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

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

Neuronal Processing of Complex Mixtures Establishes a Unique Odor Representation in the Moth Antennal Lobe

Animals typically perceive natural odor cues in their olfactory environment as a complex mixture of chemically diverse components. In insects, the initial representation of an odor mixture occurs in the first olfactory center of the brain, the antennal lobe (AL). The contribution of single neurons to the processing of complex mixtures in insects, and in particular moths, is still largely unknown. Using a novel multicomponent stimulus system to equilibrate component and mixture concentrations according to vapor pressure, we performed intracellular recordings of projection and interneurons in an attempt to quantitatively characterize mixture representation and integration properties of single AL neurons in the moth. We found that the fine spatiotemporal representation of 2–7 component mixtures among single neurons in the AL revealed a highly combinatorial, non-linear process for coding host mixtures presumably shaped by the AL network: 82% of mixture responding projection neurons and local interneurons showed non-linear spike frequencies in response to a defined host odor mixture, exhibiting an array of interactions including suppression, hypoadditivity, and synergism. Our results indicate that odor mixtures are represented by each cell as a unique combinatorial representation, and there is no general rule by which the network computes the mixture in comparison to single components. On the single neuron level, we show that those differences manifest in a variety of parameters, including the spatial location, frequency, latency, and temporal pattern of the response kinetics

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

Sequential simulation-based inference for gravitational wave signals

The current and upcoming generations of gravitational wave experiments represent an exciting step forward in terms of detector sensitivity and performance. For example, key upgrades at the LIGO, Virgo and KAGRA facilities will see the next observing run (O4) probe a spatial volume around four times larger than the previous run (O3), and design implementations for, e.g., the Einstein Telescope, Cosmic Explorer, and LISA experiments are taking shape to explore a wider frequency range and probe cosmic distances. In this context, however, a number of very real data analysis problems face the gravitational wave community. For example, it will be critical to develop tools and strategies to analyze (among other scenarios) signals that arrive coincidentally in detectors, longer signals that are in the presence of nonstationary noise or other shorter transients, as well as noisy, potentially correlated, coherent stochastic backgrounds. With these challenges in mind, we develop peregrine, a new sequential simulation-based inference approach designed to study broad classes of gravitational wave signal. In this work, we describe the method and implementation, before demonstrating its accuracy and robustness through direct comparison with established likelihood-based methods. Specifically, we show that we are able to fully reconstruct the posterior distributions for every parameter of a spinning, precessing compact binary coalescence using one of the most physically detailed and computationally expensive waveform approximants (SEOBNRv4PHM). Crucially, we are able to do this using only 2% of the waveform evaluations that are required in, e.g., nested sampling approaches. Finally, we provide some outlook as to how this level of simulation efficiency and flexibility in the statistical analysis could allow peregrine to tackle these current and future gravitational wave data analysis problems