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In this paper, we use Clifford algebra $Cl_{2,0}$ to find the 2D orbit of Hydrogen electron under a Coulomb force and a perturbing circularly polarized electric field of light at angular frequency~$\omega$, which is turned on at time $t = 0$ via a unit step switch. Using a coordinate system co-rotating with the electron's unperturbed circular orbit at angular frequency $\omega_0$, we derive the complex nonlinear differential equation for the perturbation which is similar to but different from the Lorentz oscillator equation: (1) the acceleration terms are similar, (2) the damping term coefficient is not real but imaginary due to Coriolis force, (3) the term similar to spring force is not positive but negative, (3) there is a complex conjugate of the perturbation term which has no Lorentz analog but which makes the equation nonlinear, and (4) the angular frequency of the forcing term is not $\omega$ but $\omega - \omega_0$. By imposing that the position and velocity of the electron are continuous at time $t = 0$, we show that the orbit of the electron is a sum of five exponential Fourier terms with frequencies 0, $\omega_0$, $2\omega_0$, $(2\omega_0 - \omega)$, and $\omega$, which correspond to the eccentric, deferent, and three epicycles in Copernican astronomy. We show that at the three resonant light frequencies $0$, $\omega_0$, and $2\omega_0$, the electron's orbit is divergent, but approximates a Keplerian ellipse. At other light frequencies, the orbits are nondivergent with periods that are integer multiples of $\pi/\omega_0$ depending on the frequency ratio $\omega/\omega_0$. And as $\omega/\omega_0\rightarrow \pm\infty$, the orbit approaches the electron's unperturbed circular orbit., Comment: 50 pages, 200+ equations

We introduce a diagrammatic monoidal category, the spin Brauer category, that plays the same role for the spin and pin groups as the Brauer category does for the orthogonal groups. In particular, there is a full functor from the spin Brauer category to the category of finite-dimensional modules for the spin and pin groups. This functor becomes essentially surjective after passing to the Karoubi envelope, and its kernel is the tensor ideal of negligible morphisms. In this way, the spin Brauer category can be thought of as an interpolating category for the spin and pin groups. We also define an affine version of the spin Brauer category, which acts on categories of modules for the pin and spin groups via translation functors., Comment: 46 pages. v2: Changed order of Sections 7 and 8, other improvements and corrections throughout

We discuss some consequences of the invertibility of Rickard complexes in a categorified quantum group. Results include a description of reflection functors for quiver Hecke algebras and a theory of restricting categorical representations along a face., Comment: 20pp

Kang, Kashiwara, Kim and Oh have proved that cluster monomials lie in the dual canonical basis, under a symmetric type assumption. This involves constructing a monoidal categorification of a quantum cluster algebra using representations of KLR algebras. We use a folding technique to generalise their results to all Lie types., Comment: 29pp. Comments welcome

Traditional finite element approaches are well-known to introduce spurious oscillations when applied to advection-dominated problems. We explore alleviation of this issue from the perspective of a generalized finite element formulation, which enables stabilization through an enrichment process. The presented work uses solution-tailored enrichments for the numerical solution of the one-dimensional, unsteady Burgers equation. Mainly, generalizable exponential and hyperbolic tangent enrichments effectively capture local, steep boundary layer/shock features. Results show natural alleviation of oscillations and return smooth numerical solutions over coarse grids. Additionally, significantly improved error levels are observed compared to Lagrangian finite element methods., Comment: 26 pages, 34 figures

We describe an algorithm which pattern embeds, in the sense of Woo-Yong, any Bruhat interval of a symmetric group into an interval whose extremes lie in the same right Kazhdan-Lusztig cell. This apparently harmless fact has applications in finding examples of reducible associated varieties of $\mathfrak{sl}_n$-highest weight modules, as well as in the study of $W$-graphs for symmetric groups, and in comparing various bases of irreducible representations of the symmetric group or its Hecke algebra. For example, we are able to systematically produce many negative answers to a question from the 1980s of Borho-Brylinski and Joseph, which had been settled by Williamson via computer calculations only in 2014.

We give examples of non-perverse parity sheaves on Schubert varieties for all primes., Comment: 10pp. minor revisions for v4. Remains a significantly more detailed proof than the initial posting with Theorem 5.1 possibly of independent interest

Kato has constructed reflection functors for KLR algebras which categorify the braid group action on a quantum group by algebra automorphisms. We prove that these reflection functors are monoidal., Comment: 6pp

We study the question of when geometric extension algebras are polynomial quasihereditary. Our main theorem is that under certain assumptions, a geometric extension algebra is polynomial quasihereditary if and only if it arises from an even resolution. We give an application to the construction of reflection functors for quiver Hecke algebras., Comment: v3, 19pp

This paper develops the theory of KLR algebras with a Dynkin diagram automorphism. This is foundational material intended to allow folding techniques in the theory of KLR algebras., Comment: 24pp, changes in response to referee comments, main theorem ever so slightly weakened

Simple representations of KLR algebras can be used to realize the infinity crystal for the corresponding symmetrizable Kac-Moody algebra. It was recently shown that, in finite and affine types, certain sub-categories of cuspidal representations realize crystals for sub Kac-Moody algebras. Here we put that observation an a firmer categorical footing by exhibiting a functor between the category of representations of the KLR algebra for the sub Kac-Moody algebra and the category of cuspidal representations of the original KLR algebra., Comment: 26 pages, significant revisions

We studied the orbit of an electron revolving around an infinitely massive nucleus of a large classical Hydrogen atom subject to an AC electric field oscillating perpendicular to the electron's circular orbit. Using perturbation theory in geometric algebra, we show that the equation of motion of the electron perpendicular to the unperturbed orbital plane satisfies a forced simple harmonic oscillator equation found in Lorentz dispersion law in Optics. We show that even though we did not introduce a damping term, the initial orbital position and velocity of the electron results to a solution whose absorbed energies are finite at the dominant resonant frequency $\omega=\omega_0$; the electron slowly increases its amplitude of oscillation until it becomes ionized. We computed the average power absorbed by the electron both at the perturbing frequency and at the electron's orbital frequency. We graphed the trace of the angular momentum vector at different frequencies. We showed that at different perturbing frequencies, the angular momentum vector traces epicyclical patterns., Comment: 12 pages, 8 figures