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We explore the possibility of using machine learning to identify interesting mathematical structures by using certain quantities that serve as fingerprints. In particular, we extract features from integer sequences using two empirical laws:…
We discuss conjectures related to the following two conjectures: (1) for each complex numbers x_1,...,x_n there exist rationals y_1,...,y_n \in [-2^{n-1},2^{n-1}] such that \forall i \in {1,...,n} (x_i=1 \Rightarrow y_i=1) \forall i,j,k \in…
We generalize the study of higher-form-symmetries to theories with supersymmetry. Using a supergeometry formulation, we find that ordinary higher-form-symmetries nicely combine with supersymmetry to give rise to a much larger spectrum of…
We present an algebraic framework which simultaneously generalizes the notion of linear subspaces, matroids, valuated matroids, oriented matroids, and regular matroids. To do this, we first introduce algebraic objects called tracts which…
Introducing the notion of a rational system of measure preserving transformations and proving a recurrence result for such systems, we give sufficient conditions in order a subset of rational numbers to contain arbitrary long arithmetic…
Recent years have seen a surprising connection between the physics of scattering amplitudes and a class of mathematical objects--the positive Grassmannian, positive loop Grassmannians, tree and loop Amplituhedra--which have been loosely…
Theory of motivic superpolynomials is developed, including its extension to algebraic links colored by rows, relations to $L$-functions of plane curve singularities, the justification of the motivic versions of Weak Riemann Hypothesis, and…
Recently, Z. W. Sun introduced a new kind of numbers $S_n$ and also posed a conjecture on ratio monotonicity of combinatorial sequences related to $S_n$. In this paper, by investigating some arithmetic properties of $S_n$, we give an…
The aim of this article is to define some new families of the special numbers. These numbers provide some further motivation for computation of combinatorial sums involving binomial coefficients and the Euler kind numbers of negative order.…
Loop torsors over Laurent polynomial rings in characteristic 0 were originally introduced in relation to infinite dimensional Lie theory. Applications to other areas require a theory that can yields results in positive characteristic, and…
The $abc$ conjecture is a very deep concept in number theory with wide application to many areas of number theory. In this article we introduce the conjecture and give examples of its applications. In particular we apply the $abc$…
Using the basic ingredient of supersymmetry, we develop a simple alternative approach to perturbation theory in one-dimensional non-relativistic quantum mechanics. The formulae for the energy shifts and wave functions do not involve tedious…
We present a collection of results concerning the location and distribution of very triangular numbers among triangular numbers, including the twin very triangular number theorem, the existence of arbitrarily long gaps between -- and an…
We will see that key concepts of number theory can be defined for arbitrary operations. We give a generalized distributivity for hyperoperations (usual arithmetic operations and operations going beyond exponentiation) and a generalization…
The text deals with generalizations of the Markoff equation in number theory, arising from continued fractions. It gives the method for the complete resolution of such new equations, and their interpretation in algebra and algebraic…
The goal of this paper is to sketch a broader outline of the mathematical structures present in the Nonlinear Maxwell Theory in continuation of work presented in my previous articles. In particular, I display new types of both dynamic and…
We exhibit a connection between two statistics on set partitions, the intertwining number and the depth-index. In particular, results link the intertwining number to the algebraic geometry of Borel orbits. Furthermore, by studying the…
Wigner's "unreasonable effectiveness of mathematics" in physics can be understood as a reflection of a deep and unexpected unity between the fundamental structures of mathematics and of physics. Some of the history of evidence for this is…
We consider the canonical ensemble of $N$ particles admitting a strange Hamiltonian description. Each of the particles obeys a set of Newtonian equation of motion, which can also be described by the standard canonical Hamiltonian mechanics.…
We develop techniques to deal with monotonicity of sequences z_{n+1}/z_n and \sqrt[n]{z_n}. A series of conjectures of Zhi-Wei Sun and of Amdeberhan et al. are verified in certain unified approaches.