Abstract We explore the properties of polynomial Lagrangians for chiral p -forms previously proposed by last named author, and in particular, provide a self-contained treatment symmetries equations motion that shows great economy simplicity this formalism. further use analogous techniques to construct democratic general where electric magnetic potentials appear on equal footing as explicit dyna...

We define two symmetric $q,t$-Catalan polynomials in terms of the area and depth statistic dinv statistics. prove symmetry using an involution on plane trees. The same proves Tutte polynomials. also provide a combinatorial proof remark by Garsia et al. regarding parking functions number connected graphs fixed vertices.

Motivated by the study of ribbon knots we explore symmetric unions, a beautiful construction introduced by Kinoshita and Terasaka in 1957. We develop a twovariable refinement WD(s,t) of the Jones polynomial that is invariant under symmetric Reidemeister moves. If D is a symmetric union diagram, representing a ribbon knot K, then the polynomial WD(s,t) nicely reflects their topological propertie...

We construct chain maps between the bar and Koszul resolutions for a quantum symmetric algebra (skew polynomial ring). This construction uses a recursive technique involving explicit formulae for contracting homotopies. We use these chain maps to compute the Gerstenhaber bracket, obtaining a quantum version of the Schouten-Nijenhuis bracket on a symmetric algebra (polynomial ring). We compute b...

A symmetric basis of trigonometric polynomial space is presented. Based on the basis, symmetric trigonometric polynomial approximants like Bernstein polynomials are constructed. Two kinds of nodes are given to show that the trigonometric polynomial sequence is uniformly convergent. The convergence of the derivative of the trigonometric polynomials is shown. Trigonometric quasi-interpolants of r...