We give a precise mathematical formulation of a variational model for the irreversible quasi-static evolution of a brittle fracture proposed by G.A. Francfort and J.-J. Marigo, and based on Griffith’s theory of crack growth. In the two-dimensional case we prove an existence result for the quasi-static evolution and show that the total energy is an absolutely continuous function of time, althoug...

Using path integral Monte Carlo we simulate a 3D system of up to 1000 magnetic flux lines by mapping it onto interacting bosons in (2+1)D. With increasing temperatures we find first order melting from an ordered solid to an entangled liquid signaled by a finite entropy jump and sharp discontinuities of the defect density and the structure factor S(G). For a particular density of strong columnar...

We present a second order accurate, geometrically flexible and easy to implement method for solving the variable coefficient Poisson equation with interfacial discontinuities or on irregular domains, handling both cases with the same approach. We discretize the equations using an embedded approach on a uniform Cartesian grid employing virtual nodes at interfaces and boundaries. A variational me...

Image denoising is important in image analysis. It is often used for pre-processing images so that subsequent image analysis is more reliable. Besides noise removal, one important requirement for image denoising procedures is that they should preserve true image structures, such as edges. This paper proposes a novel denoising procedure which can preserve edges and major edge features (e.g., ang...

We present a fully automated framework to estimate derivatives nonparametrically without estimating the regression function. Derivative estimation plays an important role in the exploration of structures in curves (jump detection and discontinuities), comparison of regression curves, analysis of human growth data, etc. Hence, the study of estimating derivatives is equally important as regressio...

In this paper we present a new construction of generalized hierarchical bases (GHB) for symmetric positive definite matrices arising from discontinuous Galerkin discretization of secondorder partial differential equations (PDE) with highly varying coefficients. In the well-established theory of hierarchical basis multilevel methods one basic assumption is that the PDE coefficients are smooth fu...

Finding strongly standard complete axiomatizations for t-norm based fuzzy logics (i.e. complete for deductions with infinite sets of premises w.r.t. semantics on the real unit interval [0, 1]) is still an open problem in general, even though results are already available for some particular cases like some infinitary logics based on a continuous t-norm or certain expansions of Monoidal t-norm b...

A novel strategy for the Hybridizable Discontinuous Galerkin (HDG) solution of heat bimaterial problems is proposed. It is based on eXtended Finite Element philosophy, together with a level set description of interfaces. Heaviside enrichment on cut elements and cut faces is used to represent discontinuities across the interface. A suitable weak form for the HDG local problem on cut elements is ...

We give a functional integral representation of the semigroup generated by the spin-boson Hamiltonian by making use of a Poisson point process and a Euclidean field. We present a method of constructing Gibbs path measures indexed by the full real line which can be applied also to more general stochastic processes with jump discontinuities. Using these tools we then show existence and uniqueness...

The paper considers a class of additive neural networks where the neuron activations are modeled by discontinuous functions or by continuous non-Lipschitz functions. Some tools are developed which enable us to apply a Lyapunov-like approach to differential equations with discontinuous right-hand side modeling the neural network dynamics. The tools include a chain rule for computing the time der...