3,184 research outputs found
A Reduced Form for Linear Differential Systems and its Application to Integrability of Hamiltonian Systems
Let with be a differential linear
system. We say that a matrix is a {\em reduced
form} of if and there exists such that . Such a form is
often the sparsest possible attainable through gauge transformations without
introducing new transcendants. In this article, we discuss how to compute
reduced forms of some symplectic differential systems, arising as variational
equations of hamiltonian systems. We use this to give an effective form of the
Morales-Ramis theorem on (non)-integrability of Hamiltonian systems.Comment: 28 page
The cultural epigenetics of psychopathology: The missing heritability of complex diseases found?
We extend a cognitive paradigm for gene expression based on the asymptotic limit theorems of information theory to the epigenetic epidemiology of mental disorders. In particular, we recognize the fundamental role culture plays in human biology, another heritage mechanism parallel to, and interacting with, the more familiar genetic and epigenetic systems. We do this via a model through which culture acts as another tunable epigenetic catalyst that both directs developmental trajectories, and becomes convoluted with individual ontology, via a mutually-interacting crosstalk mediated by a social interaction that is itself culturally driven. We call for the incorporation of embedding culture as an essential component of the epigenetic regulation of human mental development and its dysfunctions, bringing what is perhaps the central reality of human biology into the center of biological psychiatry. Current US work on gene-environment interactions in psychiatry must be extended to a model of gene-environment-culture interaction to avoid becoming victim of an extreme American individualism that threatens to create paradigms particular to that culture and that are, indeed, peculiar in the context of the world's cultures. The cultural and epigenetic systems of heritage may well provide the 'missing' heritability of complex diseases now under so much intense discussion
Methods in Mathematica for Solving Ordinary Differential Equations
An overview of the solution methods for ordinary differential equations in
the Mathematica function DSolve is presented.Comment: 13 page
On Abel's problem and Gauss congruences
A classical problem due to Abel is to determine if a differential equation
admits a non-trivial solution algebraic over
when is a given algebraic function over . Risch designed
an algorithm that, given , determines whether there exists an algebraic
solution or not. In this paper, we adopt a different point of view when
admits a Puiseux expansion with rational coefficients at some point in , which can be assumed to be 0 without loss of generality. We
prove the following arithmetic characterization: there exists a non-trivial
algebraic solution of if and only if the coefficients of the
Puiseux expansion of at satisfy Gauss congruences for almost all
prime numbers. We then apply our criterion to hypergeometric series: we
completely determine the equations with an algebraic solution when
is an algebraic hypergeometric series with rational parameters, and
this enables us to prove a prediction Golyshev made using the theory of
motives. We also present three other applications, in particular to diagonals
of rational fractions and to directed two-dimensional walks
On globally nilpotent differential equations
In a previous work of the authors, a middle convolution operation on the
category of Fuchsian differential systems was introduced. In this note we show
that the middle convolution of Fuchsian systems preserves the property of
global nilpotence. This leads to a globally nilpotent Fuchsian system of rank
two which does not belong to the known classes of globally nilpotent rank two
systems. Moreover, we give a globally nilpotent Fuchsian system of rank seven
whose differential Galois group is isomorphic to the exceptional simple
algebraic group of type $G_2.
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