A group of elementary associativity operators is introduced so that the bracketing graphs which are the skeletons of Stasheff’s associahedra become orbits and can be constructed as subgraphs of the Cayley graph of this group. A very simple proof of Mac Lane’s coherence theorem is given, as well as an oriented version of this result. We also sketch a more general theory and compare the cases of ...

We discuss parabolic versions of Euler’s identitye = cos t + i sin t.A purely algebraic approach based on dual numbers is known to produce a verytrivial relation e = 1 + εt. Therefore we use a geometric setup of parabolicrotations to recover the corresponding non-trivial algebraic framework. Our maintool is Möbius transformations which turn out to be closely related to induc...

The algebraic reformulation of the BMV conjecture is equivalent to a family of dimensionfree tracial inequalities involving positive semidefinite matrices. Sufficient conditions for these to hold in the form of algebraic identities involving polynomials in noncommuting variables have been given by Markus Schweighofer and the second author. Later the existence of these certificates has been sett...

We show that a pair of matrices satisfying a certain algebraic identity, reminiscent of the binomial theorem, must have the same characteristic polynomial. This is a generalization of Problem 4 (11th grade) from the Romanian National Mathematical Olympiad 2011.

The dispersionless differential Fay identity is shown to be equivalent to a kernel expansion providing a universal algebraic characterization and solution of the dispersionless Hirota equations. Some calculations based on D-bar data of the action are also indicated.

following the categorical approach to universal algebra through algebraic theories, proposed by f.~w.~lawvere in his phd thesis, this paper aims at introducing a similar setting for general topology. the cornerstone of the new framework is the notion of emph{categorically-algebraic} (emph{catalg}) emph{topological theory}, whose models induce a category of topological structures. we introduce t...

In this paper, we give a new short proof of the dual of Spivey’s Bell number identity due to Mező. Our approach follows from basic manipulations involving a fundamental identity representing factorials in the Zeon algebra. This work, along with a previous one due to the author and dos Anjos, shows that Spivey’s and Mező’s identities have at their root a common underlying algebraic origin.

Abstract. We give a general multiplication-convolution identity for the multivariate and bivariate rank generating polynomial of a matroid. The bivariate rank generating polynomial is transformable to and from the Tutte polynomial by simple algebraic operations. Several identities, almost all already known in some form, are specialization of this identity. Combinatorial or probabilistic interpr...

Algebraic approach to the integrability condition called shape invariance is briefly reviewed. Various applications of shape-invariance available in the literature are listed. A class of shape-invariant bound-state problems which represent two-level systems are examined. These generalize the Jaynes-Cummings Hamiltonian. Coherent states associated with shape-invariant systems are discussed. For ...