In recent work (Maierhofer and Huybrechs, 2022, Adv. Comput. Math.), the authors showed that least-squares oversampling can improve convergence properties of collocation methods for boundary integral equations involving operators certain pseudo-differential form. The underlying principle is discrete method approximates a Bubnov–Galerkin in suitable sense. present work, we extend this analysis t...

Geometric interpretation of the Hirota equation is presented as equation describing the Laplace sequence of two-dimensional quadrilateral lattices. Different forms of the equation are given together with their geometric interpretation: (i) the discrete coupled Volterra system for the coefficients of the Laplace equation, (ii) the gauge invariant form of the Hirota equation for projective invari...

Based on recent results for 2-D continuous-discrete systems, this paper develops 2-D Laplace-z transform, which can be used to analyze 2-D continuous-discrete signals and system in Laplace-z hybrid domain. Current 1-D Laplace transformation and z transform can be combined into the new 2-D s-z transform. However, 2-D s-z transformation is not a simple extension of 1D transform, in 2-D case, we n...

Implications of a formal context (G,M, I) have a minimal implication basis, called Duquenne-Guigues basis or stem base. It is shown that the problem of deciding whether a set of attributes is a premise of the stem base is in coNP and determining the size of the stem base is polynomially Turing equivalent to a #P-complete problem.

We show that the Laplace approximation of a supremum by L-norms has interesting consequences in optimization. For instance, the logarithmic barrier functions (LBF) of a primal convex problem P and its dual P∗ appear naturally when using this simple approximation technique for the value function g of P or its Legendre-Fenchel conjugate g∗. In addition, minimizing the LBF of the dual P∗ is just e...

We prove the Anderson localization near the bottom of the spectrum for two dimensional discrete Schrödinger operators with a class of random vector potentials and no scalar potentials. Main lemmas are the Lifshitz tail and the Wegner estimate on the integrated density of states. Then, the Anderson localization, i.e., the pure point spectrum with exponentially decreasing eigenfunctions, is prove...

Consider two perfectly conducting spheres in a homogeneous medium where the current-electric field relation is the power law. Electric field E blows up in the L∞-norm as δ, the distance between the conductors, tends to zero. We give here a concise rigorous justification of the rate of this blow-up in terms of δ. If the current-electric field relation is linear, see similar results obtained earl...

In this article, we first show the existence of a positive solution to { −Δpu−αΔu = λ(u− f (u)) in Ω, u = 0 on ∂Ω, by the method of lower and upper solutions and then under certain conditions on f , we show the stability of positive solution. Mathematics subject classification (2010): 35J92, 35B35, 35B05.

In this paper we show the existence of at least three nontrivial solutions to the following quasilinear elliptic equation −∆pu = |u| ∗ u + λf(x, u) in a smooth bounded domain Ω of R with homogeneous Dirichlet boundary conditions on ∂Ω, where p = Np/(N − p) is the critical Sobolev exponent and ∆pu = div(|∇u|∇u) is the p−laplacian. The proof is based on variational arguments and the classical con...