Two-Rowed Hecke Algebra Representations at Roots of Unity

In this paper, we initiate a study into the explicit construction of irreducible representations of the Hecke algebra $H_n(q)$ of type $A_{n-1}$ in the non-generic case where $q$ is a root of unity. The approach is via the Specht modules of $H_n(q)$ which are irreducible in the generic case, and possess a natural basis indexed by Young tableaux. The general framework in which the irreducible non-generic $H_n(q)$-modules are to be constructed is set up and, in particular, the full set of modules corresponding to two-part partitions is described. Plentiful examples are given.Comment: LaTeX, 9 pages. Submitted for the Proceedings of the 4th International Colloquium ``Quantum Groups and Integrable Systems,'' Prague, 22-24 June 199

On the Representation Theory of an Algebra of Braids and Ties

We consider the algebra ${\cal E}_n(u)$ introduced by F. Aicardi and J. Juyumaya as an abstraction of the Yokonuma-Hecke algebra. We construct a tensor space representation for ${\cal E}_n(u)$ and show that this is faithful. We use it to give a basis for ${\cal E}_n(u)$ and to classify its irreducible representations.Comment: 24 pages. Final version. To appear in Journal of Algebraic Combinatorics

hapbin: An Efficient Program for performing Haplotype-Based Scans for Positive Selection in Large Genomic Datasets

Understanding how the genome is shaped by selective processes forms an integral part of modern biology. However, as genomic datasets continue to grow larger it is becoming increasingly difficult to apply traditional statistics for detecting signatures of selection to these cohorts. There is therefore a pressing need for the development of the next generation of computational and analytical tools for detecting signatures of selection in large genomic datasets. Here, we present hapbin, an efficient multithreaded implementation of extended haplotype homzygosity-based statistics for detecting selection, which is up to 3,400 times faster than the current fastest implementations of these algorithms

First-order cosmological phase transitions in the radiation dominated era

We consider first-order phase transitions of the Universe in the radiation-dominated era. We argue that in general the velocity of interfaces is non-relativistic due to the interaction with the plasma and the release of latent heat. We study the general evolution of such slow phase transitions, which comprise essentially a short reheating stage and a longer phase equilibrium stage. We perform a completely analytical description of both stages. Some rough approximations are needed for the first stage, due to the non-trivial relations between the quantities that determine the variation of temperature with time. The second stage, instead, is considerably simplified by the fact that it develops at a constant temperature, close to the critical one. Indeed, in this case the equations can be solved exactly, including back-reaction on the expansion of the Universe. This treatment also applies to phase transitions mediated by impurities. We also investigate the relations between the different parameters that govern the characteristics of the phase transition and its cosmological consequences, and discuss the dependence of these parameters with the particle content of the theory.Comment: 38 pages, 3 figures; v2: Minor changes, references added; v3: several typos correcte

Assessment of various continual reassessment method models for dose-escalation phase 1 oncology clinical trials : using real clinical data and simulation studies

Background The continual reassessment method (CRM) identifies the maximum tolerated dose (MTD) more efficiently and identifies the true MTD more frequently compared to standard methods such as the 3 + 3 method. An initial estimate of the dose-toxicity relationship (prior skeleton) is required, and there is limited guidance on how to select this. Previously, we compared the CRM with six different skeletons to the 3 + 3 method by conducting post-hoc analysis on a phase 1 oncology study (AZD3514), each CRM model reduced the number of patients allocated to suboptimal and toxic doses. This manuscript extends this work by assessing the ability of the 3 + 3 method and the CRM with different skeletons in determining the true MTD of various “true” dose-toxicity relationships. Methods One thousand studies were simulated for each “true” dose toxicity relationship considered, four were based on clinical trial data (AZD3514, AZD1208, AZD1480, AZD4877), and four were theoretical. The 3 + 3 method and 2-stage extended CRM with six skeletons were applied to identify the MTD, where the true MTD was considered as the largest dose where the probability of experiencing a dose limiting toxicity (DLT) is ≤33%. Results For every true dose-toxicity relationship, the CRM selected the MTD that matched the true MTD in a higher proportion of studies compared to the 3 + 3 method. The CRM overestimated the MTD in a higher proportion of simulations compared to the 3 + 3 method. The proportion of studies where the correct MTD was selected varied considerably between skeletons. For some true dose-toxicity relationships, some skeletons identified the true MTD in a higher proportion of scenarios compared to the skeleton that matched the true dose-toxicity relationship. Conclusion Through simulation, the CRM generally outperformed the 3 + 3 method for the clinical and theoretical true dose-toxicity relationships. It was observed that accurate estimates of the true skeleton do not always outperform a generic skeleton, therefore the application of wide confidence intervals may enable a generic skeleton to be used. Further work is needed to determine the optimum skeleton

Diagonalization of the XXZ Hamiltonian by Vertex Operators

We diagonalize the anti-ferroelectric XXZ-Hamiltonian directly in the thermodynamic limit, where the model becomes invariant under the action of affine U_q( sl(2) ). Our method is based on the representation theory of quantum affine algebras, the related vertex operators and KZ equation, and thereby bypasses the usual process of starting from a finite lattice, taking the thermodynamic limit and filling the Dirac sea. From recent results on the algebraic structure of the corner transfer matrix of the model, we obtain the vacuum vector of the Hamiltonian. The rest of the eigenvectors are obtained by applying the vertex operators, which act as particle creation operators in the space of eigenvectors. We check the agreement of our results with those obtained using the Bethe Ansatz in a number of cases, and with others obtained in the scaling limit --- the $su(2)$-invariant Thirring model.Comment: 65 page

Loop-Generated Bounds on Changes to the Graviton Dispersion Relation

We identify the effective theory appropriate to the propagation of massless bulk fields in brane-world scenarios, to show that the dominant low-energy effect of asymmetric warping in the bulk is to modify the dispersion relation of the effective 4-dimensional modes. We show how such changes to the graviton dispersion relation may be bounded through the effects they imply, through loops, for the propagation of standard model particles. We compute these bounds and show that they provide, in some cases, the strongest constraints on nonstandard gravitational dispersions. The bounds obtained in this way are the strongest for the fewest extra dimensions and when the extra-dimensional Planck mass is the smallest. Although the best bounds come for warped 5-D scenarios, for which the 5D Planck Mass is O(TeV), even in 4 dimensions the graviton loop can lead to a bound on the graviton speed which is comparable with other constraints.Comment: 18 pages, LaTeX, 4 figures, uses revte