4,181 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
Rate-Constrained Wireless Networks with Fading Channels: Interference-Limited and Noise-Limited Regimes
A network of wireless communication links is considered in a Rayleigh
fading environment. It is assumed that each link can be active and transmit
with a constant power or remain silent. The objective is to maximize the
number of active links such that each active link can transmit with a constant
rate . An upper bound is derived that shows the number of active links
scales at most like . To obtain a lower bound, a
decentralized link activation strategy is described and analyzed. It is shown
that for small values of , the number of supported links by this
strategy meets the upper bound; however, as grows, this number
becomes far below the upper bound. To shrink the gap between the upper bound
and the achievability result, a modified link activation strategy is proposed
and analyzed based on some results from random graph theory. It is shown that
this modified strategy performs very close to the optimum. Specifically, this
strategy is \emph{asymptotically almost surely} optimum when
approaches or 0. It turns out the optimality results are obtained in
an interference-limited regime. It is demonstrated that, by proper selection of
the algorithm parameters, the proposed scheme also allows the network to
operate in a noise-limited regime in which the transmission rates can be
adjusted by the transmission powers. The price for this flexibility is a
decrease in the throughput scaling law by a multiplicative factor of .Comment: Submitted to IEEE Trans. Information Theor
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
Quantum Analogy of Poisson Geometry, Related Dendriform Algebras and Rota-Baxter Operators
We will introduce an associative (or quantum) version of Poisson structure
tensors. This object is defined as an operator satisfying a "generalized"
Rota-Baxter identity of weight zero. Such operators are called generalized
Rota-Baxter operators. We will show that generalized Rota-Baxter operators are
characterized by a cocycle condition so that Poisson structures are so. By
analogy with twisted Poisson structures, we propose a new operator "twisted
Rota-Baxter operators" which is a natural generalization of generalized
Rota-Baxter operators. It is known that classical Rota-Baxter operators are
closely related with dendriform algebras. We will show that twisted Rota-Baxter
operators induce NS-algebras which is a twisted version of dendriform algebra.
The twisted Poisson condition is considered as a Maurer-Cartan equation up to
homotopy. We will show the twisted Rota-Baxter condition also is so. And we
will study a Poisson-geometric reason, how the twisted Rota-Baxter condition
arises.Comment: 18 pages. Final versio
Characterization of Rate Region in Interference Channels with Constrained Power
In this paper, an -user Gaussian interference channel, where the power of
the transmitters are subject to some upper-bounds is studied. We obtain a
closed-form expression for the rate region of such a channel based on the
Perron-Frobenius theorem. While the boundary of the rate region for the case of
unconstrained power is a well-established result, this is the first result for
the case of constrained power. We extend this result to the time-varying
channels and obtain a closed-form solution for the rate region of such
channels.Comment: 21 Pages, The Conference Version is Submitted to IEEE International
Symposium on Information Theory (ISIT2007
Time-ordering and a generalized Magnus expansion
Both the classical time-ordering and the Magnus expansion are well-known in
the context of linear initial value problems. Motivated by the noncommutativity
between time-ordering and time derivation, and related problems raised recently
in statistical physics, we introduce a generalization of the Magnus expansion.
Whereas the classical expansion computes the logarithm of the evolution
operator of a linear differential equation, our generalization addresses the
same problem, including however directly a non-trivial initial condition. As a
by-product we recover a variant of the time ordering operation, known as
T*-ordering. Eventually, placing our results in the general context of
Rota-Baxter algebras permits us to present them in a more natural algebraic
setting. It encompasses, for example, the case where one considers linear
difference equations instead of linear differential equations
Subdeterminant Maximization via Nonconvex Relaxations and Anti-concentration
Several fundamental problems that arise in optimization and computer science
can be cast as follows: Given vectors and a
constraint family , find a set that
maximizes the squared volume of the simplex spanned by the vectors in . A
motivating example is the data-summarization problem in machine learning where
one is given a collection of vectors that represent data such as documents or
images. The volume of a set of vectors is used as a measure of their diversity,
and partition or matroid constraints over are imposed in order to ensure
resource or fairness constraints. Recently, Nikolov and Singh presented a
convex program and showed how it can be used to estimate the value of the most
diverse set when corresponds to a partition matroid. This result was
recently extended to regular matroids in works of Straszak and Vishnoi, and
Anari and Oveis Gharan. The question of whether these estimation algorithms can
be converted into the more useful approximation algorithms -- that also output
a set -- remained open.
The main contribution of this paper is to give the first approximation
algorithms for both partition and regular matroids. We present novel
formulations for the subdeterminant maximization problem for these matroids;
this reduces them to the problem of finding a point that maximizes the absolute
value of a nonconvex function over a Cartesian product of probability
simplices. The technical core of our results is a new anti-concentration
inequality for dependent random variables that allows us to relate the optimal
value of these nonconvex functions to their value at a random point. Unlike
prior work on the constrained subdeterminant maximization problem, our proofs
do not rely on real-stability or convexity and could be of independent interest
both in algorithms and complexity.Comment: in FOCS 201
- âŠ