Regular characters of classical groups over complete discrete valuation rings

Let $\mathfrak{o}$ be a complete discrete valuation ring with finite residue field $\mathsf{k}$ of odd characteristic, and let G be a symplectic or special orthogonal group scheme over $\mathfrak{o}$. For any $?\in \mathbb{N}$ let $G^{?}$ denote the ?-th principal congruence subgroup of $\mathbf{G}(\mathfrak{o})$. An irreducible character of the group $\mathbf{G}(\mathfrak{o})$ is said to be regular if it is trivial on a subgroup $G^{?+1}$ for some ?, and if its restriction to $G^{?}/G^{?+1}?\mathrm{Lie}(\mathbf{G})(\mathsf{k})$ consists of characters of minimal $\mathbf{G}({\mathsf{k}}^{\mathrm{alg}})$-stabilizer dimension. In the present paper we consider the regular characters of such classical groups over $\mathfrak{o}$, and construct and enumerate all regular characters of $\mathbf{G}(\mathfrak{o})$, when the characteristic of $\mathsf{k}$ is greater than two. As a result, we compute the regular part of their representation zeta function.     

Moser–Trudinger inequality involving the anisotropic Dirichlet norm (∫ΩFN(∇u)dx)1N on W01,N(Ω)

We investigate a sharp Moser–Trudinger inequality which involves the anisotropic Dirichlet norm ${({\int }_{\mathrm{\Omega }}{F}^N(\mathrm{?}u)dx)}^{\frac{1}{N}}$ on $W_0^{1,N}(\mathrm{\Omega })$ for $N\ge 2$. Here F is convex and homogeneous of degree 1, and its polar $F^o$ represents a Finsler metric on ${\mathbb{R}}^N$. Under this anisotropic Dirichlet norm, we establish the Lions type concentration-compactness alternative. Then by using a blow-up procedure, we obtain the existence of extremal functions for this sharp geometric inequality.     

Positive solutions for a class of singular quasilinear Schrödinger equations with critical Sobolev exponent

We prove the existence of positive solutions of the following singular quasilinear Schrödinger equations at critical growth

$?\mathrm{\Delta }u?\lambda c(x)u?\kappa \alpha (\mathrm{\Delta }(|u{|}^{2\alpha }))|u{|}^{2\alpha ?2}u=|u{|}^{q?2}u+|u{|}^{2^{?}?2}u,u\in D^{1,2}({\mathbb{R}}^N),$

via variational methods, where $\lambda \ge 0$, $c:{\mathbb{R}}^N\to {\mathbb{R}}^{+}$, $\kappa >0$, $0<\alpha <1/2$, $2<q<2^{?}$. It is interesting that we do not need to add a weight function to control $|u{|}^{q?2}u$.     

Classification of metabelian 2-groups G with Gab ≃ (2,2n),n ≥ 2, and rank d(G′)=2; Applications to real quadratic number fields

We characterize all finite metabelian 2-groups G whose abelianizations $G^{\mathrm{ab}}$ are of type $(2,2^n)$, with $n\ge 2$, and for which their commutator subgroups $G^{\prime }$ have $\text{rank}=2$. This is given in terms of the order of the abelianizations of the maximal subgroups and the structure of the abelianizations of those normal subgroups of index 4 in G. We then translate these group theoretic properties to give a characterization of number fields k with 2-class group ${\mathrm{Cl}}_2(k)?(2,2^n)$, $n\ge 2$, such that the rank of ${\mathrm{Cl}}_2(k^1)=2$ where $k^1$ is the Hilbert 2-class field of k. In particular, we apply all this to real quadratic number fields whose discriminants are a sum of two squares.     

Finite computable dimension and degrees of categoricity

We first give an example of a rigid structure of computable dimension 2 such that the unique isomorphism between two non-computably isomorphic computable copies has Turing degree strictly below ${\mathbf{0}}^{″}$, and not above ${\mathbf{0}}^{\prime }$. This gives a first example of a computable structure with a degree of categoricity that does not belong to an interval of the form $[{\mathbf{0}}^{(\alpha )},{\mathbf{0}}^{(\alpha +1)}]$ for any computable ordinal α. We then extend the technique to produce a rigid structure of computable dimension 3 such that if ${\mathbf{d}}_0$, ${\mathbf{d}}_1$, and ${\mathbf{d}}_2$ are the degrees of isomorphisms between distinct representatives of the three computable equivalence classes, then each ${\mathbf{d}}_i<{\mathbf{d}}_0\oplus {\mathbf{d}}_1\oplus {\mathbf{d}}_2$. The resulting structure is an example of a structure that has a degree of categoricity, but not strongly.     

Existence of nontrivial solutions for modified nonlinear fourth-order elliptic equations with indefinite potential

In this paper, we investigate the following modified nonlinear fourth-order elliptic equations$\{\begin{array}{cc}{\mathrm{\Delta }}^{2}u?\mathrm{\Delta }u+V(x)u?\frac{1}{2}u\mathrm{\Delta }(u^2)=g(u),\hfill & \text{in}{\mathbb{R}}^N,\hfill \\ u\in H^2({\mathbb{R}}^N)\hfill \end{array}$ where ${\mathrm{\Delta }}^2=\mathrm{\Delta }(\mathrm{\Delta })$ is the biharmonic operator, V is an indefinite potential, g grows subcritically and satisfies the Ambrosetti-Rabinowitz type condition $g(t)t\ge \mu G(t)\ge 0$ with $\mu >3$. Using Morse theory, we obtain nontrivial solutions of the above equations. Our result complements recent results in [17], where g has to be 3-superlinear at infinity.     

A note on the distribution of weights of fixed-rank matrices over the binary field

Let M be a random $m\times n$ rank-r matrix over the binary field ${\mathbb{F}}_2$, and let $\mathrm{wt}(\mathbf{M})$ be its Hamming weight, that is, the number of nonzero entries of M.We prove that, as $m,n\to +\infty$ with r fixed and $m/n$ tending to a constant, we have that$\frac{\mathrm{wt}(\mathbf{M})-\frac{1-2^{-r}}{2}mn}{\sqrt{\frac{2^{-r}(1-2^{-r})}{4}(m+n)mn}}$ converges in distribution to a standard normal random variable.     

SL(m,C)-equivariant and translation covariant continuous tensor valuations

The space of continuous, $\mathrm{SL}(m,\mathbb{C})$-equivariant, $m\ge 2$, and translation covariant valuations taking values in the space of real symmetric tensors on ${\mathbb{C}}^m?{\mathbb{R}}^{2m}$ of rank $r\ge 0$ is completely described. The classification involves the moment tensor valuation for $r\ge 1$ and is analogous to the known classification of the corresponding tensor valuations that are $\mathrm{SL}(2m,\mathbb{R})$-equivariant, although the method of proof cannot be adapted.     

An improvement to the Hilton-Zhao vertex-splitting conjecture

For a simple graph G, denote by n, $\mathrm{\Delta }(G)$, and ${\chi }^{\prime }(G)$ its order, maximum degree, and chromatic index, respectively. A graph G is edge-chromatic critical if ${\chi }^{\prime }(G)=\mathrm{\Delta }(G)+1$ and ${\chi }^{\prime }(H)<{\chi }^{\prime }(G)$ for every proper subgraph H of G. Let G be an n-vertex connected regular class 1 graph, and let $G^{?}$ be obtained from G by splitting one vertex of G into two vertices. Hilton and Zhao in 1997 conjectured that $G^{?}$ must be edge-chromatic critical if $\mathrm{\Delta }(G)>n/3$, and they verified this when $\mathrm{\Delta }(G)\ge \frac{n}{2}(\sqrt{7}?1)\approx 0.82n$. In this paper, we prove it for $\mathrm{\Delta }(G)\ge 0.75n$.     

Long-time behavior for 3D MHD equations with nonlinear damping

The existence of global attractors is proved for the MHD equations with damping terms $|u{|}^{\alpha ?1}u$ and $|B{|}^{\beta ?1}B$ $(\alpha ,\beta ?1)$ on a bounded domain $\mathrm{\Omega }?{\mathbb{R}}^3$. First we establish the well-posedness of strong solutions. Then, the continuity of the corresponding semigroup is verified under the assumption $\alpha ,\beta <5$, which is guided by Gagliardo-Nirenberg inequality. Finally, the system is shown to possess an $(\mathbb{V},\mathbb{V})$-global attractor and an $(\mathbb{V},{\mathbf{\text{H}}}^2)$-global attractor.