Rhombus filtrations and Rauzy algebras

Peach introduced rhombal algebras associated to quivers given by tilings of the plane by rhombi. We develop general techniques to analyze rhombal algebras, including a filtration by what we call rhombus modules. We introduce a way to relate the infinite-dimensional rhombal algebra corresponding to a complete tiling of the plane to finite-dimensional algebras corresponding to finite portions of the tiling. Throughout, we apply our general techniques to the special case of the Rauzy tiling, which is built in stages reflecting an underlying self-similarity. Exploiting this self-similar structure allows us to uncover interesting features of the associated finite-dimensional algebras, including some of the tree classes in the stable Auslander-Reiten quiver.     

Warning’s second theorem with relaxed outputs

We present a generalization of Warning’s second theorem to polynomial systems over a finite local principal ring with restricted input and relaxed output variables. This generalizes a recent result with Forrow and Schmitt (and gives a new proof of that result). Applications to additive group theory, graph theory and polynomial interpolation are pursued in detail.     

ABC and the Hasse principle for quadratic twists of hyperelliptic curves

Conditionally on the ABC conjecture, we apply work of Granville to show that a hyperelliptic curve $C_{/\mathbb{Q}}$ of genus at least three has infinitely many quadratic twists that violate the Hasse Principle iff it has no $\mathbb{Q}$-rational hyperelliptic branch points.     

The intermediate disorder regime for a directed polymer model on a hierarchical lattice

We study a directed polymer model defined on a hierarchical diamond lattice, where the lattice is constructed recursively through a recipe depending on a branching number $b\in \mathbb{N}$ and a segment number $s\in \mathbb{N}$. When $b\le s$ it is known that the model exhibits strong disorder for all positive values of the inverse temperature $\beta$, and thus weak disorder reigns only for $\beta =0$ (infinite temperature). Our focus is on the so-called intermediate disorder regime in which the inverse temperature $\beta \equiv {\beta }_n$ vanishes at an appropriate rate as the size $n$ of the system grows. Our analysis requires separate treatment for the cases $b<s$ and $b=s$. In the case $b<s$ we prove that when the inverse temperature is taken to be of the form ${\beta }_n=\stackrel{?}{\beta }{(b/s)}^{n/2}$ for $\stackrel{?}{\beta }>0$, the normalized partition function of the system converges weakly as $n\to \infty$ to a distribution $L\left(\stackrel{?}{\beta }\right)$ and does so universally with respect to the initial weight distribution. We prove the convergence using renormalization group type ideas rather than the standard Wiener chaos analysis. In the case $b=s$ we find a critical point in the behavior of the model when the inverse temperature is scaled as ${\beta }_n=\stackrel{?}{\beta }/n$; for an explicitly computable critical value ${\kappa }_b>0$ the variance of the normalized partition function converges to zero with large $n$ when $\stackrel{?}{\beta }\le {\kappa }_b$ and grows without bound when $\stackrel{?}{\beta }>{\kappa }_b$. Finally, we prove a central limit theorem for the normalized partition function when $\stackrel{?}{\beta }\le {\kappa }_b$.     

Diagonal-preserving ring ⁎-isomorphisms of Leavitt path algebras

The graph groupoids of directed graphs E and F are topologically isomorphic if and only if there is a diagonal-preserving ring ?-isomorphism between the Leavitt path algebras of E and F.     

Mathematical Analysis and Numerical Simulation of a Nonlinear Thermoelastic System

Abstract

In this paper, we give a theoretical and numerical analysis of a model for small vertical vibrations of an elastic membrane coupled with a heat equation and subject to linear mixed boundary conditions. We establish the existence, uniqueness, and a uniform decay rate for global solutions to this nonlinear non-local thermoelastic coupled system with linear boundary conditions. Furthermore, we introduced a numerical method based on finite element discretization in a spatial variable and finite difference scheme in time which results in a nonlinear system to be solved by Newton’s method. Numerical experiments for one-dimensional and two-dimensional cases are presented in order to estimate the rate of convergence of the numerical solution that confirm our analysis and show the efficiency of the method.     

Using Inquiry to Improve Pedagogy through K‐12/University Partnerships

This article focuses on the impact a collaborative project between university graduate fellows and K‐12 classroom teachers had on improved pedagogy in the classroom and in the future at the university. Nine teams participated in a yearlong professional development project to improve pedagogy and communication skills of the participants. This study shows that the participants, the fellows and the K‐12 teacher partners, made changes in planning, implementation, and even motivation for using inquiry‐based methods in their classroom. External observations of the teams further support the individuals' claims of improved pedagogy using inquiry and impact on student conceptual understanding. The Horizon Classroom Observation instrument was used for these observations. The teams showed an overall increase in scores, as well as overall effective and exemplarily implementation of their planning. The program design, the implementation, and the results of this three‐year study will be elaborated in this article.