5,570 research outputs found
Electron loss of fast heavy projectiles in collision with neutral targets
The multiple electron loss of heavy projectiles in fast ion-atom collisions
has been studied in the framework of the sudden perturbation approximation.
Especially, a model is developed to calculate the cross sections for the loss
of any number of electrons from the projectile ion, including the ionization of
a single electron and up to the complete stripping of the projectile. For a
given collision system, that is specified by the (type and charge state of the)
projectile and target as well as the collision energy, in fact, the
experimental cross sections for just three final states of the projectile are
required by this model in order to predict the loss of any number, , of
electrons for the same collision system, or for any similar system that differs
only in the energy or the initial charge state of the projectile ion. The model
is simple and can be utilized for both, the projectile and target ionization,
and without that large computer resources are requested. Detailed computation
have been carried out for the multiple electron loss of Xe and
U projectiles in collision with neutral Ar and Ne gas
targets
Nijenhuis geometry IV: conservation laws, symmetries and integration of certain non-diagonalisable systems of hydrodynamic type in quadratures
The paper contains two lines of results: the first one is a study of
symmetries and conservation laws of gl-regular Nijenhuis operators. We prove
the splitting Theorem for symmetries and conservation laws of Nijenhuis
operators, show that the space of symmetries of a gl-regular Nijenhuis operator
forms a commutative algebra with respect to (pointwise) matrix multiplication.
Moreover, all the elements of this algebra are strong symmetries of each other.
We establish a natural relationship between symmetries and conservation laws of
a gl-regular Nijenhuis operator and systems of the first and second companion
coordinates. Moreover, we show that the space of conservation laws is naturally
related to the space of symmetries in the sense that any conservation laws can
be obtained from a single conservation law by multiplication with an
appropriate symmetry. In particular, we provide an explicit description of all
symmetries and conservation laws for gl-regular operators at algebraically
generic points. The second line of results contains an application of the
theoretical part to a certain system of partial differential equations of
hydrodynamic type, which was previously studied by different authors, but
mainly in the diagonalisable case. We show that this system is integrable in
quadratures, i.e., its solutions can be found for almost all initial curves by
integrating closed 1-forms and solving some systems of functional equations.
The system is not diagonalisable in general, and construction and integration
of such systems is an actively studied and explicitly stated problem in the
literature
Orthogonal separation of variables for spaces of constant curvature
We construct all orthogonal separating coordinates in constant curvature
spaces of arbitrary signature. Further, we construct explicit transformation
between orthogonal separating and flat or generalised flat coordinates, as well
as explicit formulas for the corresponding Killing tensors and the St\"ackel
matrices.Comment: 28 pages, one figure. Comments are welcom
Nijenhuis Geometry
This work is the first, and main, of the series of papers in progress
dedicated to Nienhuis operators, i.e., fields of endomorphisms with vanishing
Nijenhuis tensor. It serves as an introduction to Nijenhuis Geometry that
should be understood in much wider context than before: from local description
at generic points to singularities and global analysis. The goal of the present
paper is to introduce terminology, develop new important techniques (e.g.,
analytic functions of Nijenhuis operators, splitting theorem and
linearisation), summarise and generalise basic facts (some of which are already
known but we give new self-contained proofs), and more importantly, to
demonstrate that the research programme proposed in the paper is realistic by
proving a series of new, not at all obvious, results
Applications of Nijenhuis geometry II: maximal pencils of multihamiltonian structures of hydrodynamic type
We connect two a priori unrelated topics, theory of geodesically equivalent
metrics in differential geometry, and theory of compatible infinite dimensional
Poisson brackets of hydrodynamic type in mathematical physics. Namely, we prove
that a pair of geodesically equivalent metrics such that one is flat produces a
pair of such brackets. We construct Casimirs for these brackets and the
corresponding commuting flows. There are two ways to produce a large family of
compatible Poisson structures from a pair of geodesically equivalent metrics
one of which is flat. One of these families is dimensional; we
describe it completely and show that it is maximal. Another has dimension and is, in a certain sense, polynomial. We show that a nontrivial
polynomial family of compatible Poisson structures of dimension is unique
and comes from a pair of geodesically equivalent metrics. In addition, we
generalise a result of Sinjukov (1961) from constant curvature metrics to
arbitrary Einstein metrics.Comment: comments are welcom
Applications of Nijenhuis Geometry V: geodesically equivalent metrics and finite-dimensional reductions of certain integrable quasilinear systems
We describe all metrics geodesically compatible with a gl-regular Nijenhuis
operator . The set of such metrics is large enough so that a generic local
curve is a geodesic for a suitable metric from this set. Next, we
show that a certain evolutionary PDE system of hydrodynamic type constructed
from preserves the property of to be a -geodesic. This implies
that every metric geodesically compatible with gives us a finite
dimensional reduction of this PDE system. We show that its restriction onto the
set of -geodesics is naturally equivalent to the Poisson action of
on the cotangent bundle generated by the integrals coming from
geodesic compatibility
Applications of Nijenhuis geometry III: Frobenius pencils and compatible non-homogeneous Poisson structures
We consider multicomponent local Poisson structures of the form , under the assumption that the third order term
is Darboux-Poisson and non-degenerate, and study the Poisson compatibility of
two such structures. We give an algebraic interpretation of this problem in
terms of Frobenius algebras and reduce it to classification of Frobenius
pencils, i.e. of linear families of Frobenius algebras. Then, we completely
describe and classify Frobenius pencils under minor genericity conditions. In
particular we show that each such Frobenuis pencil is a subpencil of a certain
maximal pencil. These maximal pencils are uniquely determined by some
combinatorial object, a directed rooted in-forest with edges labeled by numbers
's and vertices labeled by natural numbers whose sum is the
dimension of the manifold. These pencils are naturally related to certain
(polynomial, in the most nondegenerate case) pencils of Nijenhuis operators. We
show that common Frobenius coordinate systems admit an elegant invariant
description in terms of the Nijenhuis pencil.Comment: In Version v2, Theorem 4 and its proof are improved, and a mistake in
Theorem 5 is corrected. In Version v3 we changed the signs in certain
formulas for cosmetical reasons, to avoid multiple use of , and to
make the paper better compatible with arXiv:2212.01605, and also updated the
reference
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