We explore various 4d Yang-Mills gauge theories (YM) living as boundary conditions of 5d gapped short/long-range entangled (SRE/LRE) topological states. Specifically, we explore 4d time-reversal symmetric pure YM of an SU(2) gauge group with a second-Chern-class topological term at θ=π (SU(2)θ=π YM). Its higher 't Hooft anomalies of generalized global symmetries indicate that the 4d SU(2)θ=π YM, in order to realize all global symmetries locally, necessarily couples to a 5d higher symmetry-protected topological state (SPTs, as an invertible TQFT, or as a 5d 1-form-center-symmetry-protected interacting "topological superconductor" in condensed matter). We revisit the 4d SU(2)θ=π YM-5d SRE-higher-SPTs coupled systems in [arXiv:1812.11968] and find their "Fantastic Four Siblings" with four sets of new higher anomalies associated with the Kramers singlet/doublet and bosonic/fermionic properties of Wilson lines. Following Weyl's gauge principle, by dynamically gauging the 1-form center symmetry, we transform a 5d bulk SRE SPTs into an LRE symmetry-enriched topologically ordered state (SETs); thus we obtain the 4d SO(3)θ=π YM-5d LRE-higher-SETs coupled system with dynamical higher-form gauge fields. Apply the tool introduced in [arXiv:1612.09298], we derive new exotic anyonic statistics of extended objects such as 2-worldsheet of strings and 3-worldvolume of branes, which physically characterize the 5d SETs. We discover new triple and quadruple link invariants potentially associated with the underlying 5d higher-gauge TQFTs, hinting a new intrinsic relation between non-supersymmetric 4d pure YM and topological links in 5d. We provide lattice simplicial complex regularizations and "condensed matter" realizations.

Let $K/k$ be a finite Galois extension and $\pi = \fn{Gal}(K/k)$. An algebraic torus $T$ defined over $k$ is called a $\pi$-torus if $T\times_{\fn{Spec}(k)} \fn{Spec}(K)\simeq \bm{G}_{m,K}^n$ for some integer $n$. The set of all algebraic $\pi$-tori defined over $k$ under the stably birational equivalence forms a semigroup, denoted by $T(\pi)$. We will give a complete proof of the following theorem due to Endo and Miyata \cite{EM4}. Theorem. Let $\pi$ be a finite group. Then $T(\pi)\simeq C(\Omega_{\bm{Z}\pi})$ where $\Omega_{\bm{Z}\pi}$ is a maximal $\bm{Z}$-order in $\bm{Q}\pi$ containing $\bm{Z}\pi$ and $C(\Omega_{\bm{Z}\pi})$ is the locally free class group of $\Omega_{\bm{Z}\pi}$, provided that $\pi$ is isomorphic to one of the following four types of groups : $C_n$ ($n$ is any positive integer), $D_m$ ($m$ is any odd integer $\ge 3$), $C_{q^f}\times D_m$ ($m$ is any odd integer $\ge 3$, $q$ is an odd prime number not dividing $m$, $f\ge 1$, and $(\bm{Z}/q^f\bm{Z})^{\times}=\langle \bar{p}\rangle$ for any prime divisor $p$ of $m$), $Q_{4m}$ ($m$ is any odd integer $\ge 3$, $p\equiv 3 \pmod{4}$ for any prime divisor $p$ of $m$).

We analyse the spectrum of additive finite-rank deformations of $N \times N$ Wigner matrices $H$. The spectrum of the deformed matrix undergoes a transition, associated with the creation or annihilation of an outlier, when an eigenvalue $d_i$ of the deformation crosses a critical value $\pm 1$. This transition happens on the scale $|d_i| - 1 \sim N^{-1/3}$. We allow the eigenvalues $d_i$ of the deformation to depend on $N$ under the condition $|\abs{d_i} - 1| \geq (\log N)^{C \log \log N} N^{-1/3}$. We make no assumptions on the eigenvectors of the deformation. In the limit $N \to \infty$, we identify the law of the outliers and prove that the non-outliers close to the spectral edge have a universal distribution coinciding with that of the extremal eigenvalues of a Gaussian matrix ensemble. A key ingredient in our proof is the \emph{isotropic local semicircle law}, which establishes optimal high-probability bounds on the quantity $< v,[(H - z)^{-1} - m(z) 1] w >$, where $m(z)$ is the Stieltjes transform of Wigner's semicircle law and $v, w$ are arbitrary deterministic vectors.

For decades, the widely used finite difference method on staggered grids, also known as the marker and cell (MAC) method, has been one of the simplest and most effective numerical schemes for solving the Stokes equations and Navier–Stokes equations. Its superconvergence on uniform meshes has been observed by Nicolaides (SIAM J Numer Anal 29(6):1579–1591, 1992), but the rigorous proof is never given. Its behavior on non-uniform grids is not well studied, since most publications only consider uniform grids. In this work, we develop the MAC scheme on non-uniform rectangular meshes, and for the first time we theoretically prove that the superconvergence phenomenon (i.e., second order convergence in the L2 norm for both velocity and pressure) holds true for the MAC method on non-uniform rectangular meshes. With a careful and accurate analysis of various sources of errors, we observe that even though the local truncation errors are only first order in terms of mesh size, the global errors after summation are second order due to the amazing cancellation of local errors. This observation leads to the elegant superconvergence analysis even with non-uniform meshes. Numerical results are given to verify our theoretical analysis.

This paper deals with the derivation and analysis of the the Hall Magneto-Hydrodynamic equations. We first provide a derivation of this system from a two-fluids Euler-Maxwell system for electrons and ions, through a set of scaling limits. We also propose a kinetic formulation for the Hall-MHD equations which contains as fluid closure different variants of the Hall-MHD model. Then, we prove the existence of global weak solutions for the incompressible viscous resistive Hall-MHD model. We use the particular structure of the Hall term which has zero contribution to the energy identity. Finally, we discuss particular solutions in the form of axisymmetric purely swirling magnetic fields and propose some regularization of the Hall equation.