Related papers: Mathematical Monsters
Axial algebras of Monster type are a class of non-associative algebras which generalise the Griess algebra, whose automorphism group is the largest sporadic simple group, the Monster. The $2$-generated algebras, which are the building…
Wigner's famous and influential claim that mathematics is "unreasonably effective" in physics is founded on unreasonable assumptions about the nature of mathematics and its independence of physics. Here I argue that what is surprising is…
Statistics is a uniquely difficult field to convey to the uninitiated. It sits astride the abstract and the concrete, the theoretical and the applied. It has a mathematical flavor and yet it is not simply a branch of mathematics. Its core…
As a contribution to an eventual solution of the problem of determination of the maximal subgroups of the Monster we show that there is a unique conjugacy class of subgroups isomorphic to $PSU_3(8)$. The argument depends on some…
Dumont has conjectured a marvellous identity, which generalizes, in particular, the classical results of Lagrange, Gauss, Jacobi and Kronecker on the sums of two, three and four squares. We give a combinatorial proof of Dumont's conjecture.
Biology is data-rich, and it is equally rich in concepts and hypotheses. Part of trying to understand biological processes and systems is therefore to confront our ideas and hypotheses with data using statistical methods to determine the…
The paper deals with topical issues of modern mathematical theory of dynamical chaos and its applications. At present, it is customary to assume that dynamical chaos in finitedimensional smooth systems can exist in three different forms.…
We study last-layer outlier dimensions, i.e. dimensions that display extreme activations for the majority of inputs. We show that outlier dimensions arise in many different modern language models, and trace their function back to the…
The famous Laplace's Demon is not only of strict physical determinism, but also related to the power of differential equations. When deterministically extended structures are taken into consideration, it is admissible that fractals are…
Chaotic iterations have been introduced on the one hand by Chazan, Mi- ranker [6] and Miellou [10] in a numerical analysis context, and on the other hand by Robert [12] and Pellegrin [11] in the discrete dynamical systems frame- work. In…
The paper discusses fundamental problems in mathematical description of social systems based on physical concepts, with so-called statistical social systems being the main subject of consideration. Basic properties of human beings and human…
Mathematical notation, i.e., the writing system used to communicate concepts in mathematics, encodes valuable information for a variety of information search and retrieval systems. Yet, mathematical notations remain mostly unutilized by…
Thirty original and collected problems, puzzles, and paradoxes in mathematics and physics are explained in this paper, taught by the author to the elementary and high school teachers at the University of New Mexico - Gallup in 1997-8 and…
Many have wondered how mathematics, which appears to be the result of both human creativity and human discovery, can possibly exhibit the degree of success and seemingly-universal applicability to quantifying the physical world as…
Discussions surrounding the nature of the infinite in mathematics have been underway for two millennia. Mathematicians, philosophers, and theologians have all taken part. The basic question has been whether the infinite exists only in…
This paper reviews the maxims used by three early modern fictional detectives: Monsieur Lecoq, C. Auguste Dupin and Sherlock Holmes. It find similarities between these maxims and Bayesian thought. Poe's Dupin uses ideas very similar to…
In this paper we propose a new perspective on the evolution and history of the idea of mathematical proof. Proofs will be studied at three levels: syntactical, semantical and pragmatical. Computer-assisted proofs will be give a special…
The sciences of complexity present some recurrent themes: the emergence of qualitatively new behaviors in dissipative systems out of equilibrium, the aparent tendency of complex system to lie at the border of phase transitions and…
Back in 1755, Euler explored an interesting array of numbers that now frequently appears in polynomial identities, combinatorial problems, and finite calculus, among other places. These numbers share a strong connection with well-known…
In this article, I explain why I abandoned my original plan to write a novel about a mathematician proving Fermat's Last Theorem, letting my protagonist solve the entirely fictional Wild Number Problem instead. I enumerate the steps that…