59 research outputs found
Diagonal Ising susceptibility: elliptic integrals, modular forms and Calabi-Yau equations
We give the exact expressions of the partial susceptibilities
and for the diagonal susceptibility of the Ising model in terms
of modular forms and Calabi-Yau ODEs, and more specifically,
and hypergeometric functions. By solving the connection problems we
analytically compute the behavior at all finite singular points for
and . We also give new results for .
We see in particular, the emergence of a remarkable order-six operator, which
is such that its symmetric square has a rational solution. These new exact
results indicate that the linear differential operators occurring in the
-fold integrals of the Ising model are not only "Derived from Geometry"
(globally nilpotent), but actually correspond to "Special Geometry"
(homomorphic to their formal adjoint). This raises the question of seeing if
these "special geometry" Ising-operators, are "special" ones, reducing, in fact
systematically, to (selected, k-balanced, ...) hypergeometric
functions, or correspond to the more general solutions of Calabi-Yau equations.Comment: 35 page
Canonical decomposition of linear differential operators with selected differential Galois groups
We revisit an order-six linear differential operator having a solution which
is a diagonal of a rational function of three variables. Its exterior square
has a rational solution, indicating that it has a selected differential Galois
group, and is actually homomorphic to its adjoint. We obtain the two
corresponding intertwiners giving this homomorphism to the adjoint. We show
that these intertwiners are also homomorphic to their adjoint and have a simple
decomposition, already underlined in a previous paper, in terms of order-two
self-adjoint operators. From these results, we deduce a new form of
decomposition of operators for this selected order-six linear differential
operator in terms of three order-two self-adjoint operators. We then generalize
the previous decomposition to decompositions in terms of an arbitrary number of
self-adjoint operators of the same parity order. This yields an infinite family
of linear differential operators homomorphic to their adjoint, and, thus, with
a selected differential Galois group. We show that the equivalence of such
operators is compatible with these canonical decompositions. The rational
solutions of the symmetric, or exterior, squares of these selected operators
are, noticeably, seen to depend only on the rightmost self-adjoint operator in
the decomposition. These results, and tools, are applied on operators of large
orders. For instance, it is seen that a large set of (quite massive) operators,
associated with reflexive 4-polytopes defining Calabi-Yau 3-folds, obtained
recently by P. Lairez, correspond to a particular form of the decomposition
detailed in this paper.Comment: 40 page
Cosmic Shear Analysis with CFHTLS Deep data
We present the first cosmic shear measurements obtained from the T0001
release of the Canada-France-Hawaii Telescope Legacy Survey. The data set
covers three uncorrelated patches (D1, D3 and D4) of one square degree each
observed in u*, g', r', i' and z' bands, out to i'=25.5. The depth and the
multicolored observations done in deep fields enable several data quality
controls. The lensing signal is detected in both r' and i' bands and shows
similar amplitude and slope in both filters. B-modes are found to be
statistically zero at all scales. Using multi-color information, we derived a
photometric redshift for each galaxy and separate the sample into medium and
high-z galaxies. A stronger shear signal is detected from the high-z subsample
than from the low-z subsample, as expected from weak lensing tomography. While
further work is needed to model the effects of errors in the photometric
redshifts, this results suggests that it will be possible to obtain constraints
on the growth of dark matter fluctuations with lensing wide field surveys. The
various quality tests and analysis discussed in this work demonstrate that
MegaPrime/Megacam instrument produces excellent quality data. The combined Deep
and Wide surveys give sigma_8= 0.89 pm 0.06 assuming the Peacock & Dodds
non-linear scheme and sigma_8=0.86 pm 0.05 for the halo fitting model and
Omega_m=0.3. We assumed a Cold Dark Matter model with flat geometry.
Systematics, Hubble constant and redshift uncertainties have been marginalized
over. Using only data from the Deep survey, the 1 sigma upper bound for w_0,
the constant equation of state parameter is w_0 < -0.8.Comment: 14 pages, 16 figures, accepted A&
New Travelling Wave solutions of the Korteweg De Vries Equation by - (G'/G') Expansion method.
Exact hyperbolic, trigonometric and rational function travelling wave solutions to the Korteweg De Vries (KDV) equation using the (G'/G) expansion method are presented in this paper. More travelling wave solutions to the KDV equation were obtained with Liu's theorem. The solutions obtained were verified by putting them back into the equation with the aid of Mathematica. This shows that the (G'/G) expansion method is a powerful and effective tool for obtaining exact solutions to nonlinear partial differential equations in physics, mathematics and other applications
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