We show that any finite affinely independent set can be isometrically embedded into a regular polygonal torus, is, the product of vertex sets some polygons. apply this result in context Euclidean Ramsey Theory, highlighting connection between two most significant results field which were previously thought to independent. In particular with straightforward application Kříž’s theorem we give an ...

Abstract: We prove an existence and uniqueness result for quasilinear Stochastic PDEs with obstacle (OSPDE in short). Our method is based on analytical technics coming from the parabolic potential theory. The solution is expressed as a pair (u, ν) where u is a predictable continuous process which takes values in a proper Sobolev space and ν is a random regular measure satisfying minimal Skohoro...

We consider the problem of estimating the state of a diffusion process, based on discrete time observations in singular noise. We reduce the problem to a static problem, and we show that the solution is provided by the area or co–area formula of ge~ metric measure theory, provided the observed value is a regular value of the observation function. In order to address the case of singular values,...

We are interested in the asymptotic analysis of singular solutions with blowup boundary for a class of quasilinear logistic equations with indefinite potential. Under natural assumptions, we study the competition between the growth of the variable weight and the behaviour of the nonlinear term, in order to establish the blow-up rate of the positive solution. The proofs combine the Karamata regu...

In this article, we are concerned with the existence, uniqueness and asymptotic behavior of a positive classical solution to the semilinear boundaryvalue problem −∆u = a(x)u in D, lim |x|→1 u(x) = lim |x|→∞ u(x) = 0. Here D is the complement of the closed unit ball of Rn (n ≥ 3), σ < 1 and the function a is a nonnegative function in C loc(D), 0 < γ < 1, satisfying some appropriate assumptions r...