Type-Two Well-Ordering Principles, Admissible Sets, and Pi^1_1-Comprehension

This thesis introduces a well-ordering principle of type two, which we call the Bachmann-Howard principle. The main result states that the Bachmann-Howard principle is equivalent to the existence of admissible sets and thus to Pi^1_1-comprehension. This solves a conjecture of Rathjen and Montalbán. The equivalence is interesting because it relates "concrete" notions from ordinal analysis to "abstract" notions from reverse mathematics and set theory. A type-one well-ordering principle is a map T which transforms each well-order X into another well-order T[X]. If T is particularly uniform then it is called a dilator (due to Girard). Our Bachmann-Howard principle transforms each dilator T into a well-order BH(T). The latter is a certain kind of fixed-point: It comes with an "almost" monotone collapse theta:T[BH(T)]->BH(T) (we cannot expect full monotonicity, since the order-type of T[X] may always exceed the order-type of X). The Bachmann-Howard principle asserts that such a collapsing structure exists. In fact we define three variants of this principle: They are equivalent but differ in the sense in which the order BH(T) is "computed". On a technical level, our investigation involves the following achievements: a detailed discussion of primitive recursive set theory as a basis for set-theoretic reverse mathematics; a formalization of dilators in weak set theories and second-order arithmetic; a functorial version of the constructible hierarchy; an approach to deduction chains (Schütte) and beta-completeness (Girard) in a set-theoretic context; and a beta-consistency proof for Kripke-Platek set theory. Independently of the Bachmann-Howard principle, the thesis contains a series of results connected to slow consistency (introduced by S.-D. Friedman, Rathjen and Weiermann): We present a slow reflection statement and investigate its consistency strength, as well as its computational properties. Exploiting the latter, we show that instances of the Paris-Harrington principle can only have extremely long proofs in certain fragments of arithmetic

Resolving the inner jet structure of 1924-292 with the EVENT HORIZON TELESCOPE

We present the first 1.3 mm (230 GHz) very long baseline interferometry model image of an AGN jet using closure phase techniques with a four-element array. The model image of the quasar 1924-292 was obtained with four telescopes at three observatories: the James Clerk Maxwell Telescope (JCMT) on Mauna Kea in Hawaii, the Arizona Radio Observatory's Submillimeter Telescope (SMT) in Arizona, and two telescopes of the Combined Array for Research in Millimeterwave Astronomy (CARMA) in California in April 2009. With the greatly improved resolution compared with previous observations and robust closure phase measurement, the inner jet structure of 1924-292 was spatially resolved. The inner jet extends to the northwest along a position angle of $-53^\circ$ at a distance of 0.38\,mas from the tentatively identified core, in agreement with the inner jet structure inferred from lower frequencies, and making a position angle difference of $\sim 80^{\circ}$ with respect to the cm-jet. The size of the compact core is 0.15\,pc with a brightness temperature of $1.2\times10^{11}$\,K. Compared with those measured at lower frequencies, the low brightness temperature may argue in favor of the decelerating jet model or particle-cascade models. The successful measurement of closure phase paves the way for imaging and time resolving Sgr A* and nearby AGN with the Event Horizon Telescope.Comment: 6 pages, 4 figures, accepted for publication in ApJ

Detection of intrinsic source structure at ~3 Schwarzschild radii with Millimeter-VLBI observations of SAGITTARIUS A*

We report results from very long baseline interferometric (VLBI) observations of the supermassive black hole in the Galactic center, Sgr A*, at 1.3 mm (230 GHz). The observations were performed in 2013 March using six VLBI stations in Hawaii, California, Arizona, and Chile. Compared to earlier observations, the addition of the APEX telescope in Chile almost doubles the longest baseline length in the array, provides additional {\it uv} coverage in the N-S direction, and leads to a spatial resolution of $\sim$30 $\mu$as ($\sim$3 Schwarzschild radii) for Sgr A*. The source is detected even at the longest baselines with visibility amplitudes of $\sim$4-13% of the total flux density. We argue that such flux densities cannot result from interstellar refractive scattering alone, but indicate the presence of compact intrinsic source structure on scales of $\sim$3 Schwarzschild radii. The measured nonzero closure phases rule out point-symmetric emission. We discuss our results in the context of simple geometric models that capture the basic characteristics and brightness distributions of disk- and jet-dominated models and show that both can reproduce the observed data. Common to these models are the brightness asymmetry, the orientation, and characteristic sizes, which are comparable to the expected size of the black hole shadow. Future 1.3 mm VLBI observations with an expanded array and better sensitivity will allow a more detailed imaging of the horizon-scale structure and bear the potential for a deep insight into the physical processes at the black hole boundary.Comment: 11 pages, 5 figures, accepted to Ap

230 GHz VLBI observations of M87: event-horizon-scale structure at the enhanced very-high-energy γ-ray state in 2012

We report on 230 GHz (1.3 mm) VLBI observations of M87 with the Event Horizon Telescope using antennas on Mauna Kea in Hawaii, Mt. Graham in Arizona and Cedar Flat in California. For the first time, we have acquired 230 GHz VLBI interferometric phase information on M87 through measurement of closure phase on the triangle of long baselines. Most of the measured closure phases are consistent with 0◦ as expected by physically-motivated models for 230 GHz structure such as jet models and accretion disk models. The brightness temperature of the event-horizon-scale structure is ∼ 1 × 1010 K derived from the compact flux density of ∼ 1 Jy and the angular size of ∼ 40 µas ∼ 5.5 Rs, which is broadly consistent with the peak brightness of the radio cores at 1-86 GHz located within ∼ 102 Rs. Our observations occurred in the middle of an enhancement in very-high-energy (VHE) γ-ray flux, presumably originating in the vicinity of the central black hole. Our measurements, combined with results of multi-wavelength observations, favor a scenario in which the VHE region has an extended size of ∼20-60 Rs.Astronom

Search for dark matter produced in association with bottom or top quarks in √s = 13 TeV pp collisions with the ATLAS detector

A search for weakly interacting massive particle dark matter produced in association with bottom or top quarks is presented. Final states containing third-generation quarks and miss- ing transverse momentum are considered. The analysis uses 36.1 fb−1 of proton–proton collision data recorded by the ATLAS experiment at √s = 13 TeV in 2015 and 2016. No significant excess of events above the estimated backgrounds is observed. The results are in- terpreted in the framework of simplified models of spin-0 dark-matter mediators. For colour- neutral spin-0 mediators produced in association with top quarks and decaying into a pair of dark-matter particles, mediator masses below 50 GeV are excluded assuming a dark-matter candidate mass of 1 GeV and unitary couplings. For scalar and pseudoscalar mediators produced in association with bottom quarks, the search sets limits on the production cross- section of 300 times the predicted rate for mediators with masses between 10 and 50 GeV and assuming a dark-matter mass of 1 GeV and unitary coupling. Constraints on colour- charged scalar simplified models are also presented. Assuming a dark-matter particle mass of 35 GeV, mediator particles with mass below 1.1 TeV are excluded for couplings yielding a dark-matter relic density consistent with measurements

Measurement of nuclear modification factors of gamma(1S)), gamma(2S), and gamma(3S) mesons in PbPb collisions at root s(NN)=5.02 TeV

The cross sections for ϒ(1S), ϒ(2S), and ϒ(3S) production in lead-lead (PbPb) and proton-proton (pp) collisions at √sNN = 5.02 TeV have been measured using the CMS detector at the LHC. The nuclear modification factors, RAA, derived from the PbPb-to-pp ratio of yields for each state, are studied as functions of meson rapidity and transverse momentum, as well as PbPb collision centrality. The yields of all three states are found to be significantly suppressed, and compatible with a sequential ordering of the suppression, RAA(ϒ(1S)) > RAA(ϒ(2S)) > RAA(ϒ(3S)). The suppression of ϒ(1S) is larger than that seen at √sNN = 2.76 TeV, although the two are compatible within uncertainties. The upper limit on the RAA of ϒ(3S) integrated over pT, rapidity and centrality is 0.096 at 95% confidence level, which is the strongest suppression observed for a quarkonium state in heavy ion collisions to date. © 2019 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3.Peer reviewe