Modular Invariant of Quantum Tori II: The Golden Mean

In our first article in this series ("Modular Invariant of Quantum Tori I: Definitions Nonstandard and Standard" arXiv:0909.0143) a modular invariant of quantum tori was defined. In this paper, we consider the case of the quantum torus associated to the golden mean. We show that the modular invariant is approximately 9538.249655644 by producing an explicit formula for it involving weighted versions of the Rogers-Ramanujan functions

Consequences of Supergravity with Gauged U(1)R\rm U(1)_R Symmetry

The structure of gauged R supergravity Lagrangians is reviewed, and we consider models with a hidden sector plus light fields of the MSSM. A simple potential for the hidden sector is presented which has a global minimum with zero cosmological constant and spontaneously broken SUSY and R-symmetry. The $\rm U(1)_R$ vector multiplet acquires a Planck scale mass through the Higgs mechanism, and it decouples at low energy. Due to very interesting cancellations, the $\rm U(1)_R$ D-terms also drop out at low energy. Thus no direct effects of the gauging of R-symmetry remain in the low energy effective Lagrangian, and this result is model independent, requiring only that R-symmetry be broken at the Planck scale and $= 0$, where $D$ is the auxiliary field of the $\rm U(1)_R$ vector multiplet. The low energy theory is fairly conventional with soft SUSY breaking terms for the MSSM fields. As a remnant of the gauging of R-symmetry, it also contains light fields, some required to cancel R-anomalies and others from the hidden sector.Comment: 36 pages, plain LaTeX, all macros included, no figure

On Petersson's partition limit formula

For each prime $p\equiv 1\pmod{4}$ consider the Legendre character $\chi=(\frac{\cdot}{p})$. Let $p_\pm(n)$ be the number of partitions of $n$ into parts $\lambda>0$ such that $\chi(\lambda)=\pm 1$. Petersson proved a beautiful limit formula for the ratio of $p_+(n)$ to $p_-(n)$ as $n\to\infty$ expressed in terms of important invariants of the real quadratic field $\mathbb{Q}(\sqrt{p})$. But his proof is not illuminating and Grosswald conjectured a more natural proof using a Tauberian converse of the Stolz-Ces\`aro theorem. In this paper we suggest an approach to address Grosswald's conjecture. We discuss a monotonicity conjecture which looks quite natural in the context of the monotonicity theorems of Bateman-Erd\H{o}s

Enfermedades del frijol

Diseases are one of the limiting factors to bean production in L.A. The prevalency and severity of the disease depends on the quantity of inoculum existing in the region, environmental conditions and the quality of the seed used. Data are given on geographic distribution, favorable climatic conditions for development, symptomatology, vectors, type of damage caused and control measures, including var. resistant to (1) fungal diseases, the most numerous and causing the most damage: rust, anthracnose, angular leaf spot, root rot, web blight, gray spot, powdery mildew; (2) virus diseases, some of economic importance transmitted by aphids (BCMV, BYMV); transmitted by whiteflies (BGMV, bean chlorotic mottle, euphorbia mosaic); transmitted by chrysomelids (BRMV, pod mottle, BSMV, yellow mottle); (3) bacterial diseases are few but can be a serious limitation to the crop: common and fuscous blights, halo blight; and (4) nematodes: Meloidogyne sp., Trichodorus spp., Pratylenchus spp., Heterodera spp. (CIAT

Conicoid Mirrors

The first order equation relating object and image location for a mirror of arbitrary conic-sectional shape is derived. It is also shown that the parabolic reflecting surface is the only one free of aberration and only in the limiting case of distant sources.Comment: 9 page