Are neutrino masses modular forms?

We explore a new class of supersymmetric models for lepton masses and mixing angles where the role of flavour symmetry is played by modular invariance. The building blocks are modular forms of level N and matter supermultiplets, both transforming in representations of a finite discrete group Gamma_N. In the simplest version of these models, Yukawa couplings are just modular forms and the only source of flavour symmetry breaking is the vacuum expectation value of a single complex field, the modulus. In the special case where modular forms are constant functions the whole construction collapses to a supersymmetric flavour model invariant under Gamma_N, the case treated so far in the literature. The framework has a number of appealing features. Flavon fields other than the modulus might not be needed. Neutrino masses and mixing angles are simultaneously constrained by the modular symmetry. As long as supersymmetry is exact, modular invariance determines all higher-dimensional operators in the superpotential. We discuss the general framework and we provide complete examples of the new construction. The most economical model predicts neutrino mass ratios, lepton mixing angles, Dirac and Majorana phases uniquely in terms of the modulus vacuum expectation value, with all the parameters except one within the experimentally allowed range. As a byproduct of the general formalism we extend the notion of non-linearly realised symmetries to the discrete case.Comment: 40 pages, 3 figures; added comments and a new section with an example of normal ordering of neutrino masses; to appear in the book "From my vast repertoire: the legacy of Guido Altarelli", S. Forte, A. Levy and G. Ridolfi, ed

Sigma Model BPS Lumps on Torus

We study doubly periodic Bogomol'nyi-Prasad-Sommerfield (BPS) lumps in supersymmetric CP^{N-1} non-linear sigma models on a torus T^2. Following the philosophy of the Harrington-Shepard construction of calorons in Yang-Mills theory, we obtain the n-lump solutions on compact spaces by suitably arranging the n-lumps on R^2 at equal intervals. We examine the modular invariance of the solutions and find that there are no modular invariant solutions for n=1,2 in this construction.Comment: 15 pages, 3 figures, published versio

Generalized string compactifications with spontaneously broken supersymmetry

The Narain lattice construction of string compactifications is generalized to include spontaneously broken supersymmetry. Consistency conditions from modular invariance and Lorentz symmetry are solved in full generality. This framework incorporates models where supersymmetry breaking is inversely proportional to the radii of compact dimensions. The enhanced lattice description, however, might allow for models with a different geometrical or even non-geometrical interpretation.Comment: 15 pages, LaTeX, no figure

Self-similar planar graphs as models for complex networks

In this paper we introduce a family of planar, modular and self-similar graphs which have small-world and scale-free properties. The main parameters of this family are comparable to those of networks associated to complex systems, and therefore the graphs are of interest as mathematical models for these systems. As the clustering coefficient of the graphs is zero, this family is an explicit construction that does not match the usual characterization of hierarchical modular networks, namely that vertices have clustering values inversely proportional to their degrees.Comment: 10 pages, submitted to 19th International Workshop on Combinatorial Algorithms (IWOCA 2008

Long life, low cost ball valve, with lifted seals and cartridge type construction

Ball valve design incorporates modular construction in easy-to-install, easy-to-replace cartridge housing, and a system of cams to lift upstream and downstream seals away from the ball during rotation. Tests conducted on nitrogen tetroxide prove new valve to be more efficient than previous models in preventing leakage

A Vision-based Scheme for Kinematic Model Construction of Re-configurable Modular Robots

Re-configurable modular robotic (RMR) systems are advantageous for their reconfigurability and versatility. A new modular robot can be built for a specific task by using modules as building blocks. However, constructing a kinematic model for a newly conceived robot requires significant work. Due to the finite size of module-types, models of all module-types can be built individually and stored in a database beforehand. With this priori knowledge, the model construction process can be automated by detecting the modules and their corresponding interconnections. Previous literature proposed theoretical frameworks for constructing kinematic models of modular robots, assuming that such information was known a priori. While well-devised mechanisms and built-in sensors can be employed to detect these parameters automatically, they significantly complicate the module design and thus are expensive. In this paper, we propose a vision-based method to identify kinematic chains and automatically construct robot models for modular robots. Each module is affixed with augmented reality (AR) tags that are encoded with unique IDs. An image of a modular robot is taken and the detected modules are recognized by querying a database that maintains all module information. The poses of detected modules are used to compute: (i) the connection between modules and (ii) joint angles of joint-modules. Finally, the robot serial-link chain is identified and the kinematic model constructed and visualized. Our experimental results validate the effectiveness of our approach. While implementation with only our RMR is shown, our method can be applied to other RMRs where self-identification is not possible