behavior related to the character pillars of Character Counts! as linked by the independent evaluators. Questions asked whether there were differences in middle-school student scores from 2001 to 2003 as measured by the YRBS, between males and females, and whether there was a relationship between the students\u27 scores on the four character pillars and risk behaviors on the YRBS in middle-school students. Another question asked if there were differences in the character pillars based on four risk factors. Significant differences were found between the student scores in 2001 and 2003. Flowever, the scores reported a decrease in the number of positive behaviors from 2001 to 2003. No differences were found between males and females in the two survey reports. A significant negative relationship was found between the character pillars and risk behaviors. The findings suggested that positive behavior changes were difficult to quantify. Recommendations include integrating character education into the entire school program with appropriate staff development for teachers and administrators

On the word problem for SP-categories, and the properties of two-way communication

International audienceThe word problem for categories with free products and coproducts (sums), SP-categories, is directly related to the problem of determining the equivalence of certain processes. Indeed, the maps in these categories may be directly interpreted as processes which communicate by two-way channels. The maps of an SP-category may also be viewed as a proof theory for a simple logic with a game theoretic intepretation. The cut-elimination procedure for this logic determines equality only up to certain permuting conversions. As the equality classes under these permuting conversions are finite, it is easy to see that equality between cut-free terms (even in the presence of the additive units) is decidable. Unfortunately, this does not yield a tractable decision algorithm as these equivalence classes can contain exponentially many terms. However, the rather special properties of these free categories -- and, thus, of two-way communication -- allow one to devise a tractable algorithm for equality. We show that, restricted to cut-free terms s,t : X --> A, the decision procedure runs in time polynomial on |X||A|, the product of the sizes of the domain and codomain type

Abstract Tensor Systems as Monoidal Categories

The primary contribution of this paper is to give a formal, categorical treatment to Penrose's abstract tensor notation, in the context of traced symmetric monoidal categories. To do so, we introduce a typed, sum-free version of an abstract tensor system and demonstrate the construction of its associated category. We then show that the associated category of the free abstract tensor system is in fact the free traced symmetric monoidal category on a monoidal signature. A notable consequence of this result is a simple proof for the soundness and completeness of the diagrammatic language for traced symmetric monoidal categories.Comment: Dedicated to Joachim Lambek on the occasion of his 90th birthda

Calculating Colimits Compositionally

We show how finite limits and colimits can be calculated compositionally using the algebras of spans and cospans, and give as an application a proof of the Kleene Theorem on regular languages

Random quantum channels I: graphical calculus and the Bell state phenomenon

This paper is the first of a series where we study quantum channels from the random matrix point of view. We develop a graphical tool that allows us to compute the expected moments of the output of a random quantum channel. As an application, we study variations of random matrix models introduced by Hayden \cite{hayden}, and show that their eigenvalues converge almost surely. In particular we obtain for some models sharp improvements on the value of the largest eigenvalue, and this is shown in a further work to have new applications to minimal output entropy inequalities.Comment: Several typos were correcte

A categorical framework for the quantum harmonic oscillator

This paper describes how the structure of the state space of the quantum harmonic oscillator can be described by an adjunction of categories, that encodes the raising and lowering operators into a commutative comonoid. The formulation is an entirely general one in which Hilbert spaces play no special role. Generalised coherent states arise through the hom-set isomorphisms defining the adjunction, and we prove that they are eigenstates of the lowering operators. Surprisingly, generalised exponentials also emerge naturally in this setting, and we demonstrate that coherent states are produced by the exponential of a raising morphism acting on the zero-particle state. Finally, we examine all of these constructions in a suitable category of Hilbert spaces, and find that they reproduce the conventional mathematical structures.Comment: 44 pages, many figure

Equational reasoning with context-free families of string diagrams

String diagrams provide an intuitive language for expressing networks of interacting processes graphically. A discrete representation of string diagrams, called string graphs, allows for mechanised equational reasoning by double-pushout rewriting. However, one often wishes to express not just single equations, but entire families of equations between diagrams of arbitrary size. To do this we define a class of context-free grammars, called B-ESG grammars, that are suitable for defining entire families of string graphs, and crucially, of string graph rewrite rules. We show that the language-membership and match-enumeration problems are decidable for these grammars, and hence that there is an algorithm for rewriting string graphs according to B-ESG rewrite patterns. We also show that it is possible to reason at the level of grammars by providing a simple method for transforming a grammar by string graph rewriting, and showing admissibility of the induced B-ESG rewrite pattern.Comment: International Conference on Graph Transformation, ICGT 2015. The final publication is available at Springer via http://dx.doi.org/10.1007/978-3-319-21145-9_

Virtual reality as a screening tool for sports concussion in adolescents

PRIMARY OBJECTIVE: There is controversy surrounding the cognitive effects of sports concussion. This study aimed to verify whether the technique of virtual reality could aid in the identification of attention and inhibition deficits in adolescents. STUDY DESIGN: A prospective design was used to assess 25 sports-concussed and 25 non-sports-concussed adolescents enrolled in a sport and education programme. METHODS AND PROCEDURES: Participants were evaluated in immersive virtual reality via ClinicaVR: Classroom-CPT and in real life via the traditional VIGIL-CPT. MAIN OUTCOMES AND RESULTS: The neuropsychological assessment using virtual reality showed greater sensitivity to the subtle effects of sports concussion compared to the traditional test, which showed no difference between groups. The results also demonstrated that the sports concussion group reported more symptoms of cybersickness and more intense cybersickness than the control group. CONCLUSIONS: Sports concussion was associated with subtle deficits in attention and inhibition. However, further studies are needed to support these results