Related papers: Circle patterns with obtuse exterior intersection …
We study the systems of ordinary differential equations which are implicit with respect to the higher derivatives, appearing in the linear form, and their solutions near the singular points. The invertibility of the higher derivatives…
A rough structure theorem is proved for graphs $G$ containing no copy of a bounded degree tree $T$: from any such $G$, one can delete $o(|G||T|)$ edges in order to get a subgraph all of whose connected components have a cover of order…
A method is developed to construct the solutions of one and many variable, linear differential equations of arbitrary order. Using this, the $N$-particle Sutherland model, with pair-wise inverse sine-square interactions among the particles,…
A formula for the radii and positions of four circles in the plane for an arbitrary linearly independent circle configuration is found. Among special cases is the recent extended Descartes Theorem on the Descartes configuration and an…
We provide a complete structure theorem for involutory matrices. This yields a new approach to principal angles between subspaces and provide a series of nice formulae for these angles.
The Tait-Kneser theorem states that the osculating circles of a plane curve with monotonic curvature are pairwise disjoint and nested. We discuss this theorem and a number of its variations.
Circle geometries are incidence structures that capture the geometry of circles on spheres, cones and hyperboloids in 3-dimensional space. In a previous paper, the author characterised the largest intersecting families in finite ovoidal…
The aim of Part II is to explore the technique of invariance of tautological equations in the realm of Gromov--Witten theory. The main result is a proof of Invariance Theorem (Invariance Conjecture~1 in [14]), via the techniques from…
Classical Sturm non-oscillation and comparison theorems as well as the Sturm theorem on zeros for solutions of second order differential equations have a natural symplectic version, since they describe the rotation of a line in the phase…
We develop a Thurston-like theory to characterize geometrically finite rational maps, then apply it to study pinching and plumbing deformations of rational maps. We show that in certain conditions the pinching path converges uniformly and…
Guo and Luo introduced generalized circle patterns on surfaces and proved their rigidity. In this paper, we prove the existence of Guo-Luo's generalized circle patterns with prescribed generalized intersection angles on surfaces with cusps,…
We discuss the higher-order topological field theory and response of topological crystalline insulators with no other symmetries. We show how the topology and geometry of the system is organised in terms of the elasticity tetrads which are…
We show that the cycle relation between Dehn twists about curves in a circuit detects whether the circuit bounds an embedded disc. This is done by determining the isomorphism type of the group generated by said Dehn twists for various…
Motivated by Guo-Luo's generalized circle packings on surfaces with boundary \cite{GL2}, we introduce the generalized Thurston's sphere packings on 3-dimensional manifolds with boundary. Then we investigate the rigidity of the generalized…
We introduce a new method for studying length spectrum rigidity problems based on a combination of ideas from dynamical systems and geometric group theory. This allows us to compare the marked length spectrum of metrics and distance-like…
The chapter presents mathematical models intended for creating a topological drawing of a non-separable non-planar graph based on the methods of G. Ringel's vertex rotation theory. The induced system of cycles generates a topological…
We suggest a new definition for discrete minimal surfaces in terms of sphere packings with orthogonally intersecting circles. These discrete minimal surfaces can be constructed from Schramm's circle patterns. We present a variational…
We prove the No Invariant Line Fields conjecture for a class of generalized postcritically-finite branched covers on higher-dimensional Riemannian manifolds. Moreover, we establish a quasisymmetric uniformization theorem for this class of…
We study the classical result by Bruijn and Erd\H os regarding the bound on the number of lines determined by a $n$-point configuration in the plane, and in the light of the recently proven Tropical Sylvester-Gallai theorem, come up with a…
A rigidity theory is developed for frameworks in a metric space with two types of distance constraints. Mixed sparsity graph characterisations are obtained for the infinitesimal and continuous rigidity of completely regular bar-joint…