Related papers: Stable orbital integrals for classical Lie algebra…

We prove a myriad of results related to the stabilizer in an algebraic group $G$ of a generic vector in a representation $V$ of $G$ over an algebraically closed field $k$. Our results are on the level of group schemes, which carries more…

In a bounded domain $\Omega$, we consider a positive solution of the problem $\Delta u+f(u)=0$ in $\Omega$, $u=0$ on $\partial\Omega$, where $f:\mathbb{R}\to\mathbb{R}$ is a locally Lipschitz continuous function. Under sufficient conditions…

A classical result due to Frank and Seiringer asserts that for $1\leq p<\frac Ns$, there exists a sharp constant $\mathcal{C}_{N,s,p}>0$ such that $$…

We consider delay differential equations of the form $ y'(t)=-ay(t)+bf(y(t-1)) $ with positive parameters $a,b$ and a unimodal $f:[0,\infty)\to [0,1]$. It is assumed that the nonlinear $f$ is close to a function $g:[0,\infty)\to [0,1]$ with…

For a field K, rational function phi in K(z) of degree at least two, and alpha in P^1(K), we study the polynomials in K[z] whose roots are given by the solutions to phi^n(z) = alpha, where phi^n denotes the nth iterate of phi. When the…

We study stable solutions to fractional semilinear equations $(-\Delta)^s u = f(u)$ in $\Omega \subset \mathbb{R}^n$, for convex nonlinearities $f$, and under the Dirichlet exterior condition $u=g$ in $\mathbb{R}^n \setminus \Omega$ with…

Let G_R be a Lie group acting on an oriented manifold M, and let $\omega$ be an equivariantly closed form on M. If both G_R and M are compact, then the integral $\int_M \omega$ is given by the fixed point integral localization formula…

We give the stabilisation of local orbital integrals and the trace formula over a global field for $SL(2)$ with proofs valid in any characteristic. New features appear in characteristic 2. We obtain, via the stabilisation, an asymptotic…

A $(\gamma,n)$-gonal pair is a pair $(S,f)$, where $S$ is a closed Riemann surface and $f:S \to R$ is a degree $n$ holomorphic map onto a closed Riemann surface $R$ of genus $\gamma$. If the signature of $(S,f)$ is of hyperbolic type, then…

Let $G$ be a real semisimple Lie group with finite center and let $\mathfrak g=\mathfrak k \oplus \mathfrak p$ be a Cartan decomposition of its Lie algebra. Let $K$ be a maximal compact subgroup of $G$ with Lie algebra $\mathfrak k$ and let…

In this note, we prove that if $(G,V)$ is a prehomogeneous vector space over any field $k$ such that the stabilizer of a generic point is reductive, the set of semi-stable points is a single orbit over the separable closure of $k$.

We introduce a new flatness index for the boundary of an open subset $\Omega$ of $\mathbb{R}^n$, $n\ge 2$. This index provides a necessary condition for $\partial\Omega$ to be a harmonic pseudosphere and sufficient conditions for a harmonic…

We prove the existence of the set of ground states in a suitable energy space $\Sigma^s=\{u: \int_{\mathbb{R}^N} \bar{u}(-\Delta+m^2)^s u+V |u|^2<\infty\}$, $s\in (0,\frac{N}{2})$ for the mass-subcritical nonlinear fractional Hartree…

We consider solving the surface Helmholtz equation on a smooth two dimensional surface embedded into a three dimensional space meshed with tetrahedra. The mesh does not respect the surface and thus the surface cuts through the elements. We…

We study the existence of homoclic solutions for reversible Hamiltonian systems taking the family of differential equations u^4+au^2-u+f(u,b)=0 as a model. Here f is an analytic function and a, b real parameters. These equations are…

Let $\mathfrak{h}_3$ be the Heisenberg algebra and let $\mathfrak g$ be the 3-dimensional Lie algebra having $[e_1,e_2]=e_1\,(=-[e_2,e_1])$ as its only non-zero commutation relations. We describe the closure of the orbit of a vector of…

A time-fractional Fokker-Planck initial-boundary value problem is considered, with differential operator $u_t-\nabla\cdot(\partial_t^{1-\alpha}\kappa_\alpha\nabla u-\textbf{F}\partial_t^{1-\alpha}u)$, where $0<\alpha <1$. The forcing…

For an algebraic group $Q$ with $\mathsf{Lie\,} Q=\mathfrak q$, we develop a method for estimating the index of a subalgebra $\mathfrak h$ in $\mathfrak q$ via the use of coadjoint $Q$-orbits in $\mathfrak q^*$. Let $\mathfrak q^\xi$ denote…

We consider functionals of the form $$\mathcal{F}(u):=\int_\Omega\!F(x,u,\nabla u)\,\mathrm{d} x,$$ where $\Omega\subseteq\mathbb{R}^n$ is open and bounded. The integrand $F\colon\Omega\times\mathbb{R}^N\times\mathbb{R}^{N\times…

This is a sequel to the paper [Cas]. Here, we extend the methods of Farb-Wolfson using the theory of FI_G-modules to obtain stability of equivariant Galois representations of the etale cohomology of orbit configuration spaces. We establish…