Spelling suggestions: "subject:"picard groups"" "subject:"ricard groups""
1 |
The action of the picard group on hyperbolic 3-space and complex continued fractionsHayward, Grant Paul 11 August 2014 (has links)
A dissertation submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of the requirements for the degree of Master of Science. Johannesburg, 2013. / Continued fractions have been extensively studied in number theoretic ways.
These continued fractions are expressed as compositions of M¨obius
maps in the Picard group PS L(2;C) that act, by Poincar´e’s extension, as isometries
on H3. We investigate the Picard group with its generators and derive the fundamental
domain using a direct method. From the fundamental domain, we produce
an ideal octahedron, O0, that generates the Farey tessellation of H3. We explore
the properties of Farey neighbours, Farey geodesics and Farey triangles that arise
from the Farey tessellation and relate these to Ford spheres. We consider the Farey
addition of two rationals in R as a subdivision of an interval and hence are able
to generalise this notion to a subdivision of a Farey triangle with Gaussian Farey
neighbour vertices. This Farey set allows us to revisit the Farey triangle subdivision
given by Schmidt [44] and interpret it as a theorem about adjacent octahedra in
the Farey tessellation of H3. We consider continued fraction algorithms with Gaussian
integer coe cients. We introduce an analogue of Series [45] cutting sequence
across H2 in H3. We derive a continued fraction expansion based on this cutting
sequence generated by a geodesic in H3 that ends at the point in C that passes
through O0.
|
2 |
Über die Picard'schen Gruppen aus dem Zahlkörper der dritten und der vierten EinheitswurzelBohler, Otto. January 1905 (has links)
Thesis (doctoral)--Universität Zürich, 1905.
|
3 |
On the admissible pairs of rational homogeneous manifolds of Picard number 1 and geometric structures defined by their varieties of minimal rational tangentsZhang, Yunxin, 张云鑫 January 2014 (has links)
In a series of works, Jun-Muk Hwang and Ngaiming Mok have developed a geometric theory of uniruled projective manifolds, especially those of Picard Number 1, relying on the study of Varieties of Minimal Rational Tangents (VMRT) from both the algebro-geometric and the G-structure perspectives. Based on this theory, Ngaiming Mok and Jaehyun Hong studied the standard embedding between two Rational Homogeneous Spaces (RHS) associated to long simple roots which are of different dimensions. In this thesis, I consider admissible pairs of RHS (X0, X) of Picard number 1 and locally closed complex submanifolds S ⊂ X inheriting VMRT sub-structures modeled on X0 = G0/P0 ⊂ X = G/P de_ned by taking intersections of VMRT of X with tangent space of S. Moreover, if any such S modeled on (X0, X) is necessarily the image of a standard embedding i : X0 → X, (X0, X) is said to be rigid. In this thesis, it is proved that an admissible pair (X0, X) is rigid whenever X is associated to a long simple root and X0 is non-linear and de_ned by a marked Dynkin sub-diagram. In the case of the pair (S0, S) of compact Hermitian Symmetric Spaces (cHSS), all the admissible pairs (S0, S) are completely classified. Based on this classification, a sufficient condition for the pair (S0, S) to be non-rigid is established through explicitly constructing a submanifold S ⊂ S such that S can never be obtained from the image of any standard embedding i : S0 → S. Besides, the term special pair is coined for those (S0; S) sorted out through classification, and the algebraicity of submanifolds modeled on special pairs is confirmed by checking a modified form of the non-degeneracy condition defined by Hong and Mok is satisfied. However, the question as to whether these special pairs are rigid, as pointed out in this thesis, remains to be investigated. Finally, pairs of hyperquadrics (Q^n, Q^m) are studied separately. Since non-rigidity is trivial, in these cases it is interesting to establish a characterization of the standard embedding i : Q^n→Q^m under some stronger condition. In this thesis, the latter problem is solved in terms of the partial vanishing of second fundamental forms. / published_or_final_version / Mathematics / Doctoral / Doctor of Philosophy
|
4 |
Floquet theory for picard-type systems of differential equations /Sticka, Wilhelm Michael, January 1996 (has links)
Thesis (Ph. D.)--University of Missouri-Columbia, 1996. / Typescript. Vita. Includes bibliographical references (leaves 94-95). Also available on the Internet.
|
5 |
Floquet theory for picard-type systems of differential equationsSticka, Wilhelm Michael, January 1996 (has links)
Thesis (Ph. D.)--University of Missouri-Columbia, 1996. / Typescript. Vita. Includes bibliographical references (leaves 94-95). Also available on the Internet.
|
6 |
Über die Picard'schen gruppen aus dem zahlkörper der dritten und der vierten einheitswurzel ...Bohler, Otto. January 1905 (has links)
Inaug.-diss.--Zürich. / Part of the Cornell Digital Library Math Collection.
|
7 |
On the Picard functor in formal-rigid geometryLi, Shizhang January 2019 (has links)
In this thesis, we report three preprints [Li17a] [Li17b] and [HL17] the author wrote (the last one was written jointly with D. Hansen) during his pursuing of PhD at Columbia.
We study smooth proper rigid varieties which admit formal models whose special fibers are projective. The main theorem asserts that the identity components of the associated rigid Picard varieties will automatically be proper. Consequently, we prove that non-archimedean Hopf varieties do not have a projective reduction. The proof of our main theorem uses the theory of moduli of semistable coherent sheaves.
Combine known structure theorems for the relevant Picard varieties, together with recent advances in p-adic Hodge theory, We then prove several related results on the low-degree Hodge numbers of proper smooth rigid analytic varieties over p-adic fields.
|
Page generated in 0.0289 seconds