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132
result(s) for
"Möbius transformations."
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Circular regression
A new model for an angular regression link function is introduced. The model employs an angular scale parameter, incorporates proper and improper rotations as special cases, and is equivalent to the Möbius circle mapping for complex variables. Desirable properties of the circle mapping carry over to angular regression. Parameter estimation and inferential methods are developed and illustrated.
Journal Article
Inscribable Order Types
We call an order type inscribable if it is realized by a point configuration where all extreme points are all on a circle. In this paper, we investigate inscribability of order types. We first construct an infinite family of minimally uninscribable order types. The proof of uninscribability mainly uses Möbius transformations and the Frantz ellipse. We further show that every simple order type with at most two interior points is inscribable, and that the number of such order types is Θ(4nn3/2). We also suggest open problems around inscribability.
Journal Article
Möbius Transformations in the Second Symmetric Product of ℂ
Let F2(C) denote the second symmetric product of the complex plane C endowed with the Hausdorff topology, i.e., F2(C)=A⊂C:|A|≤2,A≠∅. In this paper, we extended the concept of Möbius transformations to F2(C). More precisely, given a Möbius transformation T of C, we define the map T˜(z,w)=T(z),T(w) within F2(C). We describe some general properties of these maps, including the structure of their generators, characteristics related to transitivity, and the geometry of the conjugacy classes.
Journal Article
Incidences of Möbius Transformations in Fp
We develop the methods used by Rudnev and Wheeler (2022) to prove an incidence theorem between arbitrary sets of Möbius transformations and point sets in Fp2. We also note some asymmetric incidence results, and give applications of these results to various problems in additive combinatorics and discrete geometry. For instance, we give an improvement to a result of Shkredov concerning the number of representations of a non-zero product defined by a set with small sum-set, and a version of Beck’s theorem for Möbius transformations.
Journal Article
Structured backward error analysis of linearized structured polynomial eigenvalue problems
We start by introducing a new class of structured matrix polynomials, namely, the class of \\mathbf{M}_A-structured matrix polynomials, to provide a common framework for many classes of structured matrix polynomials that are important in applications: the classes of (skew-)symmetric, (anti-)palindromic, and alternating matrix polynomials. Then, we introduce the families of \\mathbf{M}_A-structured strong block minimal bases pencils and of \\mathbf{M}_A-structured block Kronecker pencils, which are particular examples of block minimal bases pencils recently introduced by Dopico, Lawrence, Pérez and Van Dooren, and show that any \\mathbf{M}_A-structured odd-degree matrix polynomial can be strongly linearized via an \\mathbf{M}_A-structured block Kronecker pencil. Finally, for the classes of (skew-)symmetric, (anti-)palindromic, and alternating odd-degree matrix polynomials, the \\mathbf{M}_A-structured framework allows us to perform a global and structured backward stability analysis of complete structured polynomial eigenproblems, regular or singular, solved by applying to a \\mathbf{M}_A-structured block Kronecker pencil a structurally backward stable algorithm that computes its complete eigenstructure, like the palindromic-QR algorithm or the structured versions of the staircase algorithm. This analysis allows us to identify those \\mathbf{M}_A-structured block Kronecker pencils that yield a computed complete eigenstructure which is the exact one of a slightly perturbed structured matrix polynomial. These pencils include (modulo permutations) the well-known block-tridiagonal and block-anti-tridiagonal structure-preserving linearizations. Our analysis incorporates structure to the recent (unstructured) backward error analysis performed for block Kronecker linearizations by Dopico, Lawrence, Pérez and Van Dooren, and share with it its key features, namely, it is a rigorous analysis valid for finite perturbations, i.e., it is not a first order analysis, it provides precise bounds, and it is valid simultaneously for a large class of structure-preserving strong linearizations.
Journal Article
Small-Determinant Directional Dilation Matrices for Anisotropic Multiresolution Analysis
Dilation matrices are important in multiple subdivision and multiple multiresolution analysis, as they govern the process of data refinement and play a crucial role in capturing directional features. One common limitation in the existing methods is the relatively large determinant of their dilation matrices, leading to high computational and storage costs. To address this issue, this paper proposes a novel family of pairs of directional dilation matrices with determinant 3. Such dilation matrices satisfy the joint expansion property and directional sensitivity. The joint expansion property is verified via the joint spectral radius, while by connecting the action of the matrices to certain elliptic elements of PSL(2,R), their directional adaptability can be established. Compared to most of the existing dilation matrices, the proposed ones achieve a balance between determinant and directional adaptability and provide a new insight into the construction of directional dilation matrices. This makes them suitable for addressing practical anisotropic problems.
Journal Article
Change of Polytope Volumes Under Möbius Transformations and the Circumcenter Of Mass
The circumcenter of mass of a simplicial polytope P is defined as follows: triangulate P, assign to each simplex its circumcenter taken with weight equal to the volume of the simplex, and then find the center of mass of the resulting system of point masses. The so obtained point is independent of the triangulation. The aim of the present note is to give a definition of the circumcenter of mass that does not rely on a triangulation. To do so we investigate how volumes of polytopes change under Möbius transformations.
Journal Article
CR-Selfdual Cubic Curves
We introduce a special class of cubic curves whose defining parameter satisfies a linear or quadratic equation provided by the values of a cross ratio. There are only seven such cubics and several properties of the real cubics in this class (some of them being elliptic curves) are discussed. Using the Möbius transformation, we extend this self-duality and obtain new families of remarkable complex cubics. In addition, we study (from the differential geometric viewpoint) the surface parameterized by all real cubic curves and we derive its curvature functions. As a by-product, we find a new classification of real Möbius transformations and some estimates for the number of vertices of real cubic curves.
Journal Article
The Apollonian structure of Bianchi groups
We study the orbit of R^\\widehat {\\mathbb {R}} under the Möbius action of the Bianchi group PSL2(OK)\\rm {PSL}_2(\\mathcal {O}_K) on C^\\widehat {\\mathbb {C}}, where OK\\mathcal {O}_K is the ring of integers of an imaginary quadratic field KK. The orbit SK{\\mathcal {S}}_K, called a Schmidt arrangement, is a geometric realisation, as an intricate circle packing, of the arithmetic of KK. We give a simple geometric characterisation of certain subsets of SK{\\mathcal {S}}_K generalizing Apollonian circle packings, and show that SK{\\mathcal {S}}_K, considered with orientations, is a disjoint union of all primitive integral such KK-Apollonian packings. These packings are described by a new class of thin groups of arithmetic interest called KK-Apollonian groups. We make a conjecture on the curvatures of these packings, generalizing the local-to-global conjecture for Apollonian circle packings.
Journal Article