On Robin's criterion for the Riemann Hypothesis

Robin's criterion states that the Riemann Hypothesis (RH) is true if and only if Robin's inequality sum_{d|n}d=5041, where gamma denotes the Euler(-Mascheroni) constant. We show by elementary methods that if n>=37 does not satisfy Robin's criterion it must be even and is neither squarefree nor squarefull. Using a bound of Rosser and Schoenfeld we show, moreover, that n must be divisible by a fifth power >1. As a consequence we infer that RH holds true if and only if every natural number divisible by a fifth power >1 satisfies Robin's inequality.Comment: 15 pages. Corrected version. In the first version in Theorem 5 (main result) it was falsely asserted that n must be superabundant, invalidating the proof. An alternative proof is provided in this version, some typos have been corrected, and the presentation has been improve

Testing hypotheses of the cause of peripheral thinning of the Greenland Ice Sheet: is land-terminating ice thinning at anomalously high rates?

Recent observations have shown that the periphery of the Greenland ice sheet (GrIS) is thinning rapidly and that this thinning is greatest around marine-terminating outlet glaciers. Several theories have been proposed which provide a link between climate and ice thinning. We present surface elevation change (<i>dh/dt</i>) data from NASA's Program for Arctic Regional Climate Assessment (PARCA) laser altimetry surveys for fourteen and eleven of the largest outlet glaciers in Southern Greenland from 1993 to 1998 and 1998 to 2006 respectively to test the applicability of these theories to the GrIS. <br><br> Initially, outlet glacier <i>dh/dt</i> data are compared with data from concurrent surveys over inland ice (slow flowing ice that is not obviously draining into an outlet glacier) to confirm the effect of ice flow on surface thinning rates. Land-terminating and marine-terminating outlet glacier <i>dh/dt</i> data are then compared from 1993 to 1998 and from 1998 to 2006. Finally, ablation anomalies (the difference between the "normal" ablation rate from 1970 to 2000 and the ablation rate in the time period of interest) calculated with a positive degree day model are compared to both marine-terminating and land-terminating outlet glacier <i>dh/dt</i> data. <br><br> Our results support earlier conclusions that certain marine-terminating outlet glaciers have thinned much more than land-terminating outlet glaciers during both time periods. Furthermore we show that these differences are not limited to the largest, fastest-flowing outlet glaciers – almost all marine-terminating outlet glaciers are thinning more than land-terminating outlet glaciers. There was a four fold increase in mean marine-terminating outlet glacier thinning rates below 1000 m elevation between the periods 1993 to 1998 and 1998 to 2006, while thinning rates of land-terminating outlet glaciers remained statistically unchanged. This suggests that a change in a controlling mechanism specific to the thinning rates of marine-terminating outlet glaciers occurred in the late 1990s and that this change did not affect thinning rates of land-terminating outlet glaciers. <br><br> Thinning rates of land-terminating outlet glaciers are statistically the same as ablation anomalies, while thinning rates of marine-terminating outlet glaciers are not. Thinning of land-terminating outlet glaciers therefore seems to be a response to changes in local mass balance (principally increases in air temperature) while thinning of marine-terminating outlet glaciers is principally controlled by ice dynamics. The mechanism by which this dynamic thinning occurs is still not clear although its association with marine-terminating outlet glaciers suggests perturbations at marine termini (calving) as the likely cause

Sustainable Design: Using Physical Prototypes to Most Benefit Design Students and Environment?

After reviewing the main environmental areas of concern today, this paper will focus on waste management within the area of sustainability. For many years, as part of the design process, physical prototypes were a necessary. This was in order to prove a design‘s functionality and safety. They were expensive, time consuming and, by todays standards wastful. Often, once production began, they were scrapped. The necessity of a physical prototype outweighed all environmental considerations. In the 21st century, with the growth in Computer Aided Engineering (CAE), physical prototypes have nearly been replaced with virtual prototypes. The evidence is overwhelming as to the benfits of virtual prototypes to designers, designs and the environment. This paper will research and identify an area where physical prototypes are still beneficial, that of educating design engineers. A survey of mechanical engineering students over 5 years will identify a significant difference in the basic engineering knowledge of full-time engineering students when compared with their part-time colleagues. The use of physical prototypes can help reduce this difference. This paper argues that physical prototypes, under certain conditions, can reduce waste and still be sustainable

Liquid phase epitaxy and optical investigation of KYb(WO4)2 thin layers

In recent years, Yb3+ has attracted much attention as an activating ion because of its small quantum defect for laser emission from 2F5/2 to 2F7/2 at ~1.03 µm [1], which provides high efficiency and reduced heat generation. Of high practical interest is the thin-disk laser concept [2], which possesses a tremendous advantage over rod lasers because of its axial-cooling approach and consequent weak thermal lensing and good beam quality.\ud A promising material for Yb3+ thin-disk lasers is KYb(WO4)2 (KYbW) [3]. It can be grown from high-temperature solutions [4]. Nevertheless, the growth of high-quality, single-crystalline layers with thickness in the range of the absorption length of ~13 µm at 981 nm has as yet not been reported. A suitable substrate material is KY(WO4)2 (KYW), but the relatively large differences in the thermal expansion coefficients between KYW and KYbW along the [100], [001], and especially [010] directions [5] favor low temperatures for the hetero-epitaxial growth.\ud For the first time, we demonstrate liquid phase epitaxy (LPE) of KYbW layers. The layers were grown at start temperatures as low as 520°C, which is favorable in order to decrease the thermal stresses due to the differences in the thermal expansion coefficients of substrate and layer. Moreover, the choice of [010]-oriented substrates bypasses the large difference in the thermal expansion coefficient along the [010] direction. KY1-xYbx(WO4)2 layers with varying x = 0.03-1.00 were grown by LPE. The chloride solvent consisted of the eutectic composition [6] 24.4 mol.% KCl, 30.4 mol.% NaCl, and 42.2 mol.% CsCl. The growth temperature spanned the range from 580 to 500°C and the cooling rate was 0.67-1.00 Kh-1. Crack-free, transparent KYbW layers were grown on (010) substrates.\ud Spectroscopic investigations have shown that the lifetime of ~250 µs measured in our LPE-grown KYbW layers is dominated by radiative decay and is very similar to that measured in top-seeded-solution-grown bulk samples [4]. Fast energy migration among the Yb3+ ions and energy transfer to small amounts of Tm3+ and Er3+ ions present in the YbCl3 reagent lead to visible upconversion luminescence in the layers under 981-nm excitation.\ud \ud [1] T.Y. Fan, IEEE J. Quantum Electron. 29, 1457 (1993).\ud [2] A. Giesen, H. Hügel, A. Voss, K. Wittig, U. Brauch, H. Opower, Appl. Phys. B 58, 365 (1994).\ud [3] P. Klopp, U. Griebner, V. Petrov, X. Mateos, M.A. Bursukova, M.C. Pujol, R. Solé, J. Gavaldà, M. Aguiló, F. Güell, J. Massons, T. Kirilov, F. Díaz, Appl. Phys. B 74, 185 (2002).\ud [4] M.C. Pujol, M.A. Bursukova, F. Güell, X. Mateos, R. Solé, J. Gavaldà, M. Aguiló, J. Massons, F. Díaz, P. Klopp, U. Griebner, V. Petrov, Phys. Rev. B 65, 165121 (2002).\ud [5] M.C. Pujol, X. Mateos, R. Solé, J. Massons, J. Gavaldà, F. Díaz, M. Aguiló, Mater. Sci. Forum 378-381, 710 (2001).\ud [6] D. Ehrentraut, M. Pollnau, S. Kück, Appl. Phys. B 75, 59 (2002)

Seasonal variations in Greenland Ice Sheet motion : Inland extent and behaviour at higher elevations

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Theoretical Characterization of the Interface in a Nonequilibrium Lattice System

The influence of nonequilibrium bulk conditions on the properties of the interfaces exhibited by a kinetic Ising--like model system with nonequilibrium steady states is studied. The system is maintained out of equilibrium by perturbing the familiar spin--flip dynamics at temperature T with completely--random flips; one may interpret these as ideally simulating some (dynamic) impurities. We find evidence that, in the present case, the nonequilibrium mechanism adds to the basic thermal one resulting on a renormalization of microscopic parameters such as the probability of interfacial broken bonds. On this assumption, we develop theory for the nonequilibrium "surface tension", which happens to show a non--monotonous behavior with a maximum at some finite T. It ensues, in full agreement with Monte Carlo simulations, that interface fluctuations differ qualitatively from the equilibrium case, e.g., the interface remains rough at zero--T. We discuss on some consequences of these facts for nucleation theory, and make some explicit predictions concerning the nonequilibrium droplet structure.Comment: 10 pages, 7 figures, submitted to Phys. Re