On the structure and applications of the Bondi-Metzner-Sachs group

This work is a pedagogical review dedicated to a modern description of the Bondi-Metzner-Sachs group. The curved space-times that will be taken into account are the ones that suitably approach, at infinity, Minkowski space-time. In particular we will focus on asymptotically flat space-times. In this work the concept of asymptotic symmetry group of those space-times will be studied. In the first two sections we derive the asymptotic group following the classical approach which was basically developed by Bondi, van den Burg, Metzner and Sachs. This is essentially the group of transformations between coordinate systems of a certain type in asymptotically flat space-times. In the third section the conformal method and the notion of asymptotic simplicity are introduced, following mainly the works of Penrose. This section prepares us for another derivation of the Bondi-Metzner-Sachs group which will involve the conformal structure, and is thus more geometrical and fundamental. In the subsequent sections we discuss the properties of the Bondi-Metzner-Sachs group, e.g. its algebra and the possibility to obtain as its subgroup the Poincar\'e group, as we may expect. The paper ends with a review of the Bondi-Metzner-Sachs invariance properties of classical gravitational scattering discovered by Strominger, that are finding application to black hole physics and quantum gravity in the literature.Comment: 62 pages, 9 figures. Misprints have been amended and two important references have been adde

A Katsylo theorem for sheets of spherical conjugacy classes

We show that, for a sheet or a Lusztig stratum S containing spherical conjugacy classes in a connected reductive algebraic group G over an algebraically closed field in good characteristic, the orbit space S/G is isomorphic to the quotient of an affine subvariety of G modulo the action of a finite abelian 2-group. The affine subvariety is a closed subset of a Bruhat double coset and the abelian group is a finite subgroup of a maximal torus of G. We show that sheets of spherical conjugacy classes in a simple group are always smooth and we list which strata containing spherical classes are smooth

Credit risk tools: an overview

This document presents several Credit Risk tools which have been developed for the Credit Derivatives Risk Management. The models used in this context are suitable for the pricing, sensitivity/scenario analysis and the derivation of risk measures for plain vanilla credit default swaps (CDS), standardized and bespoke collateralized debt obligations (CDO) and, in general, for any credit risk exposed A/L portfolio.\\ In this brief work we compute the market implied probability of default (PD) from market spreads and the theoretical CDS spreads from historical default frequencies. The loss given default (LGD) probability distribution has been constructed for a large pool portfolio of credit obligations exploiting a single-factor gaussian copula with a direct convolution algorithm computed at several default correlation parameters. Theoretical CDO tranche prices have been calculated. We finally design stochastic cash-flow stream model simulations to test fair pricing, compute credit value at risk (CV@R) and to evaluate the one year total future potential exposure (FPE) and derive the value at risk (V@R) for a CDO equity tranche exposure.interest rate swap, spot rate term structure, credit default swap, probability of default, copula function, direct convolution, loss given default, collateralized debt obligation, exposure at default, stochastic cash-flow stream model, value at risk, credit value at risk, future potential exposure, Monte Carlo simulation.

A note on Fontaine theory using different Lubin-Tate groups

Using different Lubin-Tate groups, we compare $(\phi, \Gamma)$ modules associated to a Galois representation via Fontaine's theory

Geometry of quiver Grassmannians of Kronecker type and canonical basis of cluster algebras

We study quiver Grassmannians associated with indecomposable representations of the Kronecker quiver. We find a cellular decomposition of them and we compute their Betti numbers. As an application, we give a geometric realization of the "canonical basis" of cluster algebras of Kronecker type (found by Sherman and Zelevinsky) and of type $A_2^{(1)}$.Comment: 21 page

User-friendly mathematical model for the design of sulfate reducing H2/CO2 fed bioreactors

The paper presents three steady-state mathematical models for the design of H2/CO2 fed gas-lift reactors aimed at biological sulfate reduction to remove sulfate from wastewater. Models 1A and 1B are based on heterotrophic sulfate reducing bacteria (HSRB), while Model 2 is based on autotrophic sulfate reducing bacteria (ASRB) as the dominant group of sulfate reducers in the gas-lift reactor. Once the influent wastewater characteristics are known and the desired sulfate removal efficiency is fixed, all models give explicit mathematical relationships to determine the bioreactor volume and the effluent concentrations of substrates and products. The derived explicit relationships make application of the models very easy, fast and no iterative procedures are required. Model simulations show that the size of the H2/CO2 fed gas-lift reactors aimed at biological sulfate removal from wastewater highly depends on the number and type of trophic groups growing in the bioreactor. In particular, if the biological sulfate reduction is performed in a bioreactor where ASRB prevail, the required bioreactor volume is much smaller than that needed with HSRB. This is because ASRB can out-compete methanogenic archarea (MA) for H2 (assuming sulfate concentrations are not limiting), whereas HSRB do not necessarily out-compete MA due to their dependence on homoacetogenic bacteria (HB) for organic carbon. The reactor sizes to reach the same sulfate removal efficiency by HSRB and ASRB are only comparable when methanogenesis is inhibited. Moreover, model results indicate that acetate supply to the reactor influent does not affect the HSRB biomass required in the reactor, but favours the dominance of MA on HB as a consequence of a lower HB requirement for acetate supply