Combinatorics and Graph Theory

1910 Submissions

[3] viXra:1910.0283 [pdf] replaced on 2019-11-11 20:44:29

A Computing Method About How Many `comparable' Pairs of Elements Exist in a Certain Set

Authors: Yasushi Ieno
Comments: 17 Pages.

Given two sets, one consisting of variables representing distinct positive n numbers, the other set `a kind of power set' of this n-element set. I got interested in the fact that for the latter set, depending on the values of two elements, it can occur that not every pair of elements is `comparable', that is to say, it is not always uniquely determined which of two elements is smaller. By proving theorems in order to go ahead with our research, we show a table which describes for how many `comparable' cases exist, for several n's.
Category: Combinatorics and Graph Theory

[2] viXra:1910.0245 [pdf] replaced on 2024-07-29 17:05:47

Invitation to Hadamard Matrices

Authors: Teo Banica
Comments: 400 Pages.

An Hadamard matrix is a square matrix $Hin M_N(pm1)$ whose rows and pairwise orthogonal. More generally, we can talk about the complex Hadamard matrices, which are the square matrices $Hin M_N(mathbb C)$ whose entries are on the unit circle, $|H_{ij}|=1$, and whose rows and pairwise orthogonal. The main examples are the Fourier matrices, $F_N=(w^{ij})$ with $w=e^{2pi i/N}$, and at the level of the general theory, the complex Hadamard matrices can be thought of as being some sort of exotic, generalized Fourier matrices. We discuss here the basic theory of the Hadamard matrices, real and complex, with emphasis on the complex matrices, and their geometric and analytic aspects.
Category: Combinatorics and Graph Theory

[1] viXra:1910.0230 [pdf] replaced on 2019-11-03 22:12:39

Another Method to Solve the Grasshopper Problem (The International Mathematical Olympiad)

Authors: Yasushi Ieno
Comments: 31 Pages.

The 6th problem of the 50th International Mathematical Olympiad (IMO), held in Germany, 2009, is called 'the grasshopper problem'. To this problem Kos developed theory from unique viewpoints by reference of Noga Alon’s combinatorial Nullstellensatz. We have tried to solve this problem by an original method inspired by a polynomial function that Kos defined, then examined for n=3, 4 and 5. For almost cases the claim of this problem follows, but there remains imperfection due to 'singularity'.
Category: Combinatorics and Graph Theory