624 research outputs found
Action-Angle variables for the Gel'fand-Dikii flows
Using the scattering transform for order linear scalar operators,
the Poisson bracket found by Gel'fand and Dikii, which generalizes the Gardner
Poisson bracket for the KdV hierarchy, is computed on the scattering side.
Action-angle variables are then constructed. Using this, complete integrability
is demonstrated in the strong sense. Real action-angle variables are
constructed in the self-adjoint case
Hierarchy of general invariants for bivariate LPDOs
We study invariants under gauge transformations of linear partial
differential operators on two variables. Using results of BK-factorization, we
construct hierarchy of general invariants for operators of an arbitrary order.
Properties of general invariants are studied and some examples are presented.
We also show that classical Laplace invariants correspond to some particular
cases of general invariants.Comment: to appear in J. "Theor.Math.Phys." in May 200
Exact quantum query complexity of
In the exact quantum query model a successful algorithm must always output
the correct function value. We investigate the function that is true if exactly
or of the input bits given by an oracle are 1. We find an optimal
algorithm (for some cases), and a nontrivial general lower and upper bound on
the minimum number of queries to the black box.Comment: 19 pages, fixed some typos and constraint
A Riemann-Hilbert Problem for an Energy Dependent Schr\"odinger Operator
\We consider an inverse scattering problem for Schr\"odinger operators with
energy dependent potentials. The inverse problem is formulated as a
Riemann-Hilbert problem on a Riemann surface. A vanishing lemma is proved for
two distinct symmetry classes. As an application we prove global existence
theorems for the two distinct systems of partial differential equations
for suitably restricted,
complementary classes of initial data
Constructive factorization of LPDO in two variables
We study conditions under which a partial differential operator of arbitrary
order in two variables or ordinary linear differential operator admits a
factorization with a first-order factor on the left. The factorization process
consists of solving, recursively, systems of linear equations, subject to
certain differential compatibility conditions. In the generic case of partial
differential operators one does not have to solve a differential equation. In
special degenerate cases, such as ordinary differential, the problem is finally
reduced to the solution of some Riccati equation(s). The conditions of
factorization are given explicitly for second- and, and an outline is given for
the higher-order case.Comment: 16 pages, to be published in Journal "Theor. Math. Phys." (2005
On the Caudrey-Beals-Coifman System and the Gauge Group Action
The generalized Zakharov-Shabat systems with complex-valued Cartan elements
and the systems studied by Caudrey, Beals and Coifman (CBC systems) and their
gauge equivalent are studies. This includes: the properties of fundamental
analytical solutions (FAS) for the gauge-equivalent to CBC systems and the
minimal set of scattering data; the description of the class of nonlinear
evolutionary equations solvable by the inverse scattering method and the
recursion operator, related to such systems; the hierarchies of Hamiltonian
structures.Comment: 12 pages, no figures, contribution to the NEEDS 2007 proceedings
(Submitted to J. Nonlin. Math. Phys.
Scalability of quantum computation with addressable optical lattices
We make a detailed analysis of error mechanisms, gate fidelity, and
scalability of proposals for quantum computation with neutral atoms in
addressable (large lattice constant) optical lattices. We have identified
possible limits to the size of quantum computations, arising in 3D optical
lattices from current limitations on the ability to perform single qubit gates
in parallel and in 2D lattices from constraints on laser power. Our results
suggest that 3D arrays as large as 100 x 100 x 100 sites (i.e.,
qubits) may be achievable, provided two-qubit gates can be performed with
sufficiently high precision and degree of parallelizability. Parallelizability
of long range interaction-based two-qubit gates is qualitatively compared to
that of collisional gates. Different methods of performing single qubit gates
are compared, and a lower bound of is determined on the
error rate for the error mechanisms affecting Cs in a blue-detuned
lattice with Raman transition-based single qubit gates, given reasonable limits
on experimental parameters.Comment: 17 pages, 5 figures. Accepted for publication in Physical Review
Mothers' complex talk when sharing books with their toddlers: Book genre matters
This study investigated the influence of book genre (narrative or didactic) on mothers' language use during a book sharing interaction with their 18- to 25-month-olds. Mother-child dyads were videotaped sharing both a narrative and a didactic book, adapted from two commercially available books, and matched in terms of length, quantity of text, and target content. A greater proportion of mothers' talk was complex (i.e., predictions, text-to-life comparisons) during narrative book sharing than during didactic book sharing. Mothers also used a greater variety of verb tenses and referenced more mental states during narrative book sharing. These results differ from findings from previous studies with older children where it has been concluded that didactic books offer greater opportunities for complex talk than narrative books. The results also highlight the importance of taking situational factors into account when investigating parent-child communicative interactions. © The Author(s) 2013
Young children's cognitive achievement: home learning environment, language and ethnic background
For decades, research has shown differences in cognitive assessment scores between White and minority ethnic group(s) learners as well as differences across different minority ethnic groups. More recent data have indicated that the home learning environment and languages spoken can impact cognitive assessment and other corollary outcomes. This study uses the Millennium Cohort Study to jointly assess how minority ethnic group, home learning environment and home languages predict child cognitive assessment scores. Regression analyses were conducted using two assessment measures. The following is hypothesised: (1) cognitive achievement scores vary by minority ethnic group, (2) more home learning environment in early childhood leads to higher cognitive development scores and (3) English only in the home yields the highest cognitive scores while no English in the home yields the lowest. Findings reveal that there are differences in cognitive scores along ethnic group categories although there are also some unexpected findings. Home learning environment does not play as large a role as was predicted in raising the assessment scores overall for learners while speaking English in the home does, irrespective of ethnic background
Local Isometric immersions of pseudo-spherical surfaces and evolution equations
The class of differential equations describing pseudo-spherical surfaces,
first introduced by Chern and Tenenblat [3], is characterized by the property
that to each solution of a differential equation, within the class, there
corresponds a 2-dimensional Riemannian metric of curvature equal to . The
class of differential equations describing pseudo-spherical surfaces carries
close ties to the property of complete integrability, as manifested by the
existence of infinite hierarchies of conservation laws and associated linear
problems. As such, it contains many important known examples of integrable
equations, like the sine-Gordon, Liouville and KdV equations. It also gives
rise to many new families of integrable equations. The question we address in
this paper concerns the local isometric immersion of pseudo-spherical surfaces
in from the perspective of the differential equations that give
rise to the metrics. Indeed, a classical theorem in the differential geometry
of surfaces states that any pseudo-spherical surface can be locally
isometrically immersed in . In the case of the sine-Gordon
equation, one can derive an expression for the second fundamental form of the
immersion that depends only on a jet of finite order of the solution of the
pde. A natural question is to know if this remarkable property extends to
equations other than the sine-Gordon equation within the class of differential
equations describing pseudo-spherical surfaces. In an earlier paper [11], we
have shown that this property fails to hold for all other second order
equations, except for those belonging to a very special class of evolution
equations. In the present paper, we consider a class of evolution equations for
of order describing pseudo-spherical surfaces. We show that
whenever an isometric immersion in exists, depending on a jet of
finite order of , then the coefficients of the second fundamental forms are
functions of the independent variables and only.Comment: Fields Institute Communications, 2015, Hamiltonian PDEs and
Applications, pp.N
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