Error-Correction Coding and Decoding: Bounds, Codes, Decoders, Analysis and Applications

Coding; Communications; Engineering; Networks; Information Theory; Algorithm

Numerical calculation of automorphic functions for finite index subgroups of triangle groups

We present a new method to calculate automorphic functions for finite index subgroups of triangle groups. Since automorphic functions are holomorphic, it is well known that the real and the imaginary part are both harmonic. The central idea of my advisor Monien was to look at the two parts separately. We solve the Laplace equation to find the real and imaginary part of an automorphic function. This solution can be calculated using numerical methods. To each finite index subgroup of a triangle group we can associate a Belyi function and a dessin d'enfant. The zeros of this Belyi function are the values of the automorphic function we calculated at elliptic points. Hence, we can find an approximation for the coefficients of the Belyi function. The precision of this approximation is increased by the use of Newton's method. Once we have an approximation with high accuracy, we find the correct algebraic number using the LLL algorithm. From the exact Belyi function we can reconstruct the exact automorphic function. In order to handle finite index subgroups of triangle groups, we introduce the notion of generalized Farey symbols. These symbols are a generalization of the classical Farey symbols for the modular group. They are used to do efficient calculations with subgroups of Hecke groups

Theory for the FCC-ee : Report on the 11th FCC-ee Workshop

The Future Circular Collider (FCC) at CERN, a proposed 100-km circular facility with several colliders in succession, culminates with a 100 TeV proton-proton collider. It offers a vast new domain of exploration in particle physics, with orders of magnitude advances in terms of Precision, Sensitivity and Energy. The implementation plan foresees, as a first step, an Electroweak Factory electron-positron collider. This high luminosity facility, operating between 90 and 365 GeV centre-of-mass energy, will study the heavy particles of the Standard Model, Z, W, Higgs, and top with unprecedented accuracy. The Electroweak Factory $e^+e^-$ collider constitutes a real challenge to the theory and to precision calculations, triggering the need for the development of new mathematical methods and software tools. A first workshop in 2018 had focused on the first FCC-ee stage, the Tera-Z, and confronted the theoretical status of precision Standard Model calculations on the Z-boson resonance to the experimental demands. The second workshop in January 2019, which is reported here, extended the scope to the next stages, with the production of W-bosons (FCC-ee-W), the Higgs boson (FCC-ee-H) and top quarks (FCC-ee-tt). In particular, the theoretical precision in the determination of the crucial input parameters, alpha_QED, alpha_QCD, M_W, m_t at the level of FCC-ee requirements is thoroughly discussed. The requirements on Standard Model theory calculations were spelled out, so as to meet the demanding accuracy of the FCC-ee experimental potential. The discussion of innovative methods and tools for multi-loop calculations was deepened. Furthermore, phenomenological analyses beyond the Standard Model were discussed, in particular the effective theory approaches. The reports of 2018 and 2019 serve as white papers of the workshop results and subsequent developments