701 research outputs found
Asymptotically fast polynomial matrix algorithms for multivariable systems
We present the asymptotically fastest known algorithms for some basic
problems on univariate polynomial matrices: rank, nullspace, determinant,
generic inverse, reduced form. We show that they essentially can be reduced to
two computer algebra techniques, minimal basis computations and matrix fraction
expansion/reconstruction, and to polynomial matrix multiplication. Such
reductions eventually imply that all these problems can be solved in about the
same amount of time as polynomial matrix multiplication
Algebraic approach to q-deformed supersymmetric variants of the Hubbard model with pair hoppings
We construct two quantum spin chains Hamiltonians with quantum sl(2|1)
invariance. These spin chains define variants of the Hubbard model and describe
electron models with pair hoppings. A cubic algebra that admits the
Birman-Wenzl-Murakami algebra as a quotient allows exact solvability of the
periodic chain. The two Hamiltonians, respectively built using the
distinguished and the fermionic bases of U_q(sl(2|1)) differ only in the
boundary terms. They are actually equivalent, but the equivalence is non local.
Reflection equations are solved to get exact solvability on open chains with
non trivial boundary conditions. Two families of diagonal solutions are found.
The centre and the Scasimirs of the quantum enveloping algebra of sl(2|1)
appear as tools for the construction of exactly solvable Hamiltonians.Comment: 22 pages, LaTeX2e, uses amsfonts; some references added and typos
correcte
The Multivariate Resultant is NP-hard in any Characteristic
The multivariate resultant is a fundamental tool of computational algebraic
geometry. It can in particular be used to decide whether a system of n
homogeneous equations in n variables is satisfiable (the resultant is a
polynomial in the system's coefficients which vanishes if and only if the
system is satisfiable). In this paper we present several NP-hardness results
for testing whether a multivariate resultant vanishes, or equivalently for
deciding whether a square system of homogeneous equations is satisfiable. Our
main result is that testing the resultant for zero is NP-hard under
deterministic reductions in any characteristic, for systems of low-degree
polynomials with coefficients in the ground field (rather than in an
extension). We also observe that in characteristic zero, this problem is in the
Arthur-Merlin class AM if the generalized Riemann hypothesis holds true. In
positive characteristic, the best upper bound remains PSPACE.Comment: 13 page
On the Voting Time of the Deterministic Majority Process
In the deterministic binary majority process we are given a simple graph
where each node has one out of two initial opinions. In every round, every node
adopts the majority opinion among its neighbors. By using a potential argument
first discovered by Goles and Olivos (1980), it is known that this process
always converges in rounds to a two-periodic state in which every node
either keeps its opinion or changes it in every round.
It has been shown by Frischknecht, Keller, and Wattenhofer (2013) that the
bound on the convergence time of the deterministic binary majority
process is indeed tight even for dense graphs. However, in many graphs such as
the complete graph, from any initial opinion assignment, the process converges
in just a constant number of rounds.
By carefully exploiting the structure of the potential function by Goles and
Olivos (1980), we derive a new upper bound on the convergence time of the
deterministic binary majority process that accounts for such exceptional cases.
We show that it is possible to identify certain modules of a graph in order
to obtain a new graph with the property that the worst-case
convergence time of is an upper bound on that of . Moreover, even
though our upper bound can be computed in linear time, we show that, given an
integer , it is NP-hard to decide whether there exists an initial opinion
assignment for which it takes more than rounds to converge to the
two-periodic state.Comment: full version of brief announcement accepted at DISC'1
Complete controllability of quantum systems
Complete controllability is a fundamental issue in the field of control of quantum systems because of its implications for dynamical realizability of the kinematical bounds on the optimization of observables. Of special importance is the question of complete controllability of Morse and harmonic oscillators since they serve as basic models for many physical systems. We prove that most non-decomposable, anharmonic quantum systems, including the N-level Morse oscillator, are completely controllable with a single control. Furthermore, we establish sufficient conditions for complete controllability of systems with equally spaced energy levels and show that they are satisfied by the standard N-level harmonic oscillator
Bi-criteria Pipeline Mappings for Parallel Image Processing
Mapping workflow applications onto parallel platforms is a challenging
problem, even for simple application patterns such as pipeline graphs. Several
antagonistic criteria should be optimized, such as throughput and latency (or a
combination). Typical applications include digital image processing, where
images are processed in steady-state mode. In this paper, we study the mapping
of a particular image processing application, the JPEG encoding. Mapping
pipelined JPEG encoding onto parallel platforms is useful for instance for
encoding Motion JPEG images. As the bi-criteria mapping problem is NP-complete,
we concentrate on the evaluation and performance of polynomial heuristics
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