182 research outputs found
Dynamical systems defining Jacobi's theta-constants
We propose a system of equations that defines Weierstrass--Jacobi's eta- and
theta-constant series in a differentially closed way. This system is shown to
have a direct relationship to a little-known dynamical system obtained by
Jacobi. The classically known differential equations by Darboux--Halphen,
Chazy, and Ramanujan are the differential consequences or reductions of these
systems. The proposed system is shown to admit the Lagrangian, Hamiltonian, and
Nambu formulations. We explicitly construct a pencil of nonlinear Poisson
brackets and complete set of involutive conserved quantities. As byproducts of
the theory, we exemplify conserved quantities for the Ramamani dynamical system
and quadratic system of Halphen--Brioschi.Comment: Final version. Major changes; LaTeX, 23 pages (was 17), no figure
Detection and construction of an elliptic solution to the complex cubic-quintic Ginzburg-Landau equation
In evolution equations for a complex amplitude, the phase obeys a much more
intricate equation than the amplitude. Nevertheless, general methods should be
applicable to both variables. On the example of the traveling wave reduction of
the complex cubic-quintic Ginzburg-Landau equation (CGL5), we explain how to
overcome the difficulties arising in two such methods: (i) the criterium that
the sum of residues of an elliptic solution should be zero, (ii) the
construction of a first order differential equation admitting the given
equation as a differential consequence (subequation method).Comment: 12 pages, no figure, to appear, Theoretical and Mathematical Physic
Projective dynamics and classical gravitation
Given a real vector space V of finite dimension, together with a particular
homogeneous field of bivectors that we call a "field of projective forces", we
define a law of dynamics such that the position of the particle is a "ray" i.e.
a half-line drawn from the origin of V. The impulsion is a bivector whose
support is a 2-plane containing the ray. Throwing the particle with a given
initial impulsion defines a projective trajectory. It is a curve in the space
of rays S(V), together with an impulsion attached to each ray. In the simplest
example where the force is identically zero, the curve is a straight line and
the impulsion a constant bivector. A striking feature of projective dynamics
appears: the trajectories are not parameterized.
Among the projective force fields corresponding to a central force, the one
defining the Kepler problem is simpler than those corresponding to other
homogeneities. Here the thrown ray describes a quadratic cone whose section by
a hyperplane corresponds to a Keplerian conic. An original point of view on the
hidden symmetries of the Kepler problem emerges, and clarifies some remarks due
to Halphen and Appell. We also get the unexpected conclusion that there exists
a notion of divergence-free field of projective forces if and only if dim V=4.
No metric is involved in the axioms of projective dynamics.Comment: 20 pages, 4 figure
Meromorphic traveling wave solutions of the complex cubic-quintic Ginzburg-Landau equation
We look for singlevalued solutions of the squared modulus M of the traveling
wave reduction of the complex cubic-quintic Ginzburg-Landau equation. Using
Clunie's lemma, we first prove that any meromorphic solution M is necessarily
elliptic or degenerate elliptic. We then give the two canonical decompositions
of the new elliptic solution recently obtained by the subequation method.Comment: 14 pages, no figure, to appear, Acta Applicandae Mathematica
Elliptic Solitons and Groebner Bases
We consider the solution of spectral problems with elliptic coefficients in
the framework of the Hermite ansatz. We show that the search for exactly
solvable potentials and their spectral characteristics is reduced to a system
of polynomial equations solvable by the Gr\"obner bases method and others. New
integrable potentials and corresponding solutions of the Sawada-Kotera,
Kaup-Kupershmidt, Boussinesq equations and others are found.Comment: 18 pages, no figures, LaTeX'2
Analytic and Asymptotic Methods for Nonlinear Singularity Analysis: a Review and Extensions of Tests for the Painlev\'e Property
The integrability (solvability via an associated single-valued linear
problem) of a differential equation is closely related to the singularity
structure of its solutions. In particular, there is strong evidence that all
integrable equations have the Painlev\'e property, that is, all solutions are
single-valued around all movable singularities. In this expository article, we
review methods for analysing such singularity structure. In particular, we
describe well known techniques of nonlinear regular-singular-type analysis,
i.e. the Painlev\'e tests for ordinary and partial differential equations. Then
we discuss methods of obtaining sufficiency conditions for the Painlev\'e
property. Recently, extensions of \textit{irregular} singularity analysis to
nonlinear equations have been achieved. Also, new asymptotic limits of
differential equations preserving the Painlev\'e property have been found. We
discuss these also.Comment: 40 pages in LaTeX2e. To appear in the Proceedings of the CIMPA Summer
School on "Nonlinear Systems," Pondicherry, India, January 1996, (eds) B.
Grammaticos and K. Tamizhman
Colloquium: Mechanical formalisms for tissue dynamics
The understanding of morphogenesis in living organisms has been renewed by
tremendous progressin experimental techniques that provide access to
cell-scale, quantitative information both on theshapes of cells within tissues
and on the genes being expressed. This information suggests that
ourunderstanding of the respective contributions of gene expression and
mechanics, and of their crucialentanglement, will soon leap forward.
Biomechanics increasingly benefits from models, which assistthe design and
interpretation of experiments, point out the main ingredients and assumptions,
andultimately lead to predictions. The newly accessible local information thus
calls for a reflectionon how to select suitable classes of mechanical models.
We review both mechanical ingredientssuggested by the current knowledge of
tissue behaviour, and modelling methods that can helpgenerate a rheological
diagram or a constitutive equation. We distinguish cell scale ("intra-cell")and
tissue scale ("inter-cell") contributions. We recall the mathematical framework
developpedfor continuum materials and explain how to transform a constitutive
equation into a set of partialdifferential equations amenable to numerical
resolution. We show that when plastic behaviour isrelevant, the dissipation
function formalism appears appropriate to generate constitutive equations;its
variational nature facilitates numerical implementation, and we discuss
adaptations needed in thecase of large deformations. The present article
gathers theoretical methods that can readily enhancethe significance of the
data to be extracted from recent or future high throughput
biomechanicalexperiments.Comment: 33 pages, 20 figures. This version (26 Sept. 2015) contains a few
corrections to the published version, all in Appendix D.2 devoted to large
deformation
The \u3cem\u3eChlamydomonas\u3c/em\u3e Genome Reveals the Evolution of Key Animal and Plant Functions
Chlamydomonas reinhardtii is a unicellular green alga whose lineage diverged from land plants over 1 billion years ago. It is a model system for studying chloroplast-based photosynthesis, as well as the structure, assembly, and function of eukaryotic flagella (cilia), which were inherited from the common ancestor of plants and animals, but lost in land plants. We sequenced the âŒ120-megabase nuclear genome of Chlamydomonas and performed comparative phylogenomic analyses, identifying genes encoding uncharacterized proteins that are likely associated with the function and biogenesis of chloroplasts or eukaryotic flagella. Analyses of the Chlamydomonas genome advance our understanding of the ancestral eukaryotic cell, reveal previously unknown genes associated with photosynthetic and flagellar functions, and establish links between ciliopathy and the composition and function of flagella
A comparison of delamination models: Modeling, properties, and applications
This contribution presents recent results in the modeling and the analysis of delamination problems. It addresses adhesive contact, brittle, and cohesive zone models both in a quasistatic and a viscous, dynamic setting for the bulk part. Also different evolution laws for the delaminating surface are discussed
Pasquale del Pezzo, Duke of Caianello, Neapolitan mathematician
This article is dedicated to a reconstruction of some events and achievements, both personal and scientific, in the life of the Neapolitan mathematician Pasquale del Pezzo, Duke of Caianello
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