Related papers: Analytical calculation of the solid angle defined …
We present an approach for the study and design of reflectors with rotational or translational symmetry that redirect light from a point source into any desired radiant intensity distribution. This method is based on a simple conformal map…
We prove a simple relation for a special case of Carlson's elliptic integral $R_D$. The findings are applied to derive explicit formulae for the asymptotics of certain moments of the angular central Gaussian distribution in terms of the…
The study of vortex flows in the vicinity of multiple solid obstacles is of considerable theoretical interest and practical importance. In particular, the case of flows past a circular cylinder placed above a plane wall has attracted a lot…
The electrostatics problem of a point charge next to a conducting plane is best solved by placing an image charge placed on the opposite side. For a charge between two parallel planes this can be solved with image charges outside the planes…
Several projects in radioastronomy plan to use large static cylindrical reflectors with an extended lobe sampling a sector of the rotating sky. This study provides the exact mathematical expression of the transit time of a celestial object…
The polygonal distributions are a class of distributions that can be defined via the mixture of triangular distributions over the unit interval. The class includes the uniform and trapezoidal distributions, and is an alternative to the beta…
Here we present a comprehensive derivation of the analytical expression for the spatiotemporal acoustic pressure generated by photoacoustic sources with spherically symmetric initial pressure distributions. Starting from the fundamental…
We give a new perspective on the Lorentzian OPE inversion formula of arXiv:1703.00278, building on arXiv:2302.06469. We introduce an ``auxiliary'' fourpoint function that can be related to the traditionally defined ones via a Radon…
A cyclic random motion at finite velocity with orthogonal directions is considered in the plane and in $\mathbb{R}^3$. We obtain in both cases the explicit conditional distributions of the position of the moving particle when the number of…
Segregation is a popular phenomenon. It has considerable effects on material performance. To the author's knowledge, there is still no automated objective quantitative indicator for segregation. In order to full fill this task, segregation…
Square-tiled surfaces can be classified by their number of squares and their cylinder diagrams (also called realizable separatrix diagrams). For the case of $n$ squares and two cone points with angle $4 \pi$ each, we set up and parametrize…
We consider a square expanding with constant speed seen from an observer moving away with constant acceleration and study the distribution of angles between rays from the observer towards the lattice points in the square. We prove the…
We examine a class of geometric theorems on cyclic 2n-gons. We prove that if we take n disjoint pairs of sides, each pair separated by an even number of polygon sides, then there is a linear combination of the angles between those sides…
If we label the vertices of a triangle with 1, 2 and 4, and the orthocentre with 7, then any of the four numbers 1, 2, 4, 7 is the nim-sum of the other three and is their orthocentre. Regard the triangle as an orthocentric quadrangle.…
We provide a robust and general algorithm for computing distribution functions associated to induced orthogonal polynomial measures. We leverage several tools for orthogonal polynomials to provide a spectrally-accurate method for a broad…
The formula for the dihedral angle of the simplex of n dimensions, arccos(1/n), is derived using classical geometry.
This article presents and compares four approaches for computing the rotation of a point about an axis by an angle in $\mathbb{R}^3$. We illustrate these methods by computing, by hand, the rotation of point $P=(1,0,1)^T$ about axis…
In the paper we consider the Maxwell operator in a three-dimensional cylinder with coefficients periodic along the axis of a cylinder. It is proved that for cylinders with circular and rectangular cross-section the spectrum of the Maxwell…
Consider a random $d$-dimensional simplex whose vertices are $d+1$ random points sampled independently and uniformly from the unit sphere in $\mathbb R^d$. We show that the expected sum of solid angles at the vertices of this random simplex…
We consider the solvability of the direct scattering problem of an obliquely incident time-harmonic electromagnetic wave by a piecewise constant inhomogeneous, penetrable and infinitely long cylinder. We prove the existence and uniqueness…