The FireBird Mission – A Scientific Mission for Earth Observation and Hot SpotDetection

More than 10 years ago the first specialized small satellite for hot spot recognition and fire observation was designed, built and operated by several DLR departments. This BIRD (Bi-spectral Infra Red Detection) satellite demonstrated the capability of fire monitoring from space by using a dedicated small satellite and sensor system. On the other hand it has shown that DLR is capable to manage nearly a complete space mission “in house”. The comparison of typical BIRD data with the well-known MODIS fire products led to the label “fire zoom” for BIRD data. It is due to the high geometric and radiometric resolution of BIRD fire products. Typically small fires with a diameter of 4m could be detected. The precise estimation of fire parameters was successfully shown without problems like false alarms. The success of BIRD opened the doors for next steps. The scientific DLR Earth observation mission “FireBird” will continue the fire monitoring topic by using two small satellites (TET-1, launched June 2012, BIROS launch planed 2014). The paper shall present this mission. It will finally be focused on possible interfaces for a desired worldwide international scientific cooperation within this running space mission

Melting and Freezing Lines for a Mixture of Charged Colloidal Spheres with Spindle-Type Phase Diagram

We have measured the phase behavior of a binary mixture of like-charged colloidal spheres with a size ratio of 0.9 and a charge ratio of 0.96 as a function of particle number density n and composition p. Under exhaustively deionized conditions the aqueous suspension forms solid solutions of body centered cubic structure for all compositions. The freezing and melting lines as a function of composition show opposite behavior and open a wide, spindle shaped coexistence region. Lacking more sophisticated treatments, we model the interaction in our mixtures as an effective one-component pair energy accounting for number weighted effective charge and screening constant. Using this description, we find that within experimental error the location of the experimental melting points meets the range of melting points predicted for monodisperse, one component Yukawa systems made in several theoretical approaches. We further discuss that a detailed understanding of the exact phase diagram shape including the composition dependent width of the coexistence region will need an extended theoretical treatment.Comment: 25 pages, 4 figure

Control problems for nonlocal set evolutions

In this paper, we extend fundamental notions of control theory to evolving compact subsets of the Euclidean space. Dispensing with any restriction of regularity, shapes can be interpreted as nonempty compact subsets of the Euclidean space. Their family, however, does not have any obvious linear structure, but in combination with the popular Pompeiu-Hausdorff distance, it is a metric space. Here Aubin's framework of morphological equations is used for extending ordinary differential equations beyond vector spaces, namely to the metric space of nonempty compact subsets of the Euclidean space supplied with Pompeiu-Hausdorff distance. Now various control problems are formulated for compact sets depending on time: open-loop, relaxed and closed-loop control problems – each of them with state constraints. Using the close relation to morphological inclusions with state constraints, we specify sufficient conditions for the existence of compact-valued solutions

Evolution equations in ostensible metric spaces : Definitions and existence.

The primary aim is to unify the definition of solution for completely different types of evolutions. Such a common approach is to lay the foundations for solving systems like, for example, a semilinear evolution equation (of parabolic type) in combination with a first order geometric evolution. In regard to geometric evolutions, this concept is to fulfill 3 conditions : First, consider nonempty compact subsets K(t) of R^N without a priori restrictions on the regularity of the boundary. Second, the evolution of K(t) might depend on nonlocal properties of the set K(t) and its normal cones. Last, but not least, no inclusion principle. The approach here is based on generalizing the mutational equations of Aubin for metric spaces in two respects : Replacing the metric by a countable family of (possibly nonsymmetric) distances (called ostensible metrics) and extending the basic idea of distributions

Evolution equations in ostensible metric spaces. II. Examples in Banach spaces and of free boundaries.

In part I, generalizing mutational equations of Aubin in metric spaces has led to so-called right-hand forward solutions in a nonempty set with a countable family of (possibly nonsymmetric) ostensible metrics. Now this concept is applied to two different types of evolutions that have motivated the definitions : semilinear evolution equations (of parabolic type) in a reflexive Banach space and compact subsets of R^N whose evolution depend on nonlocal properties of both the set and their limiting normal cones at the boundary. For verifying that reachable sets of differential inclusions are appropriate transitions for first-order geometric evolutions, their regularity at the boundary is studied in the appendix

Quasilinear continuity equations of measures for bounded BV vector fields

The focus of interest here is a quasilinear form of the conservative continuity equation d/dt v + D·(f(v, ·) v) = 0 (in R^N× ]0, T[) together with its measure-valued distributional solutions. On the basis of Ambrosio’s results about the nonautonomous linear equation, the existence and uniqueness of solutions are investigated for coefficients being bounded vector fields with bounded spatial variation and Lebesgue absolutely continuous divergence in combination with positive measures absolutely continuous with respect to Lebesgue measure. The step towards the nonlinear problem here relies on a further generalization of Aubin's mutational equations that is extending the notions of distribution-like solutions and "weak compactness" to a set supplied with a countable family of (possibly non–symmetric) distance functions (so–called ostensible metrics)

Radon measures solving the Cauchy problem of the nonlinear transport equation

The focus of interest is the Cauchy problem of the nonlinear transport equation d_t u + div (f(u, ·) u) = g(u, ·) u together with its distributional solutions u(·) whose values are positive Radon measures on the Euclidean space with compact support. The coefficients f(u, t), g(u, t) are assumed to be uniformly bounded and Lipschitz continuous vector fields on the Euclidean space. Sufficient conditions on the coefficients for existence, uniqueness and even for stability of these distributional solutions are presented. Starting from the well-known results about the corresponding linear problem, the step towards the nonlinear problem here relies on Aubin's mutational equations, i.e. dynamical systems in a metric space (with a new slight modification)

Generalizing evolution equations in ostensible metric spaces: Timed right-hand sleek solutions provide uniqueness of first-order geometric examples.

The mutational equations of Aubin extend ordinary differential equations to metric spaces (with compact balls). In first-order geometric evolutions, however, the topological boundary need not be continuous in the sense of Painleve–Kuratowski. So this paper suggests a generalization of Aubin’s mutational equations that extends classical notions of dynamical systems and functional analysis beyond the traditional border of vector spaces: Distribution– like solutions are introduced in a set just supplied with a countable family of (possibly non-symmetric) distance functions. Moreover their existence is proved by means of Euler approximations and a form of “weak” sequential compactness (although no continuous linear forms are available beyond topological vector spaces). This general framework is applied to a first-order geometric example, i.e. compact subsets of the Euclidean space evolving according to the nonlocal properties of both the current set and its proximal normal cones. Here neither regularity assumptions about the boundaries nor the inclusion principle are required. In particular, we specify sufficient conditions for the uniqueness of these solutions

Evolution equations in ostensible metric spaces: First-order evolutions of nonsmooth sets with nonlocal terms

Similarly to funnel equations of Panasyuk, the so-called mutational equations of Aubin provide a generalization of ordinary differential equations to locally compact metric spaces. Here we present their extension to a nonempty set with a possibly nonsymmetric distance. A distribution-like approach leads to so-called right-hand forward solutions. This concept is applied to a type of geometric evolution having motivated the definitions : compact subsets of the Euclidean space evolve according to nonlocal properties of both the set and their limiting normal cones at the boundary. The existence of a solution is based on Euler method using reachable sets of differential inclusions as "elementary deformations" (called forward transitions). Thus, the regularity of these reachable sets at the topological boundaries is studied extensively in the appendix

A viability theorem for morphological inclusions

The aim of this paper is to adapt the Viability Theorem from differential inclusions (governing the evolution of vectors in a finite dimensional space) to so-called morphological inclusions (governing the evolution of nonempty compact subsets of the Euclidean space). In this morphological framework, the evolution of compact subsets of the Euclidean space is described by means of flows along bounded Lipschitz vector fields (similarly to the velocity method alias speed method in shape analysis). Now for each compact subset, more than just one vector field is admitted - correspondingly to the set-valued map of a differential inclusion in finite dimensions. We specify sufficient conditions on the given data such that for every initial compact set, at least one of these compact-valued evolutions satisfies fixed state constraints in addition. The proofs follow an approximative track similar to the standard approach for differential inclusions in finite dimensions, but they use tools about weak compactness and weak convergence of Banach-valued functions. Finally an application to shape optimization under state constraints is sketched