33,369 research outputs found
Three-term recurrence relations of minimal affinizations of type
Minimal affinizations form a class of modules of quantum affine algebras
introduced by Chari. We introduce a system of equations satisfied by the
-characters of minimal affinizations of type which we call the
M-system of type . The M-system of type contains all minimal
affinizations of type and only contains minimal affinizations. The
equations in the M-system of type are three-term recurrence relations.
The M-system of type is much simpler than the extended T-system of type
obtained by Mukhin and the second author. We also interpret the
three-term recurrence relations in the M-system of type as exchange
relations in a cluster algebra constructed by Hernandez and Leclerc.Comment: 23 pages. The original name of the paper is: Cluster algebras and
minimal affinizations of representations of the quantum group of type $G_2
Maximal Scheduling in Wireless Networks with Priorities
We consider a general class of low complexity distributed scheduling
algorithms in wireless networks, maximal scheduling with priorities, where a
maximal set of transmitting links in each time slot are selected according to
certain pre-specified static priorities. The proposed scheduling scheme is
simple, which is easily amendable for distributed implementation in practice,
such as using inter-frame space (IFS) parameters under the ubiquitous 802.11
protocols. To obtain throughput guarantees, we first analyze the case of
maximal scheduling with a fixed priority vector, and formulate a lower bound on
its stability region and scheduling efficiency. We further propose a low
complexity priority assignment algorithm, which can stabilize any arrival rate
that is in the union of the lower bound regions of all priorities. The
stability result is proved using fluid limits, and can be applied to very
general stochastic arrival processes. Finally, the performance of the proposed
prioritized maximal scheduling scheme is verified by simulation results.Comment: 10 pages, 6 figure
Linear algebraic analogues of the graph isomorphism problem and the Erd\H{o}s-R\'enyi model
A classical difficult isomorphism testing problem is to test isomorphism of
p-groups of class 2 and exponent p in time polynomial in the group order. It is
known that this problem can be reduced to solving the alternating matrix space
isometry problem over a finite field in time polynomial in the underlying
vector space size. We propose a venue of attack for the latter problem by
viewing it as a linear algebraic analogue of the graph isomorphism problem.
This viewpoint leads us to explore the possibility of transferring techniques
for graph isomorphism to this long-believed bottleneck case of group
isomorphism.
In 1970's, Babai, Erd\H{o}s, and Selkow presented the first average-case
efficient graph isomorphism testing algorithm (SIAM J Computing, 1980).
Inspired by that algorithm, we devise an average-case efficient algorithm for
the alternating matrix space isometry problem over a key range of parameters,
in a random model of alternating matrix spaces in vein of the Erd\H{o}s-R\'enyi
model of random graphs. For this, we develop a linear algebraic analogue of the
classical individualisation technique, a technique belonging to a set of
combinatorial techniques that has been critical for the progress on the
worst-case time complexity for graph isomorphism, but was missing in the group
isomorphism context. As a consequence of the main algorithm, we establish a
weaker linear algebraic analogue of Erd\H{o}s and R\'enyi's classical result
that most graphs have the trivial automorphism group. We finally show that
Luks' dynamic programming technique for graph isomorphism (STOC 1999) can be
adapted to slightly improve the worst-case time complexity of the alternating
matrix space isometry problem in a certain range of parameters.Comment: 32 pages, 2 figures. The implication to group enumeration correcte
Representations of Rota-Baxter algebras and regular singular decompositions
There is a Rota-Baxter algebra structure on the field
with being the projection map onto . We study the representation
theory and regular-singular decompositions of any finite dimensional -vector
space. The main result shows that the category of finite dimensional
representations is semisimple and consists of exactly three isomorphism classes
of irreducible representations which are all one-dimensional. As a consequence,
the number of -orbits in the set of all regular-singular
decompositions of an -dimensional -vector space is .
We also use the result to compute the generalized class number, i.e., the
number of the -isomorphism classes of finitely generated
-submodules of
Representations of Polynomial Rota-Baxter Algebras
A Rota--Baxter operator is an algebraic abstraction of integration, which is
the typical example of a weight zero Rota-Baxter operator. We show that
studying the modules over the polynomial Rota--Baxter algebra is
equivalent to studying the modules over the Jordan plane, and we generalize the
direct decomposability results for the -modules in [Iy] from
algebraically closed fields of characteristic zero to fields of characteristic
zero. Furthermore, we provide a classification of Rota--Baxter modules up to
isomorphism based on indecomposable -modules
Mean Li-Yorke chaos along some good sequences
If a topological dynamical system has positive topological entropy,
then it is multivariant mean Li-Yorke chaotic along a sequence
of positive integers which is "good" for pointwise
ergodic convergence with a mild condition; more specifically, there exists a
Cantor subset of such that for every and pairwise distinct
points in we have
and
Examples are given for the classic sequences of
primes and generalized polynomials.Comment: 18 page
Greedy Maximal Scheduling in Wireless Networks
In this paper we consider greedy scheduling algorithms in wireless networks,
i.e., the schedules are computed by adding links greedily based on some
priority vector. Two special cases are considered: 1) Longest Queue First (LQF)
scheduling, where the priorities are computed using queue lengths, and 2)
Static Priority (SP) scheduling, where the priorities are pre-assigned. We
first propose a closed-form lower bound stability region for LQF scheduling,
and discuss the tightness result in some scenarios. We then propose an lower
bound stability region for SP scheduling with multiple priority vectors, as
well as a heuristic priority assignment algorithm, which is related to the
well-known Expectation-Maximization (EM) algorithm. The performance gain of the
proposed heuristic algorithm is finally confirmed by simulations.Comment: 6 pages, 3 figures. A shorter version will appear in IEEE Globecom
201
Scheduling in Wireless Networks under Uncertainties: A Greedy Primal-Dual Approach
This paper proposes a dynamic primal-dual type algorithm to solve the optimal
scheduling problem in wireless networks subject to uncertain parameters, which
are generated by stochastic network processes such as random packet arrivals,
channel fading, and node mobilities. The algorithm is a generalization of the
well-known max-weight scheduling algorithm proposed by Tassiulas et al., where
only queue length information is used for computing the schedules when the
arrival rates are uncertain. Using the technique of fluid limits, sample path
convergence of the algorithm to an arbitrarily close to optimal solution is
proved, under the assumption that the Strong Law of Large Numbers (SLLN)
applies to the random processes which generate the uncertain parameters. The
performance of the algorithm is further verified by simulation results. The
method may potentially be applied to other applications where dynamic
algorithms for convex problems with uncertain parameters are needed.Comment: 8 pages, 2 figures, submitted to Allerton 2010. The title and
abstract are updated compared to version
A Necessary and Sufficient Criterion for the Separability of Quantum State
Quantum entanglement has been regarded as one of the key physical resources
in quantum information sciences. However, the determination of whether a mixed
state is entangled or not is generally a hard issue, even for the bipartite
system. In this work we propose an operational necessary and sufficient
criterion for the separability of an arbitrary bipartite mixed state, by virtue
of the multiplicative Horn's problem. The work follows the work initiated by
Horodecki {\it et. al.} and uses the Bloch vector representation introduced to
the separability problem by J. De Vicente. In our criterion, a complete and
finite set of inequalities to determine the separability of compound system is
obtained, which may be viewed as trade-off relations between the quantumness of
subsystems. We apply the obtained result to explicit examples, e.g. the
separable decomposition of arbitrary dimension Werner state and isotropic
state.Comment: 33 pages; published in Scientific Report
Reformulating the Quantum Uncertainty Relation
Uncertainty principle is one of the cornerstones of quantum theory. In the
literature, there are two types of uncertainty relations, the operator form
concerning the variances of physical observables and the entropy form related
to entropic quantities. Both these forms are inequalities involving pairwise
observables, and are found to be nontrivial to incorporate multiple
observables. In this work we introduce a new form of uncertainty relation which
may give out complete trade-off relations for variances of observables in pure
and mixed quantum systems. Unlike the prevailing uncertainty relations, which
are either quantum state dependent or not directly measurable, our bounds for
variances of observables are quantum state independent and immune from the
"triviality" problem of having zero expectation values. Furthermore, the new
uncertainty relation may provide a geometric explanation for the reason why
there are limitations on the simultaneous determination of different
observables in -dimensional Hilbert space.Comment: 15 pages, 2 figures; published in Scientific Report
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