Search results for "ryhmäteoria"

Space of signatures as inverse limits of Carnot groups

2021

We formalize the notion of limit of an inverse system of metric spaces with 1-Lipschitz projections having unbounded fibers. The construction is applied to the sequence of free Carnot groups of fixed rank n and increasing step. In this case, the limit space is in correspondence with the space of signatures of rectifiable paths in ℝn, as introduced by Chen. Hambly-Lyons’s result on the uniqueness of signature implies that this space is a geodesic metric tree. As a particular consequence we deduce that every path in ℝn can be approximated by projections of some geodesics in some Carnot group of rank n, giving an evidence that the complexity of sub-Riemannian geodesics increases with the step.…

Local minimizers and gamma-convergence for nonlocal perimeters in Carnot groups

2020

We prove the local minimality of halfspaces in Carnot groups for a class of nonlocal functionals usually addressed as nonlocal perimeters. Moreover, in a class of Carnot groups in which the De Giorgi's rectifiability Theorem holds, we provide a lower bound for the $\Gamma$-liminf of the rescaled energy in terms of the horizontal perimeter.

Viidennen asteen yhtälön ratkaisukaavan olemassaolon mahdottomuus Galois'n teorian pohjalta

2014

Gromov-hyperboliset ryhmät

2016

Meeri Martimo, Gromov-hyperboliset ryhmät (engl. Gromov-hyperbolic groups), matematiikan pro gradu -tutkielma, 51 s., Jyväskylän yliopisto, Matematiikan ja tilastotieteen laitos, syksy 2016. Tässä tutkielmassa käsitellään Gromov-hyperbolisia ryhmiä, jotka ovat geometrisen ryhmäteorian tutkimuskohde. Geometrinen ryhmäteoria on melko uusi matematiikan suuntaus, ja 1980-luvulla Gromov-hyperboliset ryhmät kehittänyt ranskalaisvenäläinen matemaatikko Mikhail Gromov yksi sen uranuurtajista. Gromov-hyperbolisuus määritellään ensin metrisille avaruuksille tietyllä tavalla ohuiden kolmioiden avulla. Kolmiot ovat vaaditulla tavalla ohuita esimerkiksi hyperbolisissa avaruuksissa, mutta eivät reaaliaks…

Monitoring the complexities: Nuclear power and public opinion

2009

Interaction between organizations and stakeholders today takes place on virtual and physical ‘issue arenas’. This study examined opinions on nuclear power and asked who are the players discussing nuclear power in Finland? Through content analysis, surveys and interviews, the study concluded that politicians, power companies and regulators were the ones with voice, as NGOs and citizens were hardly heard. The paper suggests the future role of PR practitioners to be to find the right issue arenas and facilitate corporate voice and reputation on those arenas. peerReviewed

Polynomial and horizontally polynomial functions on Lie groups

2022

We generalize both the notion of polynomial functions on Lie groups and the notion of horizontally affine maps on Carnot groups. We fix a subset $S$ of the algebra $\mathfrak g$ of left-invariant vector fields on a Lie group $\mathbb G$ and we assume that $S$ Lie generates $\mathfrak g$. We say that a function $f:\mathbb G\to \mathbb R$ (or more generally a distribution on $\mathbb G$) is $S$-polynomial if for all $X\in S$ there exists $k\in \mathbb N$ such that the iterated derivative $X^k f$ is zero in the sense of distributions. First, we show that all $S$-polynomial functions (as well as distributions) are represented by analytic functions and, if the exponent $k$ in the previous defini…

Integral binary Hamiltonian forms and their waterworlds

2018

We give a graphical theory of integral indefinite binary Hamiltonian forms $f$ analogous to the one by Conway for binary quadratic forms and the one of Bestvina-Savin for binary Hermitian forms. Given a maximal order $\mathcal O$ in a definite quaternion algebra over $\mathbb Q$, we define the waterworld of $f$, analogous to Conway's river and Bestvina-Savin's ocean, and use it to give a combinatorial description of the values of $f$ on $\mathcal O\times\mathcal O$. We use an appropriate normalisation of Busemann distances to the cusps (with an algebraic description given in an independent appendix), and the $\operatorname{SL}_2(\mathcal O)$-equivariant Ford-Voronoi cellulation of the real …

Counting and equidistribution in quaternionic Heisenberg groups

2020

AbstractWe develop the relationship between quaternionic hyperbolic geometry and arithmetic counting or equidistribution applications, that arises from the action of arithmetic groups on quaternionic hyperbolic spaces, especially in dimension 2. We prove a Mertens counting formula for the rational points over a definite quaternion algebra A over ${\mathbb{Q}}$ in the light cone of quaternionic Hermitian forms, as well as a Neville equidistribution theorem of the set of rational points over A in quaternionic Heisenberg groups.

Area of intrinsic graphs and coarea formula in Carnot Groups

2020

AbstractWe consider submanifolds of sub-Riemannian Carnot groups with intrinsic $$C^1$$ C 1 regularity ($$C^1_H$$ C H 1 ). Our first main result is an area formula for $$C^1_H$$ C H 1 intrinsic graphs; as an application, we deduce density properties for Hausdorff measures on rectifiable sets. Our second main result is a coarea formula for slicing $$C^1_H$$ C H 1 submanifolds into level sets of a $$C^1_H$$ C H 1 function.

Metric equivalences of Heintze groups and applications to classifications in low dimension

2021

We approach the quasi-isometric classification questions on Lie groups by considering low dimensional cases and isometries alongside quasi-isometries. First, we present some new results related to quasi-isometries between Heintze groups. Then we will see how these results together with the existing tools related to isometries can be applied to groups of dimension 4 and 5 in particular. Thus we take steps towards determining all the equivalence classes of groups up to isometry and quasi-isometry. We completely solve the classification up to isometry for simply connected solvable groups in dimension 4, and for the subclass of groups of polynomial growth in dimension 5.