38,004 research outputs found
Analytic connections on Riemann surfaces and orbifolds
We give a differentially closed description of the uniformizing
representation to the analytical apparatus on Riemann surfaces and orbifolds of
finite analytic type. Apart from well-known automorphic functions and Abelian
differentials it involves construction of the connection objects. Like
functions and differentials, the connection, being also the fundamental object,
is described by algorithmically derivable ODEs. Automorphic properties of all
of the objects are associated to different discrete groups, among which are
excessive ones. We show, in an example of the hyperelliptic curves, how can the
connection be explicitly constructed. We study also a relation between
classical/traditional `linearly differential' viewpoint (principal Fuchsian
equation) and uniformizing -representation of the theory. The latter is
shown to be supplemented with the second (to the principal) Fuchsian equation.Comment: Final version. LaTeX, 16 pages. No figure
Thomas decompositions of parametric nonlinear control systems
This paper presents an algorithmic method to study structural properties of
nonlinear control systems in dependence of parameters. The result consists of a
description of parameter configurations which cause different control-theoretic
behaviour of the system (in terms of observability, flatness, etc.). The
constructive symbolic method is based on the differential Thomas decomposition
into disjoint simple systems, in particular its elimination properties
Towards a Model Theory for Transseries
The differential field of transseries extends the field of real Laurent
series, and occurs in various context: asymptotic expansions, analytic vector
fields, o-minimal structures, to name a few. We give an overview of the
algebraic and model-theoretic aspects of this differential field, and report on
our efforts to understand its first-order theory.Comment: Notre Dame J. Form. Log., to appear; 33 p
The resultant on compact Riemann surfaces
We introduce a notion of resultant of two meromorphic functions on a compact
Riemann surface and demonstrate its usefulness in several respects. For
example, we exhibit several integral formulas for the resultant, relate it to
potential theory and give explicit formulas for the algebraic dependence
between two meromorphic functions on a compact Riemann surface. As a particular
application, the exponential transform of a quadrature domain in the complex
plane is expressed in terms of the resultant of two meromorphic functions on
the Schottky double of the domain.Comment: 44 page
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
Initial-Boundary Value Problems for Linear and Soliton PDEs
Evolution PDEs for dispersive waves are considered in both linear and
nonlinear integrable cases, and initial-boundary value problems associated with
them are formulated in spectral space. A method of solution is presented, which
is based on the elimination of the unknown boundary values by proper
restrictions of the functional space and of the spectral variable complex
domain. Illustrative examples include the linear Schroedinger equation on
compact and semicompact n-dimensional domains and the nonlinear Schroedinger
equation on the semiline.Comment: 18 pages, LATEX, submitted to the proccedings of NEEDS 2001 - Special
Issue, to be published in the Journal of Theoretical and Mathematical Physic
Automatic Classification of Restricted Lattice Walks
We propose an experimental mathematics approach leading to the
computer-driven discovery of various structural properties of general counting
functions coming from enumeration of walks
- …