On complete reducibility for infinite-dimensional Lie algebras

In this paper we study the complete reducibility of representations of infinite-dimensional Lie algebras from the perspective of representation theory of vertex algebras.     

A geometric approach to Mathon maximal arcs

In 1969 Denniston [3] gave a construction of maximal arcs of degree d in Desarguesian projective planes of even order q, for all d dividing q. In 2002 Mathon [8] gave a construction method generalizing the one of Denniston. We will give a new geometric approach to these maximal arcs. This will allow us to count the number of isomorphism classes of Mathon maximal arcs of degree 8 in PG(2,h²), h prime.     

A geometric approach to generalized Stokes conjectures

We consider the Stokes conjecture concerning the shape of extreme 2-dimensional water waves. By new geometric methods including a non-linear frequency formula, we prove the Stokes conjecture in the original variables. Our results do not rely on structural assumptions needed in previous results such as isolated singularities, symmetry and monotonicity. Part of our results extends to the mathematical problem in higher dimensions.     

A sufficient condition for the complete reducibility of the regular representation

The “Mackey machine” is heavily employed to prove the following theorem. Let G be a separable locally compact group. Suppose that every positive definite function p on G which vanishes at infinity is associated with the regular representation R, i.e., p(g) = (R_g?, ?) for some L² function ?. Then R decomposes into a direct sum of irreducible representations. This generalizes the theorem of Figà-Talamanca for unimodular groups. Although we use his result several times, our techniques are basically very different, the most difficult part occurring in a connected Lie group context.     

A geometric approach to duality of metric entropy

We introduce a new geometric approach for studying the Duality of Metric Entropy (of operators, or of convex sets). We demonstrate such duality, up to a logarithmic factor, in one important case, for the hitherto not-too-well understood low `levels of resolution'.     

A geometric approach to the quadratic reciprocity law

A geometric viewpoint of much broader potential yields a unified proof of the quadratic reciprocity law based on counting points on quadratic surfaces over finite prime fields.     

A geometric approach to characterize rigidity in proteins

Kinematic analysis, in contrast to sophisticated molecular dynamics simulations, can provide high-level insights into conformational diversity of proteins and other biomolecules, with broad implications for human health. Here, we model a protein as a kinematic linkage and present a new geometric method to characterize molecular rigidity. While existing combinatorial constraint counting is limited to generic structures, our geometric approach is also valid for non-generic linkages. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A geometric approach to equal sums of fifth powers

Most of the known results about the Diophantine equation x⁵ + y⁵ + z⁵ = u⁵ + v⁵ + w⁵ are shown to be particular instances of a simple geometrical construction. By studying a K3 surface contained in the fourfold, we show that there are an infinity of parametric solutions also satisfying x + y + z = u + v + w, x ? y = u ? v; and we show that these may be effectively determined.     

Linear algebra approach to geometric graphs

We introduce a new linear algebra approach for studying extremal problems in geometric graphs. We give alternative proofs to well-established facts on geometric graphs, as well as new results about triangulations.     

A geometric approach for Hermite subdivision

We present a non-stationary, non-uniform scheme for two-point Hermite subdivision. The novelty of this approach relies on a geometric interpretation of the subdivision steps—related to generalized Bernstein bases—which permits to overcome the usually unavoidable analytical difficulties. The main advantages consist in extra smoothness conditions, which in turn produce highly regular limit curves, and in an elegant structure of the subdivision—described by three de Casteljau type matrices. As a by-product, the scheme is inherently shape preserving.     

A geometric approach to the cascade approximation operator for wavelets

This paper is devoted to an approximation problem for operators in Hilbert space, that appears when one tries to study geometrically thecascade algorithm in wavelet theory. Let be a Hilbert space, and let be a representation ofL ( ) on . LetR be a positive operator inL ( ) such thatR(1) =1, where1 denotes the constant function 1. We study operatorsM on (bounded, but noncontractive) such that