29 research outputs found
Punctured polygons and polyominoes on the square lattice
We use the finite lattice method to count the number of punctured staircase
and self-avoiding polygons with up to three holes on the square lattice. New or
radically extended series have been derived for both the perimeter and area
generating functions. We show that the critical point is unchanged by a finite
number of punctures, and that the critical exponent increases by a fixed amount
for each puncture. The increase is 1.5 per puncture when enumerating by
perimeter and 1.0 when enumerating by area. A refined estimate of the
connective constant for polygons by area is given. A similar set of results is
obtained for finitely punctured polyominoes. The exponent increase is proved to
be 1.0 per puncture for polyominoes.Comment: 36 pages, 11 figure
Globally nilpotent differential operators and the square Ising model
We recall various multiple integrals related to the isotropic square Ising
model, and corresponding, respectively, to the n-particle contributions of the
magnetic susceptibility, to the (lattice) form factors, to the two-point
correlation functions and to their lambda-extensions. These integrals are
holonomic and even G-functions: they satisfy Fuchsian linear differential
equations with polynomial coefficients and have some arithmetic properties. We
recall the explicit forms, found in previous work, of these Fuchsian equations.
These differential operators are very selected Fuchsian linear differential
operators, and their remarkable properties have a deep geometrical origin: they
are all globally nilpotent, or, sometimes, even have zero p-curvature. Focusing
on the factorised parts of all these operators, we find out that the global
nilpotence of the factors corresponds to a set of selected structures of
algebraic geometry: elliptic curves, modular curves, and even a remarkable
weight-1 modular form emerging in the three-particle contribution
of the magnetic susceptibility of the square Ising model. In the case where we
do not have G-functions, but Hamburger functions (one irregular singularity at
0 or ) that correspond to the confluence of singularities in the
scaling limit, the p-curvature is also found to verify new structures
associated with simple deformations of the nilpotent property.Comment: 55 page
Difference system for Selberg correlation integrals
The Selberg correlation integrals are averages of the products
with respect to the Selberg
density. Our interest is in the case , , when this
corresponds to the -th moment of the corresponding characteristic
polynomial. We give the explicit form of a matrix linear
difference system in the variable which determines the average, and we
give the Gauss decomposition of the corresponding matrix.
For a positive integer the difference system can be used to efficiently
compute the power series defined by this average.Comment: 21 page
Diagonals of rational functions, pullbacked 2F1 hypergeometric functions and modular forms (unabrigded version)
We recall that diagonals of rational functions naturally occur in lattice
statistical mechanics and enumerative combinatorics. We find that a
seven-parameter rational function of three variables with a numerator equal to
one (reciprocal of a polynomial of degree two at most) can be expressed as a
pullbacked 2F1 hypergeometric function. This result can be seen as the simplest
non-trivial family of diagonals of rational functions. We focus on some
subcases such that the diagonals of the corresponding rational functions can be
written as a pullbacked 2F1 hypergeometric function with two possible rational
functions pullbacks algebraically related by modular equations, thus showing
explicitely that the diagonal is a modular form. We then generalise this result
to eight, nine and ten parameters families adding some selected cubic terms at
the denominator of the rational function defining the diagonal. We finally show
that each of these previous rational functions yields an infinite number of
rational functions whose diagonals are also pullbacked 2F1 hypergeometric
functions and modular forms.Comment: 39 page
A generic housing grammar for the generation of different housing languages: a generic housing shape grammar for Palladian villas, Prairie and Malagueira houses
Shape grammars have traditionally described a design language and replicated it using a procedure. In the majority of existing studies, one language corresponded to one grammar and vice versa; the generative procedure was univocal and language specific. Generic grammars, which are capable of describing multiple design languages, potentially allow greater flexibility and help describe not only languages but relationships between languages. This study proposes a generic housing process based on a parametric shape grammar, and uses this to investigate relationships between several grammars or families of designs. A study case of three single housing grammars was selected using the Palladian villas, Prairie and Malagueira houses. Specific parameterisation confers the sense of style required to define a language. From the generated corpora two methods were exercised to explore two research questions: 1. A qualitative method tested how the parametric space of a shape grammar corresponded with our intuition of similarities and differences amongst designs. This was performed using a set of questionnaires posed to both laymen and expert observers. 2. A quantitative method was used to test how well the parametric space of a shape grammar coincided with the design space expressed by the different corpora. Principal Components Analysis was used to inform if the set of parameters used to design the solutions would group into clusters. Results indicate that the expected relationships between individual designs are captured by the generic grammar. The design solutions generated by the generic grammar were also naturally perceived by observers and clustering was identified amongst language related design solutions. A tool such as a generic shape grammar captures the principles of design as described by the generative shape rules and its parameterisation, which can be used in academia, practice or analysis to explore design