686 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
Generalized Chaplygin gas as geometrical dark energy
The generalized Chaplygin gas provides an interesting candidate for the
present accelerated expansion of the universe. We explore a geometrical
explanation for the generalized Chaplygin gas within the context of brane world
theories where matter fields are confined to the brane by means of the action
of a confining potential. We obtain the modified Friedmann equations,
deceleration parameter and age of the universe in this scenario and show that
they are consistent with the present observational data.Comment: 11 pages, 3 figures, to appear in PR
Localization of gravity in brane world with arbitrary extra dimensions
We study the induced 4-dimensional linearized Einstein field equations in an
m-dimensional bulk space by means of a confining potential. It is shown that in
this approach the mass of graviton is quantized. The cosmological constant
problem is also addressed within the context of this approach. We show that the
difference between the values of the cosmological constant in particle physics
and cosmology stems from our measurements in two different scales, small and
large.Comment: 8 pages. arXiv admin note: substantial text overlap with
arXiv:gr-qc/0408004, arXiv:gr-qc/0607067, arXiv:0704.1035, arXiv:0707.3558,
arXiv:0710.266
On CSP and the Algebraic Theory of Effects
We consider CSP from the point of view of the algebraic theory of effects,
which classifies operations as effect constructors or effect deconstructors; it
also provides a link with functional programming, being a refinement of Moggi's
seminal monadic point of view. There is a natural algebraic theory of the
constructors whose free algebra functor is Moggi's monad; we illustrate this by
characterising free and initial algebras in terms of two versions of the stable
failures model of CSP, one more general than the other. Deconstructors are
dealt with as homomorphisms to (possibly non-free) algebras.
One can view CSP's action and choice operators as constructors and the rest,
such as concealment and concurrency, as deconstructors. Carrying this programme
out results in taking deterministic external choice as constructor rather than
general external choice. However, binary deconstructors, such as the CSP
concurrency operator, provide unresolved difficulties. We conclude by
presenting a combination of CSP with Moggi's computational {\lambda}-calculus,
in which the operators, including concurrency, are polymorphic. While the paper
mainly concerns CSP, it ought to be possible to carry over similar ideas to
other process calculi
Shuffle relations for regularised integrals of symbols
We prove shuffle relations which relate a product of regularised integrals of
classical symbols to regularised nested (Chen) iterated integrals, which hold
if all the symbols involved have non-vanishing residue. This is true in
particular for non-integer order symbols. In general the shuffle relations hold
up to finite parts of corrective terms arising from renormalisation on tensor
products of classical symbols, a procedure adapted from renormalisation
procedures on Feynman diagrams familiar to physicists. We relate the shuffle
relations for regularised integrals of symbols with shuffle relations for
multizeta functions adapting the above constructions to the case of symbols on
the unit circle.Comment: 40 pages,latex. Changes concern sections 4 and 5 : an error in
section 4 has been corrected, and the link between section 5 and the previous
ones has been precise
Comment on: Modular Theory and Geometry
In this note we comment on part of a recent article by B. Schroer and H.-W.
Wiesbrock. Therein they calculate some new modular structure for the
U(1)-current-algebra (Weyl-algebra). We point out that their findings are true
in a more general setting. The split-property allows an extension to
doubly-localized algebras.Comment: 13 pages, corrected versio
An algebraic scheme associated with the noncommutative KP hierarchy and some of its extensions
A well-known ansatz (`trace method') for soliton solutions turns the
equations of the (noncommutative) KP hierarchy, and those of certain
extensions, into families of algebraic sum identities. We develop an algebraic
formalism, in particular involving a (mixable) shuffle product, to explore
their structure. More precisely, we show that the equations of the
noncommutative KP hierarchy and its extension (xncKP) in the case of a
Moyal-deformed product, as derived in previous work, correspond to identities
in this algebra. Furthermore, the Moyal product is replaced by a more general
associative product. This leads to a new even more general extension of the
noncommutative KP hierarchy. Relations with Rota-Baxter algebras are
established.Comment: 59 pages, relative to the second version a few minor corrections, but
quite a lot of amendments, to appear in J. Phys.
Towards Bottom-Up Analysis of Social Food
in ACM Digital Health Conference 201
Overview of (pro-)Lie group structures on Hopf algebra character groups
Character groups of Hopf algebras appear in a variety of mathematical and
physical contexts. To name just a few, they arise in non-commutative geometry,
renormalisation of quantum field theory, and numerical analysis. In the present
article we review recent results on the structure of character groups of Hopf
algebras as infinite-dimensional (pro-)Lie groups. It turns out that under mild
assumptions on the Hopf algebra or the target algebra the character groups
possess strong structural properties. Moreover, these properties are of
interest in applications of these groups outside of Lie theory. We emphasise
this point in the context of two main examples: The Butcher group from
numerical analysis and character groups which arise from the Connes--Kreimer
theory of renormalisation of quantum field theories.Comment: 31 pages, precursor and companion to arXiv:1704.01099, Workshop on
"New Developments in Discrete Mechanics, Geometric Integration and
Lie-Butcher Series", May 25-28, 2015, ICMAT, Madrid, Spai
Fast computation by block permanents of cumulative distribution functions of order statistics from several populations
The joint cumulative distribution function for order statistics arising from
several different populations is given in terms of the distribution function of
the populations. The computational cost of the formula in the case of two
populations is still exponential in the worst case, but it is a dramatic
improvement compared to the general formula by Bapat and Beg. In the case when
only the joint distribution function of a subset of the order statistics of
fixed size is needed, the complexity is polynomial, for the case of two
populations.Comment: 21 pages, 3 figure
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