Let 0＜γ＜π be a fixed pythagorean angle. We study the abelian group H_{r} of primitive integral triangles (a,b,c) for which the angle opposite side c is γ. Addition in H_{r } is defined by adding the anglesβopposite side b and modding out by π-γ. The only H_{r} for which the structure is known is H_{π}_{/}_{2}, which is free abelian. We prove that for generalγ, H_{r} has an element of order two iff 2(1-

Abstract:
We introduce the M\"obius polynomial $ M_n(x) = \sum_{d|n} \mu\left( \frac nd \right) x^d $, which gives the number of aperiodic bracelets of length $n$ with $x$ possible types of gems, and therefore satisfies $M_n(x) \equiv 0$ (mod $n$) for all $x \in \mathbb Z$. We derive some key properties, analyze graphs in the complex plane, and then apply M\"obius polynomials combinatorially to juggling patterns, irreducible polynomials over finite fields, and Euler's totient theorem.

Abstract:
We analyze the homothety types of associative bilinear forms that can occur on a Hopf algebra or on a local Frobenius \(k\)-algebra \(R\) with residue field \(k\). If \(R\) is symmetric, then there exists a unique form on \(R\) up to homothety iff \(R\) is commutative. If \(R\) is Frobenius, then we introduce a norm based on the Nakayama automorphism of \(R\). We show that if two forms on \(R\) are homothetic, then the norm of the unit separating them is central, and we conjecture the converse. We show that if the dimension of \(R\) is even, then the determinant of a form on \(R\), taken in \(\dot k/\dot k^2\), is an invariant for \(R\). \textit{Key words}: bilinear form, Frobenius algebra, homothety, Hopf algebra, isometry, local algebra, Nakayama automorphism, Ore extension, symmetric algebra

Abstract:
We show that the Nakayama automorphism of a Frobenius algebra $R$ over a field $k$ is independent of the field (Theorem 4). Consequently, the $k$-dual functor on left $R$-modules and the bimodule isomorphism type of the $k$-dual of $R$, and hence the question of whether $R$ is a symmetric $k$-algebra, are independent of $k$. We give a purely ring-theoretic condition that is necessary and sufficient for a finite-dimensional algebra over an infinite field to be a symmetric algebra (Theorem 7). Key words: Nakayama automorphism, Frobenius algebra, Frobenius ring, symmetric algebra, dual module, dual functor, bimodule, Brauer Equivalence.

Cognitive
disorders following hypoxic ischemic brain injury involve a variety of
disorders including consciousness, behavior, mood and affect, impairment of
attention, and memory dysfunction. The case of a 45-year-old former military
aviator and engineer, now a physician in residency training, presenting with
cognitive difficulties, is described. The patient described having difficulty
remembering medical knowledge and feeling fatigued. After almost nine months
without any medical intervention and the patient’s deteriorating condition, the
patient was finally evaluated medically. It was ultimately discovered that the
patient suffered from a variety of neurologic impairments that were the direct
result of exposures to various toxic substances during his military service. Significant
diagnoses included hypoxic ischemic brain damage, severe mixed sleep apnea, and
cognitive disorder NOS. Relevant literature about the application of
neurocognitive rehabilitation and retraining to treating patients suffering
from brain injuries is discussed. The overlap of the neuroscience of emotion
with cognitive learning and how emotion and affect impacts learning and
education is presented. This case also serves to demonstrate the application of
learning and cognition to individual differences and disabilities. Further
research is needed to evaluate whether this result is reproducible and
generalizable to other patients with similar presenting signs and symptoms.

This paper is divided into three parts. In the
first part, we review the historical background of a system
of logic devised by Henry S. Leonard to allow for reasoning using existence as
a predicate. In the second part, we consider various directions
in which his logic could be further developed, syntactically, semantically, and
as an adjunct to quantifier elimination and set theory. In the third and final
part, we develop proofs of some underlying results of his logic, using modern
notation but retaining his axioms and rules of inference.

Abstract:
High maternal, gestational weight gain is associated with high birthweight, large-for-gestational-age birthweights, cesarean delivery, child overweight, and short- and long-term postpartum weight retention. In this phenomenological study, the meaning and experiences of weight gain for pregnant women with high gestational weight gain were investigated. Data were collected through interviews with pregnant women from Atlantic Canada. van Manen’s method of phenomenology was utilized. The data analysis revealed four patterns or major themes: being caught off guard; losing your bearings; hanging on for dear life; and hoping for health. The participants experienced their gestational weight gain as an unexpected “wild ride” that they could not control. The findings highlight the need for health care professionals to provide pregnant women with more support concerning gestational weight gain.

Abstract:
This research critically examines traditional quantitative measurements of alcohol establishments and assaults. In doing so, the research first performs a quantitative, spatially-lagged regression model measuring the relationship between location of alcohol establishment and assaults, using traditional measurements of liquor license designation and reports to the police. It then examines the same phenomenon using qualitative measurements. This includes creating a designation of “criminogenic” establishments through atmosphere assessments and employee perception of assault. Results indicate that qualitative analysis, while not wildly divergent in results, may allow more nuanced operationalization of some of the concepts suggested in Routine Activity Theory. The implications of these definitions are considered in terms of their probable impact on previous research results, and future direction in developing accurate measurement in this area is discussed.

Abstract:
In this paper we undertake to examine how algebra, its tools and its methods, can be used to formulate the mathematics used in applications. We give particular attention to the mathematics used in application to physics. We suggest that methods first proposed by Henry Siggins Leonard are well suited to such an examination.

Abstract:
A number of statistical tests are proposed for the purpose of change-point detection in a general nonparametric regression model under mild conditions. New proofs are given to prove the weak convergence of the underlying processes which assume remove the stringent condition of bounded total variation of the regression function and need only second moments. Since many quantities, such as the regression function, the distribution of the covariates and the distribution of the errors, are unspecified, the results are not distribution-free. A weighted bootstrap approach is proposed to approximate the limiting distributions. Results of a simulation study for this paper show good performance for moderate samples sizes.