A short proof of the Mock Theta Conjectures using Maass forms

A celebrated work of D. Hickerson gives a proof of the Mock Theta Conjectures using Hecke-type identities discovered by G. Andrews. Here, we respond to a remark by K. Bringmann, K. Ono and R. Rhoades and provide a short proof of the Mock Theta Conjectures by realizing each side of the identities as the holomorphic projection of a harmonic weak Maass form.

     

Duality involving the mock theta function f(q)

We show that the coefficients of Ramanujan's mock theta functionf(q) are the first non-trivial coefficients of a canonical sequenceof modular forms. This fact follows from a duality which equatescoefficients of the holomorphic projections of certain weight1/2 Maass forms with coefficients of certain weight 3/2 modularforms. This work depends on the theory of Poincaré series,and a modification of an argument of Goldfeld and Sarnak onKloosterman–Selberg zeta functions.     

Auditory brain stem responses from human infants: pure-tone masking profiles for clicks and filtered clicks

The effects of simultaneous pure-tone maskers on ABR wave V latency and amplitude were examined in three-month-old infants as a means of delineating the frequency specificity of these responses in the immature auditory system. Masking profiles at two intensities (60 and 40 dBn HL) were obtained for click, as well as 4000- and 1000-Hz filtered-click stimuli. Infant profiles, obtained by measuring both latency and amplitude shifts as a result of the discrete-frequency maskers, were compared to adult data obtained under an identical masking paradigm. Both latency and amplitude analyses showed masking profiles for infants which reveal greater low-frequency contribution to responses than found in adult profiles. Additionally, the infant profiles reveal clear differences in the degree of high-frequency spread of masking when comparisons are made to the adult data.     

Auditory brain stem responses from human adults and infants: wave V tuning curves

Decrement in ABR wave V amplitude was measured in the presence of simultaneous tonal maskers. Probe stimuli were 1.0, 4.0, and 8.0-kHz third-octave-filtered clicks. Adults and 3-month-old infants served as subjects. The resultant amplitude-decrement functions for each tonal masker were fit with regression lines. The sound pressure level (SPL) required to reduce wave V to 50% of the unmasked probe amplitude was plotted for each masker to develop tuning curves. The tuning curves were quantified by calculations of tip-to-tail difference, Q 10, and SPL at maximum masker frequency (MMF). Tuning curves for adult and infant subjects were similar for the 1.0-kHz probe. For the high-frequency probes (4.0 and 8.0 kHz), smaller tip-to-tail differences and lower Q 10 values were observed for the infant subjects. Ranges of MMF level were similar across adult and infant subjects. For the 8.0-kHz probe, tuning curves from infant subjects consistently showed maximum masker frequencies which were lower than the probe.     

Mock and mixed mock modular forms in the lower half-plane

We study mock and mixed mock modular forms in the lower half-plane. In particular, our results apply to Zwegers’ three-variable mock Jacobi form \({\mu(u,v;\tau)}\), three-variable generalizations of the universal mock modular partition rank generating function, and the quantum and mock modular strongly unimodal sequence rank generating function. We do not rely upon the analytic properties of these functions; we establish our results concisely using the theory of q-hypergeometric series and partial theta functions. We extend related results of Ramanujan, Hikami, and prior work of the author with Bringmann and Rhoades, and also incorporate more recent aspects of the theory pertaining to quantum modular forms and the behavior of these functions at rational numbers when viewed as functions of \({\tau}\) (or equivalently, at roots of unity when viewed as functions of q).     

Partial theta functions and mock modular forms as q-hypergeometric series

Ramanujan studied the analytic properties of many q-hypergeometric series. Of those, mock theta functions have been particularly intriguing, and by work of Zwegers, we now know how these curious q-series fit into the theory of automorphic forms. The analytic theory of partial theta functions however, which have q-expansions resembling modular theta functions, is not well understood. Here we consider families of q-hypergeometric series which converge in two disjoint domains. In one domain, we show that these series are often equal to one another, and define mock theta functions, including the classical mock theta functions of Ramanujan, as well as certain combinatorial generating functions, as special cases. In the other domain, we prove that these series are typically not equal to one another, but instead are related by partial theta functions.     

Quantum Jacobi forms and finite evaluations of unimodal rank generating functions

In this paper, we introduce the notion of a quantum Jacobi form, and offer the two-variable combinatorial generating function for ranks of strongly unimodal sequences as an example. We then use its quantum Jacobi properties to establish a new, simpler expression for this function as a two-variable Laurent polynomial when evaluated at pairs of rational numbers. Our results also yield a new expression for radial limits associated to the partition rank and crank functions previously studied by Ono, Rhoades, and Folsom.     

The Controlled Oxidation of Organic Sulfides to Sulfoxides with the Help of O-Iodosylbenzoic Acid

The oxidation procedure plus a simple preparation of oiodosylbenzoic acid are described.