Weak order for the discretization of the stochastic heat equation driven by impulsive noise

Considering a linear parabolic stochastic partial differential equation driven by impulsive space time noise, dX_t+AX_t dt= Q^{1/2}dZ_t, X_0=x_0\in H, t\in [0,T], we approximate the distribution of X_T. (Z_t)_{t\in[0,T]} is an impulsive cylindrical process and Q describes the spatial covariance structure of the noise; Tr(A^{-\alpha})0 and A^\beta Q is bounded for some \beta\in(\alpha-1,\alpha]. A discretization (X_h^n)_{n\in\{0,1,...,N\}} is defined via the finite element method in space (parameter h>0) and a \theta-method in time (parameter \Delta t=T/N). For \phi\in C^2_b(H;R) we show an integral representation for the error |E\phi(X^N_h)-E\phi(X_T)| and prove that |E\phi(X^N_h)-E\phi(X_T)|=O(h^{2\gamma}+(\Delta t)^{\gamma}) where \gamma<1-\alpha+\beta.Comment: 29 pages; Section 1 extended, new results in Appendix

Density of quasismooth hypersurfaces in simplicial toric varieties

This paper investigates the density of hypersurfaces in a projective normal simplicial toric variety over a finite field having a quasismooth intersection with a given quasismooth subscheme. The result generalizes the formula found by B. Poonen for smooth projective varieties. As an application, we further analyze the density of hypersurfaces with bounds on their number of singularities and on the length of their singular schemes.Comment: extended and accepted versio

The Future of low Energy Photon Experiments

"Light-shining-through-a-wall" experiments search for Weakly Interacting Sub-eV Particles (WISPs). The necessity and status of such enterprises as well as their future potential are sketched.Comment: 9 pages, 4 figures, contribution to the conference PHOTON 2009 (Hamburg, May 2009

Circulant matrices: norm, powers, and positivity

In their recent paper "The spectral norm of a Horadam circulant matrix", Merikoski, Haukkanen, Mattila and Tossavainen study under which conditions the spectral norm of a general real circulant matrix ${\bf C}$ equals the modulus of its row/column sum. We improve on their sufficient condition until we have a necessary one. Our results connect the above problem to positivity of sufficiently high powers of the matrix ${\bf C^\top C}$. We then generalize the result to complex circulant matrices

Theoretical basis for a solution to the cosmic coincidence problem

Following a short discussion of some unresolved issues in the standard model of cosmology (considered to be a generic $\Lambda$CDM model with flat geometry and an early period of inflation), an update on the current state of research regarding the problem of negative energy is provided. Arguments are then given to the effect that traditional assumptions concerning the behavior of negative action matter give rise to violations of both the principle of relativity and the principle of inertia. An alternative set of axioms is proposed that would govern the behavior of negative action matter if it is to be considered a viable element of physical theories upon which cosmological models are build. A simple framework, based on general relativity and the proposed axioms, is elaborated which enables the formulation of quantitative predictions concerning the interaction between positive and negative action bodies. Based on those developments, a solution is proposed to the problem of the discrepancy between current experimental and theoretical values of vacuum energy density (in any cosmological model), which may also constitute a solution to the problem of the unexplained coincidence between the (model dependent) experimental values of vacuum energy density and present day average matter energy density. It is also shown how irreversibility naturally arises in cosmological models derived in this context.Comment: 56 pages report with some clarifications and added comments concerning later development