192 research outputs found
Higher analogues of the discrete-time Toda equation and the quotient-difference algorithm
The discrete-time Toda equation arises as a universal equation for the
relevant Hankel determinants associated with one-variable orthogonal
polynomials through the mechanism of adjacency, which amounts to the inclusion
of shifted weight functions in the orthogonality condition. In this paper we
extend this mechanism to a new class of two-variable orthogonal polynomials
where the variables are related via an elliptic curve. This leads to a `Higher
order Analogue of the Discrete-time Toda' (HADT) equation for the associated
Hankel determinants, together with its Lax pair, which is derived from the
relevant recurrence relations for the orthogonal polynomials. In a similar way
as the quotient-difference (QD) algorithm is related to the discrete-time Toda
equation, a novel quotient-quotient-difference (QQD) scheme is presented for
the HADT equation. We show that for both the HADT equation and the QQD scheme,
there exists well-posed -periodic initial value problems, for almost all
\s\in\Z^2. From the Lax-pairs we furthermore derive invariants for
corresponding reductions to dynamical mappings for some explicit examples.Comment: 38 page
Mathematical Properties of a New Levin-Type Sequence Transformation Introduced by \v{C}\'{\i}\v{z}ek, Zamastil, and Sk\'{a}la. I. Algebraic Theory
\v{C}\'{\i}\v{z}ek, Zamastil, and Sk\'{a}la [J. Math. Phys. \textbf{44}, 962
- 968 (2003)] introduced in connection with the summation of the divergent
perturbation expansion of the hydrogen atom in an external magnetic field a new
sequence transformation which uses as input data not only the elements of a
sequence of partial sums, but also explicit estimates
for the truncation errors. The explicit
incorporation of the information contained in the truncation error estimates
makes this and related transformations potentially much more powerful than for
instance Pad\'{e} approximants. Special cases of the new transformation are
sequence transformations introduced by Levin [Int. J. Comput. Math. B
\textbf{3}, 371 - 388 (1973)] and Weniger [Comput. Phys. Rep. \textbf{10}, 189
- 371 (1989), Sections 7 -9; Numer. Algor. \textbf{3}, 477 - 486 (1992)] and
also a variant of Richardson extrapolation [Phil. Trans. Roy. Soc. London A
\textbf{226}, 299 - 349 (1927)]. The algebraic theory of these transformations
- explicit expressions, recurrence formulas, explicit expressions in the case
of special remainder estimates, and asymptotic order estimates satisfied by
rational approximants to power series - is formulated in terms of hitherto
unknown mathematical properties of the new transformation introduced by
\v{C}\'{\i}\v{z}ek, Zamastil, and Sk\'{a}la. This leads to a considerable
formal simplification and unification.Comment: 41 + ii pages, LaTeX2e, 0 figures. Submitted to Journal of
Mathematical Physic
Explicit solutions for second order operator differential equations with two boundary value conditions
AbstractBoundary value problems for second order operator differential equations with two boundary value conditions are studied. Explicit expressions of the solutions in terms of data problems are given. By means of the application of algebraic techniques, analogous expressions to the ones known for the scalar case are obtained
Acceleration of generalized hypergeometric functions through precise remainder asymptotics
We express the asymptotics of the remainders of the partial sums {s_n} of the
generalized hypergeometric function q+1_F_q through an inverse power series z^n
n^l \sum_k c_k/n^k, where the exponent l and the asymptotic coefficients {c_k}
may be recursively computed to any desired order from the hypergeometric
parameters and argument. From this we derive a new series acceleration
technique that can be applied to any such function, even with complex
parameters and at the branch point z=1. For moderate parameters (up to
approximately ten) a C implementation at fixed precision is very effective at
computing these functions; for larger parameters an implementation in higher
than machine precision would be needed. Even for larger parameters, however,
our C implementation is able to correctly determine whether or not it has
converged; and when it converges, its estimate of its error is accurate.Comment: 36 pages, 6 figures, LaTeX2e. Fixed sign error in Eq. (2.28), added
several references, added comparison to other methods, and added discussion
of recursion stabilit
Hypothetical Reasoning via Provenance Abstraction
Data analytics often involves hypothetical reasoning: repeatedly modifying
the data and observing the induced effect on the computation result of a
data-centric application. Previous work has shown that fine-grained data
provenance can help make such an analysis more efficient: instead of a costly
re-execution of the underlying application, hypothetical scenarios are applied
to a pre-computed provenance expression. However, storing provenance for
complex queries and large-scale data leads to a significant overhead, which is
often a barrier to the incorporation of provenance-based solutions.
To this end, we present a framework that allows to reduce provenance size.
Our approach is based on reducing the provenance granularity using user defined
abstraction trees over the provenance variables; the granularity is based on
the anticipated hypothetical scenarios. We formalize the tradeoff between
provenance size and supported granularity of the hypothetical reasoning, and
study the complexity of the resulting optimization problem, provide efficient
algorithms for tractable cases and heuristics for others. We experimentally
study the performance of our solution for various queries and abstraction
trees. Our study shows that the algorithms generally lead to substantial
speedup of hypothetical reasoning, with a reasonable loss of accuracy
Asymptotic Improvement of Resummation and Perturbative Predictions in Quantum Field Theory
The improvement of resummation algorithms for divergent perturbative
expansions in quantum field theory by asymptotic information about perturbative
coefficients is investigated. Various asymptotically optimized resummation
prescriptions are considered. The improvement of perturbative predictions
beyond the reexpansion of rational approximants is discussed.Comment: 21 pages, LaTeX, 3 tables; title shortened; typographical errors
corrected; minor changes of style; 2 references adde
The structure of iterative methods for symmetric linear discrete ill-posed problems
The iterative solution of large linear discrete ill-posed problems with an error contaminated data vector requires the use of specially designed methods in order to avoid severe error propagation. Range restricted minimal residual methods have been found to be well suited for the solution of many such problems. This paper discusses the structure of matrices that arise in a range restricted minimal residual method for the solution of large linear discrete ill-posed problems with a symmetric matrix. The exploitation of the structure results in a method that is competitive with respect to computer storage, number of iterations, and accuracy.Acknowledgments We would like to thank the referees for comments. The work of F. M. was supported
by Dirección General de Investigación Científica y Técnica, Ministerio de Economía y Competitividad of
Spain under grant MTM2012-36732-C03-01. Work of L. R. was supported by Universidad Carlos III de
Madrid in the Department of Mathematics during the academic year 2010-2011 within the framework of
the Chair of Excellence Program and by NSF grant DMS-1115385
The staircase method: integrals for periodic reductions of integrable lattice equations
We show, in full generality, that the staircase method provides integrals for
mappings, and correspondences, obtained as traveling wave reductions of
(systems of) integrable partial difference equations. We apply the staircase
method to a variety of equations, including the Korteweg-De Vries equation, the
five-point Bruschi-Calogero-Droghei equation, the QD-algorithm, and the
Boussinesq system. We show that, in all these cases, if the staircase method
provides r integrals for an n-dimensional mapping, with 2r<n, then one can
introduce q<= 2r variables, which reduce the dimension of the mapping from n to
q. These dimension-reducing variables are obtained as joint invariants of
k-symmetries of the mappings. Our results support the idea that often the
staircase method provides sufficiently many integrals for the periodic
reductions of integrable lattice equations to be completely integrable. We also
study reductions on other quad-graphs than the regular 2D lattice, and we prove
linear growth of the multi-valuedness of iterates of high-dimensional
correspondences obtained as reductions of the QD-algorithm.Comment: 40 pages, 23 Figure
Mathematical Modelling of Optical Coherence Tomography
In this chapter a general mathematical model of Optical Coherence Tomography
(OCT) is presented on the basis of the electromagnetic theory. OCT produces
high resolution images of the inner structure of biological tissues. Images are
obtained by measuring the time delay and the intensity of the backscattered
light from the sample considering also the coherence properties of light. The
scattering problem is considered for a weakly scattering medium located far
enough from the detector. The inverse problem is to reconstruct the
susceptibility of the medium given the measurements for different positions of
the mirror. Different approaches are addressed depending on the different
assumptions made about the optical properties of the sample. This procedure is
applied to a full field OCT system and an extension to standard (time and
frequency domain) OCT is briefly presented.Comment: 28 pages, 5 figures, book chapte
Regional Trajectories of Entrepreneurship and Growth
The development of regions is considerably shaped by their history. We review research that finds significant persistence of regional levels of entrepreneurship over longer periods of time. It is argued that the long term persistence of regional entrepreneurship indicates the presence and effect of a culture of entrepreneurship that is conducive to new business formation and regional growth. Hence, regional development is characterized by long term trajectories of entrepreneurship. We derive a number of policy implications and propose avenues for further research.</p
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