Related papers: Deriving Pi from Colliding Blocks
We estimate the cross section for quasi-elastic double pion exchange in high energy proton-proton collisions. Total and elastic $\pi\pi$ cross sections are calculated in an equivalent pion approximation, with pion-baryon vertices taken from…
We consider a rigid body acted upon by two forces, a constant force and the collective force of interaction with a continuum of particles. We assume that some of the particles that collide with the body reflect elastically (specularly),…
A simple way is shown to construct the length $\pi$ from the unit length with 4 digits accuracy.
We derive a formula for $p(n)$ (the number of partitions of $n$) in terms of the partial Bell polynomials using Fa\`{a} di Bruno's formula and Euler's pentagonal number theorem.
We pose and answer several questions concerning the number of ways to fold a polygon to a polytope, and how many polytopes can be obtained from one polygon; and the analogous questions for unfolding polytopes to polygons. Our answers are,…
We consider a dimer formed by two particles with an attractive contact interaction in one dimension, colliding with a hard wall. We compute the scattering phase shifts and the reflection coefficients for various collision energies and…
We give an algorithm for approximating a given plane curve segment by a planar elastic curve. The method depends on an analytic representation of the space of elastic curve segments, together with a geometric method for obtaining a good…
An $N$-dimensional parallelepiped will be called a bar if and only if there are no more than $k$ different numbers among the lengths of its sides (the definition of bar depends on $k$). We prove that a parallelepiped can be dissected into…
We present a genuinely non-radial quantum-mechanical route by which $\pi$ emerges from equatorial localization on the sphere. For the highest-weight branch of spherical harmonics, this localization is captured by a natural geometric…
In this Note, we start off with the primary representation of e and from there present an elementary short proof for the Wallis formula for $\pi$.
Particle-particle and particle-wall collisions occur in many natural and industrial applications such as sedimentation, agglomeration, and granular flows. To accurately predict the behavior of particulate flows, fundamental knowledge of the…
A construction of polytopes is given based on integers. These geometries are constructed through a mapping to pure numbers and have multiple applications, including statistical mechanics and computer science. The number form is useful in…
Adders are key building blocks of many error-tolerant applications. Leveraging the application-level error tolerance, a number of approximate adders were proposed recently. Many of them belong to the category of block-based approximate…
A graph is called (generically) rigid in $\mathbb{R}^d$ if, for any choice of sufficiently generic edge lengths, it can be embedded in $\mathbb{R}^d$ in a finite number of distinct ways, modulo rigid transformations. Here we deal with the…
In areas as diverse as contemporary art, play structures, climbing equipment, and modular construction toys, we see the presence of building block-like polyhedral complexes, which are generalizations of the pieces in the game Tetris. We…
Let $\Pi$ be a convex decomposition of a set $P$ of $n\geq 3$ points in general position in the plane. If $\Pi$ consists of more than one polygon, then either $\Pi$ contains a deletable edge or $\Pi$ contains a contractible edge.
We consider loci of points such that their sum of distances or sum of squared distances to each of the sides of a given triangle is constant. These loci are inspired by Viviani's theorem and its extension. The former locus is a line segment…
I study the contact between a rigid solid with a randomly rough surface and an elastic block with a flat surface. I derive a relation between the (average) interfacial separation $u$ and the applied normal squeezing pressure $p$. I show…
The impact of a two-dimensional elastic disk with a wall is numerically studied. It is clarified that the coefficient of restitution (COR) decreases with the impact velocity. The result is not consistent with the recent quasi-static theory…
Computing the distance function to some surface or line is a problem that occurs very frequently. There are several ways of computing a relevant approximation of this function, using for example technique originating from the approximation…