Jacobi forms and differential operators

We affirmatively answer a question due to S. Bocherer concerning the feasibility of removing one differential operator from the standard collection of m + 1 of them used to embed the space of Jacobi forms of weight 2 and index m into several pieces of elliptic modular forms. (C) 2014 Elsevier Inc. All rights reserved.

Congruences in modular, Jacobi, Siegel, and mock modular forms with applications

We study congruences in the coefficients of modular and other automorphic forms. Ramanujan famously found congruences for the partition function like p(5n+4) = 0 mod 5. For a wide class of modular forms, we classify the primes for which there can be analogous congruences in the coefficients of the Fourier expansion. We have several applications. We describe the Ramanujan congruences in the counting functions for overparitions, overpartition pairs, crank differences, and Andrews' two-coloured generalized Frobenius partitions. We also study Ramanujan congruences in the Fourier coefficients of certain ratios of Eisenstein series. We also determine the exact number of holomorphic modular forms with Ramanujan congruences when the weight is large enough. In a chapter based on joint work with Olav Richter, we study Ramanujan congruences in the coefficients of Jacobi forms and Siegel modular forms of degree two. Finally, the last chapter contains a completely unrelated result about harmonic weak Maass forms.

Numerical Computation of a Certain Dirichlet Series Attached to Siegel Modular Forms of Degree Two

The Rankin convolution type Dirichlet series D-F,D-G(s) of Siegel modular forms F and G of degree two, which was introduced by Kohnen and the second author, is computed numerically for various F and G. In particular, we prove that the series D-F,D-G(s), which shares the same functional equation and analytic behavior with the spinor L-functions of eigenforms of the same weight are not linear combinations of those. In order to conduct these experiments a numerical method to compute the Petersson scalar products of Jacobi Forms is developed and discussed in detail.

Comportamento assintótico dos polinômios ortogonais de Sobolev-Jacobi e Sobolev-Laguerre

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

New infinite families of exact sums of squares formulas, Jacobi elliptic functions, and Ramanujan’s tau function

In this paper, we give two infinite families of explicit exact formulas that generalize Jacobi’s (1829) 4 and 8 squares identities to 4n2 or 4n(n + 1) squares, respectively, without using cusp forms. Our 24 squares identity leads to a different formula for Ramanujan’s tau function τ(n), when n is odd. These results arise in the setting of Jacobi elliptic functions, Jacobi continued fractions, Hankel or Turánian determinants, Fourier series, Lambert series, inclusion/exclusion, Laplace expansion formula for determinants, and Schur functions. We have also obtained many additional infinite families of identities in this same setting that are analogous to the η-function identities in appendix I of Macdonald’s work [Macdonald, I. G. (1972) Invent. Math. 15, 91–143]. A special case of our methods yields a proof of the two conjectured [Kac, V. G. and Wakimoto, M. (1994) in Progress in Mathematics, eds. Brylinski, J.-L., Brylinski, R., Guillemin, V. & Kac, V. (Birkhäuser Boston, Boston, MA), Vol. 123, pp. 415–456] identities involving representing a positive integer by sums of 4n2 or 4n(n + 1) triangular numbers, respectively. Our 16 and 24 squares identities were originally obtained via multiple basic hypergeometric series, Gustafson’s Cℓ nonterminating 6φ5 summation theorem, and Andrews’ basic hypergeometric series proof of Jacobi’s 4 and 8 squares identities. We have (elsewhere) applied symmetry and Schur function techniques to this original approach to prove the existence of similar infinite families of sums of squares identities for n2 or n(n + 1) squares, respectively. Our sums of more than 8 squares identities are not the same as the formulas of Mathews (1895), Glaisher (1907), Ramanujan (1916), Mordell (1917, 1919), Hardy (1918, 1920), Kac and Wakimoto, and many others.

Report a review of the concepts and definitions of the various forms of relational contracting

Partnering has been defined in many ways. It can be considered as an individual project mechanism or can be considered as a long term strategy. Alliancing is normally assumed to be a long term business strategy linking together client, contractor and supply chain. Relational contracting goes further than this and brings in the whole philosophy of the value chain and the linking of the interdependent parts within the construction project as a key business objective. This document aims to review existing definitions of these three concepts and present and overview of the current state of-the-art in terms of their use and implementation. The document should be useful for all of those project team members looking to sharpen their understanding of the various concepts and will also provide a platform for debating the current state of the definitions and implementations being used in Main Roads and Public Works Departments.

Jacobi quartic curves revisited

This paper provides new results about efficient arithmetic on Jacobi quartic form elliptic curves, y 2 = d x 4 + 2 a x 2 + 1. With recent bandwidth-efficient proposals, the arithmetic on Jacobi quartic curves became solidly faster than that of Weierstrass curves. These proposals use up to 7 coordinates to represent a single point. However, fast scalar multiplication algorithms based on windowing techniques, precompute and store several points which require more space than what it takes with 3 coordinates. Also note that some of these proposals require d = 1 for full speed. Unfortunately, elliptic curves having 2-times-a-prime number of points, cannot be written in Jacobi quartic form if d = 1. Even worse the contemporary formulae may fail to output correct coordinates for some inputs. This paper provides improved speeds using fewer coordinates without causing the above mentioned problems. For instance, our proposed point doubling algorithm takes only 2 multiplications, 5 squarings, and no multiplication with curve constants when d is arbitrary and a = ±1/2.

Digital storytelling, podcasts, blogs and vlogs : exploring a range of new media texts and forms in English

Curriculum initiatives in Australia emphasise the use of technologies and new media in classrooms. Some English teachers might fear this deployment of technologies because we are not all ‘digital natives’ like our students. If we embrace new media forms such as podcasts, blogs, vodcasts, and digital stories, a whole new world of possibilities open up for literary response and recreative texts, with new audiences and publication spaces. This article encourages English teachers to embrace these new digital forms and how shows we can go about it.

Measuring episodic memory: A novel approach with an indefinite number of alternative forms

Both clinical practice and clinical research settings can require successive administrations of a memory test, particularly when following the trajectory of suspected memory decline in older adults. However, relatively few verbal episodic memory tests have alternative forms. We set out to create a broad based memory test to allow for the use of an essentially unlimited number of alternative forms. Four tasks for inclusion in such a test were developed. These tasks varied the requirement for recall as opposed to recognition, the need to form an association between unrelated words, and the need to discriminate the most recent list from earlier lists, all of which proved useful. A total of 115 participants completed the battery of tests and were used to show that the test could differentiate between older and younger adults; a sub-sample of 73 participants completed alternative forms of the tests to determine test-retest reliability and the amount of learning to learn.