Degree-regular triangulations of torus and Klein bottle

A triangulation of a connected closed surface is called weakly regular if the action of its automorphism group on its vertices is transitive. A triangulation of a connected closed surface is called degree-regular if each of its vertices have the same degree. Clearly, a weakly regular triangulation is degree-regular. In 1999, Lutz has classified all the weakly regular triangulations on at most 15 vertices. In 2001, Datta and Nilakantan have classified all the degree-regular triangulations of closed surfaces on at most 11 vertices. In this article, we have proved that any degree-regular triangulation of the torus is weakly regular. We have shown that there exists an $n$-vertex degree-regular triangulation of the Klein bottle if and only if $n$ is a composite number $\geq 9$. We have constructed two distinct $n$-vertex weakly regular triangulations of the torus for each $n \geq 12$ and a $(4m + 2)$-vertex weakly regular triangulation of the Klein bottle for each $m \geq 2$. For $12 \leq n \leq 15$, we have classified all the $n$-vertex degree-regular triangulations of the torus and the Klein bottle. There are exactly 19 such triangulations, 12 of which are triangulations of the torus and remaining 7 are triangulations of the Klein bottle. Among the last 7, only one is weakly regular.Comment: Revised version, 26 pages, To appear in Proceedings of Indian Academy of Sciences (Math. Sci.

Degree-regular triangulations of the double-torus

A connected combinatorial 2-manifold is called degree-regular if each of its vertices have the same degree. A connected combinatorial 2-manifold is called weakly regular if it has a vertex-transitive automorphism group. Clearly, a weakly regular combinatorial 2-manifold is degree-regular and a degree-regular combinatorial 2-manifold of Euler characteristic - 2 must contain 12 vertices. In 1982, McMullen et al. constructed a 12-vertex geometrically realized triangulation of the double-torus in \RR^3. As an abstract simplicial complex, this triangulation is a weakly regular combinatorial 2-manifold. In 1999, Lutz showed that there are exactly three weakly regular orientable combinatorial 2-manifolds of Euler characteristic - 2. In this article, we classify all the orientable degree-regular combinatorial 2-manifolds of Euler characteristic - 2. There are exactly six such combinatorial 2-manifolds. This classifies all the orientable equivelar polyhedral maps of Euler characteristic - 2.Comment: 13 pages. To appear in `Forum Mathematicum

Antipsychotic medication for childhood-onset schizophrenia

We review the development of several new approaches for extending the performance of Brillouin based slow light systems. In particular we describe the use of cavity effects to enhance the achievable delays, gain saturation to decouple the delay and associated signal gain, and the use of tailored pump beams to effect reshaping and retiming of periodic signals

New Physics in b→sμ+μ−b \to s \mu^+ \mu^- after the Measurement of RK∗R_{K^*}

The recent measurement of $R_{K^*}$ is yet another hint of new physics (NP), and supports the idea that it is present in $b\to s\mu^+\mu^-$ decays. We perform a combined model-independent and model-dependent analysis in order to deduce properties of this NP. Like others, we find that the NP must obey one of two scenarios: (I) $C_9^{\mu\mu}({\rm NP}) < 0$ or (II) $C_9^{\mu\mu}({\rm NP}) = - C_{10}^{\mu\mu}({\rm NP}) < 0$. A third scenario, (III) $C_9^{\mu\mu}({\rm NP}) = - C_{9}^{\prime \mu\mu}({\rm NP})$, is rejected largely because it predicts $R_K = 1$, in disagreement with experiment. The simplest NP models involve the tree-level exchange of a leptoquark (LQ) or a $Z'$ boson. We show that scenario (II) can arise in LQ or $Z'$ models, but scenario (I) is only possible with a $Z'$. Fits to $Z'$ models must take into account the additional constraints from $B^0_s$-${\bar B}^0_s$ mixing and neutrino trident production. Although the LQs must be heavy, O(TeV), we find that the $Z'$ can be light, e.g., $M_{Z'} = 10$ GeV or 200 MeV.Comment: 18 pages, 1 figure; final version accepted for publication in Physical Review

Fourier transform for functions of bicomplex variables

This paper examines the existence and region of convergence of Fourier transform of the functions of bicomplex variables with the help of projection on its idempotent components as auxiliary complex planes. Several basic properties of this bicomplex version of Fourier transform are examined.Comment: 1 figur

Sbottoms as probes to MSSM with nonholomorphic soft interactions

Presence of nonholomorphic soft SUSY breaking terms is known to be a possibility in the popular setup of the Minimal Supersymmetric Standard Model (MSSM). It has been shown that such a scenario known as NonHolomorphic Supersymmetric Standard Model (NHSSM) could remain `natural' ( i.e., not fine-tuned) even in the presence of a rather heavy higgsino-like LSP. However, it turns out that distinguishing such a scenario from the MSSM is unlikely to be an easy task, in particular at the Large Hadron Collider (LHC). In a first study of such a scenario at colliders (LHC), we explore a possible way that focuses on the sbottom phenomenology. This exploits the usual $\tan\beta$-dependence (enhancement) of the bottom Yukawa coupling but reinforced/altered in the presence of non-vanishing nonholomorphic soft trilinear parameter $A_b^{\prime}$. For a given set of masses of the sbottom(s) and the light electroweakinos (LSP, lighter chargino etc.) which are known from experiments, the difference between the two scenarios could manifest itself via event rate in the 2b-jets + ${\, \not \! \! E_T}$ final state, which could be characteristically different from its MSSM expectation. Impact on the phenomenology of the stops at the LHC is also touched upon.Comment: 32 pages, 17 figures, 1 table, no changes in texts/figures, three references added, version published in JHE