The Cube Recurrence

We construct a combinatorial model that is described by the cube recurrence, a nonlinear recurrence relation introduced by Propp, which generates families of Laurent polynomials indexed by points in $\mathbb{Z}^3$. In the process, we prove several conjectures of Propp and of Fomin and Zelevinsky, and we obtain a combinatorial interpretation for the terms of Gale-Robinson sequences. We also indicate how the model might be used to obtain some interesting results about perfect matchings of certain bipartite planar graphs

Computing Tropical Varieties

The tropical variety of a $d$-dimensional prime ideal in a polynomial ring with complex coefficients is a pure $d$-dimensional polyhedral fan. This fan is shown to be connected in codimension one. We present algorithmic tools for computing the tropical variety, and we discuss our implementation of these tools in the Gr\"obner fan software \texttt{Gfan}. Every ideal is shown to have a finite tropical basis, and a sharp lower bound is given for the size of a tropical basis for an ideal of linear forms.Comment: 22 pages, 2 figure

The m−m-dissimilarity map and representation theory of SLmSL_m

We give another proof that $m$-dissimilarity vectors of weighted trees are points on the tropical Grassmanian, as conjectured by Cools, and proved by Giraldo in response to a question of Sturmfels and Pachter. We accomplish this by relating $m$-dissimilarity vectors to the representation theory of $SL_m.$Comment: 11 pages, 8 figure

Q-systems, Heaps, Paths and Cluster Positivity

We consider the cluster algebra associated to the $Q$-system for $A_r$ as a tool for relating $Q$-system solutions to all possible sets of initial data. We show that the conserved quantities of the $Q$-system are partition functions for hard particles on particular target graphs with weights, which are determined by the choice of initial data. This allows us to interpret the simplest solutions of the Q-system as generating functions for Viennot's heaps on these target graphs, and equivalently as generating functions of weighted paths on suitable dual target graphs. The generating functions take the form of finite continued fractions. In this setting, the cluster mutations correspond to local rearrangements of the fractions which leave their final value unchanged. Finally, the general solutions of the $Q$-system are interpreted as partition functions for strongly non-intersecting families of lattice paths on target lattices. This expresses all cluster variables as manifestly positive Laurent polynomials of any initial data, thus proving the cluster positivity conjecture for the $A_r$ $Q$-system. We also give an alternative formulation in terms of domino tilings of deformed Aztec diamonds with defects.Comment: 106 pages, 38 figure

Computing Linear Matrix Representations of Helton-Vinnikov Curves

Helton and Vinnikov showed that every rigidly convex curve in the real plane bounds a spectrahedron. This leads to the computational problem of explicitly producing a symmetric (positive definite) linear determinantal representation for a given curve. We study three approaches to this problem: an algebraic approach via solving polynomial equations, a geometric approach via contact curves, and an analytic approach via theta functions. These are explained, compared, and tested experimentally for low degree instances.Comment: 19 pages, 3 figures, minor revisions; Mathematical Methods in Systems, Optimization and Control, Birkhauser, Base

Changes in domestic heating fuel use in Greece : Effects on atmospheric chemistry and radiation

For the past 8 years, Greece has been experiencing a major financial crisis which, among other side effects, has led to a shift in the fuel used for residential heating from fossil fuel towards biofuels, primarily wood. This study simulates the fate of the residential wood burning aerosol plume (RWB smog) and the implications on atmospheric chemistry and radiation, with the support of detailed aerosol characterization from measurements during the winter of 2013–2014 in Athens. The applied model system (TNO-MACC_II emissions and COSMO-ART model) and configuration used reproduces the measured frequent nighttime aerosol spikes (hourly PM₁₀  >  75 µg m⁻³) and their chemical profile (carbonaceous components and ratios). Updated temporal and chemical RWB emission profiles, derived from measurements, were used, while the level of the model performance was tested for different heating demand (HD) conditions, resulting in better agreement with measurements for T$_{min}$ < 9 °C. Half of the aerosol mass over the Athens basin is organic in the submicron range, of which 80 % corresponds to RWB (average values during the smog period). Although organic particles are important light scatterers, the direct radiative cooling of the aerosol plume during wintertime is found low (monthly average forcing of –0.4 W m⁻² at the surface), followed by a minor feedback to the concentration levels of aerosol species. The low radiative cooling of a period with such intense air pollution conditions is attributed to the timing of the smog plume appearance, both directly (longwave radiation increases during nighttime) and indirectly (the mild effect of the residual plume on solar radiation during the next day, due to removal and dispersion processes

Discrete integrable systems, positivity, and continued fraction rearrangements

In this review article, we present a unified approach to solving discrete, integrable, possibly non-commutative, dynamical systems, including the $Q$- and $T$-systems based on $A_r$. The initial data of the systems are seen as cluster variables in a suitable cluster algebra, and may evolve by local mutations. We show that the solutions are always expressed as Laurent polynomials of the initial data with non-negative integer coefficients. This is done by reformulating the mutations of initial data as local rearrangements of continued fractions generating some particular solutions, that preserve manifest positivity. We also show how these techniques apply as well to non-commutative settings.Comment: 24 pages, 2 figure

What influences speech-language pathologists' use of different types of language assessments for elementary school-age children?

Purpose: This study reports on data from a survey of speech-language pathologists' (SLPs) language assessment practices for elementary school-age children. The objective was to investigate the regularity with which SLPs use different types of assessments (described across data types, task types, environmental contexts, and dynamic features). This study also investigated factors that influence assessment practice, the main sources from which SLPs obtain information on language assessment and the main challenges reported by SLPs in relation to language assessment.Method: A web-based survey was used to collect information from 407 Australian SLPs regarding the types of assessments they use. Factors that influenced the regularity with which different types of assessments were used were investigated using regression analysis.Results: Most SLPs regularly used assessments that are norm-referenced, decontextualized, and conducted in a clinical context and less regularly used other types of assessments. Service agency, Australian state, and SLPs' years of experience were found to influence the regularity with which some types of assessments were used. Informal discussions with colleagues were the most frequently identified source of information on assessment practice. Main challenges related to limited time, lack of assessment materials, and lack of confidence in assessing children from culturally and linguistically diverse backgrounds.Conclusions: SLPs could improve current language assessment practice for elementary school-age children through more regular use of some types of assessments. Actions to facilitate evidence-based assessment practice should consider the contextual differences that exist between service agencies and states and address challenges that SLPs experience in relation to language assessment.Otorhinolaryngolog

Integrable structure of box-ball systems: crystal, Bethe ansatz, ultradiscretization and tropical geometry

The box-ball system is an integrable cellular automaton on one dimensional lattice. It arises from either quantum or classical integrable systems by the procedures called crystallization and ultradiscretization, respectively. The double origin of the integrability has endowed the box-ball system with a variety of aspects related to Yang-Baxter integrable models in statistical mechanics, crystal base theory in quantum groups, combinatorial Bethe ansatz, geometric crystals, classical theory of solitons, tau functions, inverse scattering method, action-angle variables and invariant tori in completely integrable systems, spectral curves, tropical geometry and so forth. In this review article, we demonstrate these integrable structures of the box-ball system and its generalizations based on the developments in the last two decades.Comment: 73 page

The solution of the quantum A1A_1 T-system for arbitrary boundary

We solve the quantum version of the $A_1$ $T$-system by use of quantum networks. The system is interpreted as a particular set of mutations of a suitable (infinite-rank) quantum cluster algebra, and Laurent positivity follows from our solution. As an application we re-derive the corresponding quantum network solution to the quantum $A_1$ $Q$-system and generalize it to the fully non-commutative case. We give the relation between the quantum $T$-system and the quantum lattice Liouville equation, which is the quantized $Y$-system.Comment: 24 pages, 18 figure