33 research outputs found
Sur les facteurs des suites de sturm
RésuméCet article a pour objet l'étude d'une construction associant à toute droite de pente p/q (p et q premiers entre eux et q⩽n) un mot de longueur n sur l'alphabet {0,1}. Nous montrons que nous obtenons par cette construction le langage constitué de tous les facteurs des suites de Sturm. Nous formulons, aprés avoir obtenu une èquation fonctionelle dont la solution est la série génératrice de ce langage, une conjecture reliant cette série génératrice à la fonction d'Euler.AbstractIn this paper, we study a construction which connects to each line with slope p/q (such that gcd(p, q) = 1 and q⩽n) a word of length n over the alphabet {0, 1}. We show that this construction yields the language of all the factors of the sturmian sequences. We first obtain a functional equation whose solution is the generating function of this language, and then we give a conjecture relating this generating function to the Euler function
Correlators for the Wigner–Smith time-delay matrix of chaotic cavities
We study the Wigner–Smith time-delay matrix Q of a ballistic quantum dot supporting N scattering channels. We compute the v-point correlators of the power traces Tr Qk for arbitrary v>1 at leading order for large N using techniques from the random matrix theory approach to quantum chromodynamics. We conjecture that the cumulants of the Tr Qkʼs are integer-valued at leading order in N and include a MATHEMATICA code that computes their generating functions recursively
Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models. V. Further Results for the Square-Lattice Chromatic Polynomial
We derive some new structural results for the transfer matrix of
square-lattice Potts models with free and cylindrical boundary conditions. In
particular, we obtain explicit closed-form expressions for the dominant (at
large |q|) diagonal entry in the transfer matrix, for arbitrary widths m, as
the solution of a special one-dimensional polymer model. We also obtain the
large-q expansion of the bulk and surface (resp. corner) free energies for the
zero-temperature antiferromagnet (= chromatic polynomial) through order q^{-47}
(resp. q^{-46}). Finally, we compute chromatic roots for strips of widths 9 <=
m <= 12 with free boundary conditions and locate roughly the limiting curves.Comment: 111 pages (LaTeX2e). Includes tex file, three sty files, and 19
Postscript figures. Also included are Mathematica files data_CYL.m and
data_FREE.m. Many changes from version 1: new material on series expansions
and their analysis, and several proofs of previously conjectured results.
Final version to be published in J. Stat. Phy
Transfer matrices and partition-function zeros for antiferromagnetic Potts models. VI. Square lattice with special boundary conditions
We study, using transfer-matrix methods, the partition-function zeros of the
square-lattice q-state Potts antiferromagnet at zero temperature (=
square-lattice chromatic polynomial) for the special boundary conditions that
are obtained from an m x n grid with free boundary conditions by adjoining one
new vertex adjacent to all the sites in the leftmost column and a second new
vertex adjacent to all the sites in the rightmost column. We provide numerical
evidence that the partition-function zeros are becoming dense everywhere in the
complex q-plane outside the limiting curve B_\infty(sq) for this model with
ordinary (e.g. free or cylindrical) boundary conditions. Despite this, the
infinite-volume free energy is perfectly analytic in this region.Comment: 114 pages (LaTeX2e). Includes tex file, three sty files, and 23
Postscript figures. Also included are Mathematica files data_Eq.m,
data_Neq.m,and data_Diff.m. Many changes from version 1, including several
proofs of previously conjectured results. Final version to be published in J.
Stat. Phy
Euclid preparation. TBD. Forecast impact of super-sample covariance on 3x2pt analysis with Euclid
Deviations from Gaussianity in the distribution of the fields probed by
large-scale structure surveys generate additional terms in the data covariance
matrix, increasing the uncertainties in the measurement of the cosmological
parameters. Super-sample covariance (SSC) is among the largest of these
non-Gaussian contributions, with the potential to significantly degrade
constraints on some of the parameters of the cosmological model under study --
especially for weak lensing cosmic shear. We compute and validate the impact of
SSC on the forecast uncertainties on the cosmological parameters for the Euclid
photometric survey, obtained with a Fisher matrix analysis, both considering
the Gaussian covariance alone and adding the SSC term -- computed through the
public code PySSC. The photometric probes are considered in isolation and
combined in the `32pt' analysis. We find the SSC impact to be
non-negligible -- halving the Figure of Merit of the dark energy parameters
(, ) in the 32pt case and substantially increasing the
uncertainties on , and for cosmic shear;
photometric galaxy clustering, on the other hand, is less affected due to the
lower probe response. The relative impact of SSC does not show significant
changes under variations of the redshift binning scheme, while it is smaller
for weak lensing when marginalising over the multiplicative shear bias nuisance
parameters, which also leads to poorer constraints on the cosmological
parameters. Finally, we explore how the use of prior information on the shear
and galaxy bias changes the SSC impact. Improving shear bias priors does not
have a significant impact, while galaxy bias must be calibrated to sub-percent
level to increase the Figure of Merit by the large amount needed to achieve the
value when SSC is not included.Comment: 22 pages, 13 figure
Equivalence of the two-dimensional directed animal problem to a one-dimensional path problem
AbstractWe introduce a one-to-one correspondence between directed animals on a square lattice and a class of one-dimensional paths. We derive very simply the formulae giving the exact number of directed animals of given size and the average width of such animals. The more surprising result is the fact that the number of compactrooted directed animals of size n is 3n− 1