If$X$is a closed subset of the real line, denote by$G$_{$X$}the supremum of the size of the gap in the Fourier spectrum of a measure, taken over all non-trivial finite complex measures supported on$X$. In this paper we attempt to find$G$_{$X$}in terms of$X$.

In [$Rong$, F., Quasi-parabolic analytic transformations of$C$^{$n$},$J. Math. Anal. Appl.$$343$(2008), 99–109], we showed the existence of “parabolic curves” for certain quasi-parabolic analytic transformations of$C$^{$n$}. Under some extra assumptions, we show the existence of “parabolic manifolds” for such transformations.

We propose a generalized multiscale finite element method (GMsFEM) based on clustering algorithm to study the elliptic PDEs with random coefficients in the multi-query setting. Our method consists of offline and online stages. In the offline stage, we construct a small number of reduced basis functions within each coarse grid block, which can then be used to approximate the multiscale finite element basis functions. In addition, we coarsen the corresponding random space through a clustering algorithm. In the online stage, we can obtain the multiscale finite element basis very efficiently on a coarse grid by using the pre-computed multiscale basis. The new GMsFEM can be applied to multiscale SPDE starting with a relatively coarse grid, without requiring the coarsest grid to resolve the smallest-scale of the solution. The new method offers considerable savings in solving multiscale SPDEs. Numerical results are

Let K=Q(√(-q)), where q is any prime number congruent to 7 modulo 8, and let O be the ring of integers of K. The prime 2 splits in K, say 2O=pp∗, and there is a unique Z_2-extension K_∞ of K which is unramified outside p. Let H be the Hilbert class field of K, and write H_∞=HK_∞. Let M(H_∞) be the maximal abelian 2-extension of H_∞ which is unramified outside the primes above p, and put X(H_∞)=Gal(M(H_∞)/H_∞). We prove that X(H_∞) is always a finitely generated Z_2-module, by an elliptic analogue of Sinnott’s cyclotomic argument. We then use this result to prove for the first time the weak p-adic Leopoldt conjecture for the compositum J_∞ of K_∞ with arbitrary quadratic extensions J of H. We also prove some new cases of the finite generation of the Mordell–Weil group E(J_∞) modulo torsion of certain elliptic curves E with complex multiplication by O.

Let Y be a closed and oriented 3-manifold. We define different versions of unfolded Seiberg-Witten Floer spectra for Y. These invariants generalize Manolescu's Seiberg-Witten Floer spectrum for rational homology 3-spheres. We also compute some examples when Y is a Seifert space.