On the dual nature of partial theta functions and Appell-Lerch sums

In recent work, Hickerson and the author demonstrated that it is useful to think of Appell--Lerch sums as partial theta functions. This notion can be used to relate identities involving partial theta functions with identities involving Appell--Lerch sums. In this sense, Appell--Lerch sums and partial theta functions appear to be dual to each other. This duality theory is not unlike that found by Andrews between various sets of identities of Rogers-Ramanujan type with respect to Baxter's solution to the hard hexagon model of statistical mechanics. As an application we construct bilateral $q$-series with mixed mock modular behaviour.Comment: To be published in Advances in Mathematic

“What I care bout dogs?” How the hegemony of the English language colonizes marginalized groups

Language has always seemed to play a pivotal role in the identity of individuals and groups who are relegated to the status of other by members of the dominant group. The misshapen identities of the others are further exacerbated by systems and structures in place, like education, which ensure the continued hegemonic status of dominant groups. Not being a part of the dominant group, and yet being required to conform to their standards, can be confusing for those on the outside. The character Celie, expressed this confusion, “Whitefolks all over them, talking bout apples and dogs. What I care bout dogs? [emphasis added] […] Look like to me only a fool would want you to talk in a way that peculiar to your mind.” (Walker, 1982, pp. 193-194). This autoethnography is informed by Critical Race Theory and Critical Research. CRT states that racism is a normal part of American life, and in terms of Critical Research, Merriam (2009) wrote that “power in combination with hegemonic social structures results in the marginalization and oppression of those without power.” (p. 35). My own memories of living in South Africa and the United States, give voice to this phenomenon. I want to explore whether language is replacing race as a social and political identifier. Finally, this ethnographic study will hopefully encourage others to come forth and add their own voices. If all dialects hold equal status in a society, would this lessen the chance of language replacing race as a social identifier? Would structures like the police then serve all people, and would the people, and their identities, begin the process of healing

Hecke-type double sums, Appell-Lerch sums, and mock theta functions (I)

By developing a connection between partial theta functions and Appell-Lerch sums, we find and prove a formula which expresses Hecke-type double sums in terms of Appell-Lerch sums and theta functions. Not only does our formula prove classical Hecke-type double sum identities such as those found in work Kac and Peterson on affine Lie Algebras and Hecke modular forms, but once we have the Hecke-type forms for Ramanujan's mock theta functions our formula gives straightforward proofs of many of the classical mock theta function identities. In particular, we obtain a new proof of the mock theta conjectures. Our formula also applies to positive-level string functions associated with admissable representations of the affine Lie Algebra $A_1^{(1)}$ as introduced by Kac and Wakimoto

Using the Information Processing Approach to Explain the Mysteries of the Black Box: Implications for Teaching Religious and Moral Education

The aim of this paper is to use the three “stage theory proposed by Atkinson and Shiffrin (1968) or the “information processing approach” to explain the cognitive processes or the mysteries of the “black box” and its implication for teaching Religious and Moral Education. The rapid creation of computers has encouraged the use of computer model to explain learning, which is termed as the information processing approach, proposed by Atkinson and Shiffrin in the late 1960’s. Thus, the information processing approach is based on the apparent similarities between the operation of the human brain and that of the computer and this gives the implication that information processing approaches are concerned with the nature of the cognitive processes. The implications of the information processing approach or the stage theory for teaching Religious and Moral Education, should place emphasis on the design of the curriculum, that is experiencing what is being taught, focusing on the developmental levels of leaners and employing the appropriate teaching and learning materials to give the right information to the learners especially in the initial stages of cognitive development. Keywords: black box, cognitive development, memory, memory system information, teachin

Further Evidence on the Great Crash, the Oil Price Shock, and the Unit Root Hypothesis

Recently Perron (1989) has carried out tests of the unit root hypothesis against the alternative hypothesis of trend stationarity with a break in the trend occurring at the Great Crash of 1929 or at the 1973 oil price shock. His analysis covers the Nelson-Plosser macroeconomic data series as well as a post-war quarter real GNP series. His tests reject the unit root null hypothesis for most of the series. This paper takes issue with the assumption used by Perron that the Great Crash and the oil price shock can be treated as exogenous events. A variation of Perron’s test is considered in which the break point is estimated rather than fixed. We argue this test is more appropriate than Perron’s, since it circumvents the problem of data-mining. The asymptotic distribution of the “estimated break point” test statistic is determined. The data series considered by Perron are reanalyzed using this test statistic. The empirical results make use of the asymptotics developed for the test statistic as well as extensive finite sample corrections obtained by simulation. The effect on the empirical results of fat-tailed and temporally dependent innovations is investigated. In brief, by treating the break point as endogenous, we find that there is less evidence against the unit root hypothesis than Perron finds for many of the data series, but stronger evidence against it for several of the series, including the Nelson-Plosser industrial production, nominal GNP, and real GNP series

Thiopurines are negatively associated with anthropometric parameters in pediatric Crohn's disease.

AimTo determine the distribution of anthropometric parameter (AP)-z-scores and characterize associations between medications/serum biomarkers and AP-z-scores in pediatric Crohn's disease (CD).MethodsCD patients [< chronological age (CA) 21 years] were enrolled in a cross-sectional study. Descriptive statistics were generated for participants' demographic characteristics and key variables of interest. Paired t-tests were used to compare AP-z-scores calculated based on CA (CA z-scores) and bone age (BA) (BA z-scores) for interpretation of AP's. Linear regression was utilized to examine associations between medications and serum biomarkers with AP-z-scores calculated based on CA (n = 82) and BA (n = 49). We reported regression coefficients as well as their corresponding p-values and 95% confidence intervals.ResultsMean CA at the time of the study visit was 15.3 ± 3.5 (SD; range = 4.8-20.7) years. Mean triceps skinfold (P = 0.039), subscapular skinfold (P = 0.002) and mid-arm circumference (MAC) (P = 0.001) BA z-scores were higher than corresponding CA z-scores. Medications were positively associated with subscapular skinfold [adalimumab (P = 0.018) and methotrexate (P = 0.027)] and BMI CA z-scores [adalimumab (P = 0.029)]. Azathioprine/6-mercaptopurine were negatively associated with MAC (P = 0.045), subscapular skinfold (P = 0.014), weight (P = 0.002) and BMI (P = 0.013) CA z-scores. ESR, CRP, and WBC count were negatively associated, while albumin and IGF-1 BA z-scores were positively associated, with specific AP z-scores (P < 0.05). Mean height CA z-scores were higher in females, not males, treated with infliximab (P = 0.038). Hemoglobin (P = 0.018) was positively associated, while platelets (P = 0.005), ESR (P = 0.003) and CRP (P = 0.039) were negatively associated with height CA z-scores in males, not females.ConclusionOur results suggest poor efficacy of thiopurines and a possible sex difference in statural growth response to infliximab in pediatric CD. Prospective longitudinal studies are required

A Disk-based Dynamical Mass Estimate for the Young Binary AK Sco

We present spatially and spectrally resolved Atacama Large Millimeter/submillimeter Array (ALMA) observations of gas and dust in the disk orbiting the pre-main sequence binary AK Sco. By forward-modeling the disk velocity field traced by CO J=2-1 line emission, we infer the mass of the central binary, $M_\ast = 2.49 \pm 0.10~M_\odot$, a new dynamical measurement that is independent of stellar evolutionary models. Assuming the disk and binary are co-planar within $\sim$2{\deg}, this disk-based binary mass measurement is in excellent agreement with constraints from radial velocity monitoring of the combined stellar spectra. These ALMA results are also compared with the standard approach of estimating masses from the location of the binary in the Hertzsprung-Russell diagram, using several common pre-main sequence model grids. These models predict stellar masses that are marginally consistent with our dynamical measurement (at $\sim 2\,\sigma$), but are systematically high (by $\sim$10%). These same models consistently predict an age of $18\pm1$ Myr for AK Sco, in line with its membership in the Upper Centaurus-Lupus association but surprisingly old for it to still host a gas-rich disk. As ALMA accumulates comparable data for large samples of pre-main sequence stars, the methodology employed here to extract a dynamical mass from the disk rotation curve should prove extraordinarily useful for efforts to characterize the fundamental parameters of early stellar evolution.Comment: Accepted to The Astrophysical Journa

Experimental Technology Prototype At Microsoft

In our project, sponsored by Microsoft, we built a proof of concept application to demonstrate the capabilities of an emerging technology when compared with a parallel implementation currently being developed using a similar technology. With technical specifications and interface mockups, we created a UI infrastructure and components such as filtered and searchable tables, dialog boxes, context menus, and navigation to allow users to manage server entities. The application makes minimal use of open source libraries for platform compatibility, but in general runs solely on a custom framework used primarily for UI layout management and for standardizing communication and XML parsing between the server and client systems

Creation of vortices in a Bose-Einstein condensate by a Raman technique

We propose a method for taking a Bose-Einstein condensate in the ground trap state simultaneously to a different atomic hyperfine state and to a vortex trap state. This can be accomplished through a Raman scheme in which one of the two copropagating laser beams has a higher-order Laguerre-Gaussian mode profile. Coefficients relating the beam waist, pulse area, and trap potentials for a complete transfer to the m = 1 vortex are calculated for a condensate in the non-interacting and strongly interacting regimes.Comment: RevTex, 4 pages, 2 PostScript figure