Singularly Perturbed Linear and Semilinear Hyperbolic Systems: Kinetic Theory Approach to Some Folk’s Theorems

The fact that a solution of a wave equation with strong damping can be approximated by a solution of an appropriate diffusion equation has been common knowledge to physicists and mathematicians for a few decades. In numerous branches of science one can find attempts to prove this results by various ad hoc methods. However, it seems to be difficult to trace a rigorous and exhaustive treatment of the subject. By using a recently developed asymptotic procedure, first applied in kinetic theory, we present a systematic approach to a wide spectrum of singularly perturbed hyperbolic problems and we also hope that our method will provide new insight into the field.     

Some remarks on Hilbertian fields (An appendix to the paper “Galois averages” by R. Massy)

The paper gives proofs of some results just claimed in [R. Massy, Galois averages, J. Number Theory 113 (2005) 244-275]. For instance, it is proved that for a finite non-trivial separable extension M/F, M≠F, of Hilbertian fields finitely generated over their prime field, the quotient group M^×/F^×, for the corresponding multiplicative groups of non-zero elements, cannot be a torsion group of finite exponent.     

Addendum to “Distribution of points on the circle” [J. Number Theory 127 (1) (2007) 127-135]

In an earlier paper we gave a result relating the value of an exponential sum with the distribution of summands in the arcs of the unit circle in the complex plane. In a marginal case, this result is superseded by a result of Lev. In this addendum we use an idea of Lev to give a new proof of our earlier result and to improve upon the marginal result.     

Corrigendum to “Orbit closures in the enhanced nilpotent cone” [Adv. Math. 219 (1) (2008) 27–62]

In this note, we point out an error in the proof of Theorem 4.7 of [P.N. Achar, A. Henderson, Orbit closures in the enhanced nilpotent cone, Adv. Math. 219 (1) (2008) 27–62], a statement about the existence of affine pavings for fibres of a certain resolution of singularities of an enhanced nilpotent orbit closure. We also give independent proofs of later results that depend on that statement, so all other results of that paper remain valid.     

Georg Klügel's dissertation (1763) now accessible on the worldwide web

Georg Klügel's dissertation (1763) is now accessible on the worldwide web in a German translation.     

Some Algebras Similar to the Enveloping Algebra of s1(2)

Abstract

Consider the algebra R = ?[A,B,H] subject to relations

where and . We study these algebras for different f and ç. We study highest weight modules and finite dimensional simple modules. We introduce the categories ^p with p ≥ 1, with an interesting decomposition in categories ^p _I and ^p _II . We also consider the BGG-category and develop new techniques towards the study of .     

Some results for probability measures on linear topological vector spaces with an application to Strassen's log log law

We prove some results regarding tight probability measures on real Frechet spaces and countable strict inductive limits of these spaces. These results are applied to Gaussian measures and to construct a Brownian motion on such spaces. We then prove the log log law of Strassen for this Brownian motion.     

Schur functions,Good's identity,and hypergeometric series well poised in SU(n)

A simple direct proof is given of a fundamental identity involving Schur functions which contains as special cases the identity responsible for Good's proof of the Dyson conjecture and the summation theorem of Biedenharn and Louck that appears frequently in dealing with the explicit matrix elements which arise in the unitary groups. By using the Weyl character formula, a general identity is obtained which implies our result involving Schur functions when a root system of type A_{n ? 1} is considered. As a further application of our general identity, explicit analogs of Good's identity are given, corresponding to the root systems of types B_n, C_n, and D_n. In addition, methods to obtain q-analogs of all of these results are briefly described.     

An improvement on Nourein's method for the simultaneous determination of the zeroes of a polynomial (An algorithm)

In a recent paper [2], Nourein derived an iteration formula, which exhibited cubic convergence for the simultaneous determination of the zeroes of a polynomial. In this paper - following quite a different appraoch - we derive a method which can be viewed as an improvement on that of [2]. The derivation is based on the approximation of the polynomial in question by a Lagrange interpolation formula. We give the algorithm in ALGOL 60. For a given real polynomial, the algorithm caters for the general case of complex zeroes.     

On Blattner's formula for the discrete series representations of SO(2n, 1)

Blattner's conjecture gives a formula for the multiplicity with which a unitary irreducible representation of the maximal compact subgroup K appears in any discrete series representation of a semisimple Lie group G. We give an elementary derivation of this formula from Harish Chandra's character formula for G ～- SO_e(2n, 1) (n ? 2). The idea is to regularize the character (on the Cartan subgroup), to show that the regularization is unique, and to derive the multiplicities by expanding the resulting distribution in a Fourier series. To prove uniqueness of the regularization one uses a priori constraints on the multiplicities (=Fourier coefficients) that follow from the subquotient theorem.