104 research outputs found
Letter graphs and geometric grid classes of permutations: characterization and recognition
In this paper, we reveal an intriguing relationship between two seemingly
unrelated notions: letter graphs and geometric grid classes of permutations. An
important property common for both of them is well-quasi-orderability,
implying, in a non-constructive way, a polynomial-time recognition of geometric
grid classes of permutations and -letter graphs for a fixed . However,
constructive algorithms are available only for . In this paper, we present
the first constructive polynomial-time algorithm for the recognition of
-letter graphs. It is based on a structural characterization of graphs in
this class.Comment: arXiv admin note: text overlap with arXiv:1108.6319 by other author
Monomer-dimer model in two-dimensional rectangular lattices with fixed dimer density
The classical monomer-dimer model in two-dimensional lattices has been shown
to belong to the \emph{``#P-complete''} class, which indicates the problem is
computationally ``intractable''. We use exact computational method to
investigate the number of ways to arrange dimers on
two-dimensional rectangular lattice strips with fixed dimer density . For
any dimer density , we find a logarithmic correction term in the
finite-size correction of the free energy per lattice site. The coefficient of
the logarithmic correction term is exactly -1/2. This logarithmic correction
term is explained by the newly developed asymptotic theory of Pemantle and
Wilson. The sequence of the free energy of lattice strips with cylinder
boundary condition converges so fast that very accurate free energy
for large lattices can be obtained. For example, for a half-filled lattice,
, while and . For , is accurate at least to 10 decimal
digits. The function reaches the maximum value at , with 11 correct digits. This is also
the \md constant for two-dimensional rectangular lattices. The asymptotic
expressions of free energy near close packing are investigated for finite and
infinite lattice widths. For lattices with finite width, dependence on the
parity of the lattice width is found. For infinite lattices, the data support
the functional form obtained previously through series expansions.Comment: 15 pages, 5 figures, 5 table
Three osculating walkers
We consider three directed walkers on the square lattice, which move
simultaneously at each tick of a clock and never cross. Their trajectories form
a non-crossing configuration of walks. This configuration is said to be
osculating if the walkers never share an edge, and vicious (or:
non-intersecting) if they never meet. We give a closed form expression for the
generating function of osculating configurations starting from prescribed
points. This generating function turns out to be algebraic. We also relate the
enumeration of osculating configurations with prescribed starting and ending
points to the (better understood) enumeration of non-intersecting
configurations. Our method is based on a step by step decomposition of
osculating configurations, and on the solution of the functional equation
provided by this decomposition
Asymptotics of Selberg-like integrals: The unitary case and Newton's interpolation formula
We investigate the asymptotic behavior of the Selberg-like integral ,
as for different scalings of the parameters and with .
Integrals of this type arise in the random matrix theory of electronic
scattering in chaotic cavities supporting channels in the two attached
leads. Making use of Newton's interpolation formula, we show that an asymptotic
limit exists and we compute it explicitly
Form Sequences to Polynomials and Back, via Operator Orderings
C.M. Bender and G. V. Dunne showed that linear combinations of words
, where and are subject to the relation , may be expressed as a polynomial in the symbol . Relations between such polynomials and linear
combinations of the transformed coefficients are explored. In particular,
examples yielding orthogonal polynomials are provided
Super congruences and Euler numbers
Let be a prime. We prove that
, where E_0,E_1,E_2,... are Euler numbers. Our new approach is of
combinatorial nature. We also formulate many conjectures concerning super
congruences and relate most of them to Euler numbers or Bernoulli numbers.
Motivated by our investigation of super congruences, we also raise a conjecture
on 7 new series for , and the constant
(with (-) the Jacobi symbol), two of which are
and
\sum_{k>0}(15k-4)(-27)^{k-1}/(k^3\binom{2k}{k}^2\binom{3k}k)=K.$
Simplifying Multiple Sums in Difference Fields
In this survey article we present difference field algorithms for symbolic
summation. Special emphasize is put on new aspects in how the summation
problems are rephrased in terms of difference fields, how the problems are
solved there, and how the derived results in the given difference field can be
reinterpreted as solutions of the input problem. The algorithms are illustrated
with the Mathematica package \SigmaP\ by discovering and proving new harmonic
number identities extending those from (Paule and Schneider, 2003). In
addition, the newly developed package \texttt{EvaluateMultiSums} is introduced
that combines the presented tools. In this way, large scale summation problems
for the evaluation of Feynman diagrams in QCD (Quantum ChromoDynamics) can be
solved completely automatically.Comment: Uses svmult.cls, to appear as contribution in the book "Computer
Algebra in Quantum Field Theory: Integration, Summation and Special
Functions" (www.Springer.com
Convergence Acceleration via Combined Nonlinear-Condensation Transformations
A method of numerically evaluating slowly convergent monotone series is
described. First, we apply a condensation transformation due to Van Wijngaarden
to the original series. This transforms the original monotone series into an
alternating series. In the second step, the convergence of the transformed
series is accelerated with the help of suitable nonlinear sequence
transformations that are known to be particularly powerful for alternating
series. Some theoretical aspects of our approach are discussed. The efficiency,
numerical stability, and wide applicability of the combined
nonlinear-condensation transformation is illustrated by a number of examples.
We discuss the evaluation of special functions close to or on the boundary of
the circle of convergence, even in the vicinity of singularities. We also
consider a series of products of spherical Bessel functions, which serves as a
model for partial wave expansions occurring in quantum electrodynamic bound
state calculations.Comment: 24 pages, LaTeX, 12 tables (accepted for publication in Comput. Phys.
Comm.
Logarithmic and complex constant term identities
In recent work on the representation theory of vertex algebras related to the
Virasoro minimal models M(2,p), Adamovic and Milas discovered logarithmic
analogues of (special cases of) the famous Dyson and Morris constant term
identities. In this paper we show how the identities of Adamovic and Milas
arise naturally by differentiating as-yet-conjectural complex analogues of the
constant term identities of Dyson and Morris. We also discuss the existence of
complex and logarithmic constant term identities for arbitrary root systems,
and in particular prove complex and logarithmic constant term identities for
the root system G_2.Comment: 26 page
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