Related papers: Some considerations on Collatz conjecture
We give a short proof of the log-concavity of the coefficients of the reduced characteristic polynomial of a matroid. The proof uses an extension of the theory of Lorentzian polynomials to convex cones, and reproves the Hodge-Riemann…
The determinants of modular Collatz graphs and the modular Conway amusical permutation graph are determined, and some interesting number theoretic properties are described.
For the simple random walk in Z^2 we study those points which are visited an unusually large number of times, and provide a new proof of the Erdos-Taylor conjecture describing the number of visits to the most visited point.
Trees or rooted trees have been generously studied in the literature. A forest is a set of trees or rooted trees. Here we give recurrence relations between the number of some kind of rooted forest with $k$ roots and that with $k+1$ roots on…
Simple and shorter proofs of two Dirac-type theorems involving connectivity are presented.
The expansion of the observed universe appears to be accelerating. A simple explanation of this phenomenon is provided by the non-vanishing of the cosmological constant in the Einstein equations. Arguments are commonly presented to the…
We note that in general there exist two basic aspects in any branch of physics, including cosmology - one dealing with the attributes of basic constituents and forces of nature, the other dealing with how structures arise from them and how…
Simple algebraic rules can produce complex networks with rich structures. These graphs are obtained when looking at a monoid operating on a ring. There are relations to dynamical systems theory and number theory. This document illustrates…
In this note, we combine ideas of several previous proofs in order to obtain a quite short proof of Gr\"otzsch theorem.
In this paper we present some observations about the well-known Goldbach conjecture. In particular we list and interpret some numerical results which allow us to formulate a relation between prime numbers and even integers. We can also…
Based on various strategies, we obtain several simple proofs of the celebrated Sharkovsky cycle coexistence theorem.
Starting with the Brezis-Browder principle, we give stronger versions of many variational principles and minimal element theorems which appeared in the recent literature. Relationships among the elements of different sets of assumptions are…
We first show that increasing trees are in bijection with set compositions, extending simultaneously a recent result on trees due to Tonks and a classical result on increasing binary trees. We then consider algebraic structures on the…
A short, fairly self-contained proof is given of the Poincar\'e Conjecture. In the previous version there was an error on Page 8. This gap has now been filled.
The generally accepted wisdom in computational circles is that pure proof verification is a solved problem and that the computationally hard elements and fertile areas of study lie in proof discovery. This wisdom presumably does hold for…
It is argued that some of the recent claims for cosmology are grossly overblown. Cosmology rests on a very small database: it suffers from many fundamental difficulties as a science (if it is a science at all) whilst observations of distant…
An important result for regular foliations is their formal semi-local triviality near simply connected leaves. We extend this result to singular foliations for all 2-connected leaves and a wide class of 1- connected leaves by proving a…
We look afresh at the deduction of the "Lorentz contraction" of a "rod" from the Lorentz transformation equations of the special theory of relativity. We show that under special conditions, which include acceleration of the "rod", length…
An order-theoretic forest is a countable partial order such that the set of elements larger than any element is linearly ordered. It is an order-theoretic tree if any two elements have an upper-bound. The order type of a branch can be any…
In this paper we analyse some questions concerning trees on $\kappa$, both for the countable and the uncountable case, and the connections with Cohen reals. In particular, we provide a proof for one of the implications left open in…