We give an Euler-Maclaurin formula with remainder for the sum of the values of a smooth function on the integral points in a simple integral polytope. We prove this formula by elementary methods

We study the representation theory of the superconformal algebra ${\mathcal{W}}_k(\mathfrak{g},f_{\theta })$ associated with a minimal gradation of $\mathfrak{g}$. Here, $\mathfrak{g}$ is a simple finite-dimensional Lie superalgebra with a nondegenerate, even supersymmetric invariant bilinear form. Thus, ${\mathcal{W}}_k(\mathfrak{g},f_{\theta })$ can be one of the well-known superconformal algebras including the Virasoro algebra, the Bershadsky-Polyakov algebra, the Neveu-Schwarz algebra, the Bershadsky-Knizhnik algebras, the ${\displaystyle N=2}$ superconformal algebra, the ${\displaystyle N=4}$ superconformal algebra, the ${\displaystyle N=3}$ superconformal algebra, and the big ${\displaystyle N=4}$ superconformal algebra. We prove the conjecture of V. G. Kac, S.-S. Roan, and M. Wakimoto [17, Conjecture 3.1B] for ${\mathcal{W}}_k(\mathfrak{g},f_{\theta })$. In fact, we show that any irreducible highest-weight character of ${\mathcal{W}}_k(\mathfrak{g},f_{\theta })$ at any level $k\in ℂ$ is determined by the corresponding irreducible highest-weight character of the Kac-Moody affinization of $\mathfrak{g}$

Let $G$ be a compact, connected Lie group acting smoothly on a manifold $M$. In their 1998 article [7], Goresky, Kottwitz, and MacPherson described a small Cartan model for the equivariant cohomology of $M$, quasi-isomorphic to the standard (large) Cartan complex of equivariant differential forms. In this article, we construct an explicit cochain map from the small Cartan model into the large Cartan model, intertwining the $(S{\mathfrak{g}}^{*}){}_{\mathrm{inv}}$-module structures and inducing an isomorphism in cohomology. The construction involves the solution of a remarkable inhomogeneous Maurer-Cartan equation. This solution has further applications to the theory of transgression in the Weil algebra and to the Chevalley-Koszul theory of the cohomology of principal bundles

In this article, we describe a natural framework for the vortex dynamics in the complex-valued parabolic Ginzburg-Landau equation in ${\displaystyle {\mathbb{R}}^2}$. This general setting does not rely on any assumption of well-preparedness and has the advantage of being valid even after collision times. We carefully analyze collisions leading to annihilation. A new phenomenon is identified, the phase-vortex interaction, which is related to the persistence of low-frequency oscillations and leads to an unexpected drift in the motion of vortices