11 research outputs found
On Differential Rota-Baxter Algebras
A Rota-Baxter operator of weight is an abstraction of both the
integral operator (when ) and the summation operator (when
). We similarly define a differential operator of weight
that includes both the differential operator (when ) and the
difference operator (when ). We further consider an algebraic
structure with both a differential operator of weight and a
Rota-Baxter operator of weight that are related in the same way that
the differential operator and the integral operator are related by the First
Fundamental Theorem of Calculus. We construct free objects in the corresponding
categories. In the commutative case, the free objects are given in terms of
generalized shuffles, called mixable shuffles. In the noncommutative case, the
free objects are given in terms of angularly decorated rooted forests. As a
byproduct, we obtain structures of a differential algebra on decorated and
undecorated planar rooted forests.Comment: 21 page
Difference operators on lattices
A differential operator of weight is the algebraic abstraction of
the difference quotient ,
including both the derivation as approaches to and the difference
operator when . Correspondingly, differential algebra of weight
extends the well-established theories of differential algebra and
difference algebra. In this paper, we initiate the study of differential
operators with weights, in particular difference operators, on lattices. We
show that differential operators of weight on a lattice coincide with
differential operators, while differential operators are special cases of
difference operators. Distributivity of a lattice is characterized by the
existence of certain difference operators. Furthermore, we characterize and
enumerate difference operators on finite chains and finite quasi-antichains.Comment: 19 page
Constructing general rough differential equations through flow approximations
The non-linear sewing lemma constructs flows of rough differential equations
from a braod class of approximations called almost flows. We consider a class
of almost flows that could be approximated by solutions of ordinary
differential equations, in the spirit of the backward error analysis. Mixing
algebra and analysis, a Taylor formula with remainder and a composition formula
are central in the expansion analysis. With a suitable algebraic structure on
the non-smooth vector fields to be integrated, we recover in a single framework
several results regarding high-order expansions for various kind of driving
paths. We also extend the notion of driving rough path. We also introduce as an
example a new family of branched rough paths, called aromatic rough paths
modeled after aromatic Butcher series.Comment: version R0 (august 4, 2020): bibliography updat
Differential Algebra Structures on Families of Trees
It is known that the vector space spanned by labeled rooted trees forms a Hopf algebra. Let k be a field and let R be a commutative k-algebra. Let H denote the Hopf algebra of rooted trees labeled using derivations D ∈ Der(R). In this paper, we introduce a construction which gives R a H-module algebra structure and show this induces a differential algebra structure of H acting on R. The work here extends the notion of a R/k-bialgebra introduced by Nichols and Weisfeiler.