53 research outputs found
On restricted families of projections in ℝ3
No full textWe study projections onto non-degenerate one-dimensional families of lines and planes in R 3 . Using the classical potential theoretic approach of R. Kaufman, one can show that the Hausdorff dimension of at most 12 -dimensional sets [Math Processing Error] is typically preserved under one-dimensional families of projections onto lines. We improve the result by an ε , proving that if [Math Processing Error], then the packing dimension of the projections is almost surely at least [Math Processing Error]. For projections onto planes, we obtain a similar bound, with the threshold 12 replaced by 1 . In the special case of self-similar sets [Math Processing Error] without rotations, we obtain a full Marstrand-type projection theorem for 1-parameter families of projections onto lines. The [Math Processing Error] case of the result follows from recent work of M. Hochman, but the [Math Processing Error] part is new: with this assumption, we prove that the projections have positive length almost surely
Isometric embeddings into Heisenberg groups
No full textWe study isometric embeddings of a Euclidean space or a Heisenberg group into a higher dimensional Heisenberg group, where both the source and target space are equipped with an arbitrary left-invariant homogeneous distance that is not necessarily sub-Riemannian. We show that if all infinite geodesics in the target are straight lines, then such an embedding must be a homogeneous homomorphism. We discuss a necessary and certain sufficient conditions for the target space to have this ‘geodesic linearity property’, and we provide various examples
Modulus method and radial stretch map in the Heisenberg group
No full textWe develop a modulus method for surface families inside a domain in the Heisenberg group and we prove that the stretch map between two Heisenberg spherical rings is a minimiser for the mean distortion among the class of contact quasiconformal maps between these rings which satisfy certain boundary conditions
Uniqueness of minimisers for a Grötzsch-Belinskiĭ type inequality in the Heisenberg group
No full textThe modulus method introduced by H. Grötzsch yields bounds for a mean distortion functional of quasiconformal maps between two annuli mapping the respective boundary components onto each other. P. P. Belinskiĭ studied these inequalities in the plane and identified the family of all minimisers. Beyond the Euclidean framework, a Grötzsch-Belinskiĭ-type inequality has been previously considered for quasiconformal maps between annuli in the Heisenberg group whose boundaries are Korányi spheres. In this note we show that--in contrast to the planar situation--the minimiser in this setting is essentially unique
Fast track to differential equations: applications-oriented, comprehensible, compact
No full textThis compact introduction to the ordinary differential equations and their applications is aimed at anyone who, in their studies, is confronted voluntarily or involuntarily with this versatile subject. Numerous examples from physics, technology, biomathematics, cosmology, economy and optimization allow a quick and motivating approach - abstract proofs and unnecessary formalism are avoided as far as possible. In the foreground is the modelling of ordinary differential equations of the 1st and 2nd order as well as their analytical and numerical solution methods, in which the theory is briefly dealt with before the application examples. In addition, codes show exemplarily how even more demanding questions can be answered and meaningfully represented with the help of a computer algebra system. In the first chapter the necessary previous knowledge from integral and differential calculus is treated. A large number of exercises including solutions round off the work. The Author Prof. (em.) Dr. Albert Fässler studied mathematics at ETH Zurich after studying mechanical engineering and received his doctorate in 1976 with a thesis on applied group theory. In addition to his teaching activities at the ETHZ and his many years of teaching at the Bern University of Applied Sciences, he has also always been internationally active - research stays have taken him to the USA and China, among other places. Since his retirement, he has repeatedly taught in China, India and Switzerland
Curve packing and modulus estimates
No full textA family of planar curves is called a Moser family if it contains an isometric copy of every rectifiable curve in of length one. The classical ``worm problem'' of L. Moser from 1966 asks for the least area covered by the curves in any Moser family. In 1979, J. M. Marstrand proved that the answer is not zero: the union of curves in a Moser family always has area at least for some small absolute constant . We strengthen Marstrand's result by showing that for , the -modulus of a Moser family of curves is at least $ c_{p} > 0
Projection and slicing theorems in Heisenberg groups
No full textAbstractWe study the behavior of the Hausdorff dimension of sets in the Heisenberg group Hn, n∈N, with a sub-Riemannian metric under projections onto horizontal and vertical subgroups, and under slicing by translates of vertical subgroups. We formulate almost sure statements in terms of a natural measure on the Grassmannian of isotropic subspaces
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