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For $G$ a simple, simply-connected complex algebraic group and two dominant integral weights $\lambda, \mu$, we consider the dimensions of weight spaces $V_\lambda(\mu)$ of weight $\mu$ in the irreducible, finite-dimensional highest weight $\lambda$ representation. For natural numbers $N$, the function $N \mapsto \dim V_{N\lambda}(N\mu)$ is quasi-polynomial in $N$, the stretched Kostka quasi-polynomial. Using methods of geometric invariant theory (GIT), we compute the degree of this quasi-polynomial, resolving a conjecture of Gao and Gao. We also discuss periods of this quasi-polynomial determined by the GIT approach, and give computational evidence supporting a geometric determination of the minimal period., Comment: 13 pages

For an untwisted affine Kac-Moody Lie algebra $\mathfrak{g}$ with Cartan and Borel subalgebras $\mathfrak{h} \subset \mathfrak{b} \subset \mathfrak{g}$, affine Demazure modules are certain $U(\mathfrak{b})$-submodules of the irreducible highest-weight representations of $\mathfrak{g}$. We introduce here the associated affine Demazure weight polytopes, given by the convex hull of the $\mathfrak{h}$-weights of such a module. Using methods of geometric invariant theory, we determine inequalities which define these polytopes; these inequalities come in three distinct flavors, specified by the standard, opposite, or semi-infinite Bruhat orders. We also give a combinatorial characterization of the vertices of these polytopes lying on an arbitrary face, utilizing the more general class of twisted Bruhat orders., Comment: 36 pages

A presumed impact of global climate change is the increase in frequency and intensity of tropical cyclones. Due to the possible destruction that occurs when tropical cyclones make landfall, understanding their formation should be of mass interest. In 2017, Kerry Emanuel modeled tropical cyclone formation by developing a low-dimensional dynamical system which couples tangential wind speed of the eye-wall with the inner-core moisture. For physically relevant parameters, this dynamical system always contains three fixed points: a stable fixed point at the origin corresponding to a non-storm state, an additional asymptotically stable fixed point corresponding to a stable storm state, and a saddle corresponding to an unstable storm state. The goal of this work is to provide insight into the underlying mechanisms that govern the formation and suppression of tropical cyclones through both analytical arguments and numerical experiments. We present a case study of both rate and noise-induced tipping between the stable states, relating to the destabilization or formation of a tropical cyclone. While the stochastic system exhibits transitions both to and from the non-storm state, noise-induced tipping is more likely to form a storm, whereas rate-induced tipping is more likely to be the way a storm is destabilized, and in fact, rate-induced tipping can never lead to the formation of a storm when acting alone. For rate-induced tipping acting as a destabilizer of the storm, a striking result is that both wind shear and maximal potential velocity have to increase, at a substantial rate, in order to effect tipping away from the active hurricane state. For storm formation through noise-induced tipping, we identify a specific direction along which the non-storm state is most likely to get activated.

We determine, in an inductive framework, the vertices of the polytope $P(s,K)$ controlling the conjugacy classes of elements which product to one in the maximal compact subgroup $K$ of a simple complex algebraic group $G$. This extends earlier work of the authors in related contexts. One feature of this work is the use of Kontsevich compactifications of the moduli of $P$-bundles (replacing the use of quot schemes in type A) which are related to semi-infinite geometry. We also obtain a quantum generalization of Fulton's conjecture valid for all $G$., Comment: 58 pages

Senjanovic and Tello have analyzed how one could determine the neutrino Dirac mass matrix in the minimal left-right model, assuming that the mass matrices for the light and heavy neutrinos could be taken as inputs. They have provided an analytical solution for the Dirac mass matrix in the case that the left-right symmetry is implemented via a generalized parity symmetry and that this symmetry remains unbroken in the Dirac Yukawa sector. We extend the work of Senjanovic and Tello to the case in which the generalized parity symmetry is broken in the Dirac Yukawa sector. In this case the elegant method outlined by Senjanovic and Tello breaks down and we need to adopt a numerical approach. Several iterative approaches are described; these are found to work in some cases but to be highly unstable in others. A stable, prescriptive numerical algorithm is described that works in all but a vanishingly small number of cases. We apply this algorithm to numerical data sets that are consistent with current experimental constraints on neutrino masses and mixings. We also provide some additional context and supporting explanations for the case in which the parity symmetry is unbroken., Comment: 34 pages, 4 figures; published version

The domain of cluster analysis is a meeting point for a very rich multidisciplinary encounter, with cluster-analytic methods being studied and developed in discrete mathematics, numerical analysis, statistics, data analysis, data science, and computer science (including machine learning, data mining, and knowledge discovery), to name but a few. The other side of the coin, however, is that the domain suffers from a major accessibility problem as well as from the fact that it is rife with division across many pretty isolated islands. As a way out, the present paper offers a thorough and in-depth review of the clustering domain as a whole under the form of an outline map based on an overarching conceptual framework and a common language. With this framework we wish to contribute to structuring the clustering domain, to characterizing methods that have often been developed and studied in quite different contexts, to identifying links between methods, and to introducing a frame of reference for optimally setting up cluster analyses in data-analytic practice., Comment: 43 pages, 4 figures

For $G$ a reductive group and $T\subset B$ a maximal torus and Borel subgroup, Demazure modules are certain $B$-submodules, indexed by elements of the Weyl group, of the finite irreducible representations of $G$. In order to describe the $T$-weight spaces that appear in a Demazure module, we study the convex hull of these weights - the Demazure polytope. We characterize these polytopes both by vertices and by inequalities, and we use these results to prove that Demazure characters are saturated, in the case that $G$ is simple of classical Lie type. Specializing to $G=GL_n$, we recover results of Fink, M\'esz\'aros, and St. Dizier, and separately Fan and Guo, on key polynomials, originally conjectured by Monical, Tokcan, and Yong., Comment: 32 pages, 6 figures. v2 added references and relation to MV polytopes, and simplified proofs in section 5. v3 minor expository edits, and corrections to the relation to the geometry of MV cycles

The distribution of eigenvalues of the wave equation in a bounded domain is known as Weyl's problem. We describe several computational projects related to the cumulative state number, defined as the number of states having wavenumber up to a maximum value. This quantity and its derivative, the density of states, have important applications in nuclear physics, degenerate Fermi gases, blackbody radiation, Bose-Einstein condensation and the Casimir effect. Weyl's theorem states that, in the limit of large wavenumbers, the cumulative state number depends only on the volume of the bounding domain and not on its shape. Corrections to this behavior are well known and depend on the surface area of the bounding domain, its curvature and other features. We describe several projects that allow readers to investigate this dependence for three bounding domains - a rectangular box, a sphere, and a circular cylinder. Quasi-one- and two-dimensional systems can be analyzed by considering various limits. The projects have applications in statistical mechanics, but can also be integrated into quantum mechanics, nuclear physics, or computational physics courses., Comment: 37 pages. This article has been accepted by the American Journal of Physics. After it is published, it will be found at: https://aapt.scitation.org/journal/ajp