Normal origamis of Mumford curves

An origami (also known as square-tiled surface) is a Riemann surface covering a torus with at most one branch point. Lifting two generators of the fundamental group of the punctured torus decomposes the surface into finitely many unit squares. By varying the complex structure of the torus one obtains easily accessible examples of Teichm\"uller curves in the moduli space of Riemann surfaces. The p-adic analogues of Riemann surfaces are Mumford curves. A p-adic origami is defined as a covering of Mumford curves with at most one branch point, where the bottom curve has genus one. A classification of all normal non-trivial p-adic origamis is presented and used to calculate some invariants. These can be used to describe p-adic origamis in terms of glueing squares.Comment: 21 pages, to appear in manuscripta mathematica (Springer

Degenerating families of dendrograms

Dendrograms used in data analysis are ultrametric spaces, hence objects of nonarchimedean geometry. It is known that there exist $p$-adic representation of dendrograms. Completed by a point at infinity, they can be viewed as subtrees of the Bruhat-Tits tree associated to the $p$-adic projective line. The implications are that certain moduli spaces known in algebraic geometry are $p$-adic parameter spaces of (families of) dendrograms, and stochastic classification can also be handled within this framework. At the end, we calculate the topology of the hidden part of a dendrogram.Comment: 13 pages, 8 figure

Introduction to Categories and Categorical Logic

The aim of these notes is to provide a succinct, accessible introduction to some of the basic ideas of category theory and categorical logic. The notes are based on a lecture course given at Oxford over the past few years. They contain numerous exercises, and hopefully will prove useful for self-study by those seeking a first introduction to the subject, with fairly minimal prerequisites. The coverage is by no means comprehensive, but should provide a good basis for further study; a guide to further reading is included. The main prerequisite is a basic familiarity with the elements of discrete mathematics: sets, relations and functions. An Appendix contains a summary of what we will need, and it may be useful to review this first. In addition, some prior exposure to abstract algebra - vector spaces and linear maps, or groups and group homomorphisms - would be helpful.Comment: 96 page

A series of coverings of the regular n-gon

We define an infinite series of translation coverings of Veech's double-n-gon for odd n greater or equal to 5 which share the same Veech group. Additionally we give an infinite series of translation coverings with constant Veech group of a regular n-gon for even n greater or equal to 8. These families give rise to explicit examples of infinite translation surfaces with lattice Veech group.Comment: A missing case in step 1 in the proof of Thm. 1 b was added. (To appear in Geometriae Dedicata.

Non-factorizable contribution in nonleptonic weak interactions of K mesons

Two pion decays of K mesons, K_L-K_S mass difference, two photon and the Dalitz decays of K_L are studied systematically by assuming that their amplitude is given by a sum of factorizable and non-factorizable ones. The former is estimated by using a naive factorization while the latter is assumed to be dominated by dynamical contributions of various hadron states.Comment: 23 pages,1 figur

Splittings of generalized Baumslag-Solitar groups

We study the structure of generalized Baumslag-Solitar groups from the point of view of their (usually non-unique) splittings as fundamental groups of graphs of infinite cyclic groups. We find and characterize certain decompositions of smallest complexity (`fully reduced' decompositions) and give a simplified proof of the existence of deformations. We also prove a finiteness theorem and solve the isomorphism problem for generalized Baumslag-Solitar groups with no non-trivial integral moduli.Comment: 20 pages; hyperlinked latex. Version 2: minor change

Next-to-Leading Order QCD corrections to BBˉB\bar B-mixing and ϵK\epsilon_K within the MSSM

We present a calculation of the QCD correction factors $\eta_{2B}$ and $\eta_{2K}$ up to Next-to-Leading Order within the MSSM. We took into account the region of low $\tan\beta$ for the Higgs- and chargino sector while neglecting the effect of gluinos and neutralinos.Comment: 38 pages, references added, typos corrected, extended discussion of gluino-mediated correction

On the Correlations between Flavour Observables in Minimal U(2)^3 Models

The stringent correlations between flavour observables in models with CMFV are consistent with the present data except for the correlation Delta M_{s,d}-epsilon_K. Motivated by the recent work of Barbieri et al, we compare the CMFV correlations with the ones present in a special class of models with an approximate global U(2)^3 flavour symmetry, constrained by a minimal set of spurions governing the breakdown of this symmetry and the assumption that only SM operators are relevant in flavour physics. This analog of CMFV to be called MU(2)^3 allows to avoid the Delta M_{s,d}-epsilon_K tension in question because of reduced flavour symmetry and implied non-MFV contributions to Delta M_{s,d}. While the patterns of flavour violation in K meson system is the same as in CMFV models, the CP-violation in B_{s,d} meson systems can deviate from the one in the SM and CMFV models. We point out a stringent triple S_{psi K_S}-S_{psi phi}-|V_ub| correlation in this class of models that could in the future provide a transparent distinction between different MU(2)^3 models and in the context of these models determine |V_ub| by means of precise measurements of S_{psi K_S} and S_{psi phi} with only small hadronic uncertainties. For fixed S_{psi K_S} the correlation between B(B^+ -> tau^+nu_tau) and S_{psi phi} follows. We also find that MU(2)^3 models could in principle accommodate a negative value of S_{psi phi}, provided |V_ub| is found to be in the ballpark of exclusive determinations and the particular MU(2)^3 model provides a 25% enhancement of epsilon_K. A supersymmetric U(2)^3 model worked out in the Barbieri-School appears to satisfy these requirements. However if B(B^+ -> tau^+nu_tau)>1.0 10^{-4} will be confirmed by future experiments only positive S_{psi phi} is allowed in this framework. We summarize briefly the pattern of flavour violation in rare K and B_{s,d} decays in MU(2)^3 models.Comment: 28 pages, 6 figures; v2: Few references and discussion on CP violation in B_s-> mu^+ mu^- added; v3: Several clarifying comments added, conclusions unchanged, version accepted for publication in JHE

An updated analysis of eps'/eps in the standard model with hadronic matrix elements from the chiral quark model

We discuss the theoretical and experimental status of the CP violating ratio eps'/eps. We revise our 1997 standard-model estimate-based on hadronic matrix elements computed in the chiral quark model up to O(p^4) in the chiral expansion-by including an improved statistical analysis of the uncertainties and updated determination of the Cabibbo-Kobayashi-Maskawa elements and other short-distance parameters. Using normal distributions for the experimental input data we find Re eps'/eps = (2.2 \pm 0.8) x 10^{-3}, whereas a flat scanning gives 0.9 x 10^{-3} < Re eps'/eps < 4.8 x 10^{-3}. Both results are in agreement with the current experimental data. The key element in our estimate is, as before, the fit of the Delta I=1/2 rule, which allows us to absorb most of the theoretical uncertainties in the determination of the model-dependent parameters in the hadronic matrix elements. Our semi-phenomenological approach leads to numerical stability against variations of the renormalization scale and scheme dependence of the short- and long-distance components. The same dynamical mechanism at work in the selection rule also explains the larger value obtained for \ratio with respect to other estimates. A coherent picture of K -> pi pi decays is thus provided.Comment: 15 pages, 11 figures, RevTeX, discussion updated, refs adde