3,653 research outputs found
Quantum Sturm-Liouville Equation, Quantum Resolvent, Quantum Integrals, and Quantum KdV : the Fast Decrease Case
We construct quantum operators solving the quantum versions of the
Sturm-Liouville equation and the resolvent equation, and show the existence of
conserved currents. The construction depends on the following input data: the
basic quantum field and the regularization .Comment: minor correction
Energetic Consistency and Momentum Conservation in the Gyrokinetic Description of Tokamak Plasmas
Gyrokinetic field theory is addressed in the context of a general
Hamiltonian. The background magnetic geometry is static and axisymmetric, and
all dependence of the Lagrangian upon dynamical variables is in the Hamiltonian
or in free field terms. Equations for the fields are given by functional
derivatives. The symmetry through the Hamiltonian with time and toroidal angle
invariance of the geometry lead to energy and toroidal momentum conservation.
In various levels of ordering against fluctuation amplitude, energetic
consistency is exact. The role of this in underpinning of conservation laws is
emphasised. Local transport equations for the vorticity, toroidal momentum, and
energy are derived. In particular, the momentum equation is shown for any form
of Hamiltonian to be well behaved and to relax to its magnetohydrodynamic (MHD)
form when long wavelength approximations are taken in the Hamiltonian. Several
currently used forms, those which form the basis of most global simulations,
are shown to be well defined within the gyrokinetic field theory and energetic
consistency.Comment: RevTeX 4, 47 pages, no figures, revised version updated following
referee comments (discussion more strictly correct/consistent, 4 references
added, results unchanged as they depend on consistency of the theory),
resubmitted to Physics of Plasma
On the algebraic structures connected with the linear Poisson brackets of hydrodynamics type
The generalized form of the Kac formula for Verma modules associated with
linear brackets of hydrodynamics type is proposed. Second cohomology groups of
the generalized Virasoro algebras are calculated. Connection of the central
extensions with the problem of quntization of hydrodynamics brackets is
demonstrated
Commuting Flows and Conservation Laws for Noncommutative Lax Hierarchies
We discuss commuting flows and conservation laws for Lax hierarchies on
noncommutative spaces in the framework of the Sato theory. On commutative
spaces, the Sato theory has revealed essential aspects of the integrability for
wide class of soliton equations which are derived from the Lax hierarchies in
terms of pseudo-differential operators. Noncommutative extension of the Sato
theory has been already studied by the author and Kouichi Toda, and the
existence of various noncommutative Lax hierarchies are guaranteed. In the
present paper, we present conservation laws for the noncommutative Lax
hierarchies with both space-space and space-time noncommutativities and prove
the existence of infinite number of conserved densities. We also give the
explicit representations of them in terms of Lax operators. Our results include
noncommutative versions of KP, KdV, Boussinesq, coupled KdV, Sawada-Kotera,
modified KdV equations and so on.Comment: 22 pages, LaTeX, v2: typos corrected, references added, version to
appear in JM
Darboux transformations for a twisted derivation and quasideterminant solutions to the super KdV equation
This paper is concerned with a generalized type of Darboux transformations
defined in terms of a twisted derivation satisfying
where is a homomorphism. Such twisted derivations include regular
derivations, difference and -difference operators and superderivatives as
special cases. Remarkably, the formulae for the iteration of Darboux
transformations are identical with those in the standard case of a regular
derivation and are expressed in terms of quasideterminants. As an example, we
revisit the Darboux transformations for the Manin-Radul super KdV equation,
studied in Q.P. Liu and M. Ma\~nas, Physics Letters B \textbf{396} 133--140,
(1997). The new approach we take enables us to derive a unified expression for
solution formulae in terms of quasideterminants, covering all cases at once,
rather than using several subcases. Then, by using a known relationship between
quasideterminants and superdeterminants, we obtain expressions for these
solutions as ratios of superdeterminants. This coincides with the results of
Liu and Ma\~nas in all the cases they considered but also deals with the one
subcase in which they did not obtain such an expression. Finally, we obtain
another type of quasideterminant solutions to the Main-Radul super KdV equation
constructed from its binary Darboux transformations. These can also be
expressed as ratios of superdeterminants and are a substantial generalization
of the solutions constructed using binary Darboux transformations in earlier
work on this topic
The inverse scattering problem at fixed energy based on the Marchenko equation for an auxiliary Sturm-Liouville operator
A new approach is proposed to the solution of the quantum mechanical inverse
scattering problem at fixed energy. The method relates the fixed energy phase
shifts to those arising in an auxiliary Sturm-Liouville problem via the
interpolation theory of the Weyl-Titchmarsh m-function. Then a Marchenko
equation is solved to obtain the potential.Comment: 6 pages, 8 eps figure
Impurity effects on optical response in a finite band electronic system coupled to phonons
The concepts, which have traditionally been useful in understanding the
effects of the electron--phonon interaction in optical spectroscopy, are based
on insights obtained within the infinite electronic band approximation and no
longer apply in finite band metals. Impurity and phonon contributions to
electron scattering are not additive and the apparent strength of the coupling
to the phonon degrees of freedom is substantially reduced with increased
elastic scattering. The optical mass renormalization changes sign with
increasing frequency and the optical scattering rate never reaches its high
frequency quasiparticle value which itself is also reduced below its infinite
band value
Mother-toddler interaction patterns associated with maternal depression.
Journal ArticleInteractive coordination was observed in laboratory play interactions of pairs of 29 clinically depressed and 14 nondepressed mothers and their 13-29-month-old children (M = 18.9 months). Nondepressed mothers and their children displayed more interactive coordination than depressed-mother dyads (p < .001). Depressed mothers were less likely to repair interrupted interactions, and their toddlers were less likely to maintain interactions than nondepressed controls. Toddlers matched their nondepressed but not their depressed mothers' negative behavior rates. Results suggested that early interventions focus on training mothers to attend to, maintain, and repair mother-child interactions to more closely approximate normal levels of interactive coordination
On the Complexity of Solving Zero-Dimensional Polynomial Systems via Projection
Given a zero-dimensional polynomial system consisting of n integer
polynomials in n variables, we propose a certified and complete method to
compute all complex solutions of the system as well as a corresponding
separating linear form l with coefficients of small bit size. For computing l,
we need to project the solutions into one dimension along O(n) distinct
directions but no further algebraic manipulations. The solutions are then
directly reconstructed from the considered projections. The first step is
deterministic, whereas the second step uses randomization, thus being
Las-Vegas.
The theoretical analysis of our approach shows that the overall cost for the
two problems considered above is dominated by the cost of carrying out the
projections. We also give bounds on the bit complexity of our algorithms that
are exclusively stated in terms of the number of variables, the total degree
and the bitsize of the input polynomials
Generalized Fock spaces and the Stirling numbers
The Bargmann-Fock-Segal space plays an important role in mathematical
physics, and has been extended into a number of directions. In the present
paper we imbed this space into a Gelfand triple. The spaces forming the
Fr\'echet part (i.e. the space of test functions) of the triple are
characterized both in a geometric way and in terms of the adjoint of
multiplication by the complex variable, using the Stirling numbers of the
second kind. The dual of the space of test functions has a topological algebra
structure, of the kind introduced and studied by the first named author and G.
Salomon.Comment: revised versio
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