This paper presents an elementary proof of the following theorem: Given {$r$_{$j$}}_{$j$}^{$m$}=1 with m=d+1, fix $$fix R \geqslant \sum\nolimits_{j = 1}^m {r_j } $$ and let Q=[−R, R]^{d}. Then any f∈ L^{2}(Q) is completely determined by its averages over cubes of side r_{j}that are completely contained in Q and have edges parallel to the coordinate axes if and only if r_{j}/r_{k}is irrational for j≠k. When$d$=2 this theorem is known as the local three squares theorem and is an example of a Pompeiu-type theorem. The proof of the theorem combines ideas in multisensor deconvolution and the theory of sampling on unions of rectangular lattices having incommensurate densities with a theorem of Young on sequences biorthogonal to exact sequences of exponentials.

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Topological qauntum field theory(TQFT) is a very powerful theoretical tool to study topological phases and phase transitions. In 2+1D, it is well known that the Chern-Simons theory captures all the universal topological data of topological phases, e.g., quasi-particle braiding statistics, chiral central charge and even provides us a deep insight for the nature of topological phase transitions. Recently, topological phases of quantum matter are also intensively studied in 3+1D and it has been shown that loop like excitation obeys the so-called three-loop-braiding statistics. In this paper, we will try to establish a TQFT framework to understand the quantum statistics of particle and loop like excitation in 3+1D. We will focus on Abelian topological phases for simplicity, however, the general framework developed here is not limited to Abelian topological phases.

For all open Riemann surface N and real number  2 (0, /2), we construct a conformal minimal immersion X = (X1,X2,X3) : N ! R3 such that X3+tan()|X1| : N ! R is positive and proper. Furthermore, X can be chosen with an arbitrarily prescribed flux map. Moreover, we produce properly immersed hyperbolic minimal surfaces with non-empty boundary in R3 lying above a negative sublinear graph.

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