Related papers: Throwing $\pi$ at a wall
We show that the number of incidences between $m$ distinct points and $n$ distinct lines in ${\mathbb R}^4$ is $O\left(2^{c\sqrt{\log m}} (m^{2/5}n^{4/5}+m) + m^{1/2}n^{1/2}q^{1/4} + m^{2/3}n^{1/3}s^{1/3} + n\right)$, for a suitable…
For a finite poset $P$, we study the expected size of the intersection of two independent uniformly random antichains. Equivalently, we evaluate the sum of $|A\cap A'|$ over all ordered pairs of antichains. For general posets this statistic…
We consider the motion of a finite though large number of particles in the whole space R n. Particles move freely until they experience pairwise collisions. We use our recent theory of divergence-controlled positive symmetric tensors in…
In the pinwheel problem, one is given an $m$-tuple of positive integers $(a_1, \ldots, a_m)$ and asked whether the integers can be partitioned into $m$ color classes $C_1,\ldots,C_m$ such that every interval of length $a_i$ has non-empty…
A rigid object moving in a viscous fluid and in close proximity with an elastic wall experiences self-generated elastohydrodynamic interactions. This has been the subject of an intense research activity, with a recent and growing attention…
We compute the strong coupling constant $G_{B^{**} B \pi} \; (G_{D^{**} D \pi})$, where $B^{**}$ ($D^{**}$) is the $0^+$ $P-$wave $b \bar q \; (c \bar q)$ state, by QCD sum rules and by light-cone sum rules. The two methods give compatible…
The present study provides a detailed description of the forces on an ellipsoidal particle in the vicinity of the rough wall. Three-dimensional numerical simulations are performed using body-fitted mesh to estimate the drag, lift, and…
There is little known about the methods used by the ancient Babylonians and Egyptians to arrive at their recorded estimates of the value of Pi. A surprisingly accurate estimate of Pi was recently revealed coded within a verse in the book of…
We consider pyramids made of one-dimensional pieces of fixed integer length a and which may have pairwise overlaps of integer length from 1 to a. We prove that the number of pyramids of size m, i.e. consisting of m pieces, equals (am-1,m-1)…
This paper deals with the complex problem of how to simulate multiparticle contacts. The collision process is responsible for the transfer and dissipation of energy in granular media. A novel model of the interaction force between particles…
We consider a slow particle with wave function $\psi_t(\vec{x})$, moving freely in some direction. A mirror is briefly switched on around a time $T$ and its position is scanned. It is shown that the measured reflection probability then…
We model large-scale ($\approx$2000km) impacts on a Mars-like planet using a Smoothed Particle Hydrodynamics code. The effects of material strength and of using different Equations of State on the post-impact material and temperature…
The definition of the intersection number of a map with a closed manifold can be extended to the case of a closed stratified set such that the difference between dimensions of its two biggest strata is greater than $1$. The set Sigma of…
We tabulate the abscissae and associated weights for numerical integration of integrals with either the singular weight function (-log x)^m for exponents m=1, 2 or 3, or the symmetric weight function cos(pi*x/2). Standard brute force…
In this note, we obtain a formula which leads to a practical and efficient method to calculate the number of partitions of n into parts not divisible by m for given natural numbers n and m.
We report numerical simulations on granular shear flows confined between two flat but frictional sidewalls. Novel regimes differing by their strain localization features are observed. They originate from the competition between dissipation…
We give a new upper bound $K_+$ on the number of totally elastic collisions of $n$ hard spheres with equal radii and equal masses in $R^d$. Our bound satisfies $\log K_+ \leq c(d) n \log n$.
A recipe for creating materials with a desired refraction coefficient is implemented numerically. The following assumptions are used: \bee \zeta_m=h(x_m)/a^\kappa,\quad d=O(a^{(2-\kappa)/3}),\quad M=O(1/a^{2-\kappa}),\quad \kappa\in(0,1),…
To study the onset of a stationary dynamo in the presence of inner or outer walls of various electromagnetic properties, we propose a simple 1D-model in which the flow is replaced by an alpha effect. The equation of dispersion of the…
We present a simple method to calculate the binary collision rate between atoms in near-resonant optical lattices. The method is based on the Monte Carlo wave function simulations and the collision rate is obtained by monitoring the quantum…