The Helgason Fourier transform for homogeneous vector bundles over compact Riemannian symmetric spaces—the local theory

The Helgason Fourier transform on noncompact Riemannian symmetric spaces G/K is generalized to the homogeneous vector bundles over the compact dual spaces U/K. The scalar theory on U/K was considered by Sherman (the local theory for U/K of arbitrary rank, and the global theory for U/K of rank one). In this paper we extend the local theory of Sherman to arbitrary homogeneous vector bundles on U/K. For U/K of rank one we also obtain a generalization of the Cartan-Helgason theorem valid for any K-type.     

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Parallel axiom in convexity lattices

The convexity lattices, introduced by Bennett and Birkhoff, generalize the lattices of convex sets. We present three forms of Parallel Axiom in such lattices and define Euclidean and two classes of non-Euclidean lattices via the number of parallel lines through a point. The paper deals with these three classes of lattices.     

Jordan structures of strictly lower triangular completions of nilpotent matrices

This work is part of a doctoral thesis, written under the supervision of Prof. A. Berman. It was supported by the Fund for Promotion of Research at the Technion.     

On symmetries and conservation laws of the equations of shallow water with an axisymmetric profile of bottom

The local symmetries and conservation laws are calculated for the equations of shallow water with an axisymmetric profile of bottom under the assumption that the corresponding generating functions may depend only on all variables and their derivatives up to the second order. It is shown that if the bottom has the form of a paraboloid of revolution, then there are many symmetries and conservation laws generalizing those for the case of plane bottom.     

Generalized potentials in variable exponent Lebesgue spaces on homogeneous spaces

We consider generalized potential operators with the kernel on bounded quasimetric measure space (X, μ, d) with doubling measure μ satisfying the upper growth condition μB(x, r) ? Kr^N, N ∈ (0, ∞). Under some natural assumptions on a(r) in terms of almost monotonicity we prove that such potential operators are bounded from the variable exponent Lebesgue space L^p(?)(X, μ) into a certain Musielak‐Orlicz space L^p(X, μ) with the N‐function Φ(x, r) defined by the exponent p(x) and the function a(r). A reformulation of the obtained result in terms of the Matuszewska‐Orlicz indices of the function a(r) is also given. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

Observer synthesis method for Lipschitz nonlinear discrete-time systems with time-delay: An LMI approach

In this paper, we address the problem of observer design for a class of Lipschitz nonlinear discrete-time systems with time-delay. The main contribution lies in the use of a new structure of the proposed observer with a novel Lyapunov-Krasovskii functional. Thanks to these designs, new nonrestrictive synthesis conditions, expressed in terms of linear matrix inequalities (LMIs), are obtained. Indeed, the obtained LMIs contain more degree of freedom than those established by the approaches available in the literature which consider a simple Luenberger observer with a simple Lyapunov function for the stability analysis. An extension of the presented result to H_∞ performance analysis is given in order to take into account the noise (if it exists) affecting the considered system.     

A retelling of Newton’s work on Kepler’s Laws

This paper details an investigation into Kepler’s Laws. Newton’s technique for deducing an inverse-square law from Kepler’s Laws is given a modern presentation, with necessary background material included. Kepler’s Laws are then deduced from the assumption of an inverse-square law. This is done in a geometric style, inspired by Newton’s work. Finally, a problem involving planetary orbits is stated and solved using the earlier results of the paper.