1,046 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
Acoustic Scattering and the Extended Korteweg deVries hierarchy
The acoustic scattering operator on the real line is mapped to a
Schr\"odinger operator under the Liouville transformation. The potentials in
the image are characterized precisely in terms of their scattering data, and
the inverse transformation is obtained as a simple, linear quadrature. An
existence theorem for the associated Harry Dym flows is proved, using the
scattering method. The scattering problem associated with the Camassa-Holm
flows on the real line is solved explicitly for a special case, which is used
to reduce a general class of such problems to scattering problems on finite
intervals.Comment: 18 page
Multipeakons and a theorem of Stieltjes
A closed form of the multi-peakon solutions of the Camassa-Holm equation is
found using a theorem of Stieltjes on continued fractions. An explicit formula
is obtained for the scattering shifts.Comment: 6 page
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
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
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
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ALPHA SPECTROMETRIC EVALUATION OF SRM-995 AS A POTENTIAL URANIUM/THORIUM DOUBLE TRACER SYSTEM FOR AGE-DATING URANIUM MATERIALS
Uranium-233 (t{sub 1/2} {approx} 1.59E5 years) is an artificial, fissile isotope of uranium that has significant importance in nuclear forensics. The isotope provides a unique signature in determining the origin and provenance of uranium-bearing materials and is valuable as a mass spectrometric tracer. Alpha spectrometry was employed in the critical evaluation of a {sup 233}U standard reference material (SRM-995) as a dual tracer system based on the in-growth of {sup 229}Th (t{sub 1/2} {approx} 7.34E3 years) for {approx}35 years following radiochemical purification. Preliminary investigations focused on the isotopic analysis of standards and unmodified fractions of SRM-995; all samples were separated and purified using a multi-column anion-exchange scheme. The {sup 229}Th/{sup 233}U atom ratio for SRM-995 was found to be 1.598E-4 ({+-} 4.50%) using recovery-corrected radiochemical methods. Using the Bateman equations and relevant half-lives, this ratio reflects a material that was purified {approx} 36.8 years prior to this analysis. The calculated age is discussed in contrast with both the date of certification and the recorded date of last purification
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
The Optimal Single Copy Measurement for the Hidden Subgroup Problem
The optimization of measurements for the state distinction problem has
recently been applied to the theory of quantum algorithms with considerable
successes, including efficient new quantum algorithms for the non-abelian
hidden subgroup problem. Previous work has identified the optimal single copy
measurement for the hidden subgroup problem over abelian groups as well as for
the non-abelian problem in the setting where the subgroups are restricted to be
all conjugate to each other. Here we describe the optimal single copy
measurement for the hidden subgroup problem when all of the subgroups of the
group are given with equal a priori probability. The optimal measurement is
seen to be a hybrid of the two previously discovered single copy optimal
measurements for the hidden subgroup problem.Comment: 8 pages. Error in main proof fixe
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