We approximate functions defined on smooth bounded domains by elements of the eigenspaces Laplacian or Stokes operator in such a way that approximations are and converge both Sobolev Lebesgue spaces. prove an abstract result referred to fractional power spaces positive, self-adjoint, compact-inverse operators Hilbert spaces, then obtain our main using explicit form these for Dirichlet operators...

In this paper, we investigate inhomogeneous and simultaneous Diophantine approximation in beta dynamical systems. For $\beta>1$ let $T_{\beta}$ be the $\beta$-transformation on $[0,1]$. We determine Lebesgue measure Hausdorff dimension of set \[\left\{(x,y)\in [0,1]^2: |T_{\beta}^nx-f(x,y)|<\varphi(n)\text{ for infinitely many }n\in\mathbb{N}\right\},\] where $f:[0,1]^2\to [0,1]$ is a Lipschitz...

In this paper we present the Multithreaded Advanced Fast Rational Approximation algorithm – MAFRA – for solving n-dimensional simultaneous Diophantine approximation problems. We show that in some particular applications the Lenstra-Lenstra-Lovász (L) algorithm can be substituted by the presented one in order to reduce their practical running time. MAFRA was implemented in the following architec...

The author uses an elementary lemma on primes dividing binomial coefficients and estimates for primes in arithmetic progressions to sharpen a theorem of J. Rickert on simultaneous approximation to pairs of algebraic numbers. In particular, it is proven that max {∣∣∣∣√2− p1 q ∣∣∣∣ , ∣∣∣∣√3− p2 q ∣∣∣∣} > 10−10q−1.8161 for p1, p2 and q integral. Applications of these estimates are briefly discussed.

Given k collections of 2SAT clauses on the same set of variables V , can we find one assignment that satisfies a large fraction of clauses from each collection? We consider such simultaneous constraint satisfaction problems, and design the first nontrivial approximation algorithms in this context. Our main result is that for every CSP F , for k ă Õplog1 nq, there is a polynomial time constant f...