Large subsets of discrete hypersurfaces in Zd\mathbb{Z}^d contain arbitrarily many collinear points

In 1977 L.T. Ramsey showed that any sequence in $\mathbb{Z}^2$ with bounded gaps contains arbitrarily many collinear points. Thereafter, in 1980, C. Pomerance provided a density version of this result, relaxing the condition on the sequence from having bounded gaps to having gaps bounded on average. We give a higher dimensional generalization of these results. Our main theorem is the following. Theorem: Let $d\in\mathbb{N}$, let $f:\mathbb{Z}^d\to\mathbb{Z}^{d+1}$ be a Lipschitz map and let $A\subset\mathbb{Z}^d$ have positive upper Banach density. Then $f(A)$ contains arbitrarily many collinear points. Note that Pomerance's theorem corresponds to the special case $d=1$. In our proof, we transfer the problem from a discrete to a continuous setting, allowing us to take advantage of analytic and measure theoretic tools such as Rademacher's theorem.Comment: 16 pages, small part of the argument clarified in light of suggestions from the refere

Microdeflectometry - a novel tool to acquire 3D microtopography with nanometer height resolution

We introduce "microdeflectometry", a novel technique for measuring the microtopography of specular surfaces. The primary data is the local slope of the surface under test. Measuring the slope instead of the height implies high information efficiency and extreme sensitivity to local shape irregularities. The lateral resolution can be better than one micron whereas the resulting height resolution is in the range of one nanometer. Microdeflectometry can be supplemented by methods to expand the depth of field, with the potential to provide quantitative 3D imaging with SEM-like features.Comment: 3 pages, 11 figures, latex, zip-file, accepted for publication at Optics Letter

A meta-analysis on the effects of IT capability toward agility and performance: New directions for information systems research

Information technology (IT) capability is an organizational capability that enables organizations to acquire, deploy, combine, and reconfigure IT resources. As such, it is often investigated in conjunction with organizational agility—an organization’s ability to sense and respond to changes—and organizational performance. Studies on IT capability distinguish between reactive and proactive IT capability and identify varying effects in relation to agility and performance. While reactive IT capability supports and enhances work processes, proactive IT capability supports and enhances business strategies. In the light of the mixed results of prior research, we conduct a meta-analytical investigation into the varying effects that reactive and proactive IT capability have on organizational agility and organizational performance. We identified 6.436 studies from multiple sources that we systematically reduced to include 72 empirical studies in our analysis. Contrary to previous results and widely held opinion, our meta-analysis neither finds support for differences in effect size between reactive (r(+) = 0.39, k = 34, 95% Confidence Interval (CI) [0.34, 0.44]) and proactive IT capability (r(+) = 0.38, k = 21, 95% CI [0.31, 0.45]) toward agility (z = 0.68, p = 0.25), nor from reactive IT capability (r(+) = 0.31, k = 43, 95% CI [0.26, 0.37]) and proactive IT capability (r(+) = 0.33, k = 25, 95% CI [0.27, 0.40]) toward performance (z = 1.11, p = 0.13). Given the importance of IT capability, we discuss possible explanations and propose four areas for future research: latency, sequence, configurational, and theoretical multiplicity of IT capability

Photon counting intensity interferometry in the blue at a 0.5 m telescope

Intensity interferometry is a re-emerging interferometry tool that alleviates some of the challenges of amplitude interferometry at the cost of reduced sensitivity. We demonstrate the feasibility of intensity interferometry with fast single photon counting detectors at small telescopes by utilising a telescope of diameter of merely $0.5$\,m. The entire measurement setup, including collimation, optical filtering, and two single photon detectors, is attached directly to the telescope without the use of optical fibres, facilitated by the large area of our single photon detectors. For digitisation and timing, we utilise a Time-To-Amplitude-Converter. Observing $\alpha$ Lyrae (Vega) for a total exposure time of $32.4$\,h over the course of six nights, an auto-correlation signal with a contrast of $(9.5 \pm 2.7) \times 10^-3$ and a coherence time of $(0.34 \pm 0.12)$ ps at a SNR of 2.8 is measured. The result fits well to preceding laboratory tests as well as expectations calculated from the optical and electronic characteristics of our measurement setup. This measurement, to our knowledge, constitutes the first time that a bunching signal with starlight was measured in the B band with single photon counting detectors. Simultaneously, this is to date the stellar intensity interferometry measurement utilising the smallest telescope. Our successful measurement shows that intensity interferometry can be adopted not only at large scale facilities, but also at readily available and inexpensive smaller telescopes

A proof of a sumset conjecture of Erd\H{o}s

In this paper we show that every set $A \subset \mathbb{N}$ with positive density contains $B+C$ for some pair $B,C$ of infinite subsets of $\mathbb{N}$, settling a conjecture of Erd\H{o}s. The proof features two different decompositions of an arbitrary bounded sequence into a structured component and a pseudo-random component. Our methods are quite general, allowing us to prove a version of this conjecture for countable amenable groups.Comment: 54 pages. Corrected proof of Theorem 3.22 and added Example 3.27 Keywords: sum sets, almost periodic functions, ultrafilter

Lines and Conics in a Family of Quartic Surfaces

42 pagesThis thesis is primarily concerned with the fact that the general quartic surface does not contain a line or a conic section. More formally, the set of quartic surfaces containing lines or conic sections has measure zero. This result follows from an important result in algebraic geometry, the Noether-Lefschetz theo-rem; however, this paper will use elemenary techniques in algebra and analysis to come to the same conclusion. In addition, this paper will introduce the idea of curves, surfaces, and their parameter spaces