We introduce an interactive tool which enables a user to quickly assemble an architectural model directly over a 3D point cloud acquired from large-scale scanning of an urban scene. The user loosely defines and manipulates simple building blocks, which we call SmartBoxes, over the point samples. These boxes quickly snap to their proper locations to conform to common architectural structures. The key idea is that the building blocks are smart in the sense that their locations and sizes are automatically adjusted on-the-fly to fit well to the point data, while at the same time respecting contextual relations with nearby similar blocks. SmartBoxes are assembled through a discrete optimization to balance between two snapping forces defined respectively by a data-fitting term and a contextual term, which together assist the user in reconstructing the architectural model from a sparse and noisy point cloud. We show that a combination of the user’s interactive guidance and high-level knowledge about the semantics of the underlying model, together with the snapping forces, allows the reconstruction of structures which are partially or even completely missing from the input.

The Wilson remainders for all primes less than 50,000 have been computed and tabulated. The distribution of the remainders divided by the corresponding primes has also been examined.

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We show the intersection of a compact almost complex subvariety of dimension 4 and a compact almost complex submanifold of codimension 2 is a J-holomorphic curve. This is a generalization of positivity of intersections for J-holomorphic curves in almost complex 4-manifolds to higher dimensions. As an application, we discuss pseudoholomorphic sections of a complex line bundle. We introduce a method to produce J-holomorphic curves using the differential geometry of almost Hermitian manifolds. When our main result is applied to pseudoholomorphic maps, we prove the singularity subset of a pseudoholomorphic map between almost complex 4-manifolds is J-holomorphic. Building on this, we show degree one pseudoholomorphic maps between almost complex 4-manifolds are actually birational morphisms in pseudoholomorphic category.

Let $f(x)=\sum_{n=0}^{\infty}\frac{1}{n!}q^{n(n-1)/2}x^n$ ($0<q<1$) be the deformed exponential function. It is known that the zeros of $f(x)$ are real and form a negative decreasing sequence $(x_k)$ ($k\ge 1$). We investigate the complete asymptotic expansion for $x_{k}$ and prove that for any $n\ge1$, as $k\to \infty$, \begin{align*} x_k=-kq^{1-k}\Big(1+\sum_{i=1}^{n}C_i(q)k^{-1-i}+o(k^{-1-n})\Big), \end{align*} where $C_i(q)$ are some $q$ series which can be determined recursively. We show that each $C_{i}(q)\in \mathbb{Q}[A_0,A_1,A_2]$, where $A_{i}=\sum_{m=1}^{\infty}m^i\sigma(m)q^m$ and $\sigma(m)$ denotes the sum of positive divisors of $m$. When writing $C_{i}$ as a polynomial in $A_0, A_1$ and $A_2$, we find explicit formulas for the coefficients of the linear terms by using Bernoulli numbers. Moreover, we also prove that $C_{i}(q)\in \mathbb{Q}[E_2,E_4,E_6]$, where $E_2$, $E_4$ and $E_6$ are the classical Eisenstein series of weight 2, 4 and 6, respectively.