For every finite nonnegative measure ft we introduce the Sobolev spaces W^'(Q,K) and we study the lower semicontinuity of functionals of the form where the integrand / is quasiconvex.

According to a theorem in [1], the conserved functionals of the KdV equation vanish on each formal Laurent series 1/x + u0 + ∑+∞ k=2 ukx . We propose a new, very simple geometrical proof for this statement.

New affine invariant functionals for convex polytopes are introduced. Some sharp affine isoperimetric inequalities are established for the new functionals. These new inequalities lead to fairly strong volume estimates for projection bodies. Two of the new affine isoperimetric inequalities are extensions of Ball’s reverse isoperimetric inequalities. If K is a convex body (i.e., a compact, convex...

Let K be a smooth convex set with volume one in R. Choose n random points in K independently according to the uniform distribution. The convex hull of these points, denoted by Kn, is called a random polytope. We prove that several key functionals of Kn satisfy the central limit theorem as n tends to infinity.

We study the lower semicontinuity for functionals of the form K → ∫ K φ(x, ν) dH defined on compact sets in R with a finite number of connected components and finite H measure and apply the result to the study of quasistatic growth of brittle fractures in linearly elastic inhomogeneous and anisotropic bodies.

a general notion of completely monotone functionals on an ordered banach algebra b into a proper h*-algebra a with an integral representation for such functionals is given. as an application of this result we have obtained a characterization for the generalized completely continuous monotone functions on weighted foundation semigroups. a generalized version of bochner’s theorem on foundation se...

Let X be a Poisson point process and K ⊂ R d a measurable set. Construct the Voronoi cells of all points x ∈ X with respect to X, and denote by vX (K) the union of all Voronoi cells with nucleus in K. For K a compact convex set the expectation of the volume difference V (vX (K)) − V (K) and the symmetric difference V (vX (K)△K) is computed. Precise estimates for the variance of both quantities ...

Charge-transfer rate constants (kCT) are required for the design of fullerene-based solar cells. We have assessed performance six long-range corrected hybrid functionals and a meta-hybrid-GGA functional calculation kCT with Marcus equation.

We study certain function algebras and their operator algebra completions on r-discrete abelian groupoids, the corresponding conditional expectations, maximal abelian subalgebras (masa) and eigen-functionals. We give a semidirect product decomposition for an abelian groupoid. This is done through a matched pair and leads to a C*-diagonal (for a special case). We use this decomposition to study ...

Schreiber and Yukich [Ann. Probab. 36 (2008) 363–396] establish an asymptotic representation for random convex polytope geometry in the unit ball Bd , d ≥ 2, in terms of the general theory of stabilizing functionals of Poisson point processes as well as in terms of generalized paraboloid growth processes. This paper further exploits this connection, introducing also a dual object termed the par...