Cost Efficient Lie Group Integrators in the RKMK Class

In this work a systematic procedure is implemented in order to minimise the computational cost of the Runge—Kutta—Munthe-Kaas (RKMK) class of Lie-group solvers. The process consists of the application of a linear transformation to the stages of the method and the analysis of a graded free Lie algebra to reduce the number of commutators involved. We consider here RKMK integration methods up to order seven based on some of the most popular Runge—Kutta schemes.This revised version was published online in October 2005 with corrections to the Cover Date.     

La formule de Plancherel pour les groupes de Lie presque algébriques réels

We give a proof of the Plancherel formula for real almost algebraic groups in the philosophy of the orbit method, following the lines of the one given by M. Duflo and M. Vergne for simply connected semisimple Lie groups. Main ingredients are: (1) Harish-Chandra's descent method which, interpreting Plancherel formula as an equality of semi-invariant generalized functions, allows one to reduce it to a neighbourhood of zero in the Lie algebra of the centralizer of any elliptic element; (2) character formula for representations constructed by M. Duflo, we recently proved; (3) Poisson-Plancherel formula near elliptic elements s in good position, a generalization of the classical Poisson summation formula expressing the Fourier transform of the sum of a series of Harish-Chandra type elliptic orbital integrals in the Lie algebra centralizing s as a generalized function supported on a set of admissible regular forms in the dual of this Lie algebra.     

An invariant measure for the loop space of a simply connected compact symmetric space

Let X denote a simply connected compact Riemannian symmetric space, U the universal covering of the identity component of the group of automorphisms of X, and LU the loop group of U. In this paper we prove the existence (and conjecture the uniqueness) of an LU-invariant probability measure on a distributional completion of the loop space of X.     

Zero-sum problems in finite abelian groups: A survey

We give an overview of zero-sum theory in finite abelian groups, a subfield of additive group theory and combinatorial number theory. In doing so we concentrate on the algebraic part of the theory and on the development since the appearance of the survey article by Y. Caro in 1996.     

Towards generating secure keys for braid cryptography

Braid cryptosystem was proposed in CRYPTO 2000 as an alternate public-key cryptosystem. The security of this system is based upon the conjugacy problem in braid groups. Since then, there have been several attempts to break the braid cryptosystem by solving the conjugacy problem in braid groups. In this article, we first survey all the major attacks on the braid cryptosystem and conclude that the attacks were successful because the current ways of random key generation almost always result in weaker instances of the conjugacy problem. We then propose several alternate ways of generating hard instances of the conjugacy problem for use braid cryptography.      

Higher generation subgroup sets and the $ \Sigma $-invariants of graph groups

We present a general condition, based on the idea of n-generating subgroup sets, which implies that a given character represents a point in the homotopical or homological -invariants of the group G. Let be a finite simplicial graph, the flag complex induced by , and the graph group, or 'right angled Artin group', defined by . We use our result on n-generating subgroup sets to describe the homotopical and homological -invariants of in terms of the topology of subcomplexes of . In particular, this work determines the finiteness properties of kernels of maps from graph groups to abelian groups. This is the first complete computation of the -invariants for a family of groups whose higher invariants are not determined - either implicitly or explicitly - by ¹. Received: October 18, 1996     

Krichever Maps,Faà di Bruno Polynomials,and Cohomology in KP Theory

We study the geometrical meaning of the Faà di Bruno polynomials in the context of KP theory. They provide a basis in a subspace W of the universal Grassmannian associated to the KP hierarchy. When W comes from geometrical data via the Krichever map, the Faà di Bruno recursion relation turns out to be the cocycle condition for (the Welters hypercohomology group describing) the deformations of the dynamical line bundle on the spectral curve together with the meromorphic sections which give rise to the Krichever map. Starting from this, one sees that the whole KP hierarchy has a similar cohomological meaning.     

Geometry of Planes over Nonassociative Algebras

In this paper the geometric interpretation of the exceptional Lie groups F₄, E₆, E₇, and E₈ is given. These groups are groups of motions of elliptic hyperbolic planes over nonassociative algebras of octaves and split octaves and their tensor products with algebras of usual and split complex numbers, quaternions and octaves. The explicit expressions of motions of these planes and their figures of symmetry are presented.     

Protection of the carbonyl groups in 1,2-indanedione: propellane versus acetal formation

The product resulting from the reaction between 1,2-indanedione and ethylene glycol under acidic catalysis is 2,5,7,10-tetraoxapropellane and not 1,2-dispirane as previously reported. Similar reactions also occur with 2-mercaptoethanol and 1,2-ethanedithiol, which form analogous propellanes and not corresponding thioacetals. This explains the difficulty of removing the protective groups under acidic conditions. These findings were corroborated by quantum chemical calculations. Under similar conditions, the longer-chain diol, 1,3-propyleneglycol and its thiol-analogue, 1,3-propanedithiol, form only mono-acetals, even when a 3-fold excess of the diol is applied. The nucleophilic attack, however, takes place at different positions: while propanedithiol forms the acetal at c-1, propylene glycol forms the acetal at c-2.     

Deficiencies of smooth manifolds

Being given a closed manifold Mⁿ, there are involutions (X²ⁿ, T) on closed manifolds of twice the dimension having fixed point set M. Kulkarni defined the deficiency of M for a class of involutions to be $\text{min}\text{(}\text{1}\text{2}\text{{}\text{dim}\text{H}{}^1\text{(X;Z}{}_{\mathrm{<mtext>2</mtext>}}\text{)?}\text{dim}\text{H}{}^1\text{(M;Z}{}_{\mathrm{<mtext>2</mtext>}}\text{)})}$ for all involutions (X, T) in the class. This paper exhibits manifolds for which the deficiency is positive for all involutions and studies the deficiencies for other classes.