Ideal Theory in Rings (Translation of "Idealtheorie in Ringbereichen" by Emmy Noether)

This paper is a translation of the paper "Idealtheorie in Ringbereichen", written by Emmy Noether in 1920, from the original German into English. It in particular brings the language used into the modern world so that it is easily understandable by the mathematicians of today. The paper itself deals with ideal theory, and was revolutionary in its field, that is modern algebra. Topics covered include: the representation of an ideal as the least common multiple of irreducible ideals; the representation of an ideal as the least common multiple of maximal primary ideals; the association of prime ideals with primary ideals; the representation of an ideal as the least common multiple of relatively prime irreducible ideals; isolated ideals; the representation of an ideal as the product of coprime irreducible ideals; equivalent concepts regarding modules.Comment: 51 pages, including a page dedicated to notes on the translation of particular term

Graph of groups decompositions of graph braid groups

We provide an explicit construction that allows one to easily decompose a graph braid group as a graph of groups. This allows us to compute the braid groups of a wide range of graphs, as well as providing two general criteria for a graph braid group to split as a non-trivial free product, answering two questions of Genevois. We also use this to distinguish certain right-angled Artin groups and graph braid groups. Additionally, we provide an explicit example of a graph braid group that is relatively hyperbolic, but is not hyperbolic relative to braid groups of proper subgraphs. This answers another question of Genevois in the negative.Comment: Version accepted for publication. Several minor corrections and clarifications have been made and new classes of examples have been added. Code is also available, implementing algorithms in this paper to compute free splittings and presentations of graph braid groups; see https://github.com/danberlyne/graph-braid-splitter and https://github.com/danberlyne/graph-braid-presente

Hierarchical hyperbolicity of graph products

We show that any graph product of finitely generated groups is hierarchically hyperbolic relative to its vertex groups. We apply this result to answer two questions of Behrstock, Hagen, and Sisto: we show that the syllable metric on any graph product forms a hierarchically hyperbolic space, and that graph products of hierarchically hyperbolic groups are themselves hierarchically hyperbolic groups. This last result is a strengthening of a result of Berlai and Robbio by removing the need for extra hypotheses on the vertex groups. We also answer two questions of Genevois about the geometry of the electrification of a graph product of finite groups.Comment: 63 pages, 12 figures. Comments welcom

Fractal Dimensions in Perceptual Color Space: A Comparison Study Using Jackson Pollock's Art

The fractal dimensions of color-specific paint patterns in various Jackson Pollock paintings are calculated using a filtering process which models perceptual response to color differences (\Lab color space). The advantage of the \Lab space filtering method over traditional RGB spaces is that the former is a perceptually-uniform (metric) space, leading to a more consistent definition of ``perceptually different'' colors. It is determined that the RGB filtering method underestimates the perceived fractal dimension of lighter colored patterns but not of darker ones, if the same selection criteria is applied to each. Implications of the findings to Fechner's 'Principle of the Aesthetic Middle' and Berlyne's work on perception of complexity are discussed.Comment: 21 pp LaTeX; two postscript figure

Ontogeny of sex differences in response to novel objects from adolescence to adulthood in lister-hooded rats

In humans, novelty-seeking behavior peaks in adolescence and is higher in males than females. Relatively, little information is available regarding age and sex differences in response to novelty in rodents. In this study, male and female Lister-hooded rats were tested at early adolescence (postnatal day, pnd, 28), mid-adolescence (pnd 40), or early adulthood (pnd 80) in a novel object recognition task (n = 12 males/females per age group). Males displayed a higher preference for the novel object than females at mid-adolescence, with no sex difference at early adolescence. Adult females interacted with the novel object more than adult males, but not when side biases were removed. Sex differences at mid-adolescence were not found in other measures, suggesting that the difference at this age was specific to situations involving choice of novelty. The results are considered in the context of age- and sex-dependent interactions between gonadal hormones and the dopamine system. © 2011 Wiley Periodicals, Inc. Dev Psychobiol 53:670–676, 2011

The tendency of the schematic structure to maintain stability can be interpreted as mental inertia

This paper incorporates schematic concepts related to mental inertia and provides an avenue for interpreting psychology using the principles of classical mechanics. Schemas find wide application in diverse fields, ranging from ergonomics to psychotherapy. Nonetheless, it is crucial to incorporate schemas themselves into a more unified and comprehensive theoretical framework. Drawing upon the free energy principle (FEP) and the second law of thermodynamics, it is evident that humans possess a natural inclination to construct and maintain consistent cognitive structures. This characteristic contributes to the stability of schemas within a defined range. The particular scope of the model is closely intertwined with its structure, leading to variations among individuals in diverse environments. The coherence of the schema within a defined range can be perceived as the magnitude of mental inertia. This psychological analogy emphasizes the importance of considering the influences exerted by the external environment and their effects on mental inertia when predicting the human mind and behavior

Information dynamics: patterns of expectation and surprise in the perception of music

This is a postprint of an article submitted for consideration in Connection Science © 2009 [copyright Taylor & Francis]; Connection Science is available online at:http://www.tandfonline.com/openurl?genre=article&issn=0954-0091&volume=21&issue=2-3&spage=8

Hierarchical Hyperbolicity of Graph Products and Graph Braid Groups

This thesis comprises three original contributions by the author concerning hierarchical hyperbolicity, a coarse geometric tool developed by Behrstock, Hagen, and Sisto to provide a common framework for studying aspects of non-positive curvature in a wide variety of groups and spaces. We show that any graph product of finitely generated groups is hierarchically hyperbolic relative to its vertex groups. We apply this to answer two questions of Genevois about the electrification of a graph product of finite groups. We also answer two questions of Behrstock, Hagen, and Sisto: we show that the syllable metric on a graph product forms a hierarchically hyperbolic space, and that graph products of hierarchically hyperbolic groups are themselves hierarchically hyperbolic groups. This last result is a strengthening of a result of Berlai and Robbio by removing the need for extra hypotheses on the vertex groups. To achieve this, we develop a technique that allows an almost hierarchically hyperbolic structure to be promoted to a hierarchically hyperbolic structure. This technique has found independent use in work of Abbott, Behrstock, and Durham, where it is used to significantly streamline their proofs. We then turn to graph braid groups, using their structure as fundamental groups of special cube complexes to endow them with a natural hierarchically hyperbolic structure. By expressing this structure in terms of the graph, we obtain characterisations of when these groups are hyperbolic or acylindrically hyperbolic. We also conjecture and partially prove a similar characterisation of relative hyperbolicity