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Tobias Kyrion

Publications and source records attributed to Tobias Kyrion.

5 recordsLinked to original sources
arXiv2026

Recurrence Relations for $β(2k)$ and $ζ(2k + 1)$

In this work we study integrals of the form $\int_{0}^{\infty} \frac{\tanh(x)}{x} \sech(x)^{L} \exp(-Tx) dx, \quad T \in \mathbb{R}_{\geq 0}.$ We show that they can be represented as a sum of Hurwitz zeta derivatives with polynomial coefficients. As an application we evaluate these integrals for $T =2 l$ with integer $l \in \mathbb{Z}_{\geq 0}$ and obtain recurrence relations for $β(2k)$ and $ζ(2k + 1)$, where $β(z)$ is the Dirichlet beta function and $ζ(z)$ is the Riemann zeta function. For the negative $T$-derivative of the above integrals $\int_{0}^{\infty} \tanh(x)\sech(x)^{L} \exp(-Tx) dx$ we give simpler representations only involving digamma function values with polynomial coefficients.

math.GM↗
arXiv2020

Recurrence Relations for Values of the Riemann Zeta Function in Odd Integers

It is commonly known that $ζ(2k) = q_{k}\frac{ζ(2k + 2)}{π^2}$ with known rational numbers $q_{k}$. In this work we construct recurrence relations of the form $\sum_{k = 1}^{\infty}r_{k}\frac{ζ(2k + 1)}{π^{2k}} = 0$ and show that series representations for the coefficients $r_{k} \in \mathbb{R}$ can be computed explicitly.

math.NT↗
arXiv2017

Robust Inversion Methods for Aerosol Spectroscopy

The Fast Aerosol Spectrometer (FASP) is a device for spectral aerosol measurements. Its purpose is to safely monitor the atmosphere inside a reactor containment. First we describe the FASP and explain its basic physical laws. Then we introduce our reconstruction methods for aerosol particle size distributions designed for the FASP. We extend known existence results for constrained Tikhonov regularization by uniqueness criteria and use those to generate reasonable models for the size distributions. We apply a Bayesian model-selection framework on these pre-generated models. We compare our algorithm with classical inversion methods using simulated measurements. We then extend our reconstruction algorithm for two-component aerosols, so that we can simultaneously retrieve their particle-size distributions and unknown volume fractions of their two components. Finally we present the results of a numerical study for the extended algorithm.

math.NA↗
arXiv2016

Reconstruction of Refractive Indices from Spectral Measurements of Monodisperse Aerosols

For the investigation of two-component aerosols one needs to know the refractive indices of the two aerosol components. One problem is that they depend on temparature and pressure, so one needs for their determination a robust measurement instrument such as the FASP device, which can cope with rigid environmental conditions. In this article we show that the FASP device is capable of measuring the needed refractive indices, if monodisperse aerosols of the pure components are provided. We determine the particle radii of the monodisperse aerosols needed for this task and investigate how accurate the measurements have to be in order to retrieve refractive indices in a sufficient quality, such that they are suitable for investigations of two-component aerosols.

math.NA↗