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## On the Dido Problem and Plane Isoperimetric Problems

### Acta Applicandae Mathematica (1999-07-01) 57: 287-338 , July 01, 1999

This paper is a continuation of a series of papers, dealing with contact sub-Riemannian metrics on R^{3}. We study the special case of contact metrics that correspond to isoperimetric problems on the plane. The purpose is to understand the nature of the corresponding optimal synthesis, at least locally. It is equivalent to studying the associated sub-Riemannian spheres of small radius. It appears that the case of generic isoperimetric problems falls down in the category of generic sub-Riemannian metrics that we studied in our previous papers (although, there is a certain symmetry). Thanks to the classification of spheres, conjugate-loci and cut-loci, done in those papers, we conclude immediately. On the contrary, for the Dido problem on a 2-d Riemannian manifold (i.e. the problem of minimizing length, for a prescribed area), these results do not apply. Therefore, we study in details this special case, for which we solve the problem generically (again, for generic cases, we compute the conjugate loci, cut loci, and the shape of small sub-Riemannian spheres, with their singularities). In an addendum, we say a few words about: (1) the singularities that can appear in general for the Dido problem, and (2) the motion of particles in a nonvanishing constant magnetic field.

## Uniform Rates of Decay in Anisotropic Thermo-Viscoelasticity

### Acta Applicandae Mathematica (1998-02-01) 50: 207-224 , February 01, 1998

We consider the anisotropic and inhomogeneous thermo-viscoelastic equation. We prove that the first and second-order energy decay exponentially as time goes to infinity provided the relaxation function also decays exponentially to zero. While if the relaxation functions decay polynomially to zero, then the energy decays also polynomially. That is, the kernel of the convolution defines the rate of decay of the solution.

## Some Local Properties of Bäcklund Transformations

### Acta Applicandae Mathematica (1998-10-01) 54: 1-25 , October 01, 1998

For Bäcklund transformations, treated as relations in the categoryof diffieties, local conditions of effectivity and normality are introduced,having implications for the solution generating properties. We check themfor the pKdV, the sine-Gordon, and the Tzitzéica equation.

## Riemannian Geometry of Grassmann Manifolds with a View on Algorithmic Computation

### Acta Applicandae Mathematica (2004-01-01) 80: 199-220 , January 01, 2004

We give simple formulas for the canonical metric, gradient, Lie derivative, Riemannian connection, parallel translation, geodesics and distance on the Grassmann manifold of *p*-planes in *R*^{n}. In these formulas, *p*-planes are represented as the column space of *n*×*p* matrices. The Newton method on abstract Riemannian manifolds proposed by Smith is made explicit on the Grassmann manifold. Two applications – computing an invariant subspace of a matrix and the mean of subspaces – are worked out.

## Extended harmonic analysis of phase space representations for the Galilei group

### Acta Applicandae Mathematica (1986-05-01) 6: 19-45 , May 01, 1986

The spectral resolution of phase space representations of the Galilei group is achieved by deriving all possible decompositions into irreducible representations corresponding to reproducing kernel Hilbert spaces. Spectral syntheses in terms of eigenfunction expansions, as well as in terms of continuous resolutions of the identity, are achieved. For the latter, the existence, uniqueness and other basic properties of resolution generators are established. This is shown to lead to systems of covariance related to measurements of stochastic phase space values performed with extended quantum test particles, whose proper wavefunctions are the aforementioned resolution generators.

## Epi-Convergence and Lower and Epi-Upper Semicontinuous Approximations in Distribution

### Acta Applicandae Mathematica (2003-08-01) 78: 243-250 , August 01, 2003

The paper introduces an extension of the epi-convergence, the lower semicontinuous approximation and the epi-upper semicontinuous approximation of random real functions in distribution. The new notions could be helpful tools for sensitivity analyzes of stochastic optimization problems. The research is evoked by S. Vogel and continues the research started by Vogel and the author.

## Finite-Type Invariants of Cubic Complexes

### Acta Applicandae Mathematica (2003-01-01) 75: 125-132 , January 01, 2003

The paper is for a general audience and may serve as a preliminary introduction to the theory of finite-type invariants.

## Poincaré δ-Lemma for Smooth Algebras

### Acta Applicandae Mathematica (1997-12-01) 49: 249-255 , December 01, 1997

The Poincaré δ-lemma (stable triviality of Spencer cohomology groups) for smooth algebras is proved.

## The Continuous Wavelet Transform and Symmetric Spaces

### Acta Applicandae Mathematica (2003-05-01) 77: 41-69 , May 01, 2003

The continuous wavelet transform has become a widely used tool in applied science during the last decade. In this article we discuss some generalizations coming from actions of closed subgroups *H* of GL(*n*,*R*) acting on *R*^{n}. If *R*^{n} has finitely many open orbits under the transposed action of *H* such that the union has full measure, then *L*^{2}(*R*^{n}) decomposes into finitely many irreducible representations, *L*^{2}(*R*^{n})≃*V*_{1}⊕⋅⋅⋅⊕*V*_{k} under the action of the semidirect product *H*×_{s}*R*^{n}. It is well known, that the space *V*_{j} contains an admissible vector if and only if the stabilizer in *H*^{t} of every point in *V*_{j} is compact. In this article we discuss the case where the stabilizer of a generic point in *R*^{n} is not compact, but a symmetric subgroup, a case that has not previously been discussed in the literature. In particular we show, that the wavelet transform can always be inverted in this case.