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We evidence a family $\mathcal{X}$ of square matrices over a field $\mathbb{K}$, whose elements will be called X-matrices. We show that this family is shape invariant under multiplication as well as transposition. We show that $\mathcal{X}$ is a (in general non-commutative) subring of $GL(n,\mathbb{K})$. Moreover, we analyse the condition for a matrix $A \in \mathcal{X}$ to be invertible in $\mathcal{X}$. We also show that, if one adds a symmetry condition called here bi-symmetry, then the set $\mathcal{X}^b$ of bi-symmetric X-matrices is a commutative subring of $\mathcal{X}$. We propose results for eigenvalue inclusion, showing that for X-matrices eigenvalues lie exactly on the boundary of Cassini ovals. It is shown that any monic polynomial on $ \mathbb{R} $ can be associated with a companion matrix in $ \mathcal{X} $.

The KArlsruhe TRItium Neutrino (KATRIN) experiment, which aims to make a direct and model-independent determination of the absolute neutrino mass scale, is a complex experiment with many components. More than 15 years ago, we published a technical design report (TDR) [https://publikationen.bibliothek.kit.edu/270060419] to describe the hardware design and requirements to achieve our sensitivity goal of 0.2 eV at 90% C.L. on the neutrino mass. Since then there has been considerable progress, culminating in the publication of first neutrino mass results with the entire beamline operating [arXiv:1909.06048]. In this paper, we document the current state of all completed beamline components (as of the first neutrino mass measurement campaign), demonstrate our ability to reliably and stably control them over long times, and present details on their respective commissioning campaigns., Comment: Added missing acknowledgement, corrected performance statement in chapter 4.2.5, updated author list and references

Shapley effects are attracting increasing attention as sensitivity measures. When the value function is the conditional variance, they account for the individual and higher order effects of a model input. They are also well defined under model input dependence. However, one of the issues associated with their use is computational cost. We present a new algorithm that offers major improvements for the computation of Shapley effects, reducing computational burden by several orders of magnitude (from $k!\cdot k$ to $2^k$, where $k$ is the number of inputs) with respect to currently available implementations. The algorithm works in the presence of input dependencies. The algorithm also makes it possible to estimate all generalized (Shapley-Owen) effects for interactions., Comment: 16 pages, 5 figures, 3 tables, 2 algorithms

We report on the neutrino mass measurement result from the first four-week science run of the Karlsruhe Tritium Neutrino experiment KATRIN in spring 2019. Beta-decay electrons from a high-purity gaseous molecular tritium source are energy analyzed by a high-resolution MAC-E filter. A fit of the integrated electron spectrum over a narrow interval around the kinematic endpoint at 18.57 keV gives an effective neutrino mass square value of $(-1.0^{+0.9}_{-1.1})$ eV$^2$. From this we derive an upper limit of 1.1 eV (90$\%$ confidence level) on the absolute mass scale of neutrinos. This value coincides with the KATRIN sensitivity. It improves upon previous mass limits from kinematic measurements by almost a factor of two and provides model-independent input to cosmological studies of structure formation.

In this work, we present the first spectroscopic measurements of conversion electrons originating from the decay of metastable gaseous $^\mathrm{83m}$Kr with the Karlsruhe Tritium Neutrino (KATRIN) experiment. The results obtained in this calibration measurement represent a major commissioning milestone for the upcoming direct neutrino mass measurement with KATRIN. The successful campaign demonstrates the functionalities of the full KATRIN beamline. The KATRIN main spectrometer's excellent energy resolution of ~ 1 eV made it possible to determine the narrow K-32 and L$_3$-32 conversion electron line widths with an unprecedented precision of ~ 1 %., Comment: Fixed affiliation of the corresponding author

The functional ANOVA expansion of a multivariate mapping plays a fundamental role in statistics. The expansion is unique once a unique distribution is assigned to the covariates. Recent investigations in the environmental and climate sciences show that analysts may not be in a position to assign a unique distribution in realistic applications. We offer a systematic investigation of existence, uniqueness, orthogonality, monotonicity and ultramodularity of the functional ANOVA expansion of a multivariate mapping when a multiplicity of distributions is assigned to the covariates. In particular, we show that a multivariate mapping can be associated with a core of probability measures that guarantee uniqueness. We obtain new results for variance decomposition and dimension distribution under mixtures. Implications for the global sensitivity analysis of computer experiments are also discussed., Comment: To Appear on SIAM/ASA Journal on Uncertainty Quantification 2018

The KATRIN experiment is a next-generation direct neutrino mass experiment with a sensitivity of 0.2 eV (90% C.L.) to the effective mass of the electron neutrino. It measures the tritium $\beta$-decay spectrum close to its endpoint with a spectrometer based on the MAC-E filter technique. The $\beta$-decay electrons are guided by a magnetic field that operates in the mT range in the central spectrometer volume; it is fine-tuned by a large-volume air coil system surrounding the spectrometer vessel. The purpose of the system is to provide optimal transmission properties for signal electrons and to achieve efficient magnetic shielding against background. In this paper we describe the technical design of the air coil system, including its mechanical and electrical properties. We outline the importance of its versatile operation modes in background investigation and suppression techniques. We compare magnetic field measurements in the inner spectrometer volume during system commissioning with corresponding simulations, which allows to verify the system's functionality in fine-tuning the magnetic field configuration. This is of major importance for a successful neutrino mass measurement at KATRIN., Comment: 32 pages, 16 figures

The KATRIN experiment will probe the neutrino mass by measuring the beta-electron energy spectrum near the endpoint of tritium beta-decay. An integral energy analysis will be performed by an electro-static spectrometer (Main Spectrometer), an ultra-high vacuum vessel with a length of 23.2 m, a volume of 1240 m^3, and a complex inner electrode system with about 120000 individual parts. The strong magnetic field that guides the beta-electrons is provided by super-conducting solenoids at both ends of the spectrometer. Its influence on turbo-molecular pumps and vacuum gauges had to be considered. A system consisting of 6 turbo-molecular pumps and 3 km of non-evaporable getter strips has been deployed and was tested during the commissioning of the spectrometer. In this paper the configuration, the commissioning with bake-out at 300{\deg}C, and the performance of this system are presented in detail. The vacuum system has to maintain a pressure in the 10^{-11} mbar range. It is demonstrated that the performance of the system is already close to these stringent functional requirements for the KATRIN experiment, which will start at the end of 2016., Comment: submitted for publication in JINST, 39 pages, 15 figures

We have carried out nonequilibrium molecular dynamics simulations of a system of crosslinked particles under shear flow conditions. As the fraction of crosslinks $p$ is increased the system approaches a gel point at which the shear viscosity $\eta$ and the first and second normal stress coefficients $\Psi_1$ and $\Psi_2$ diverge. All three quantities seem to diverge with a power law form: $\eta\sim \epsilon^{-s}$, $\Psi_{1,2}\sim\epsilon^{-\lambda}$ where $\epsilon =p_c-p$ and $s\approx 0.7$ and $\lambda\approx 3.15$ in three dimensions and $s\approx 2.0$ and $\lambda\approx 6.0$ in two dimensions., Comment: 5 pages, 9 figures

We study the onset of rigidity near the glass transition (GT) in a short-chain polymer melt modelled by a bead-spring model, where all beads interact with Lennard-Jones potentials. The properties of the system are examined above and below the GT. In order to minimize high cooling-rate effects and computational times, equilibrium configurations are reached via isothermal compression. We monitor quantities such as the heat capacity C_P, the short-time diffusion constants D, the viscosity \eta, and the shear modulus; the time-dependent shear modulus G(t) is compared with the shear modulus \mu obtained from an externally applied instantaneous shear. We give a detailed analysis of the effects of such shearing on the system, both locally and globally. It is found that the polymeric glass only displays long-time rigid behavior below a temperature T_1, where T_1

We investigate the behavior of the shear viscosity $\eta(p)$ and the mass-dependent diffusion coefficient $D(m,p)$ in the context of a simple model that, as the crosslink density $p$ is increased, undergoes a continuous transition from a fluid to a gel. The shear viscosity diverges at the gel point according to $\eta(p)\sim (p_c-p)^{-s}$ with $s\approx 0.65$. The diffusion constant shows a remarkable dependence on the mass of the clusters: $D(m,p)\sim m^{-0.69}$, not only at $p_c$ but well into the liquid phase. We also find that the Stokes-Einstein relation $D\eta\propto k_BT$ breaks down already quite far from the gel point., Comment: 15 pages, 10 figures. Submitted to PRE

We investigated mu+ decays at rest produced at the ISIS beam stop target. Lepton flavor (LF) conservation has been tested by searching for \nueb via the detection reaction p(\nueb,e+)n. No \nueb signal from LF violating mu+ decays was identified. We extract upper limits of the branching ratio for the LF violating decay mu+ -> e+ \nueb \nu compared to the Standard Model (SM) mu+ -> e+ nue numub decay: BR < 0.9(1.7)x10^{-3} (90%CL) depending on the spectral distribution of \nueb characterized by the Michel parameter rho=0.75 (0.0). These results improve earlier limits by one order of magnitude and restrict extensions of the SM in which \nueb emission from mu+ decay is allowed with considerable strength. The decay \mupdeb as source for the \nueb signal observed in the LSND experiment can be excluded., Comment: 10 pages, including 1 figure, 1 table

We study a two-component model for gelation consisting of $f$-functional monomers (the gel) and inert particles (the solvent). After equilibration as a simple liquid, the gel particles are gradually crosslinked to each other until the desired number of crosslinks has been attained. At a critical crosslink density the largest gel cluster percolates and an amorphous solid forms. This percolation process is different from ordinary lattice or continuum percolation of a single species in the sense that the critical exponents are new. As the crosslink density $p$ approaches its critical value $p_c$, the shear viscosity diverges: $\eta(p)\sim (p_c-p)^{-s}$ with $s$ a nonuniversal concentration-dependent exponent., Comment: 6 pages, 9 figures

The KARMEN experiment at the spallation neutron source ISIS used \numub from \mup--decay at rest in the search for neutrino oscillations \numubnueb in the appearance mode, with p(\nueb,e+)n as detection reaction of \nueb. In total, 15 candidates fulfill all conditions for the \nueb signature, in agreement with the background expectation of 15.8+-0.5 events, yielding no indication for oscillations. A single event based likelihood analysis leads to upper limits on the oscillation parameters: sin^2(2theta)<1.7x10e-3 for Dm^2>100 eV^2 and Dm^2<0.055 eV^2 for sin^2(2theta)=1 at 90% confidence. Thus, KARMEN does not confirm the LSND experiment and restricts significantly its favored parameter region for \numubnueb., Comment: PRD-Preprint style, 42 pages, 14 figures, submitted to Phys. Rev. D

We report on extensive molecular dynamics simulations on systems of soft spheres of functionality f, i.e. particles that are capable of bonding irreversibly with a maximum of f other particles. These bonds are randomly distributed throughout the system and imposed with probability p. At a critical concentration of bonds, p_c approximately equal to 0.2488 for f=6, a gel is formed and the shear viscosity \eta diverges according to \eta ~ (p_c-p)^{-s}. We find s is approximately 0.7 in agreement with some experiments and with a recent theoretical prediction based on Rouse dynamics of phantom chains. The diffusion constant decreases as the gel point is approached but does not display a well-defined power law., Comment: 4 pages, 4 figures

The KARMEN experiment uses the reaction 12C(\nu_e,e-)12N to measure the energy distribution of \nu_e emitted in muon decay at rest. The \nu_e analog \omega_l of the famous Michel parameter \rho has been derived from a maximum-likelihood analysis of events near the kinematic end point, E_max. The result, \omega_l = (2.7^{+3.8}_{-3.3}\pm 3.1) * 10^{-2}, is in good agreement with the standard model prediction \omega_l= 0. We deduce a 90% confidence upper limit of \omega_l <= 0.113, which corresponds to a limit of |g^S_RL + 2 g^T_RL| <= 0.78 on the interference term between scalar and tensor coupling constants., Comment: 11 pages including 3 figures, submitted to Phys. Rev. Letters

The 56 tonne high resolution liquid scintillation calorimeter KARMEN at the beam stop neutrino source ISIS has been used to search for neutrino oscillations in the disappearance channel nu_e->x. The nu_e emitted in mu+ decay at rest are detected with spectroscopic quality via the exclusive charged current reaction 12-C(nu_e,e-)12-N_g.s. almost free of background. Analysis of the spectral shape of e- from the nu_e induced reaction as well as a measurement of the absolute nu_e flux allow to investigate oscillations of the type nu_e->nu_tau and nu_e->nu_mu . The flux independent ratio R(CC/NC) of charged current events 12-C(nu_e,e-)12-N_g.s. to neutral current events 12-C(nu,nu')12-C* provides additional information in the oscillation channel nu_e->x . All three analysis methods show no evidence for oscillations. For the nu_e->nu_tau channel 90%CL limits of sin^2(2t)<0.338 for dm^2>100eV^2 and dm^2<0.77eV^2 for maximal mixing in a simple 2 flavor oscillation formalism are derived. A complete 3 flavor analysis of the experimental data from five years of measurement with respect to nu_e<->nu_tau and nu_e<->nu_mu mixing is presented., Comment: 20 pages including 13 figures, submitted to Phys.Rev.C

We discuss static and dynamic fluctuations of domain walls separating areas of constant but different slopes in steady-state configurations of crystalline surfaces both by an analytic treatment of the appropriate Langevin equation and by numerical simulations. In contrast to other situations that describe the dynamics in Ising-like systems such as models A and B, we find that the dynamic exponent z=2 that governs the domain wall relaxation function is not equal to the inverse of the exponent n=1/4 that describes the coarsening process that leads to the steady state., Comment: 4 pages, 2 Postscript figures, REVTeX accepted for publication in Phys. Rev. Lett