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 $\{s_n \}_{n=0}^{\infty}$ of partial sums, but also explicit estimates $\{\omega_n \}_{n=0}^{\infty}$ 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

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

The Use of the Health of the Nation Outcome Scales for Assessing Functional Change in Treatment Outcome Monitoring of Patients with Chronic Schizophrenia.

Schizophrenia is a severe mental disorder that is characterized not only by symptomatic severity but also by high levels of functional impairment. An evaluation of clinical outcome in treatment of schizophrenia should therefore target not only assessing symptom change but also alterations in functioning. This study aimed to investigate whether there is an agreement between functional- and symptom-based outcomes in a clinical sample of admissions with chronic forms of schizophrenia. A full 3-year cohort of consecutive inpatient admissions for schizophrenia (N = 205) was clinically rated with the Positive and Negative Symptom Scale (PANSS) and the Health of the Nation Outcome Scales (HoNOS) as measures of functioning at the time of admission and discharge. The sample was stratified twofold: first, according to the degree of PANSS symptom improvement during treatment with the sample being divided into three treatment response groups: non-response, low response, and high response. Second, achievement of remission was defined using the Remission in Schizophrenia Working Group criteria based on selected PANSS symptoms. Repeated measures analyses were used to compare the change of HoNOS scores over time across groups. More than a half of all admissions achieved a symptom reduction of at least 20% during treatment and around one quarter achieved remission at discharge. Similarly, HoNOS scores improved significantly between admission and discharge. Interaction analyses indicated higher functional improvements to be associated with increasing levels of treatment response. Functional improvement in individuals treated for schizophrenia was linked to a better clinical outcome, which implies a functional association. Thus, improvement of functioning represents an important therapeutic target in the treatment of schizophrenia

An efficient algorithm for accelerating the convergence of oscillatory series, useful for computing the polylogarithm and Hurwitz zeta functions

This paper sketches a technique for improving the rate of convergence of a general oscillatory sequence, and then applies this series acceleration algorithm to the polylogarithm and the Hurwitz zeta function. As such, it may be taken as an extension of the techniques given by Borwein's "An efficient algorithm for computing the Riemann zeta function", to more general series. The algorithm provides a rapid means of evaluating Li_s(z) for general values of complex s and the region of complex z values given by |z^2/(z-1)|<4. Alternatively, the Hurwitz zeta can be very rapidly evaluated by means of an Euler-Maclaurin series. The polylogarithm and the Hurwitz zeta are related, in that two evaluations of the one can be used to obtain a value of the other; thus, either algorithm can be used to evaluate either function. The Euler-Maclaurin series is a clear performance winner for the Hurwitz zeta, while the Borwein algorithm is superior for evaluating the polylogarithm in the kidney-shaped region. Both algorithms are superior to the simple Taylor's series or direct summation. The primary, concrete result of this paper is an algorithm allows the exploration of the Hurwitz zeta in the critical strip, where fast algorithms are otherwise unavailable. A discussion of the monodromy group of the polylogarithm is included.Comment: 37 pages, 6 graphs, 14 full-color phase plots. v3: Added discussion of a fast Hurwitz algorithm; expanded development of the monodromy v4:Correction and clarifiction of monodrom

The performance of the Health of the Nation Outcome Scales as measures of clinical severity.

The aim of this study was to examine the performance of the Health of the Nation Outcome Scales (HoNOS) against other measures of functioning and mental health in a full three-year cohort of admissions to a psychiatric hospital. A sample of N=1719 patients (35.3% females, aged 17-78 years) was assessed using observer-rated measures and self-reports of psychopathology at admission. Self-reports were available from 51.7% of the sample (34.4% females, aged 17-76 years). Functioning and psychopathology were compared across five ICD-10 diagnostic groups: substance use disorders, schizophrenia and psychotic disorders, affective disorders, anxiety/somatoform disorders and personality disorders. Associations between the measures were examined, stratifying by diagnostic subgroup. The HoNOS were strongly linked to other measures primarily in psychotic disorders (except for the behavioral subscale), while those with substance use disorders showed rather poor links. Those with anxiety/somatoform disorders showed null or only small associations. This study raises questions about the overall validity of the HoNOS. It seems to entail different levels of validity when applied to different diagnostic groups. In clinical practice the HoNOS should not be used as a stand-alone instrument to assess outcome but rather as part of a more comprehensive battery including diagnosis-specific measures

Time resolved X ray absorption spectroscopy of infrared laser induced temperature jumps in liquid water

A time resolved X ray absorption study of the structural dynamics of liquid water on a picosecond timescale is presented. We apply femtosecond midinfrared pulses to resonantly excite the intramolecular O H stretching band of liquid water and monitor the transient response in the oxygen K edge absorption spectrum with picosecond X ray pulses. In this way, structural changes in the hydrogen bond network of liquid water upon an ultrafast temperature jump of approximately 20 K are investigated. The changes of the X ray absorption as induced by such a temperature jump are about 3.2 . This demonstrates that our method serves as a sensitive probe of transient structural changes in liquid water and that combined infrared laser synchrotron experiments with substantially shorter X ray pulses, such as generated with a femtosecond slicing scheme, are possibl