152 research outputs found
Formal Proof of SCHUR Conjugate Function
The main goal of our work is to formally prove the correctness of the key
commands of the SCHUR software, an interactive program for calculating with
characters of Lie groups and symmetric functions. The core of the computations
relies on enumeration and manipulation of combinatorial structures. As a first
"proof of concept", we present a formal proof of the conjugate function,
written in C. This function computes the conjugate of an integer partition. To
formally prove this program, we use the Frama-C software. It allows us to
annotate C functions and to generate proof obligations, which are proved using
several automated theorem provers. In this paper, we also draw on methodology,
discussing on how to formally prove this kind of program.Comment: To appear in CALCULEMUS 201
A characterization of Dirac morphisms
Relating the Dirac operators on the total space and on the base manifold of a
horizontally conformal submersion, we characterize Dirac morphisms, i.e. maps
which pull back (local) harmonic spinor fields onto (local) harmonic spinor
fields.Comment: 18 pages; restricted to the even-dimensional cas
Eigenvalue Bounds for Perturbations of Schrodinger Operators and Jacobi Matrices With Regular Ground States
We prove general comparison theorems for eigenvalues of perturbed Schrodinger
operators that allow proof of Lieb--Thirring bounds for suitable non-free
Schrodinger operators and Jacobi matrices.Comment: 11 page
On one integrable system with a cubic first integral
Recently one integrable model with a cubic first integral of motion has been
studied by Valent using some special coordinate system. We describe the
bi-Hamiltonian structures and variables of separation for this system.Comment: LaTeX with AMS fonts, 9 page
Exact Analytic Solutions for the Rotation of an Axially Symmetric Rigid Body Subjected to a Constant Torque
New exact analytic solutions are introduced for the rotational motion of a
rigid body having two equal principal moments of inertia and subjected to an
external torque which is constant in magnitude. In particular, the solutions
are obtained for the following cases: (1) Torque parallel to the symmetry axis
and arbitrary initial angular velocity; (2) Torque perpendicular to the
symmetry axis and such that the torque is rotating at a constant rate about the
symmetry axis, and arbitrary initial angular velocity; (3) Torque and initial
angular velocity perpendicular to the symmetry axis, with the torque being
fixed with the body. In addition to the solutions for these three forced cases,
an original solution is introduced for the case of torque-free motion, which is
simpler than the classical solution as regards its derivation and uses the
rotation matrix in order to describe the body orientation. This paper builds
upon the recently discovered exact solution for the motion of a rigid body with
a spherical ellipsoid of inertia. In particular, by following Hestenes' theory,
the rotational motion of an axially symmetric rigid body is seen at any instant
in time as the combination of the motion of a "virtual" spherical body with
respect to the inertial frame and the motion of the axially symmetric body with
respect to this "virtual" body. The kinematic solutions are presented in terms
of the rotation matrix. The newly found exact analytic solutions are valid for
any motion time length and rotation amplitude. The present paper adds further
elements to the small set of special cases for which an exact solution of the
rotational motion of a rigid body exists.Comment: "Errata Corridge Postprint" version of the journal paper. The
following typos present in the Journal version are HERE corrected: 1)
Definition of \beta, before Eq. 18; 2) sign in the statement of Theorem 3; 3)
Sign in Eq. 53; 4)Item r_0 in Eq. 58; 5) Item R_{SN}(0) in Eq. 6
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
On the Kobayashi-Royden metric for ellipsoids
Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/46237/1/208_2005_Article_BF01446557.pd
Nonlinear Differential Equations Satisfied by Certain Classical Modular Forms
A unified treatment is given of low-weight modular forms on \Gamma_0(N),
N=2,3,4, that have Eisenstein series representations. For each N, certain
weight-1 forms are shown to satisfy a coupled system of nonlinear differential
equations, which yields a single nonlinear third-order equation, called a
generalized Chazy equation. As byproducts, a table of divisor function and
theta identities is generated by means of q-expansions, and a transformation
law under \Gamma_0(4) for the second complete elliptic integral is derived.
More generally, it is shown how Picard-Fuchs equations of triangle subgroups of
PSL(2,R) which are hypergeometric equations, yield systems of nonlinear
equations for weight-1 forms, and generalized Chazy equations. Each triangle
group commensurable with \Gamma(1) is treated.Comment: 40 pages, final version, accepted by Manuscripta Mathematic
Error bounds for the large-argument asymptotic expansions of the Hankel and Bessel functions
In this paper, we reconsider the large-argument asymptotic expansions of the
Hankel, Bessel and modified Bessel functions and their derivatives. New
integral representations for the remainder terms of these asymptotic expansions
are found and used to obtain sharp and realistic error bounds. We also give
re-expansions for these remainder terms and provide their error estimates. A
detailed discussion on the sharpness of our error bounds and their relation to
other results in the literature is given. The techniques used in this paper
should also generalize to asymptotic expansions which arise from an application
of the method of steepest descents.Comment: 32 pages, 2 figures, accepted for publication in Acta Applicandae
Mathematica
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