Families of spectral sets for Bernoulli convolutions

In this paper, we study the harmonic analysis of Bernoulli measures. We show a variety of orthonormal Fourier bases for the L^2 Hilbert spaces corresponding to certain Bernoulli measures, making use of contractive transfer operators. For other cases, we exhibit maximal Fourier families that are not orthonormal bases.Comment: 25 pages, same result

Additive spectra of the 1/4 Cantor measure

In this paper, we add to the characterization of the Fourier spectra for Bernoulli convolution measures. These measures are supported on Cantor subsets of the line. We prove that performing an odd additive translation to half the canonical spectrum for the 1/4 Cantor measure always yields an alternate spectrum. We call this set an additive spectrum. The proof works by connecting the additive set to a spectrum formed by odd multiplicative scaling.Comment: 9 pages, 1 figur

Simulating the Impact of X-ray Heating during the Cosmic Dawn

Upcoming observations of the 21-cm signal from the Epoch of Reionization will soon provide the first direct detection of this era. This signal is influenced by many astrophysical effects, including long range X-ray heating of the intergalactic gas. During the preceding Cosmic Dawn era the impact of this heating on the 21-cm signal is particularly prominent, especially before spin temperature saturation. We present the largest-volume (349\,Mpc comoving=244~$h^{-1}$Mpc) full numerical radiative transfer simulations to date of this epoch that include the effects of helium and multi-frequency heating, both with and without X-ray sources. We show that X-ray sources contribute significantly to early heating of the neutral intergalactic medium and, hence, to the corresponding 21-cm signal. The inclusion of hard, energetic radiation yields an earlier, extended transition from absorption to emission compared to the stellar-only case. The presence of X-ray sources decreases the absolute value of the mean 21-cm differential brightness temperature. These hard sources also significantly increase the 21-cm fluctuations compared the common assumption of temperature saturation. The 21-cm differential brightness temperature power spectrum is initially boosted on large scales, before decreasing on all scales. Compared to the case of the cold, unheated intergalactic medium, the signal has lower rms fluctuations and increased non-Gaussianity, as measured by the skewness and kurtosis of the 21-cm probability distribution functions. Images of the 21-cm signal with resolution around 11~arcmin still show fluctuations well above the expected noise for deep integrations with the SKA1-Low, indicating that direct imaging of the X-ray heating epoch could be feasible.Comment: 13 pages, 8 figure

Scaling by 5 on a 1/4-Cantor Measure

Each Cantor measure (\mu) with scaling factor 1/(2n) has at least one associated orthonormal basis of exponential functions (ONB) for L^2(\mu). In the particular case where the scaling constant for the Cantor measure is 1/4 and two specific ONBs are selected for L^2(\mu), there is a unitary operator U defined by mapping one ONB to the other. This paper focuses on the case in which one ONB (\Gamma) is the original Jorgensen-Pedersen ONB for the Cantor measure (\mu) and the other ONB is is 5\Gamma. The main theorem of the paper states that the corresponding operator U is ergodic in the sense that only the constant functions are fixed by U.Comment: 34 page

Reconstruction methods — P‾ANDA focussing-light guide disc DIRC

The Focussing-Lightguide Disc DIRC will provide crucial Particle Identification (PID) information for the P‾ANDA experiment at FAIR, GSI. This detector presents a challenging environment for reconstruction due to the complexity of the expected hit patterns and the operating conditions of the P‾ANDA experiment. A discussion of possible methods to reconstruct PID from this detector is given here. Reconstruction software is currently under development

The bar PANDA focussing-lightguide disc DIRC

bar PANDA will be a fixed target experiment internal to the HESR antiproton storage ring at the future FAIR complex. The ANDA detector requires excellent particle-identification capabilities in order to achieve its scientific potential. Cherenkov counters employing the DIRC principle were chosen as PID detectors for the Target Spectrometer. The proposed Focussing-Lightguide Disc DIRC will cover the forward part of the Target Spectrometer acceptance in the angular range between 5° and 22°. Its design includes a novel approach to mitigate dispersion effects in the solid radiator of a DIRC counter using optical elements. The dispersion correction will enable the Focussing-Lightguide Disc DIRC to provide pion-kaon identification for momenta well above 3.5 GeV/c