Constructing uniform designs: A heuristic integer programming method

In this paper, the wrap-around _L2$L_2$-discrepancy (WD) of asymmetrical design is represented as a quadratic form, thus the problem of constructing a uniform design becomes a quadratic integer programming problem. By the theory of optimization, some theoretic properties are obtained. Algorithms for constructing uniform designs are then studied. When the number of runs n$n$ is smaller than the number of all level-combinations m$m$, the construction problem can be transferred to a zero–one quadratic integer programming problem, and an efficient algorithm based on the simulated annealing is proposed. When n≥m$n\ge m$, another algorithm is proposed. Empirical study shows that when n$n$ is large, the proposed algorithms can generate designs with lower WD compared to many existing methods. Moreover, these algorithms are suitable for constructing both symmetrical and asymmetrical designs.     

A gap theorem on submanifolds with finite total curvature in spheres

We study a complete noncompact submanifold Mⁿ$M^n$ in a sphere S^n+p${\mathbb{S}}^{n+p}$. We prove that there admit no nontrivial L²$L^2$-harmonic 1-forms on M if the total curvature is bounded from above by a constant depending only on n. The gap theorem is a generalized version of Carron?s, Yun?s, Cavalcante?s and the first author?s results on submanifolds in Euclidean spaces and Seo?s result on submanifolds in hyperbolic space without the condition of minimality.     

BMO from dyadic BMO via expectations on product spaces of homogeneous type

Using the random dyadic lattices developed by Hytönen and Kairema, we build up a bridge between BMO and dyadic BMO, and hence one between VMO and dyadic VMO, via expectations over dyadic lattices on spaces of homogeneous type, including both the one-parameter and product cases. We also obtain a similar relationship between _Ap$A_p$ and dyadic _Ap$A_p$, as well as one between the reverse Hölder class _RHp${\mathit{RH}}_p$ and dyadic _RHp${\mathit{RH}}_p$, via geometric–arithmetic expectations. These results extend the earlier theory along this line, developed by Garnett, Jones, Pipher, Ward, Xiao and Treil, to the more general setting of spaces of homogeneous type in the sense of Coifman and Weiss.     

Remarks on a posteriori error estimation for finite element solutions

We utilize the classical hypercircle method and the lowest-order Raviart–Thomas H(div)$H(\mathrm{div})$ element to obtain a posteriori error estimates of the _P1$P_1$ finite element solutions for 2D Poisson's equation. A few other estimation methods are also discussed for comparison. We give some theoretical and numerical results to see the effectiveness of the methods.