The stack of iterated integrals of a path is embedded in a larger algebraic structure where iterated integrals are indexed by decorated rooted trees and where an extended Chen’s multiplicative property involves the Dürr-Connes-Kreimer coproduct on rooted trees. This turns out to be the natural setting for a non-geometric theory of rough paths. MSC: 60H99; 65L99

in this note by considering a complete lattice l, we define thenotion of an l-fuzzy hyperrelation on a given non-empty set x. then wedefine the concepts of (pom)l-fuzzy graph, hypergraph and subhypergroupand obtain some related results. in particular we construct the categories ofthe above mentioned notions, and give a (full and faithful) functor form thecategory of (pom)l-fuzzy subhypergroups ...

We introduce the generalized Heisenberg algebra appropriate for realizations of $\mathfrak{gl}(n)$ algebra. Linear are presented and corresponding star product, coproduct momenta twist constructed. The dual realization considered. Finally, we present a general algebra, two classes twists. These results can be applied to physical theories on noncommutative spaces type.

The facile and green synthesis of 1,1′-binaphthalene-2,2′-diamine (BINAM) derivatives was established via the anodic dehydrogenative homo-coupling 2-naphthylamines. sustainable protocol provided a series BINAMs in excellent yields up to 98% with good current efficiency (66%) H2 as sole coproduct without utilizing transition-metal reagents or stoichiometric oxidants.

A lot of combinatorial objects have algebra and coalgebra structures posets are important objects. In this paper, we construct on the vector space spanned by posets. Firstly, associativity unitary property, prove that with conjunction product is a graded algebra. Then definition free algebra, free. Finally, coassociativity counitary unshuffle coproduct coalgebra.

We apply a bootstrap procedure to two-loop MHV amplitudes in planar N = 4 super-Yang-Mills theory. We argue that the mathematically most complicated part (the ΛB2 coproduct component) of the n-particle amplitude is uniquely determined by a simple cluster algebra property together with a few physical constraints (dihedral symmetry, analytic structure, supersymmetry, and well-defined collinear li...

Two programs, feyngen and feyncop, were developed. feyngen is designed to generate high loop order Feynman graphs for Yang-Mills, QED and φ theories. feyncop can compute the coproduct of these graphs on the underlying Hopf algebra of Feynman graphs. The programs can be validated by exploiting zero dimensional field theory combinatorics and identities on the Hopf algebra which follow from the re...

We introduce a notion of “hopfish algebra” structure on an associative algebra, allowing the structure morphisms (coproduct, counit, antipode) to be bimodules rather than algebra homomorphisms. We prove that quasi-Hopf algebras are examples of hopfish algebras. We find that a hopfish structure on the algebra of functions on a finite set G is closely related to a “hypergroupoid” structure on G. ...

In the general setting of groupoid enriched categories, notions of suspender and looper of a map are introduced, formalizing a generalization of the classical homotopy-theoretic notions of suspension and loop space. The formalism enables subtle analysis of these constructs. In particular, it is shown that the suspender of a principal coaction splits as a coproduct. This result leads to the noti...