We consider the cubic nonlinear Schrödinger equation on 2-dimensional irrational tori. construct solutions which undergo growth of Sobolev norms. More concretely, for every s>0, s≠1 and almost choice spatial periods we whose Hs norms grow by any prescribed factor. Moreover, a set with positive Hausdorff dimension go from arbitrarily small to large. also provide estimates time needed norm explos...

Since the theory of spectral properties of non-self-accession differential operators on Sobolev spaces is an important field in mathematics, therefore, different techniques are used to study them. In this paper, two types of non-self-accession differential operators on Sobolev spaces are considered and their spectral properties are investigated with two different and new techniques.

This survey addresses sampling discretization and its connections with other areas of mathematics. The concentrates on norms elements finite-dimensional subspaces. We present here known results both integral the uniform norm beginning classical ending very recent achievements. also show how connects to spectral properties operator submatrices, embedding subspaces, moments marginals high-dimensi...

It is well-known that R has subspaces of dimension proportional to N on which the `1 and `2 norms are uniformly equivalent, but it is unknown how to construct them explicitly. We show that, for any δ > 0, such a subspace can be generated using only N random bits. This improves over previous constructions of Artstein-Avidan and Milman, and of Lovett and Sodin, which require O(N log N), and O(N) ...

In this paper we consider linear, time dependent Schrödinger equations of the form i∂tψ=K0ψ+V(t)ψ, where K0 is a strictly positive selfadjoint operator with discrete spectrum and constant spectral gaps, V(t) smooth in periodic potential. We give sufficient conditions on ensuring that K0+V(t) generates unbounded orbits. The main condition resonant average V(t), namely respect to flow K0, has non...

This paper develops first-order system least-squares (FOSLS) functionals for solving the pure traction problem in three-dimensional linear elasticity. It is a direct extension of an earlier paper on planar elasticity [Z. Cai, T. A. Manteuffel, S. F. McCormick, and S. V. Parter, SIAM J. Numer. Anal., 35 (1998), pp. 320–335]. Two functionals are developed, one involving L norms of the first-order...

This work focuses on the a posteriori error estimation for the reduced basis method applied to partial differential equations with quadratic nonlinearity and affine parameter dependence. We rely on natural norms—local parameter-dependent norms—to provide a sharp and computable lower bound of the inf-sup constant. We prove a formulation of the Brezzi–Rappaz–Raviart existence and uniqueness theor...