18,700 research outputs found
Prediction Properties of Aitken's Iterated Delta^2 Process, of Wynn's Epsilon Algorithm, and of Brezinski's Iterated Theta Algorithm
The prediction properties of Aitken's iterated Delta^2 process, Wynn's
epsilon algorithm, and Brezinski's iterated theta algorithm for (formal) power
series are analyzed. As a first step, the defining recursive schemes of these
transformations are suitably rearranged in order to permit the derivation of
accuracy-through-order relationships. On the basis of these relationships, the
rational approximants can be rewritten as a partial sum plus an appropriate
transformation term. A Taylor expansion of such a transformation term, which is
a rational function and which can be computed recursively, produces the
predictions for those coefficients of the (formal) power series which were not
used for the computation of the corresponding rational approximant.Comment: 34 pages, LaTe
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
Scalar Levin-Type Sequence Transformations
Sequence transformations are important tools for the convergence acceleration
of slowly convergent scalar sequences or series and for the summation of
divergent series. Transformations that depend not only on the sequence elements
or partial sums but also on an auxiliary sequence of so-called remainder
estimates are of Levin-type if they are linear in the , and
nonlinear in the . Known Levin-type sequence transformations are
reviewed and put into a common theoretical framework. It is discussed how such
transformations may be constructed by either a model sequence approach or by
iteration of simple transformations. As illustration, two new sequence
transformations are derived. Common properties and results on convergence
acceleration and stability are given. For important special cases, extensions
of the general results are presented. Also, guidelines for the application of
Levin-type sequence transformations are discussed, and a few numerical examples
are given.Comment: 59 pages, LaTeX, invited review for J. Comput. Applied Math.,
abstract shortene
Numerical Evidence that the Perturbation Expansion for a Non-Hermitian -Symmetric Hamiltonian is Stieltjes
Recently, several studies of non-Hermitian Hamiltonians having
symmetry have been conducted. Most striking about these complex Hamiltonians is
how closely their properties resemble those of conventional Hermitian
Hamiltonians. This paper presents further evidence of the similarity of these
Hamiltonians to Hermitian Hamiltonians by examining the summation of the
divergent weak-coupling perturbation series for the ground-state energy of the
-symmetric Hamiltonian recently
studied by Bender and Dunne. For this purpose the first 193 (nonzero)
coefficients of the Rayleigh-Schr\"odinger perturbation series in powers of
for the ground-state energy were calculated. Pad\'e-summation and
Pad\'e-prediction techniques recently described by Weniger are applied to this
perturbation series. The qualitative features of the results obtained in this
way are indistinguishable from those obtained in the case of the perturbation
series for the quartic anharmonic oscillator, which is known to be a Stieltjes
series.Comment: 20 pages, 0 figure
Recurrence relations for the number of solutions of a class of Diophantine equations
Recursive formulas are derived for the number of solutions of linear and
quadratic Diophantine equations with positive coefficients. This result is
further extended to general non-linear additive Diophantine equations. It is
shown that all three types of the recursion admit an explicit solution in the
form of complete Bell polynomial, depending on the coefficients of the power
series expansion of the logarithm of the generating functions for the sequences
of individual terms in the Diophantine equations.Comment: 11 pages, Latex. Dedicated to the 70-th anniversary of Apolodor
Radut
Prediction of peptide and protein propensity for amyloid formation
Understanding which peptides and proteins have the potential to undergo amyloid formation and what driving forces are responsible for amyloid-like fiber formation and stabilization remains limited. This is mainly because proteins that can undergo structural changes, which lead to amyloid formation, are quite diverse and share no obvious sequence or structural homology, despite the structural similarity found in the fibrils. To address these issues, a novel approach based on recursive feature selection and feed-forward neural networks was undertaken to identify key features highly correlated with the self-assembly problem. This approach allowed the identification of seven physicochemical and biochemical properties of the amino acids highly associated with the self-assembly of peptides and proteins into amyloid-like fibrils (normalized frequency of β-sheet, normalized frequency of β-sheet from LG, weights for β-sheet at the window position of 1, isoelectric point, atom-based hydrophobic moment, helix termination parameter at position j+1 and ΔGº values for peptides extrapolated in 0 M urea). Moreover, these features enabled the development of a new predictor (available at http://cran.r-project.org/web/packages/appnn/index.html) capable of accurately and reliably predicting the amyloidogenic propensity from the polypeptide sequence alone with a prediction accuracy of 84.9 % against an external validation dataset of sequences with experimental in vitro, evidence of amyloid formation
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