Search results for "FOS: Mathematics"
showing 10 items of 1448 documents
Metric Lie groups admitting dilations
2019
We consider left-invariant distances $d$ on a Lie group $G$ with the property that there exists a multiplicative one-parameter group of Lie automorphisms $(0, \infty)\rightarrow\mathtt{Aut}(G)$, $\lambda\mapsto\delta_\lambda$, so that $ d(\delta_\lambda x,\delta_\lambda y) = \lambda d(x,y)$, for all $x,y\in G$ and all $\lambda>0$. First, we show that all such distances are admissible, that is, they induce the manifold topology. Second, we characterize multiplicative one-parameter groups of Lie automorphisms that are dilations for some left-invariant distance in terms of algebraic properties of their infinitesimal generator. Third, we show that an admissible left-invariant distance on a Lie …
The probability that $x$ and $y$ commute in a compact group
2010
We show that a compact group $G$ has finite conjugacy classes, i.e., is an FC-group if and only if its center $Z(G)$ is open if and only if its commutator subgroup $G'$ is finite. Let $d(G)$ denote the Haar measure of the set of all pairs $(x,y)$ in $G \times G$ for which $[x,y] = 1$; this, formally, is the probability that two randomly picked elements commute. We prove that $d(G)$ is always rational and that it is positive if and only if $G$ is an extension of an FC-group by a finite group. This entails that $G$ is abelian by finite. The proofs involve measure theory, transformation groups, Lie theory of arbitrary compact groups, and representation theory of compact groups. Examples and re…
Rational normal curves and Hadamard products
2021
AbstractGiven $$r>n$$ r > n general hyperplanes in $$\mathbb P^n,$$ P n , a star configuration of points is the set of all the n-wise intersection of the hyperplanes. We introduce contact star configurations, which are star configurations where all the hyperplanes are osculating to the same rational normal curve. In this paper, we find a relation between this construction and Hadamard products of linear varieties. Moreover, we study the union of contact star configurations on a same conic in $$\mathbb P^2$$ P 2 , we prove that the union of two contact star configurations has a special h-vector and, in some cases, this is a complete intersection.
On critical behaviour in systems of Hamiltonian partial differential equations
2013
Abstract We study the critical behaviour of solutions to weakly dispersive Hamiltonian systems considered as perturbations of elliptic and hyperbolic systems of hydrodynamic type with two components. We argue that near the critical point of gradient catastrophe of the dispersionless system, the solutions to a suitable initial value problem for the perturbed equations are approximately described by particular solutions to the Painlevé-I (P $$_I$$ I ) equation or its fourth-order analogue P $$_I^2$$ I 2 . As concrete examples, we discuss nonlinear Schrödinger equations in the semiclassical limit. A numerical study of these cases provides strong evidence in support of the conjecture.
Hardy spaces and quasiconformal maps in the Heisenberg group
2023
We define Hardy spaces $H^p$, $00$ such that every $K$-quasiconformal map $f:B \to f(B) \subset \mathbb{H}^1$ belongs to $H^p$ for all $0<p<p_0(K)$. Second, we give two equivalent conditions for the $H^p$ membership of a quasiconformal map $f$, one in terms of the radial limits of $f$, and one using a nontangential maximal function of $f$. As an application, we characterize Carleson measures on $B$ via integral inequalities for quasiconformal mappings on $B$ and their radial limits. Our paper thus extends results by Astala and Koskela, Jerison and Weitsman, Nolder, and Zinsmeister, from $\mathbb{R}^n$ to $\mathbb{H}^1$. A crucial difference between the proofs in $\mathbb{R}^n$ and $\mathbb{…
$n$-harmonic coordinates and the regularity of conformal mappings
2014
This article studies the smoothness of conformal mappings between two Riemannian manifolds whose metric tensors have limited regularity. We show that any bi-Lipschitz conformal mapping or $1$-quasiregular mapping between two manifolds with $C^r$ metric tensors ($r > 1$) is a $C^{r+1}$ conformal (local) diffeomorphism. This result was proved in [12, 27, 33], but we give a new proof of this fact. The proof is based on $n$-harmonic coordinates, a generalization of the standard harmonic coordinates that is particularly suited to studying conformal mappings. We establish the existence of a $p$-harmonic coordinate system for $1 < p < \infty$ on any Riemannian manifold.
Conformality and $Q$-harmonicity in sub-Riemannian manifolds
2016
We prove the equivalence of several natural notions of conformal maps between sub-Riemannian manifolds. Our main contribution is in the setting of those manifolds that support a suitable regularity theory for subelliptic $p$-Laplacian operators. For such manifolds we prove a Liouville-type theorem, i.e., 1-quasiconformal maps are smooth. In particular, we prove that contact manifolds support the suitable regularity. The main new technical tools are a sub-Riemannian version of p-harmonic coordinates and a technique of propagation of regularity from horizontal layers.
Spatially localized solutions of the Hammerstein equation with sigmoid type of nonlinearity
2016
Abstract We study the existence of fixed points to a parameterized Hammerstein operator H β , β ∈ ( 0 , ∞ ] , with sigmoid type of nonlinearity. The parameter β ∞ indicates the steepness of the slope of a nonlinear smooth sigmoid function and the limit case β = ∞ corresponds to a discontinuous unit step function. We prove that spatially localized solutions to the fixed point problem for large β exist and can be approximated by the fixed points of H ∞ . These results are of a high importance in biological applications where one often approximates the smooth sigmoid by discontinuous unit step function. Moreover, in order to achieve even better approximation than a solution of the limit proble…
A radiation condition for the 2-D Helmholtz equation in stratified media
2009
We study the 2-D Helmholtz equation in perturbed stratified media, allowing the existence of guided waves. Our assumptions on the perturbing and source terms are not too restrictive. We prove two results. Firstly, we introduce a Sommerfeld-Rellich radiation condition and prove the uniqueness of the solution for the studied equation. Then, by careful asymptotic estimates, we prove the existence of a bounded solution satisfying our radiation condition.
A spectral approach to a constrained optimization problem for the Helmholtz equation in unbounded domains
2014
We study some convergence issues for a recent approach to the problem of transparent boundary conditions for the Helmholtz equation in unbounded domains (Ciraolo et al. in J Comput Phys 246:78–95, 2013) where the index of refraction is not required to be constant at infinity. The approach is based on the minimization of an integral functional, which arises from an integral formulation of the radiation condition at infinity. In this paper, we implement a Fourier–Chebyshev collocation method to study some convergence properties of the numerical algorithm; in particular, we give numerical evidence of some convergence estimates available in the literature (Ciraolo in Helmholtz equation in unbou…