Multivalued Maps and Multivalued Differential Equations

Everybody meets multivalued maps very early in his mathematical education, as inverses of maps which are not one-to-one, but in elementary lectures the multivalued aspect is usually suppressed by means of elementary tricks or restrictions which make sense for practical purposes; think of [see pdf for notation] which has the inverse[see pdf for notation], from [see pdf for notation] (the subsets of R); or think of a linear operator [see pdf for notation] with kernel [see pdf for notation]; in which case one uses the trick to consider [see pdf for notation], defined by [see pdf for notation] which is one-to-one and therefore has a singlevalued inverse [see pdf for notation]

A rigorous formulation of the cosmological Newtonian limit without averaging

We prove the existence of a large class of one-parameter families of cosmological solutions to the Einstein-Euler equations that have a Newtonian limit. This class includes solutions that represent a finite, but otherwise arbitrary, number of compact fluid bodies. These solutions provide exact cosmological models that admit Newtonian limits but, are not, either implicitly or explicitly, averaged

Existence and Comparison Theorems for Differential Equations in Banach Spaces

In our recent paper [3] we have studied the existence of maximal and minimal solutions to the IVP in a Banach space [see pdf for notation]. (1) [see pdf for notation] where [see pdf for notation] maps [see pdf for notation] into [see pdf for notation], with [see pdf for notation] and [see pdf for notation] a cone. The essential hypotheses have been that [see pdf for notation]f is quasimonotone with respect to [see pdf for notation] and that [see pdf for notation] and [see pdf for notation] have some natural properties. If such extremal solutions exist then it is trivial to prove the usual comparison theorems known from the finite-dimensional case. For example, if [see pdf for notation] is the minimal solution of (1) on some interval [see pdf for notation] and if [see pdf for notation] satisfies [see pdf for notation] and [see pdf for notation] then [see pdf for notation]. In the present paper we shall establish existence and comparison theorems for (1) without the hypothesis that [see pdf for notation] be quasimonotone, but under conditions which have been considered in case [see pdf for notation] in the classical paper of M. Muller [10] in 1926. This is not the first attempt to extend Muller\u27s results to infinite dimensions, since recently P. Volkmann [12] tried to do this. We shall improve the existing results considerably

On Existence of Extremal Solutions of Differential Equations in Banach Spaces

Let X be a real Banach space, [see pdf for notation] a cone, [see pdf for notation] and [see pdf for notation] continuous. We look for conditions on X, K and f such that the IVP (1) [see pdf for notation] has a maximal solution [see pdf for notation] and a minimal solution u with respect to the partial ordering induced by K. Contrary to known results, [5,6], we shall not assume that K has interior points, since the standard cones of many infinite dimensional spaces have empty interior. The second essential new feature is that f is supposed to be defined only on K and this demands that the extra conditions on f are required only with respect to points in K, and not on the whole space

Quasi-Solutions and Their Role in the Qualitative Theory of Differential Equations

In the study of comparison theorems and extremal solutions for systems of ordinary differential equations [5], one usually imposes a condition on the right hand side known as quasi-monotone nondecreasing property. This property. is also needed in proving comparison theorems for second order, boundary value problems [9] as well as for the initial boundary value problem for parabolic systems [5, 6]. Also, it is well known that in the .method of vector Lyapunov functions which provides an effective tool to investigate the stability of Large Scale Systems [1-4], an unpleasant drawback is the requirement of the quasi-monotone property for the comparison systems. In systems which represent physical situations, we rather often find that this property is not satisfied

MRI of "diffusion" in the human brain: New results using a modified CE-FAST sequence.

“Diffusion-weighted” MRI in the normal human brain and in a patient with a cerebral metastasis is demonstrated. The method employed was a modified CE-FAST sequence with imaging times of only 6-10 s using a conventional 1.5-T whole-body MRI system (Siemens Magnetom). As with previous phantom and animal studies, the use of strong gradients together with macroscopic motions in vivo causes unavoidable artifacts in diffusion-weighted images of the human brain. While these artifacts are shown to be considerably reduced by averaging of 8-16 images, the resulting diffusion contrast is compromised by unknown signal losses due to motion

Density of critical points for a Gaussian random function

Critical points of a scalar quantitiy are either extremal points or saddle points. The character of the critical points is determined by the sign distribution of the eigenvalues of the Hessian matrix. For a two-dimensional homogeneous and isotropic random function topological arguments are sufficient to show that all possible sign combinations are equidistributed or with other words, the density of the saddle points and extrema agree. This argument breaks down in three dimensions. All ratios of the densities of saddle points and extrema larger than one are possible. For a homogeneous Gaussian random field one finds no longer an equidistribution of signs, saddle points are slightly more frequent.Comment: 11 pages 1 figure, changes in list of references, corrected typo

Sample Solutions of Stochastic Boundary Value Problems

We prove existence theorems for nonlinear stochastic Sturmiouville problems which improve results from [4]. In the simplest case this is done by means of a known result about measurable selections of multivalued maps and a new fixed point theorem for stochastic nonlinear operators which is more realistic than existing ones