A criterion for <Emphasis Type="Italic">C</Emphasis><Superscript>2</Superscript>-rectifiability of sets

The following result is proved: Let be a n-dimensional C¹-submanifold of R^N which is domain of a given _nR^N-valued map of class C¹. Then the set of all points P such that (P) is non-zero, simple and enveloped by T_P is C²-rectifiable. As a corollary we get a criterion for the C²-rectifiability of a rectifiable set based on the rectifiability of some generalized Gauss lift to the Grassmanian bundle R^N×G(N,n). Mathematics Subject Classification (2000) Primary 49Q15, 53A07; Secondary 49Q20, 49N60     

“Invariance-Inducing” Control of Nonlinear Discrete-Time Dynamical Systems

The present study aims at the derivation of model-based control laws that attain the invariance objective for nonlinear skew-product discrete-time dynamical systems. The problem under consideration naturally arises in a variety of control problems pertaining to physical/chemical systems, and in the present study, it is conveniently formulated and addressed in the context of functional equations theory. In particular, the mathematical formulation of the problem of interest is realized via a system of nonlinear functional equations (NFEs), and a rather general set of necessary and sufficient conditions for solvability is derived. The solution to the above system of NFEs can be proven to be a unique locally analytic one, and this enables the development of a series solution method that is easily programmable with the aid of a symbolic software package such as MAPLE. It is also shown that, on the basis of the solution to the above system of NFEs, a locally analytic manifold and a nonlinear control law can be explicitly derived that renders the manifold invariant for the class of skew-product systems considered. Furthermore, the restriction of the system dynamics on the aforementioned invariant manifold represents exactly the target controlled system dynamics. Finally, the proposed method is applied to the HF molecular system classically modeled as a rotationless Morse oscillator in the presence of an external laser-field, where the primary objective is molecular dissociation.     

Auto-ignition of a polydisperse fuel spray

In the present paper, the effect of fuel spray polydispersity on the auto-ignition process in a fuel cloud is considered. In many engineering applications it is common practice to relate to the actual polydisperse spray as being equivalent to a monodisperse spray with all droplets therein having some average diameter. In combustion systems, the Sauter mean diameter (SMD) is frequently used for this purpose; it is based on the ratio between the total droplet volume and the total droplet surface area of all the droplets in the polydisperse spray. The main purpose of the current work is to examine qualitatively the dynamics of ignition of a truly polydisperse spray in a combustible gas medium and compare it with the dynamics of an equivalent monodisperse spray based on the SMD. Since the system of governing equations represents a multi-scale problem the method of integral manifolds is applied in order to extract the dynamical behavior. Preliminary computed results suggest that the use of the usual SMD-based monodisperse spray leads to quite a significant over-estimate of the ignition time. An alternative modified definition of the SMD, in which the overall liquid fuel volume is also conserved in the averaging process, reduces the discrepancy between the ignition time for the polydisperse spray and that of the equivalent monodisperse spray. However, it seems that some other sort of average droplet size needs to be determined to minimize the aforementioned discrepancy. These results highlight the care that must be exercised before dispensing with the behavior of the actual polydisperse spray in favor of that of an equivalent monodisperse spray, even at the expense of complexity.     

Characteristic Length in a Linear Acceleration

The role of the characteristic length that characterizes linear acceleration is studied, in order to find how does this length determine the characteristic wavelength of the radiation created by the accelerated charge. Unruh equation for the temperature observed by a detector accelerated relative to the vacuum is used to determine the wavelength distribution of the radiation emitted by a linearly accelerated charge, and it is found that this distribution is peaked close to the characteristic length that characterizes the linear acceleration, which is the radius of curvature of the curved electric field created by the acelerated charge. PACS numbers: 11.10; 41.60.m.     

Invertibility in infinite-dimensional spaces

An interesting result of Doyle and Hocking states that a topological -manifold is invertible if and only if it is a homeomorphic image of the -sphere . We shall prove that the sphere of any infinite-dimensional normed space is invertible. We shall also discuss the invertibility of other infinite-dimensional objects as well as an infinite-dimensional version of the Doyle-Hocking theorem.

     

On M-theory on real toric fibrations

Borrowing ideas from elliptic complex geometry, we approach M-theory compactifications on real fibrations. Precisely, we explore real toric equations rather than complex ones exploited in F-theory and related dual models. These geometries have been built by moving real toric manifolds over real bases. Using topological changing behaviors, we unveil certain data associated with gauge sectors relying on affine lie symmetries.     

Chaos in the Convective Flow of a Fluid Mixture in a Porous Medium

The results of the study of the global behaviour of the convective flow of a binary mixture in a porous medium are presented. Bifurcation diagram, fixed points, periodic, chaotic solutions, stable and unstable manifolds, and basins of attraction have been calculated. Different behaviours (chaos, undecidable behaviour, etc.) have been found.     

Temperature measurement of a curved surface using thermographic phosphors

An optical technique for surface temperature measurement based on the fluorescent emission of rare earth ion-doped phosphors was demonstrated in an experiment with a heated cylinder in crossflow. In this experiment, a uniform heat flux was imposed by applying a constant voltage across the thin stainless steel cylinder surface to produce surface temperatures between 24°C and 55°C. The fluorescent emission of a thermographic phosphor, lanthanum oxysulfide doped with europium (La₂0₂S:Eu³⁺) deposited on the surface, was recorded to determine the temperature distribution at the curved surface. When excited by ultraviolet radiation, the phosphor emits a spectrum containing certain emission lines, the intensities of which vary with temperature. For a single temperature-sensitive line, ratios of the intensity at a reference temperature to the intensity at different temperatures were correlated as a function of surface temperature. The use of intensity ratio correlations avoids complications due to geometric (viewing angle) effects. Digitized images of the cylinder permitted calculation of surface temperatures and local Nusselt numbers. Differences between surface temperatures measured by calibrated thermocouples and temperatures determined from the phosphor technique were at most 1.2°C.