相关论文: Francois Viete and his contribution to mathematics
Grothendieck's theory of dessins provides a bridge between algebraic numbers and combinatorics. This paper adds a new concept, called 'bias', to the bridge. This produces: (i) from a biased plane tree the construction of a sequence of…
C.H. Yang discovered a polynomial version of the classical Lagrange identity expressing the product of two sums of four squares as another sum of four squares. He used it to give short proofs of some important theorems on composition of…
The present paper deals with the life and some aspects of the scientific contributions of the mathematician Ren\'e Gateaux, killed during World War 1 at the age of 25. Though he died very young, he left interesting results in functional…
Given a finite-dimensional module, $V$, for a finite-dimensional, complex, semi-simple Lie algebra $\lie g$ and a positive integer $m$, we construct a family of graded modules for the current algebra $\lie g[t]$ indexed by simple $\CC\lie…
The notion of newtonianity is central to the study of the ordered differential field of logarithmic-exponential transseries done by Aschenbrenner, van den Dries, and van der Hoeven; see Chapter 14 of arxiv:1509.02588. We remove the…
The purpose of these notes is to provide an introduction to the Steenrod algebra in an algebraic manner avoiding any use of cohomology operations. The Steenrod algebra is presented as a subalgebra of the algebra of endomorphisms of a…
We use fermionic operators to construct toroidal Lie algebras of classical types, including in particular that of symplectic affine algebras, which is first realized by fermions.
The question of how to certify the non-negativity of a polynomial function lies at the heart of Real Algebra and it also has important applications to Optimization. In the setting of symmetric polynomials Timofte provided a useful way of…
A class of simple filtered Lie algebras of polynomial growth with increasing filtration is distinguished and presentations of these algebras are explicitely described for the simplest examples. Lie (super)algebras of this class appear in…
We examine prevailing philosophical and historical views about the origin of infinitesimal mathematics in light of modern infinitesimal theories, and show the works of Fermat, Leibniz, Euler, Cauchy and other giants of infinitesimal…
Newton, in notes that he would rather not have seen published, described a process for solving simultaneous equations that later authors applied specifically to linear equations. This method that Euler did not recommend, that Legendre…
You can invent striking and challenging problems with unique solution by building some symmetry into functional equations. Some are suitable for high school; others could generate college-level projects involving computer algebra. The…
This is a first of our papers devoted to "noncommutative topology and graph theory". Its origin is the paper math.QA/0002238 by I. Gelfand, V. Retakh, and R.L. Wilson where a new class of noncommutative algebras $Q_n$ was introduced. The…
These notes are an exposition of Galois Theory from the original Lagrangian and Galoisian point of view. A particular effort was made here to better understand the connection between Lagrange's purely combinatorial approach and Galois…
Non-commutative geometry (NCG) is a mathematical discipline developed in the 1990s by Alain Connes. It is presented as a new generalization of usual geometry, both encompassing and going beyond the Riemannian framework, within a purely…
Perhaps the most significant, if not the most important, achievements in chemistry and physics are the Periodic Table of the Elements in Chemistry and the Standard Model of Elementary Particles in Physics. A comparable achievement in…
In this article a new formulation of the Weyl C*-algebra, which has been invented by Fleischhack, in terms of C*-dynamical systems is presented. The quantum configuration variables are given by the holonomies along paths in a graph.…
This paper has three aims. First, for $n \geq 1$ we construct a family of real-rooted trigonometric polynomial maps $P : \mathbb C^n \mapsto \mathbb C^n$ whose divisors are Fourier Quasicrystals (FQ). For $n = 1$ these divisors include the…
B\'ezout's name is attached to his famous theorem. B\'ezout's Theorem states that the degree of the eliminand of a system a $n$ algebraic equations in $n$ unknowns, when each of the equations is generic of its degree, is the product of the…
The Fundamental Theorem of Algebra (FTA) asserts that every complex polynomial has as many complex roots, counted with multiplicities, as its degree. A probabilistic analogue of this theorem for real roots of real polynomials, sometimes…