The extremal problem for the functional determinant of a natural linear elliptic operator a on Riemannian manifold is studied. Viewing the determinant as a function of the Riemannian metric, we encounter non-linear geometric analytic phenomena: sharp inequalities comparing nonlinear functionals of the metric and its derivatives. The derivation and use of such inequalities in new situations, esp...

With every graph (or digraph) one can associate several different matrices. We have already seen the vertex-edge incidence matrix, the Laplacian and the adjacency matrix of a graph. Here we shall concentrate mainly on the adjacency matrix of (undirected) graphs, and also discuss briefly the Laplacian. We shall show that spectral properies (the eigenvalues and eigenvectors) of these matrices pro...

Economic shocks can have diverse effects on financial market dynamics at different time horizons, yet traditional portfolio management tools do not distinguish between shortand long-term components in alpha, beta, and covariance estimators. In this paper, we apply spectral analysis techniques to quantify stock-return dynamics across multiple time horizons. Using the Fourier transform, we decomp...

The notion of an adaptive landscape has proved to be a valuable concept in theoretical investigations of evolutionary change, combinatorial optimization, and the physics of disordered systems. Landscape theory has emerged as an attempt to devise suitable mathematical structures for describing the \static" properties of landscapes as well as their in uence on the dynamics of adaptation. Here we ...

These lecture notes are based on my course from the summer semester 2015. I kept the numbering and the contents of the results presented in the lectures (except for minor corrections and improvements). Typically, the proofs and calculations in the notes are a bit shorter than those given in the lecture. Moreover, the drawings and many additional, mostly oral remarks from the lectures are omitte...

Spectral graph theory is a fast developing field in modern discrete mathematics with important applications in computer science, chemistry and operational research. By merging combinatorial techniques with algebraic and analytical methods it creates new approaches to hard discrete problems and gives new insights in classical Linear Algebra. The proposed minisymposium will bring together leading...

Example 1 (Spectrum of Multiplication Operators) Let • (X, M, µ) be a σ–finite (actually semifinite will do) measure space, • 1 ≤ p ≤ ∞ and • a : X → C be a bounded measurable function on X. (Aϕ)(x) = a(x)ϕ(x) Point spectrum: Let λ ∈ C. Then λ1l − A is injective ⇐⇒ ϕ ∈ L p (X, M, µ), (λ − a(x))ϕ(x) = 0 a.e. =⇒ ϕ(x) = 0 a.e. ⇐⇒ λ − a(x) = 0 a.e.