27,766 research outputs found
Beyond series expansions: mathematical structures for the susceptibility of the square lattice Ising model
We first study the properties of the Fuchsian ordinary differential equations
for the three and four-particle contributions and
of the square lattice Ising model susceptibility. An analysis of some
mathematical properties of these Fuchsian differential equations is sketched.
For instance, we study the factorization properties of the corresponding linear
differential operators, and consider the singularities of the three and
four-particle contributions and , versus the
singularities of the associated Fuchsian ordinary differential equations, which
actually exhibit new ``Landau-like'' singularities. We sketch the analysis of
the corresponding differential Galois groups. In particular we provide a
simple, but efficient, method to calculate the so-called ``connection
matrices'' (between two neighboring singularities) and deduce the singular
behaviors of and . We provide a set of comments and
speculations on the Fuchsian ordinary differential equations associated with
the -particle contributions and address the problem of the
apparent discrepancy between such a holonomic approach and some scaling results
deduced from a Painlev\'e oriented approach.Comment: 21 pages Proceedings of the Counting Complexity conferenc
Holonomic functions of several complex variables and singularities of anisotropic Ising n-fold integrals
Lattice statistical mechanics, often provides a natural (holonomic) framework
to perform singularity analysis with several complex variables that would, in a
general mathematical framework, be too complex, or could not be defined.
Considering several Picard-Fuchs systems of two-variables "above" Calabi-Yau
ODEs, associated with double hypergeometric series, we show that holonomic
functions are actually a good framework for actually finding the singular
manifolds. We, then, analyse the singular algebraic varieties of the n-fold
integrals , corresponding to the decomposition of the magnetic
susceptibility of the anisotropic square Ising model. We revisit a set of
Nickelian singularities that turns out to be a two-parameter family of elliptic
curves. We then find a first set of non-Nickelian singularities for and , that also turns out to be rational or ellipic
curves. We underline the fact that these singular curves depend on the
anisotropy of the Ising model. We address, from a birational viewpoint, the
emergence of families of elliptic curves, and of Calabi-Yau manifolds on such
problems. We discuss the accumulation of these singular curves for the
non-holonomic anisotropic full susceptibility.Comment: 36 page
Gravitational collapse of barotropic spherical fluids
The gravitational collapse of spherical, barotropic perfect fluids is
analyzed here. For the first time, the final state of these systems is studied
without resorting to simplifying assumptions - such as self-similarity - using
a new approach based on non-linear o.d.e. techniques, and formation of naked
singularities is shown to occur for solutions such that the mass function is
analytic in a neighborhood of the spacetime singularity.Comment: 17 pages, LaTeX2
A thermodynamic characterization of future singularities?
In this Letter we consider three future singularities in different
Friedmann-Lema\^{\i}tre-Robertson-Walker scenarios and show that the universe
departs more and more from thermodynamic equilibrium as the corresponding
singularity is approached. Though not proven in general, this feature may
characterize future singularities of homogeneous and isotropic cosmologies.Comment: Key words: mathematical cosmology, singularities, thermodynamics; 11
pages, to be publised in Physics Letters
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