1,391 research outputs found
Free Rota-Baxter algebras and rooted trees
A Rota-Baxter algebra, also known as a Baxter algebra, is an algebra with a
linear operator satisfying a relation, called the Rota-Baxter relation, that
generalizes the integration by parts formula. Most of the studies on
Rota-Baxter algebras have been for commutative algebras. Two constructions of
free commutative Rota-Baxter algebras were obtained by Rota and Cartier in the
1970s and a third one by Keigher and one of the authors in the 1990s in terms
of mixable shuffles. Recently, noncommutative Rota-Baxter algebras have
appeared both in physics in connection with the work of Connes and Kreimer on
renormalization in perturbative quantum field theory, and in mathematics
related to the work of Loday and Ronco on dendriform dialgebras and
trialgebras.
This paper uses rooted trees and forests to give explicit constructions of
free noncommutative Rota--Baxter algebras on modules and sets. This highlights
the combinatorial nature of Rota--Baxter algebras and facilitates their further
study. As an application, we obtain the unitarization of Rota-Baxter algebras.Comment: 23 page
Rota-Baxter algebras and new combinatorial identities
The word problem for an arbitrary associative Rota-Baxter algebra is solved.
This leads to a noncommutative generalization of the classical Spitzer
identities. Links to other combinatorial aspects, particularly of interest in
physics, are indicated.Comment: 8 pages, improved versio
Levy Processes and Quasi-Shuffle Algebras
We investigate the algebra of repeated integrals of semimartingales. We prove
that a minimal family of semimartingales generates a quasi-shuffle algebra. In
essence, to fulfill the minimality criterion, first, the family must be a
minimal generator of the algebra of repeated integrals generated by its
elements and by quadratic covariation processes recursively constructed from
the elements of the family. Second, recursively constructed quadratic
covariation processes may lie in the linear span of previously constructed ones
and of the family, but may not lie in the linear span of repeated integrals of
these. We prove that a finite family of independent Levy processes that have
finite moments generates a minimal family. Key to the proof are the Teugels
martingales and a strong orthogonalization of them. We conclude that a finite
family of independent Levy processes form a quasi-shuffle algebra. We discuss
important potential applications to constructing efficient numerical methods
for the strong approximation of stochastic differential equations driven by
Levy processes.Comment: 10 page
Algebraic Structures and Stochastic Differential Equations driven by Levy processes
We construct an efficient integrator for stochastic differential systems
driven by Levy processes. An efficient integrator is a strong approximation
that is more accurate than the corresponding stochastic Taylor approximation,
to all orders and independent of the governing vector fields. This holds
provided the driving processes possess moments of all orders and the vector
fields are sufficiently smooth. Moreover the efficient integrator in question
is optimal within a broad class of perturbations for half-integer global root
mean-square orders of convergence. We obtain these results using the
quasi-shuffle algebra of multiple iterated integrals of independent Levy
processes.Comment: 41 pages, 11 figure
Generalized shuffles related to Nijenhuis and TD-algebras
Shuffle and quasi-shuffle products are well-known in the mathematics
literature. They are intimately related to Loday's dendriform algebras, and
were extensively used to give explicit constructions of free commutative
Rota-Baxter algebras. In the literature there exist at least two other
Rota-Baxter type algebras, namely, the Nijenhuis algebra and the so-called
TD-algebra. The explicit construction of the free unital commutative Nijenhuis
algebra uses a modified quasi-shuffle product, called the right-shift shuffle.
We show that another modification of the quasi-shuffle product, the so-called
left-shift shuffle, can be used to give an explicit construction of the free
unital commutative TD-algebra. We explore some basic properties of TD-operators
and show that the free unital commutative Nijenhuis algebra is a TD-algebra. We
relate our construction to Loday's unital commutative dendriform trialgebras,
including the involutive case. The concept of Rota-Baxter, Nijenhuis and
TD-bialgebras is introduced at the end and we show that any commutative
bialgebra provides such objects.Comment: 20 pages, typos corrected, accepted for publication in Communications
in Algebr
Flows and stochastic Taylor series in Ito calculus
For stochastic systems driven by continuous semimartingales an explicit
formula for the logarithm of the Ito flow map is given. A similar formula is
also obtained for solutions of linear matrix-valued SDEs driven by arbitrary
semimartingales. The computation relies on the lift to quasi-shuffle algebras
of formulas involving products of Ito integrals of semimartingales. Whereas the
Chen-Strichartz formula computing the logarithm of the Stratonovich flow map is
classically expanded as a formal sum indexed by permutations, the analogous
formula in Ito calculus is naturally indexed by surjections. This reflects the
change of algebraic background involved in the transition between the two
integration theories
Brane worlds and dark matter
Two problems related to dark matter is considered in the context of a brane
world model in which the confinement of gauge fields on the brane is achieved
by invoking a confining potential. First, we show that the virial mass
discrepancy can be addressed if the conserved geometrical term appearing in
this model is considered as an energy momentum tensor of an unknown type of
matter, the so-called X-matter whose equation of state is also obtained.
Second, the galaxy rotation curves are explained by assuming an anisotropic
energy momentum tensor for the X-matter.Comment: 13 pages, 1 figure, to appear in IJMP
Adsorption properties and third sound propagation in superfluid He films on carbon nanotubes
We consider the adsorption properties of superfluid He films on carbon
nanotubes. One major factor in the adsorption is the surface tension force
arising from the very small diameter of the nanotubes. Calculations show that
surface tension keeps the film thickness on the tubes very thin even when the
helium vapor is increased to the saturated pressure. The weakened Van der Waals
force due to the cylindrical geometry also contributes to this. Both of these
effects act to lower the predicted velocity of third sound propagation along
the tubes. It does not appear that superfluidity will be possible on
single-walled nanotubes of diameter about one nm, since the film thickness is
less than 3 atomic layers even at saturation. Superfluidity is possible on
larger-diameter nanotube bundles and multi-walled nanotubes, however. We have
observed third sound signals on nanotube bundles of average diameter 5 nm which
are sprayed onto a Plexiglass surface, forming a network of tubes.Comment: 4 pages, accepted for Journal of Physics: Conference Series
(Proceedings of LT25
Exponential renormalization
Moving beyond the classical additive and multiplicative approaches, we
present an "exponential" method for perturbative renormalization. Using Dyson's
identity for Green's functions as well as the link between the Faa di Bruno
Hopf algebra and the Hopf algebras of Feynman graphs, its relation to the
composition of formal power series is analyzed. Eventually, we argue that the
new method has several attractive features and encompasses the BPHZ method. The
latter can be seen as a special case of the new procedure for renormalization
scheme maps with the Rota-Baxter property. To our best knowledge, although very
natural from group-theoretical and physical points of view, several ideas
introduced in the present paper seem to be new (besides the exponential method,
let us mention the notions of counterfactors and of order n bare coupling
constants).Comment: revised version; accepted for publication in Annales Henri Poincar
Mixable Shuffles, Quasi-shuffles and Hopf Algebras
The quasi-shuffle product and mixable shuffle product are both
generalizations of the shuffle product and have both been studied quite
extensively recently. We relate these two generalizations and realize
quasi-shuffle product algebras as subalgebras of mixable shuffle product
algebras. As an application, we obtain Hopf algebra structures in free
Rota-Baxter algebras.Comment: 14 pages, no figure, references update
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