We start from a new theory (discussed earlier) in which the arena for physics is not spacetime, but its straightforward extension-the so called Clifford space (C-space), a manifold of points, lines, areas, etc..; physical quantities are Clifford algebra valued objects, called polyvectors. This provides a natural framework for description of supersymmetry, since spinors are just left or right minimal ideals of Clifford algebra. The geometry of curved C-space is investigated. It is shown that the curvature in C-space contains higher orders of the curvature in the underlying ordinary space. A C-space is parametrized not only by 1-vector coordinates x µ but also by the 2-vector coordinates σ µν , 3-vector coordinates σ µνρ , etc., called also holographic coordinates, since they describe the holographic projections of 1-lines, 2-loops, 3-loops, etc., onto the coordinate planes. A remarkable relation between the "area" derivative ∂/∂σ µν and the curvature and torsion is found: if a scalar valued quantity depends on the coordinates σ µν this indicates the presence of torsion, and if a vector valued quantity depends so, this implies non vanishing curvature. We argue that such a deeper understanding of the C-space geometry is a prerequisite for a further development of this new theory which in our opinion will lead us towards a natural and elegant formulation of M-theory.

In this article one discusses the controllability of a semi-discrete system obtained by discretizing in space the linear 1-D wave equation with a boundary control at one extremity. It is known that the semi-discrete models obtained with finite difference or the classical finite element method are not uniformly controllable as the discretization parameter h goes to zero (see [8]).Here we introduce a new semi-discrete model based on a mixed finite element method with two different basis functions for the position and velocity. We show that the controls obtained with these semi-discrete systems can be chosen uniformly bounded in L 2 (0, T ) and in such a way that they converge to the HUM control of the continuous wave equation, i.e. the minimal L 2 -norm control. We illustrate the mathematical results with several numerical experiments.

We borrow the minisuperspace approximation from quantum cosmology and the quenching approximation from QCD in order to derive a new form of the bosonic p-brane propagator. In this new approximation we obtain an exact description of both the collective mode deformation of the brane and the center of mass dynamics in the target space time. The collective mode dynamics is a generalization of string dynamics in terms of area variables. The final result is that the evolution of a p-brane in the quenched-minisuperspace approximation is formally equivalent to the effective motion of a particle in a spacetime where points as well as hypersurfaces are considered on the same footing as fundamental geometrical objects. This geometric equivalence leads us to define a new tension-shell condition that is a direct extension of the Klein-Gordon condition for material particles to the case of a physical p-brane.

We introduce a new optimization strategy to compute numerical approximations of minimizers for optimal control problems governed by scalar conservation laws in the presence of shocks. We focus on the 1 -d inviscid Burgers equation. We first prove the existence of minimizers and, by a T-convergence argument, the convergence of discrete minima obtained by means of numerical approximation schemes satisfying the so called onesided Lipschitz condition (OSLC). Then we address the problem of developing efficient descent algorithms. We first consider and compare the existing two possible approaches: the so-called discrete approach, based on a direct computation of gradients in the discrete problem and the so-called continuous one, where the discrete descent direction is obtained as a discrete copy of the continuous one. When optimal solutions have shock discontinuities, both approaches produce highly oscillating minimizing sequences and the effective descent rate is very weak. As a solution we propose a new method, that we shall call alternating descent method, that uses the recent developments of generalized tangent vectors and the linearization around discontinuous solutions. This method distinguishes and alternates the descent directions that move the shock and those that perturb the profile of the solution away of it producing very efficient and fast descent algorithms.

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This paper deals with the numerical computation of distributed null controls for the 1D wave equation. We consider supports of the controls that may vary with respect to the time variable. The goal is to compute approximations of such controls that drive the solution from a prescribed initial state to zero at a large enough controllability time. Assuming a geometric optic condition on the support of the controls, we first prove a generalized observability inequality for the homogeneous wave equation. We then introduce and prove the well-posedness of a mixed formulation that characterizes the controls of minimal square-integrable norm. Such mixed formulation, introduced in [Cindea and Münch, A mixed formulation for the direct approximation of the control of minimal L 2-norm for linear type wave equations], and solved in the framework of the (space-time) finite element method, is particularly well-adapted to address the case of time dependent support. Several numerical experiments are discussed.

Abstract. We consider the eigenvalue problem associated to the vibrations of a string with a rapidly oscillating bounded periodic density. It is well known that when the size of the microstructure is small enough with respect to the wavelength of the eigenfunctions 1/ √ λ , eigenvalues and eigenfunctions can be approximated by those of the limit system where the oscillating density is replaced by its average. On the other hand, it has been observed that when the size of the microstructure is of the order of the wavelength of the eigenfunctions ( ∼ 1/ √ λ ), singular phenomena may occur. In this paper we study the behavior of the eigenvalues and eigenfunctions when 1/ √ λ approaches the critical size . To do this we use the WKB approximation which allows us to find an explicit formula for eigenvalues and eigenfunctions with respect to . In particular, our analysis provides all order correction formulas for the limit eigenvalues and eigenfunctions below the critical size.