5,784 research outputs found
Minimal Coverability Set for Petri Nets: Karp and Miller Algorithm with Pruning
This paper presents the Monotone-Pruning algorithm (MP) for computing the minimal coverability set of Petri nets. The original Karp and Miller algorithm (K&M) unfolds the reachability graph of a Petri net and uses acceleration on branches to ensure termination. The MP algorithm improves the K&M algorithm by adding pruning between branches of the K&M tree. This idea was first introduced in the Minimal Coverability Tree algorithm (MCT), however it was recently shown to be incomplete. The MP algorithm can be viewed as the MCT algorithm with a slightly more aggressive pruning strategy which ensures completeness. Experimental results show that this algorithm is a strong improvement over the K&M algorithm as it dramatically reduces the exploration tree
Quasi-exactly Solvable Lie Superalgebras of Differential Operators
In this paper, we study Lie superalgebras of matrix-valued
first-order differential operators on the complex line. We first completely
classify all such superalgebras of finite dimension. Among the
finite-dimensional superalgebras whose odd subspace is nontrivial, we find
those admitting a finite-dimensional invariant module of smooth vector-valued
functions, and classify all the resulting finite-dimensional modules. The
latter Lie superalgebras and their modules are the building blocks in the
construction of QES quantum mechanical models for spin 1/2 particles in one
dimension.Comment: LaTeX2e using the amstex and amssymb packages, 24 page
Quasi-Exactly Solvable N-Body Spin Hamiltonians with Short-Range Interaction Potentials
We review some recent results on quasi-exactly solvable spin models
presenting near-neighbors interactions. These systems can be understood as
cyclic generalizations of the usual Calogero-Sutherland models. A nontrivial
modification of the exchange operator formalism is used to obtain several
infinite families of eigenfunctions of these models in closed form.Comment: This is a contribution to the Proc. of workshop on Geometric Aspects
of Integrable Systems (July 17-19, 2006; Coimbra, Portugal), published in
SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at
http://www.emis.de/journals/SIGMA
A Haldane-Shastry spin chain of BC_N type in a constant magnetic field
We compute the spectrum of the trigonometric Sutherland spin model of BC_N
type in the presence of a constant magnetic field. Using Polychronakos's
freezing trick, we derive an exact formula for the partition function of its
associated Haldane-Shastry spin chain.Comment: LaTeX, 13 page
The exactly solvable spin Sutherland model of B_N type and its related spin chain
We compute the spectrum of the su(m) spin Sutherland model of B_N type,
including the exact degeneracy of all energy levels. By studying the large
coupling constant limit of this model and of its scalar counterpart, we
evaluate the partition function of their associated spin chain of
Haldane-Shastry type in closed form. With the help of the formula for the
partition function thus obtained we study the chain's spectrum, showing that it
cannot be obtained as a limiting case of its BC_N counterpart. The structure of
the partition function also suggests that the spectrum of the Haldane-Shastry
spin chain of B_N type is equivalent to that of a suitable vertex model, as is
the case for its A_{N-1} counterpart, and that the density of its eigenvalues
is normally distributed when the number of sites N tends to infinity. We
analyze this last conjecture numerically using again the explicit formula for
the partition function, and check its validity for several values of N and m.Comment: Typeset in LaTeX (24 pages, 4 figures). arXiv admin note: text
overlap with arXiv:0909.296
Quasi-Exactly Solvable Potentials on the Line and Orthogonal Polynomials
In this paper we show that a quasi-exactly solvable (normalizable or
periodic) one-dimensional Hamiltonian satisfying very mild conditions defines a
family of weakly orthogonal polynomials which obey a three-term recursion
relation. In particular, we prove that (normalizable) exactly-solvable
one-dimensional systems are characterized by the fact that their associated
polynomials satisfy a two-term recursion relation. We study the properties of
the family of weakly orthogonal polynomials defined by an arbitrary
one-dimensional quasi-exactly solvable Hamiltonian, showing in particular that
its associated Stieltjes measure is supported on a finite set. From this we
deduce that the corresponding moment problem is determined, and that the -th
moment grows like the -th power of a constant as tends to infinity. We
also show that the moments satisfy a constant coefficient linear difference
equation, and that this property actually characterizes weakly orthogonal
polynomial systems.Comment: 22 pages, plain TeX. Please typeset only the file orth.te
The Berry-Tabor conjecture for spin chains of Haldane-Shastry type
According to a long-standing conjecture of Berry and Tabor, the distribution
of the spacings between consecutive levels of a "generic'' integrable model
should follow Poisson's law. In contrast, the spacings distribution of chaotic
systems typically follows Wigner's law. An important exception to the
Berry-Tabor conjecture is the integrable spin chain with long-range
interactions introduced by Haldane and Shastry in 1988, whose spacings
distribution is neither Poissonian nor of Wigner's type. In this letter we
argue that the cumulative spacings distribution of this chain should follow the
"square root of a logarithm'' law recently proposed by us as a characteristic
feature of all spin chains of Haldane-Shastry type. We also show in detail that
the latter law is valid for the rational counterpart of the Haldane-Shastry
chain introduced by Polychronakos.Comment: LaTeX with revtex4, 6 pages, 6 figure
- …