Related papers: Some contributions to Regular Polygons
The theorem of three circles in real algebraic geometry guarantees the termination and correctness of an algorithm of isolating real roots of a univariate polynomial. The main idea of its proof is to consider polynomials whose roots belong…
This papper aims to present and demonstrate Clifford's version for a generalization of Miquel's theorem with the use of Euclidean geometry arguments only.
The problem of covering a region of the plane with a fixed number of minimum-radius identical balls is studied in the present work. An explicit construction of bi-Lipschitz mappings is provided to model small perturbations of the union of…
Although it is important both in theory as well as in applications, a theory of Birkhoff interpolation with main emphasis on the shape of the set of nodes is still missing. Although we will consider various shapes (e.g. we find all the…
In this paper I survey the sources of inspiration for my own and co-authored work in trying to develop a general theory of graph polynomials. I concentrate on meta-theorems, i.e., theorem which depend only on the form infinite classes of…
The aim of this paper is to find a general formula to generate any row of Pascal's triangle as an extension of the concept of $\left(11\right)^{n}$. In this study, the visualization of each row of Pascal's triangle has been presented by…
The parametric geometry of numbers has allowed to visualize the simultaneous approximation properties of a collection of real numbers through the combined graph of the related successive minima functions. Several inequalities among…
We prove that the Newton product of efficient polynomial projectors is still efficient. Various polynomial approximation theorems are established involving Newton product projectors on spaces of holomorphic functions on a neighborhood of a…
Starting from any given rational-sided, right triangle, for example the $(3,4,5)$-triangle with area $6$, we use Euclidean geometry to show that there are infinitely many other rational-sided, right triangles of the same area. We show…
Some idea, which leads to a non-trivial solution of the quantum four-simplex equation, is exposed in this paper. We call this idea "pentagonal algebra". Few examples of the realisation of this idea are given here, and thus few examples of…
In this survey we present applications of the ideas of complement and neighborhood in the theory embeddings of manifolds into Euclidean space (in codimension at least three). We describe how the combination of these ideas gives a reduction…
We extend the notion of triangle to "imaginary triangles" with complex valued sides and angles, and parametrize families of such triangles by plane algebraic curves. We study in detail families of triangles with two commensurable angles,…
In Euclidean geometry, the Pythagorean theorem is presented as an equation involving three squares. This paper explores how analogous expressions may be identified in spherical and hyperbolic geometries.
Science about optimization methods is rapidly developing today. In machine learning, computer vision, biology, medicine, construction and in many other different areas optimization methods have vast popularity and they appear as important…
This article shines new light on the classical problem of tiling rectangles with squares efficiently with a novel method. With a twist on the traditional approach of resistor networks, we provide new and improved results on the matter using…
The aim of this article is to promote the use of probabilistic methods in the study of problems in mathematical general relativity. Two new and simple singularity theorems, whose features are different from the classical singularity…
This is a work in progress, far from being in its final form whose purpose is to investigate thoroughly the structure of Berkovich analytic curves and its relation with the semi-stable reduction theorem (of which a new proof is given here,…
The aim of this note is to describe the Poisson boundary of the group of invertible triangular matrices with coefficients in a number field. It generalizes to any dimension and to any number field a result of Brofferio concerning the…
The natural duality between "topological" and "regular," both considered as convergence space properties, extends naturally to p-regular convergence spaces, resulting in the new concept of a p-topological convergence space. Taking advantage…
In this paper we present a self-contained combinatorial proof of the lower bound theorem for normal pseudomanifolds, including a treatment of the cases of equality in this theorem. We also discuss McMullen and Walkup's generalised lower…