Related papers: Caustics by Refraction of Circles and Lines
Mathematical diffraction theory is concerned with the analysis of the diffraction image of a given structure and the corresponding inverse problem of structure determination. In recent years, the understanding of systems with continuous and…
The aim of this work is to continue the analysis, started in arXiv:2105.02108, of the dynamics of a point-mass particle $P$ moving in a galaxy with an harmonic biaxial core, in whose center sits a Keplerian attractive center (e.g. a Black…
The partial derivatives and Laplacians of the Zernike circle polynomials occur in various places in the literature on computational optics. In a number of cases, the expansion of these derivatives and Laplacians in the circle polynomials…
The late infall of dark matter onto a galaxy produces structure (such as caustics) in the distribution of dark matter in the halo. We argue that such structure is likely to occur generically on length scales proportional to $l \sim t_0…
The late infall of cold dark matter onto an isolated galaxy, such as our own, produces streams and caustics in its halo. The outer caustics are topological spheres whereas the inner caustics are rings. The self-similar model of galactic…
Percolation refers to an interesting class of problems related to the properties of disordered systems, usually formulated in terms of objects randomly placed on an underlying lattice or continuum. Despite the simplicity of the setup, most…
These lecture notes present a computation driven pathway from classical complex analysis to the theory of compact Riemann surfaces and their connections to algebraic geometry. The exposition follows a compute first then abstract philosophy,…
In this paper we relate some classical normal forms for complex elliptic curves in terms of 4-point sets in the Riemann sphere. Our main result is an alternative proof that every elliptic curve is isomorphic as a Riemann surface to one in…
We prove that any given function can be smoothly approximated by functions lying in the kernel of a linear operator involving at least one fractional component. The setting in which we work is very general, since it takes into account…
We prove that the topological recursion formalism can be used to quantize any generic classical spectral curve with smooth ramification points and simply ramified away from poles. For this purpose, we build both the associated quantum…
The process of gravitational collapse excites the fields propagating in the background geometry and gives rise to thermal radiation. We demonstrate by explicit calculations that the density matrix corresponding to such radiation actually…
A proposal of an algebraic model for the relation between a quantum environment and certain classical particle system is given. The quantum environment is described by a category of possible quantum states, the initial particle system is…
In this paper, we revisit foundations of umbral calculus using a straightforward approach based on an explicit matrix realization of binomial convolution. We construct an umbral duality of Wronskian type for rational curves in echelon form,…
We discuss the mechanism through which classicalization may occur during the collapse of a spherical field configuration modelled as a wavepacket. We demonstrate that the phenomenon is associated with the dynamical change of the equation of…
An extension of the Gutzwiller trace formula is given that includes diffraction effects due to hard wall scatterers or other singularities. The new trace formula involves periodic orbits which have arcs on the surface of singularity and…
Calculations are presented detailing the gravitational lens diffraction due to the steep brightness gradient of the limb of a stellar source. The lensing case studied is the fold caustic crossing. The limb diffraction signal greatly exceeds…
Contraction in Wasserstein 1-distance with explicit rates is established for generalized Hamiltonian Monte Carlo with stochastic gradients under possibly nonconvex conditions. The algorithms considered include splitting schemes of kinetic…
Designing freeform surfaces to control light based on real-world illumination patterns is challenging, as existing caustic lens designs often assume oversimplified point or parallel light sources. We propose representing surface light…
Shape inference is classically ill-posed, because it involves a map from the (2D) image domain to the (3D) world. Standard approaches regularize this problem by either assuming a prior on lighting and rendering or restricting the domain,…
After Galvez, Martinez and Milan discovered a (Weierstrass-type) holomorphic representation formula for flat surfaces in hyperbolic 3-space, the first, third and fourth authors here gave a framework for complete flat fronts with…