On the statistics of the minimal solution of a linear Diophantine equation and uniform distribution of the real part of orbits in hyperbolic spaces

We study a variant of a problem considered by Dinaburg and Sinai on the statistics of the minimal solution to a linear Diophantine equation. We show that the signed ratio between the Euclidean norms of the minimal solution and the coefficient vector is uniformly distributed modulo one. We reduce the problem to an equidistribution theorem of Anton Good concerning the orbits of a point in the upper half-plane under the action of a Fuchsian group.Comment: Minor changes. Final version to appear in proceedings of the conference on the occasion of Sunada's 60th birthday. Contemp.Math. series of Amer. Math.So

Live-timer method of automatic dead-time correction for precision counting

Automatic correction for dead time losses in nuclear counting experiments is implemented by a simple live timer arrangement in which each counting interval is extended for compensation for the dead time during that interval. this method eliminates repetitious manual calculations, source of error, and dependence upon paralysis shifts

Observed Faraday Effects in Damped Lyman-Alpha Absorbers and Lyman Limit Systems: The Magnetised Environment of Galactic Building Blocks at Redshift=2

Protogalactic environments are typically identified using quasar absorption lines, and these galactic building blocks can manifest as Damped Lyman-Alpha Absorbers (DLAs) and Lyman Limit Systems (LLSs). We use radio observations of Faraday effects to test whether DLAs and LLSs host a magnetised medium, by combining DLA and LLS detections throughout the literature with 1.4 GHz polarization data from the NRAO VLA Sky Survey (NVSS). We obtain a control, a DLA, and a LLS sample consisting of 114, 19, and 27 lines-of-sight respectively - all of which are polarized at $\ge8\sigma$ to ensure Rician bias is negligible. Using a Bayesian framework, we are unable to detect either coherent or random magnetic fields in DLAs: the regular coherent magnetic fields within the DLAs must be $\le2.8$ $\mu$G, and the lack of depolarization is consistent with the weakly magnetised gas in DLAs being non-turbulent and quiescent. However, we find mild suggestive evidence that LLSs have coherent magnetic fields: after controlling for the redshift-distribution of our data, we find a 71.5% probability that LLSs have a higher RM than a control sample. We also find strong evidence that LLSs host random magnetic fields, with a 95.5% probability that LLS lines-of-sight have lower polarized fractions than a control sample. The regular coherent magnetic fields within the LLSs must be $\le2.4$ $\mu$G, and the magnetised gas must be highly turbulent with a typical scale on the order of $\approx5$-20 pc, which is similar to that of the Milky Way. This is consistent with the standard dynamo pedagogy, whereby magnetic fields in protogalaxies increase in coherence and strength as a function of cosmic time. Our results are consistent with a hierarchical galaxy formation scenario, with the DLAs, LLSs, and strong magnesium II (MgII) systems exploring three different stages of magnetic field evolution in galaxies.Comment: Submitted to Ap

Testing the Modern Merger Hypothesis via the Assembly of Massive Blue Elliptical Galaxies in the Local Universe

The modern merger hypothesis offers a method of forming a new elliptical galaxy through merging two equal-mass, gas-rich disk galaxies fuelling a nuclear starburst followed by efficient quenching and dynamical stabilization. A key prediction of this scenario is a central concentration of young stars during the brief phase of morphological transformation from highly-disturbed remnant to new elliptical galaxy. To test this aspect of the merger hypothesis, we use integral field spectroscopy to track the stellar Balmer absorption and 4000\AA\ break strength indices as a function of galactic radius for 12 massive (${\rm M_{*}}\ge10^{10}{\rm M_{\odot}}$), nearby (${\rm z}\le0.03$), visually-selected plausible new ellipticals with blue-cloud optical colours and varying degrees of morphological peculiarities. We find that these index values and their radial dependence correlate with specific morphological features such that the most disturbed galaxies have the smallest 4000\AA\ break strengths and the largest Balmer absorption values. Overall, two-thirds of our sample are inconsistent with the predictions of the modern merger hypothesis. Of these eight, half exhibit signatures consistent with recent minor merger interactions. The other half have star formation histories similar to local, quiescent early-type galaxies. Of the remaining four galaxies, three have the strong morphological disturbances and star-forming optical colours consistent with being remnants of recent, gas-rich major mergers, but exhibit a weak, central burst consistent with forming $\sim5\%$ of their stars. The final galaxy possesses spectroscopic signatures of a strong, centrally-concentrated starburst and quiescent core optical colours indicative of recent quenching (i.e., a post-starburst signature) as prescribed by the modern merger hypothesis.Comment: 25 pages, 37 figures, accepted to MNRA

Federal Tax Concepts as a Guide for State Apportionment of Dividends

In ASARCO, Inc. v. Idaho State Tax Commission, the Supreme Court rejected Idaho’s definition of income, and limited the apportionability of dividends, interest, and capital gains. The Court held that a unitary business relationship did not exist between ASARCO and the dividend payors, but did not address whether such a relationship was a sufficient precondition or a necessary one. This article addresses two federal tax doctrines that may provide insight into state taxation of dividends: (1) the “effectively connected” doctrine, and (2) the Corn Products doctrine. The article first provides a detailed examination of the facts in ASARCO, and discusses the Court’s holding. The article explains that Idaho’s sweeping definition of income, which conflated business and nonbusiness income, invited the Court’s rejection. Next, the article examines the applicability of Mobil Oil Corp. v. Commissioner of Taxes of Vermont, and how that holding is not dispositive regarding the apportionability of dividends in general. Finally, the article discusses the two aforementioned federal tax doctrines. These two doctrines do not expressly address the characterization of dividends as business income, but implicate the same underlying issue. The article concludes by explaining that the tax treatment of dividends and capital gains as business income is not unique to state taxation

High resolution Ge/Li/ spectrometer reduces rate-dependent distortions at high counting rates

Modified spectrometer system with a low-noise preamplifier reduces rate-dependent distortions at high counting rates, 25,000 counts per second. Pole-zero cancellation minimizes pulse undershoots due to multiple time constants, baseline restoration improves resolution and prevents spectral shifts