Search results for " Mathematica"
showing 10 items of 689 documents
Plenty of big projections imply big pieces of Lipschitz graphs
2020
I prove that a closed $n$-regular set $E \subset \mathbb{R}^{d}$ with plenty of big projections has big pieces of Lipschitz graphs. This answers a question of David and Semmes.
Reciprocal lower bound on modulus of curve families in metric surfaces
2019
We prove that any metric space $X$ homeomorphic to $\mathbb{R}^2$ with locally finite Hausdorff 2-measure satisfies a reciprocal lower bound on modulus of curve families associated to a quadrilateral. More precisely, let $Q \subset X$ be a topological quadrilateral with boundary edges (in cyclic order) denoted by $\zeta_1, \zeta_2, \zeta_3, \zeta_4$ and let $\Gamma(\zeta_i, \zeta_j; Q)$ denote the family of curves in $Q$ connecting $\zeta_i$ and $\zeta_j$; then $\text{mod} \Gamma(\zeta_1, \zeta_3; Q) \text{mod} \Gamma(\zeta_2, \zeta_4; Q) \geq 1/\kappa$ for $\kappa = 2000^2\cdot (4/\pi)^2$. This answers a question concerning minimal hypotheses under which a metric space admits a quasiconfor…
Accessible parts of boundary for simply connected domains
2018
For a bounded simply connected domain $\Omega\subset\mathbb{R}^2$, any point $z\in\Omega$ and any $0<\alpha<1$, we give a lower bound for the $\alpha$-dimensional Hausdorff content of the set of points in the boundary of $\Omega$ which can be joined to $z$ by a John curve with a suitable John constant depending only on $\alpha$, in terms of the distance of $z$ to $\partial\Omega$. In fact this set in the boundary contains the intersection $\partial\Omega_z\cap\partial\Omega$ of the boundary of a John sub-domain $\Omega_z$ of $\Omega$, centered at $z$, with the boundary of $\Omega$. This may be understood as a quantitative version of a result of Makarov. This estimate is then applied to obta…
Fundamental isomorphism theorems for quantum groups
2017
The lattice of subgroups of a group is the subject of numerous results revolving around the central theme of decomposing the group into "chunks" (subquotients) that can then be compared to one another in various ways. Examples of results in this class would be the Noether isomorphism theorems, Zassenhaus' butterfly lemma, the Schreier refinement theorem for subnormal series of subgroups, the Dedekind modularity law, and last but not least the Jordan-H\"older theorem. We discuss analogues of the above-mentioned results in the context of locally compact quantum groups and linearly reductive quantum groups. The nature of the two cases is different: the former is operator algebraic and the latt…
Tailoring a pair of pants
2021
Abstract We show how to deform the map Log : ( C ⁎ ) n → R n such that the image of the complex pair of pants P ∘ ⊂ ( C ⁎ ) n is the tropical hyperplane by showing an (ambient) isotopy between P ∘ ⊂ ( C ⁎ ) n and a natural polyhedral subcomplex of the product of the two skeleta S × Σ ⊂ A × C of the amoeba A and the coamoeba C of P ∘ . This lays the groundwork for having the discriminant to be of codimension 2 in topological Strominger-Yau-Zaslow torus fibrations.
On invariant measures of finite affine type tilings
2006
In this paper, we consider tilings of the hyperbolic 2-space, built with a finite number of polygonal tiles, up to affine transformation. To such a tiling T, we associate a space of tilings: the continuous hull Omega(T) on which the affine group acts. This space Omega(T) inherits a solenoid structure whose leaves correspond to the orbits of the affine group. First we prove the finite harmonic measures of this laminated space correspond to finite invariant measures for the affine group action. Then we give a complete combinatorial description of these finite invariant measures. Finally we give examples with an arbitrary number of ergodic invariant probability measures.
A Quantitative Analysis of Metrics on Rn with Almost Constant Positive Scalar Curvature, with Applications to Fast Diffusion Flows
2017
We prove a quantitative structure theorem for metrics on $\mathbf{R}^n$ that are conformal to the flat metric, have almost constant positive scalar curvature, and cannot concentrate more than one bubble. As an application of our result, we show a quantitative rate of convergence in relative entropy for a fast diffusion equation in $\mathbf{R}^n$ related to the Yamabe flow.
Geometry and quasisymmetric parametrization of Semmes spaces
2014
We consider decomposition spaces R/G that are manifold factors and admit defining sequences consisting of cubes-with-handles. Metrics on R/G constructed via modular embeddings of R/G into Euclidean spaces promote the controlled topology to a controlled geometry. The quasisymmetric parametrizability of the metric space R/G×R by R for any m ≥ 0 imposes quantitative topological constraints, in terms of the circulation and the growth of the cubes-with-handles, to the defining sequences for R/G. We give a necessary condition and a sufficient condition for the existence of parametrization. The necessary condition answers negatively a question of Heinonen and Semmes on quasisymmetric parametrizabi…
Donnan phenomena in membranes with charge due to ion adsorption. Effects of the interaction between adsorbed charged groups
1993
A physical model for the modified Donnan phenomenon associated with ion adsorption on localized membrane sites is presented. This model accounts for the dependence of the concentration of adsorbed ions on electrolyte concentration and pH as it is influenced by the electrostatic interaction between adsorbed ions. The equilibrium thermodynamic concepts employed are based on the Donnan formalism for the ion equilibria between membrane and solution, and the Bragg–Williams approximation for an adsorption isotherm that incorported interaction between adsorbed ions. Our results include the concentration of charged groups in the membrane, the pH of the membrane phase solution, and the Donnan potent…
Dynamical Casimir-Polder force between an excited atom and a conducting wall
2016
We consider the dynamical atom-surface Casimir-Polder force in the non-equilibrium configuration of an atom near a perfectly conducting wall, initially prepared in an excited state with the field in its vacuum state. We evaluate the time-dependent Casimir-Polder force on the atom, and find that it shows an oscillatory behavior from attractive to repulsive both in time and in space. We also investigate the asymptotic behavior in time of the dynamical force and of related local field quantities, showing that the static value of the force, as obtained by a time-independent approach, is recovered for times much larger than the timescale of the atomic self-dressing, but smaller than the atomic d…