Recurrence and transience for suspension flows

We study the thermodynamic formalism for suspension flows over countable Markov shifts with roof functions not necessarily bounded away from zero. We establish conditions to ensure the existence and uniqueness of equilibrium measures for regular potentials. We define the notions of recurrence and transience of a potential in this setting. We define the "renewal flow", which is a symbolic model for a class of flows with diverse recurrence features. We study the corresponding thermodynamic formalism, establishing conditions for the existence of equilibrium measures and phase transitions. Applications are given to suspension flows defined over interval maps having parabolic fixed points.Comment: In this version of the paper some typos have been corrected and some references updated. Note that the former title of this paper was "Parabolic suspension flows

Transience and multifractal analysis

We study dimension theory for dissipative dynamical systems, proving a conditional variational principle for the quotients of Birkhoff averages restricted to the recurrent part of the system. On the other hand, we show that when the whole system is considered (and not just its recurrent part) the conditional variational principle does not necessarily hold. Moreover, we exhibit the first example of a topologically transitive map having discontinuous Lyapunov spectrum. The mechanism producing all these pathological features on the multifractal spectra is transience, that is, the non-recurrent part of the dynamics.Comment: Some updates following referee suggestion

Comparing Powers of Edge Ideals

Given a nontrivial homogeneous ideal $I\subseteq k[x_1,x_2,\ldots,x_d]$, a problem of great recent interest has been the comparison of the $r$th ordinary power of $I$ and the $m$th symbolic power $I^{(m)}$. This comparison has been undertaken directly via an exploration of which exponents $m$ and $r$ guarantee the subset containment $I^{(m)}\subseteq I^r$ and asymptotically via a computation of the resurgence $\rho(I)$, a number for which any $m/r > \rho(I)$ guarantees $I^{(m)}\subseteq I^r$. Recently, a third quantity, the symbolic defect, was introduced; as $I^t\subseteq I^{(t)}$, the symbolic defect is the minimal number of generators required to add to $I^t$ in order to get $I^{(t)}$. We consider these various means of comparison when $I$ is the edge ideal of certain graphs by describing an ideal $J$ for which $I^{(t)} = I^t + J$. When $I$ is the edge ideal of an odd cycle, our description of the structure of $I^{(t)}$ yields solutions to both the direct and asymptotic containment questions, as well as a partial computation of the sequence of symbolic defects.Comment: Version 2: Revised based on referee suggestions. Lemma 5.12 was added to clarify the proof of Theorem 5.13. To appear in the Journal of Algebra and its Applications. Version 1: 20 pages. This project was supported by Dordt College's undergraduate research program in summer 201

Cross-middleware Interoperability in Distributed Concurrent Engineering

Secure, distributed collaboration between different organizations is a key challenge in Grid computing today. The GDCD project has produced a Grid-based demonstrator Virtual Collaborative Facility (VCF) for the European Space Agency. The purpose of this work is to show the potential of Grid technology to support fully distributed concurrent design, while addressing practical considerations including network security, interoperability, and integration of legacy applications. The VCF allows domain engineers to use the concurrent design methodology in a distributed fashion to perform studies for future space missions. To demonstrate the interoperability and integration capabilities of Grid computing in concurrent design, we developed prototype VCF components based on ESA’s current Excel-based Concurrent Design Facility (a non-distributed environment), using a STEP-compliant database that stores design parameters. The database was exposed as a secure GRIA 5.1 Grid service, whilst a .NET/WSE3.0-based library was developed to enable secure communication between the Excel client and STEP database

Expansion of kk-Schur functions for maximal kk-rectangles within the affine nilCoxeter algebra

We give several explicit combinatorial formulas for the expansion of k-Schur functions indexed by maximal rectangles in terms of the standard basis of the affine nilCoxeter algebra. Using our result, we also show a commutation relation of k-Schur functions corresponding to rectangles with the generators of the affine nilCoxeter algebra.Comment: to appear in Journal of Combinatorics, 28 page

Refining the Gatekeeping Metaphor for Local Television News

A book review of Refining the Gatekeeping Metaphor for Local Television News by Dan Berkowitz

Understanding the Jobs-Affordable Housing Balance in the Richmond Region

The mismatch between location of jobs and housing has a significant impact on the efficiency and quality of life within metropolitan areas. A well-planned region strives to be a “community of short distances.” A wide range of housing choices located close to employment centers could shorten commuting distances and substantially reduce government outlays for transportation facilities, reduce household transportation expenses, and increase feasibility of pedestrian movement. These needs are particularly important to families earning modest wages. CURA, with support from The Community Foundation Serving Richmond and Central Virginia and the Richmond Association of Realtors, has analyzed the spatial pattern of lower-wage jobs and lower-cost housing within the Richmond Metropolitan Statistical Area (MSA). The analysis reveals how low-cost housing and modest-wage jobs in the Richmond region are not well-balanced. Few areas in which modest-wage jobs cluster have comparable levels of low-cost housing. The established suburban areas north, west, and south of Richmond’s urban center have a large number of retail and service jobs that pay modest wages, yet these areas provide few affordable-dwelling units for these wage earners. The second part of this study addressed a major obstacle to the construction of new, affordable-housing units: fears. Many new, affordable dwelling units, by financial necessity, will be built at higher densities and smaller size to reduce cost. Homeowners in nearby neighborhoods often oppose construction of these units over fear of reduced property values, higher crime, and other factors. Six higher-density, 3 lower-cost housing projects were studied for their impact on the nearby middle-income neighborhoods. Documentation of home sale prices, assessment values, and crime rates before and after construction of the more affordable dwelling units did not reveal any notable long-term impact on crime rates, property values, or property sales

Is what you see what you get? representations, metaphors and tools in mathematics didactics

This paper is exploratory in character. The aim is to investigate ways in which it is possible to use the theoretical concepts of representations, tools and metaphors to try to understand what learners of mathematics ‘see’ during classroom interactions (in their widest sense) and what they might get from such interactions. Through an analysis of a brief classroom episode, the suggestion is made that what learners see may not be the same as what they get. From each of several theoretical perspectives utilised in this paper, what learners ‘get’ appears to be something extra. According to our analysis, this something ‘extra’ is likely to depend on the form of technology being used and the representations and metaphors that are available to both teacher and learner

Calculation of disease dynamics in a population of households

Early mathematical representations of infectious disease dynamics assumed a single, large, homogeneously mixing population. Over the past decade there has been growing interest in models consisting of multiple smaller subpopulations (households, workplaces, schools, communities), with the natural assumption of strong homogeneous mixing within each subpopulation, and weaker transmission between subpopulations. Here we consider a model of SIRS (susceptible-infectious-recovered-suscep​tible) infection dynamics in a very large (assumed infinite) population of households, with the simplifying assumption that each household is of the same size (although all methods may be extended to a population with a heterogeneous distribution of household sizes). For this households model we present efficient methods for studying several quantities of epidemiological interest: (i) the threshold for invasion; (ii) the early growth rate; (iii) the household offspring distribution; (iv) the endemic prevalence of infection; and (v) the transient dynamics of the process. We utilize these methods to explore a wide region of parameter space appropriate for human infectious diseases. We then extend these results to consider the effects of more realistic gamma-distributed infectious periods. We discuss how all these results differ from standard homogeneous-mixing models and assess the implications for the invasion, transmission and persistence of infection. The computational efficiency of the methodology presented here will hopefully aid in the parameterisation of structured models and in the evaluation of appropriate responses for future disease outbreaks