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We consider closed chains of circles $C_1,C_2,\ldots,C_n,C_{n+1}=C_1$ such that two neighbouring circles $C_i,C_{i+1}$ intersect or touch each other with $A_i$ being a common point. We formulate conditions such that a polygon with vertices…
We prove that any sufficiently small perturbation of an isosceles triangle has a periodic billiard path. Our proof involves the analysis of certain infinite families of Fourier series that arise in connection with triangular billiards, and…
We present type $\theta$ Stokes' theorem for type $\theta$ $k$-chains which extends the fundamental theorem of calculus in higher dimensions.
We employ the pinching theorem, ensuring that some operators A admit any sequence of contractions as an operator diagonal of A, to deduce/improve two recent theorems of Kennedy-Skoufranis and Loreaux-Weiss for conditional expectations onto…
Plots of quadratic residues display some features that are analyzed mathematically.
We prove a homological stability theorem for unlinked circles in $3$-manifolds and give an application to certain groups of diffeomorphisms of 3-manifolds.
Given a right triangle and two inscribed squares, we show that the reciprocals of the hypotenuse and the sides of the squares satisfy an interesting Pythagorean equality. This gives new ways to obtain rational(integer)right triangles from a…
In this paper we present new, short and elementary proofs of the famous projection and section theorems that are used in Stochastic Calculus.
This paper lays the foundations of triangulated persistence categories (TPC), which brings together persistence modules with the theory of triangulated categories. As a result we introduce several measurements and metrics on the set of…
In this paper we present a variety of statements that are in the spirit of the famous theorem of Pascal, often referred to as the Mystic Hexagon. We give explicit equations describing the conditions for $d+4$ points to lie on rational…
We give three new proofs of the triangle inequality in Euclidean Geometry. There seems to be only one known proof at the moment. It is due to properties of triangles, but our proofs are due to circles or ellipses. We aim to prove the…
We study combinatorial principles known as stick and club. Several variants of these principles and cardinal invariants connected to them are also considered. We introduce a new kind of side-by-side product of partial orders which we call…
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 the first part of this paper I give an elementary overview about some number sequences which count various sorts of lattice paths in strips along the x-axis and compute their generating functions in terms of Fibonacci and Lucas…
We study the eleven points in the plane of a given triangle, whose pedal triangles are similar to the given one. We prove that the six points whose pedal triangles are positively oriented, lie on a single circle, while the five points,…
The notions of transposed Hom-Poisson and Hom-pre-Lie Poisson algebras are introduced. Their bimodules and matched pairs are defined and the relevant properties and theorems are given. The notion of Manin triple of transposed Hom-Poisson…
This note collects some facts and conjectures about the Hankel determinants and their generating functions of the columns of Hoggatt triangles which apparently are related to combinatorial objects such as Young tableaux and Narayana…
On objects of a triangulated category with a stability condition, we construct a topology.
Using the notion of existentially closed structures, we obtain embedding theorems for groups and Lie algebras. We also prove the existence of some groups and Lie algebras with prescribed properties.
We show upper and lower bounds for angles in iterations of trisections of certain triangulations.