Related papers: A proof of the butterfly theorem using the scale f…
We establish two direct extensions to the Butterfly Theorem on the cyclic quadrilateral along with the proofs using the projective method and analytic geometry of the Cartesian coordinate system.
This paper provides a uniform explanation of different extensions and generalizations of the butterfly theorem based on the Desargues involution theorem.
The Butterfly Theorem is explored in Taxicab Geometry.
Shuffle algebra has been employed to give a proof of the duality theorem for multiple zeta-star values of height one.
This paper proves a generalization of the Butterfly Theorem, a classical Euclidean result, which is valid in the complex projective plane.
We prove a group graded Morita equivalences version of the "butterfly theorem" on character triples. This gives a method to construct an equivalence between block extensions from another related equivalence.
A new simple proof of Stirling's formula via the partial fraction expansion for the tangent function is presented.
The term "butterfly effect" means an extreme sensitivity of a dynamical system to small perturbations: "The beating of a butterfly wing in South America can result in the considerable change of positions and force of a tropical cyclon in…
The Butterfly lemma we present can be considered a reiteration theorem for differentials generated from a complex interpolation process for families of K\"othe spaces. The lemma will be used to clarify the effect of different configurations…
The two squares theorem of Fermat is a gem in number theory, with a spectacular one-sentence "proof from the Book". Here is a formalisation of this proof, with an interpretation using windmill patterns. The theory behind involves…
In this note, we give an alternate proof of the multinomial theorem using a probabilistic approach. Although the multinomial theorem is basically a combinatorial result, our proof may be simpler for a student familiar with only basic…
We present an elementary combinatorial proof of the celebrated Friendship theorem. The proof involves looking at independent sets and constructing a bound on their size which forces a contradiction.
We present a probabilistic proof of Euler's pentagonal number theorem based on a shuffling model.
We find the two-variables generating function for the statistic which counts the number of variations in a word bounded by $1$. Thus, we refine and extend previous results concerning staircase words, which are words in which the variation…
We give here a new proof of a Tauberian Theorem of complex Laplace transform using the Theory of measure and theory of function with bounded variations. However we deduce the simple proof of Prime Number Theorem.
We give a new simpler proof of a theorem of Jayne and Rogers.
This paper proves that in Size Theory the comparison of multidimensional size functions can be reduced to the 1-dimensional case by a suitable change of variables. Indeed, we show that a foliation in half-planes can be given, such that the…
In this paper we show an index theorem for gerbes
We introduce the notion of doubly rooted plane trees and give a decomposition of these trees, called the butterfly decomposition which turns out to have many applications. From the butterfly decomposition we obtain a one-to-one…
This paper presents an adaptive randomized algorithm for computing the butterfly factorization of a $m\times n$ matrix with $m\approx n$ provided that both the matrix and its transpose can be rapidly applied to arbitrary vectors. The…