We consider a vertex model in d dimensions characterized by lines which run in a preferred direction. We show that this vertex model is soluble if the weights of vertices with intersecting lines are given by a free-fermion condition, and that a fugacity −1 is associated to each loop of lines. The solution is obtained by mapping the model into a dimer problem and by evaluating a Pfaffian. We als...

We prove the existence of fast traveling pulse solutions in excitable media with non-local coupling. Existence results had been known, until now, in the case of local, diffusive coupling and in the case of a discrete medium, with finite-range, non-local coupling. Our approach replaces methods from geometric singular perturbation theory, that had been crucial in previous existence proofs, by a P...

Weighted L for p ∈ 1,∞ and weak-type endpoint estimates with general weights are established for commutators of the Hardy-Littlewood maximal operator with BMO symbols on spaces of homogeneous type. As an application, a weighted weak-type endpoint estimate is proved for maximal operators associated with commutators of singular integral operators with BMO symbols on spaces of homogeneous type. Al...

This article develops approaches to generate dynamical reservoirs of echo state networks with desired properties reducing the amount of randomness. It is possible to create weight matrices with a predefined singular value spectrum. The procedure guarantees stability (echo state property). We prove the minimization of the impact of noise on the training process. The resulting reservoir types are...

The third, fifth and sixth Painlevé equations are studied by means of the weighted projective spaces CP (p, q, r, s) with suitable weights (p, q, r, s) determined by the Newton polyhedrons of the equations. Singular normal forms of the equations, symplectic atlases of the spaces of initial conditions, Riccati solutions and Boutroux’s coordinates are systematically studied in a unified way with ...

This talk presents work concepts and results for the determination of the fine structure constant α at the Z0 mass resonance. The problem consisting of the break-down of global duality for singular integral weights is circumvented by using a polynomial fit which mimics this weight function. This method is conservative in the sense that it is mostly independent of special assumptions. In this co...

We show that the classical Hardy inequalities with optimal constants in the Sobolev spaces W 1,p 0 and in higher-order Sobolev spaces on a bounded domain Ω ⊂ R can be refined by adding remainder terms which involve L norms. In the higher-order case further L norms with lower-order singular weights arise. The case 1 < p < 2 being more involved requires a different technique and is developed only...

Abstract In this note, we establish a new Carleman estimate with singular weights for the sub-Laplacian on Carnot group $$\mathbb G$$ G functions satisfying discrepancy assumption in (2.16) below. We use such an to derive sharp vanishing order solutions stationary Schrödinger equations.

In this paper we focus on the null controllability problem for heat equation with so-called inverse square potential and a memory term. To aim, first establish nonhomogeneous singular by new Carleman inequality weights which do not blow up at t = 0. Then property is proved under condition kernel, means of Kakutani’s fixed-point theorem.

In this paper, a special technique is studied by using the orthogonal Chebyshev polynomials to get approximate solutions for singular and hyper-singular integral equations of the first kind. A singular integral equation is converted to a system of algebraic equations based on using special properties of Chebyshev series. The error bounds are also stated for the regular part of approximate solut...