Related papers: An Etude on One Sharygin's Problem
Congruent polygons are congruent in angles as well as in edge lengths. We concentrate on the angle aspect, and investigate how tilings of the sphere by congruent pentagons can be determined by the angle information only. We also investigate…
We automatically verify the crucial steps in the original proof of correctness of an algorithm which, given a geometric graph satisfying certain additional properties removes edges in a systematic way for producing a connected graph in…
Geometric properties of schemes obtained by gluing algebras of monoids, including separation and finiteness properties, irreducibility, normality, catenarity, dimension, and Serre's properties (S_k) and (R_k), are investigated. This is used…
Obtaining complete information about the shape of an object by looking at it from a single direction is impossible in general. In this paper, we theoretically study obtaining differential geometric information of an object from orthogonal…
We study plane algebraic curves defined over a field k of arbitrary characteristic as coverings of the the projective line and the problem of enumerating branched coverings of $\mathbb{P}^{1}$ by using combinatorial methods.
This paper investigates the geometric constraints imposed on a domain by overdetermined problems for partial differential equations. Serrin's symmetry results are extended to overdetermined problems with potentially degenerate ellipticity…
Using parametrized curves (Section 1) or parametrized sheets (Section 3), and suitable metrics, we treat the jet bundle of order one as a semi-Riemann manifold. This point of view allows the description of solutions of DEs as pregeodesics…
The paper concerns discrete versions of the three well-known results of projective differential geometry: the four vertex theorem, the six affine vertex theorem and the Ghys theorem on four zeroes of the Schwarzian derivative. We study…
Combination of a construction of unambiguous quantum conditions out of the conventional one and a simultaneous quantization of the positions, momenta, angular momenta and Hamiltonian leads to the geometric potential given by the so-called…
We discuss the principle tools and results and state a few open problems concerning the classification and topology of plane sextics and trigonal curves in ruled surfaces.
We re-evaluate universal computation based on the synthesis of Turing machines. This leads to a view of programs as singularities of analytic varieties or, equivalently, as phases of the Bayesian posterior of a synthesis problem. This new…
Constructions and exploration of plane algebraic curves has received a new push with the development of automated methods, whose algorithms are continuously improved and implemented in various software packages. We use them to explore the…
Ideas from deformation quantization applied to algebras with one generator lead to methods to treat a nonlinear flat connection. It provides us elements of algebras to be parallel sections. The moduli space of the parallel sections is…
A new formula is obtained in algebraic topology, in terms of Betti numbers, and a new method, called the spinal method, is suggested and developed for generating quadrangulations of closed orientable surfaces. Those surfaces arise as the…
We analyse the axioms of Euclidean geometry according to standard object-oriented software development methodology. We find a perfect match: the main undefined concepts of the axioms translate to object classes. The result is a suite of C++…
We study linear systems cut out by cones of fixed degree on a smooth complex curve $C\subset\mathbb{P}^{3}$. We develop a systematic study of the families of such systems, considering their limits, their infinitesimal behaviour and some…
Synthetic algebraic geometry is a new approach to algebraic geometry. It consists in using homotopy type theory extended with three axioms, together with the interpretation of these in a higher version of the Zariski topos, in order to do…
Let $(\Sigma,p)$ be a pointed Riemann surface of genus $g\geq 1$. For any integer $k\geq 1$, we parametrize the space of meromorphic quadratic differentials on $\Sigma$ with a pole of order $(k+2)$ at $p$, having a connected critical graph…
In this paper we review nine previous proposed and solved problems of elementary 2D geometry, and we extend them either from triangles to polygons or polyhedrons, or from circles to spheres (from 2D-space to 3D-space) and make some…
Certain topics on polygons are extended from Euclidean to hyperbolic geometry. This first part deals with uniqueness and existence of cocyclic polygons with prescribed sidelengths. The non-Euclidean versions are more difficult due to the…