62 research outputs found
Weyl asymptotics: From closed to open systems
We present microwave experiments on the symmetry reduced 5-disk billiard
studying the transition from a closed to an open system. The measured microwave
reflection signal is analyzed by means of the harmonic inversion and the
counting function of the resulting resonances is studied. For the closed system
this counting function shows the Weyl asymptotic with a leading exponent equal
to 2. By opening the system successively this exponent decreases smoothly to an
non-integer value. For the open systems the extraction of resonances by the
harmonic inversion becomes more challenging and the arising difficulties are
discussed. The results can be interpreted as a first experimental indication
for the fractal Weyl conjecture for resonances.Comment: 9 pages, 7 figure
Vector continued fraction algorithms.
We consider the construction of rational approximations to given power series whose coefficients are vectors. The approximants are in the form of vector-valued continued fractions which may be used to obtain vector Padeapproximants using recurrence relations. Algorithms for the determination of the vector elements of these fractions have been established using Clifford algebras. We devise new algorithms based on these which involve operations on vectors and scalars only â a desirable characteristic for computations involving vectors of large dimension. As a consequence, we are able to form new expressions for the numerator and denominator polynomials of these approximants as products of vectors, thus retaining their Clifford nature
Resummation of projectile-target multiple scatterings and parton saturation
In the framework of a toy model which possesses the main features of QCD in
the high energy limit, we conduct a numerical study of scattering amplitudes
constructed from parton splittings and projectile-target multiple interactions,
in a way that unitarizes the amplitudes without however explicit saturation in
the wavefunction of the incoming states. This calculation is performed in two
different ways. One of these formulations, the closest to field theory,
involves the numerical resummation of a factorially divergent series, for which
we develop appropriate numerical tools. We accurately compare the properties of
the resulting amplitudes with what would be expected if saturation were
explicitly included in the evolution of the states. We observe that the
amplitudes have similar properties in a small but finite range of rapidity in
the beginning of the evolution, as expected. Some of the features of
reaction-diffusion processes are already present in that range, even when
saturation is left out of the model.Comment: 14 pages, 16 figure
The impact of Stieltjes' work on continued fractions and orthogonal polynomials
Stieltjes' work on continued fractions and the orthogonal polynomials related
to continued fraction expansions is summarized and an attempt is made to
describe the influence of Stieltjes' ideas and work in research done after his
death, with an emphasis on the theory of orthogonal polynomials
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
Pade approximants for functions with branch points - strong asymptotics of Nuttall-Stahl polynomials
Let f be a germ of an analytic function at infinity that can be analytically
continued along any path in the complex plane deprived of a finite set of
points, f \in\mathcal{A}(\bar{\C} \setminus A), \sharp A <\infty. J. Nuttall
has put forward the important relation between the maximal domain of f where
the function has a single-valued branch and the domain of convergence of the
diagonal Pade approximants for f. The Pade approximants, which are rational
functions and thus single-valued, approximate a holomorphic branch of f in the
domain of their convergence. At the same time most of their poles tend to the
boundary of the domain of convergence and the support of their limiting
distribution models the system of cuts that makes the function f single-valued.
Nuttall has conjectured (and proved for many important special cases) that this
system of cuts has minimal logarithmic capacity among all other systems
converting the function f to a single-valued branch. Thus the domain of
convergence corresponds to the maximal (in the sense of minimal boundary)
domain of single-valued holomorphy for the analytic function f
\in\mathcal{A}(\bar{\C} \setminus A). The complete proof of Nuttall's
conjecture (even in a more general setting where the set A has logarithmic
capacity zero) was obtained by H. Stahl. In this work, we derive strong
asymptotics for the denominators of the diagonal Pade approximants for this
problem in a rather general setting. We assume that A is a finite set of branch
points of f which have the algebraic character and which are placed in a
generic position. The last restriction means that we exclude from our
consideration some degenerated "constellations" of the branch points.Comment: 47 pages, 8 figure
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