Related papers: Meta Math! The Quest for Omega
In this note some philosophical thoughts and observations about mathematics are expressed, arranged as challenges to some common claims.
Let omega(n) be the number of prime divisors of an integer n. Let n be an integer taken at random between 1 and N. What can be said about the value then taken by omega(n)? What is its expected value? What is its distribution in the limit?…
Meta-analysis is an important tool for combining results from multiple studies and has been widely used in evidence-based medicine for several decades. This paper reports, for the first time, an interesting and valuable paradox in…
This short note introduces a formal system of truth and paradoxicality, outlining the main motivation, and proving its $\omega$-consistency. The system is called TP, for 'Truth and Paradoxicality'.
This work sets out to answer a single question: what is the orthodox interpretation of quantum mechanics? However, we adopt a different approach to that normally used. Rather than surveying physicists, or poring over the precise details of…
The halting probabilities of universal prefix-free machines are universal for the class of reals with computably enumerable left cut (also known as left-c.e. reals), and coincide with the Martin-Loef random elements of this class. We study…
This is the foreword to the book we edited on the origins and early development of supersymmetry, which has been just issued by World Scientific. This book presents a view on the discovery of supersymmetry and pioneering investigations…
We make use of information provided in the titles and abstracts of over half a million publications that were published by the American Physical Society during the past 119 years. By identifying all unique words and phrases and determining…
Mathematical reasoning is a cornerstone of human intelligence and a key benchmark for advanced capabilities in large language models (LLMs). However, the research community still lacks an open, large-scale, high-quality corpus tailored to…
Proposed in 1937, the Collatz conjecture has remained in the spotlight for mathematicians and computer scientists alike due to its simple proposal, yet intractable proof. In this paper, we propose several novel theorems, corollaries, and…
Metaheuristic algorithms are becoming an important part of modern optimization. A wide range of metaheuristic algorithms have emerged over the last two decades, and many metaheuristics such as particle swarm optimization are becoming…
At the beginning of the 20th century the classical electron theory (or, perhaps more appropriately, the classical electromagnetic mass theory) - the first physical theory that dared ask the question of what inertia and mass were - was…
This paper has two parts. The first part surveys Euler's work on the constant gamma=0.57721... bearing his name, together with some of his related work on the gamma function, values of the zeta function and divergent series. The second part…
To explore the limitation of a class of quantum algorithms originally proposed for the Hilbert's tenth problem, we consider two further classes of mathematically non-decidable problems, those of a modified version of the Hilbert's tenth…
A method to solve the problem f(x) = 0 efficiently on any n-dimensional domain Omega under very broad hypoteses is proposed. The position of the root of f, assumed unique, is found by computing the center of mass of an Omega-shaped object…
We describe a case of an interplay between human and computer proving which played a role in the discovery of an interesting mathematical result. The unusual feature of the use of computers here was that a computer generated but human…
In this paper we discuss the impact of the last Landau paper on physics and mathematics to date and make some predictions about its likely impact on sciences in new millennium.
The goal of this paper is to review the advances that were made during the last few decades in the study of the entropy, and in particular the entropy method, for Kac's many particle system.
The main object of this paper is to find closed form expressions for finite and infinite sums that are weighted by $\omega(n)$, where $\omega(n)$ is the number of distinct prime factors of $n$. We then derive general convergence criteria…
The situation surrounding the Olympiads is paradoxical. On the one hand, considerable resources are spent on the Olympiads. On the other hand, there are widespread arguments about the harm of the Olympiads, often very strange ones. For…