Computing Stieltjes constants using complex integration
The Stieltjes constants $\gamma\_n$ are the coefficients appearing in the Laurent series of the Riemann zeta function at $s = 1$. We give a simple and efficient method to compute a $p$-bit approximation of $\gamma\_n$ with rigorous error bounds. Starting from an integral representation due to Blagouchine, we shift the contour to eliminate cancellation. The integral is then evaluated numerically in ball arithmetic using the Petras algorithm, with the use of a Taylor expansion for bounds near the saddle point. This appears to be the first algorithm for Stieltjes constants with uniformly low complexity with respect to both $n$ and $p$. An implementation is provided in the Arb library. We can, for example, compute $\gamma\_n$ to 1000 digits in a minute for any $n \le 10^{100}$.
All the trinomial roots, their powers and logarithms from the Lambert series, Bell polynomials and Fox–Wright function: illustration for genome multiplicity in survival of irradiated cells
All the roots of the general nth degree trinomial admit certain convenient representations in terms of the Lambert and Euler series for the asymmetric and symmetric cases of the trinomial equation, respectively. Previously, various methods have been used to provide the proofs for the general terms of these two series. Taking n to be any real or complex number, we presently give an alternative proof using the Bell (or exponential) polynomials. The ensuing series is summed up yielding a single, compact, explicit, analytical formula for all the trinomial roots as the confluent Fox–Wright function $${}_1\Psi _1$$1Ψ1. Moreover, we also derive a slightly different, single formula of the trinomial root raised to any power (real or complex number) as another $${}_1\Psi _1$$1Ψ1 function. Further, in this study, the logarithm of the trinomial root is likewise expressed through a single, concise series with the binomial expansion coefficients or the Pochhammer symbols. These findings are anticipated to be of considerable help in various applications of trinomial roots. Namely, several properties of the $${}_1\Psi _1$$1Ψ1 function can advantageously be employed for its implementations in practice. For example, the simple expressions for the asymptotic limits of the $${}_1\Psi _1$$1Ψ1 function at both small and large values of the independent variable can be used to readily predict, by analytical means, the critical behaviors of the studied system in the two extreme conditions. Such limiting situations can be e.g. at the beginning of the time evolution of a system, and in the distant future, if the independent variable is time, or at low and high doses when the independent variable is radiation dose, etc. The present analytical solutions for the trinomial roots are numerically illustrated in the genome multiplicity corrections for survival of synchronous cell populations after irradiation.
DEM generation from new VIR sensors: IKONOS, ASTER and Landsat-7
DEM generation is a living R&D topic since new-launched satellite images can be used to generate stereoscopic images. Using a coplanarity equation model developed at CCRS and an automatic matching method developed at PCI, DEMs are generated with new VIR stereo data. Accuracy of 4.9 m with high-resolution IKONOS images, and 33 m and 101 m with medium-resolution ASTER and Landsat-7 ETM/sup +/ images respectively are obtained. The same matching accuracy of one-pixel is translated into worse results with Landsat due to a small B/H ratio of 0.15.