Search results for "Geometry and Topology"
showing 10 items of 457 documents
IFS attractors and Cantor sets
2006
Abstract We build a metric space which is homeomorphic to a Cantor set but cannot be realized as the attractor of an iterated function system. We give also an example of a Cantor set K in R 3 such that every homeomorphism f of R 3 which preserves K coincides with the identity on K.
Convergence in discrete Cauchy problems and applications to circle patterns
2005
A lattice-discretization of analytic Cauchy problems in two dimensions is presented. It is proven that the discrete solutions converge to a smooth solution of the original problem as the mesh size ε \varepsilon tends to zero. The convergence is in C ∞ C^\infty and the approximation error for arbitrary derivatives is quadratic in ε \varepsilon . In application, C ∞ C^\infty -approximation of conformal maps by Schramm’s orthogonal circle patterns and lattices of cross-ratio minus one is shown.
Topological Decompositions of the Pauli Group and their Influence on Dynamical Systems
2021
In the present paper we show that it is possible to obtain the well known Pauli group $P=\langle X,Y,Z \ | \ X^2=Y^2=Z^2=1, (YZ)^4=(ZX)^4=(XY)^4=1 \rangle $ of order $16$ as an appropriate quotient group of two distinct spaces of orbits of the three dimensional sphere $S^3$. The first of these spaces of orbits is realized via an action of the quaternion group $Q_8$ on $S^3$; the second one via an action of the cyclic group of order four $\mathbb{Z}(4)$ on $S^3$. We deduce a result of decomposition of $P$ of topological nature and then we find, in connection with the theory of pseudo-fermions, a possible physical interpretation of this decomposition.
Uniformization with infinitesimally metric measures
2019
We consider extensions of quasiconformal maps and the uniformization theorem to the setting of metric spaces $X$ homeomorphic to $\mathbb R^2$. Given a measure $\mu$ on such a space, we introduce $\mu$-quasiconformal maps $f:X \to \mathbb R^2$, whose definition involves deforming lengths of curves by $\mu$. We show that if $\mu$ is an infinitesimally metric measure, i.e., it satisfies an infinitesimal version of the metric doubling measure condition of David and Semmes, then such a $\mu$-quasiconformal map exists. We apply this result to give a characterization of the metric spaces admitting an infinitesimally quasisymmetric parametrization.
Generalized John disks
2014
Abstract We establish the basic properties of the class of generalized simply connected John domains.
Groups whose subgroups satisfy the weak subnormalizer condition
2019
A subgroup X of a group G is said to satisfy the weak subnormalizer condition if $$N_G(Y)\le N_G(X)$$ for each non-normal subgroup Y of G such that $$X\le Y\le N_G(X)$$ . The behaviour of generalized soluble groups whose (cyclic) subgroups satisfy the weak subnormalizer condition is investigated.
Baer cones in finite projective spaces
1987
Let R and V be two skew subspaces with dimensions r and v of P=PG(d,q). If q is a square, then there is a Baer subspace V* of V, i.e. a subspace of dimension v and order √q. We call the set C(R,V*)=\(\mathop \cup \limits_p \), where the union is taken over all PeV*, aBaer cone oftype (r,v).
Lifting paths on quotient spaces
2009
Abstract Let X be a compactum and G an upper semi-continuous decomposition of X such that each element of G is the continuous image of an ordered compactum. If the quotient space X / G is the continuous image of an ordered compactum, under what conditions is X also the continuous image of an ordered compactum? Examples around the (non-metric) Hahn–Mazurkiewicz Theorem show that one must place severe conditions on G if one wishes to obtain positive results. We prove that the compactum X is the image of an ordered compactum when each g ∈ G has 0-dimensional boundary. We also consider the case when G has only countably many non-degenerate elements. These results extend earlier work of the firs…
Spaces of typen on partially ordered sets
1989
This paper contains a generalized approach to incidence geometry on partially ordered sets. A difference to the usual geometrical concepts is that points may have different size. Our main result states that a large class of spaces allows lattice theoretic characterizations. Especially, a generalized version of the Veblen-Young axiom of projective geometry has a lattice theoretic equivalent, called then-generation property (which is a generalization of the ‘Verbindungssatz’). Modularity and distributivity of a lattice of subspaces are reflected in the underlying space. Finally we give specializations and examples.
Blocking sets and partial spreads in finite projective spaces
1980
A t-blocking set in the finite projective space PG(d, q) with d≥t+1 is a set $$\mathfrak{B}$$ of points such that any (d−t)-dimensional subspace is incident with a point of $$\mathfrak{B}$$ and no t-dimensional subspace is contained in $$\mathfrak{B}$$ . It is shown that | $$\mathfrak{B}$$ |≥q t +...+1+q t−1√q and the examples of minimal cardinality are characterized. Using this result it is possible to prove upper and lower bounds for the cardinality of partial t-spreads in PG(d, q). Finally, examples of blocking sets and maximal partial spreads are given.