Relations between two-point correlations and pressure strain terms

The structure of the two-point spatial correlations (velocity-velocity, velocity-scalar, and scalar-scalar) were studied with a view to improve turbulence closure models. The linear model for the two-point correlations proposed by Naot provides a method of including the information about the turbulence structure in the turbulence models. The assumptions and adequacy of this model were tested against the homogeneous shear flow simulation data base. The model performs poorly in some details and it is suggested how it may be improved. The models were also tested for rapid pressure-strain terms in a variety of flows including axisymmetric expansion and contraction flows, homogeneous shear flow, channel flow, and boundary layer

On Range Searching with Semialgebraic Sets II

Let $P$ be a set of $n$ points in $\R^d$. We present a linear-size data structure for answering range queries on $P$ with constant-complexity semialgebraic sets as ranges, in time close to $O(n^{1-1/d})$. It essentially matches the performance of similar structures for simplex range searching, and, for $d\ge 5$, significantly improves earlier solutions by the first two authors obtained in~1994. This almost settles a long-standing open problem in range searching. The data structure is based on the polynomial-partitioning technique of Guth and Katz [arXiv:1011.4105], which shows that for a parameter $r$, $1 < r \le n$, there exists a $d$-variate polynomial $f$ of degree $O(r^{1/d})$ such that each connected component of $\R^d\setminus Z(f)$ contains at most $n/r$ points of $P$, where $Z(f)$ is the zero set of $f$. We present an efficient randomized algorithm for computing such a polynomial partition, which is of independent interest and is likely to have additional applications

Kinetic Voronoi Diagrams and Delaunay Triangulations under Polygonal Distance Functions

Let $P$ be a set of $n$ points and $Q$ a convex $k$-gon in ${\mathbb R}^2$. We analyze in detail the topological (or discrete) changes in the structure of the Voronoi diagram and the Delaunay triangulation of $P$, under the convex distance function defined by $Q$, as the points of $P$ move along prespecified continuous trajectories. Assuming that each point of $P$ moves along an algebraic trajectory of bounded degree, we establish an upper bound of $O(k^4n\lambda_r(n))$ on the number of topological changes experienced by the diagrams throughout the motion; here $\lambda_r(n)$ is the maximum length of an $(n,r)$-Davenport-Schinzel sequence, and $r$ is a constant depending on the algebraic degree of the motion of the points. Finally, we describe an algorithm for efficiently maintaining the above structures, using the kinetic data structure (KDS) framework

Largest Placement of One Convex Polygon Inside Another

Our aim was to detect tau tangles and beta amyloid plaques in retina for the early diagnosis of Alzheimers Disease (AD)

Approximate Nearest Neighbor Search Amid Higher-Dimensional Flats

We consider the Approximate Nearest Neighbor (ANN) problem where the input set consists of n k-flats in the Euclidean Rd, for any fixed parameters k 0 is another prespecified parameter. We present an algorithm that achieves this task with n^{k+1}(log(n)/epsilon)^O(1) storage and preprocessing (where the constant of proportionality in the big-O notation depends on d), and can answer a query in O(polylog(n)) time (where the power of the logarithm depends on d and k). In particular, we need only near-quadratic storage to answer ANN queries amidst a set of n lines in any fixed-dimensional Euclidean space. As a by-product, our approach also yields an algorithm, with similar performance bounds, for answering exact nearest neighbor queries amidst k-flats with respect to any polyhedral distance function. Our results are more general, in that they also provide a tradeoff between storage and query time

Union of Hypercubes and 3D Minkowski Sums with Random Sizes

Let T={triangle_1,...,triangle_n} be a set of of n pairwise-disjoint triangles in R^3, and let B be a convex polytope in R^3 with a constant number of faces. For each i, let C_i = triangle_i oplus r_i B denote the Minkowski sum of triangle_i with a copy of B scaled by r_i>0. We show that if the scaling factors r_1, ..., r_n are chosen randomly then the expected complexity of the union of C_1, ..., C_n is O(n^{2+epsilon), for any epsilon > 0; the constant of proportionality depends on epsilon and the complexity of B. The worst-case bound can be Theta(n^3). We also consider a special case of this problem in which T is a set of points in R^3 and B is a unit cube in R^3, i.e., each C_i is a cube of side-length 2r_i. We show that if the scaling factors are chosen randomly then the expected complexity of the union of the cubes is O(n log^2 n), and it improves to O(n log n) if the scaling factors are chosen randomly from a "well-behaved" probability density function (pdf). We also extend the latter results to higher dimensions. For any fixed odd value of d, we show that the expected complexity of the union of the hypercubes is O(n^floor[d/2] log n) and the bound improves to O(n^floor[d/2]) if the scaling factors are chosen from a "well-behaved" pdf. The worst-case bounds are Theta(n^2) in R^3, and Theta(n^{ceil[d/2]}) in higher dimensions

Near-Optimal Min-Sum Motion Planning for Two Square Robots in a Polygonal Environment

Let $\mathcal{W} \subset \mathbb{R}^2$ be a planar polygonal environment (i.e., a polygon potentially with holes) with a total of $n$ vertices, and let $A,B$ be two robots, each modeled as an axis-aligned unit square, that can translate inside $\mathcal{W}$. Given source and target placements $s_A,t_A,s_B,t_B \in \mathcal{W}$ of $A$ and $B$, respectively, the goal is to compute a \emph{collision-free motion plan} $\mathbf{\pi}^*$, i.e., a motion plan that continuously moves $A$ from $s_A$ to $t_A$ and $B$ from $s_B$ to $t_B$ so that $A$ and $B$ remain inside $\mathcal{W}$ and do not collide with each other during the motion. Furthermore, if such a plan exists, then we wish to return a plan that minimizes the sum of the lengths of the paths traversed by the robots, $\left|\mathbf{\pi}^*\right|$. Given $\mathcal{W}, s_A,t_A,s_B,t_B$ and a parameter $\varepsilon > 0$, we present an $n^2\varepsilon^{-O(1)} \log n$-time $(1+\varepsilon)$-approximation algorithm for this problem. We are not aware of any polynomial time algorithm for this problem, nor do we know whether the problem is NP-Hard. Our result is the first polynomial-time $(1+\varepsilon)$-approximation algorithm for an optimal motion planning problem involving two robots moving in a polygonal environment.Comment: The conference version of the paper is accepted to SODA 202