The strong thirteen spheres problem

The thirteen spheres problem is asking if 13 equal size nonoverlapping spheres in three dimensions can touch another sphere of the same size. This problem was the subject of the famous discussion between Isaac Newton and David Gregory in 1694. The problem was solved by Schutte and van der Waerden only in 1953. A natural extension of this problem is the strong thirteen spheres problem (or the Tammes problem for 13 points) which asks to find an arrangement and the maximum radius of 13 equal size nonoverlapping spheres touching the unit sphere. In the paper we give a solution of this long-standing open problem in geometry. Our computer-assisted proof is based on a enumeration of the so-called irreducible graphs.Comment: Modified lemma 2, 16 pages, 12 figures. Uploaded program packag

Gentzen’s Original Consistency Proof and the Bar Theorem

in December after receiving criticism and, in particular, the criticism that the proof used the Fan Theorem, a criticism that, as the references just cited seem to indicate, Bernays endorsed or initiated at the time but later rejected. That particular criticism is transparently false, but the argument of the paper remains nevertheless invalid from a constructive standpoint. In a letter to Bernays dated November 4, 1935, Gentzen protested this evaluation; but then, in another letter to him dated December 11, 1935, he admits that the “critical inference in my consistency proof is defective. ” The defect in question involves the application of proof by induction to certain trees, the ‘reduction trees ’ for sequents (see below and §1), of which it is only given that they are well-founded. No doubt because of his desire to reason ‘finitistically’, Gentzen nowhere in his paper explicitly speaks of reduction trees, only of reduction rules that would generate such trees; but the requirement of well-foundedness, that every path taken in accordance with the rule terminates, of course makes implicit reference to the tree. Gentzen attempted to avoid th

MicroRNA-146 and cell trauma down-regulate expression of the psoriasis-associated atypical chemokine receptor ACKR2

Chemokines are the principal regulators of leukocyte migration and are essential for initiation and maintenance of inflammation. Atypical chemokine receptor 2 (ACKR2) binds and scavenges proinflammatory CC-chemokines, regulates cutaneous T-cell positioning, and limits the spread of inflammation in vivo. Altered ACKR2 function has been implicated in several inflammatory disorders, including psoriasis, a common and debilitating T-cell–driven disorder characterized by thick erythematous skin plaques. ACKR2 expression is abnormal in psoriatic skin, with decreased expression correlating with recruitment of T-cells into the epidermis and increased inflammation. However, the molecular mechanisms that govern ACKR2 expression are not known. Here, we identified specific psoriasis-associated microRNAs (miRs) that bind ACKR2, inhibit its expression, and are active in primary cultures of human cutaneous cells. Using both in silico and in vitro approaches, we show that miR-146b and miR-10b directly bind the ACKR2 3′-UTR and reduce expression of ACKR2 transcripts and protein in keratinocytes and lymphatic endothelial cells, respectively. Moreover, we demonstrate that ACKR2 expression is further down-regulated upon cell trauma, an important trigger for the development of new plaques in many psoriasis patients (the Koebner phenomenon). We found that tensile cell stress leads to rapid ACKR2 down-regulation and concurrent miR-146b up-regulation. Together, we provide, for the first time, evidence for epigenetic regulation of an atypical chemokine receptor. We propose a mechanism by which cell trauma and miRs coordinately exacerbate inflammation via down-regulation of ACKR2 expression and provide a putative mechanistic explanation for the Koebner phenomenon in psoriasis

Ices in Star-Forming Regions: First Results from VLT-ISAAC

The first results from a VLT-ISAAC program on L- and M-band infrared spectroscopy of deeply-embedded young stellar objects are presented. The advent of 8-m class telescopes allows high S/N spectra of low-luminosity sources to be obtained. In our first observing run, low- and medium-resolution spectra have been measured toward a dozen objects, mostly in the Vela and Chamaeleon molecular clouds. The spectra show strong absorption of H2O and CO ice, as well as weak features at `3.47' and 4.62 mu. No significant solid CH3OH feature at 3.54 mu is found, indicating that the CH3OH/H2O ice abundance is lower than toward some massive protostars. Various evolutionary diagnostics are investigated for a set of sources in Vela.Comment: 8 pages, 4 figures, to appear in The Origins of Stars and Planets: the VLT View, eds. J. Alves, M. McCaughrean (Springer Verlag

Compact intense extreme-ultraviolet source

High-intensity laser pulses covering the ultraviolet to terahertz spectral regions are nowadays routinely generated in a large number of laboratories. In contrast, intense extreme-ultraviolet (XUV) pulses have only been demonstrated using a small number of sources including free-electron laser facilities and long high-harmonic generation (HHG) beamlines. Here, we demonstrate a concept for a compact intense XUV source based on HHG that is focused to an intensity of 2×1014W/cm2, with a potential increase up to 1017W/cm2 in the future. Our approach uses tight focusing of the near-infrared (NIR) driving laser and minimizes the XUV virtual source size by generating harmonics several Rayleigh lengths away from the NIR focus. Accordingly, the XUV pulses can be refocused to a small beam waist radius of 600 nm, enabling the absorption of up to four XUV photons by a single Ar atom in a setup that fits on a modest (2 m) laser table. Our concept represents a straightforward approach for the generation of intense XUV pulses in many laboratories, providing exciting opportunities for XUV strong-field and nonlinear optics experiments, for XUV-pump XUV-probe spectroscopy and for the coherent diffractive imaging of nanoscale structures

Algebraic totality, towards completeness

Finiteness spaces constitute a categorical model of Linear Logic (LL) whose objects can be seen as linearly topologised spaces, (a class of topological vector spaces introduced by Lefschetz in 1942) and morphisms as continuous linear maps. First, we recall definitions of finiteness spaces and describe their basic properties deduced from the general theory of linearly topologised spaces. Then we give an interpretation of LL based on linear algebra. Second, thanks to separation properties, we can introduce an algebraic notion of totality candidate in the framework of linearly topologised spaces: a totality candidate is a closed affine subspace which does not contain 0. We show that finiteness spaces with totality candidates constitute a model of classical LL. Finally, we give a barycentric simply typed lambda-calculus, with booleans ${\mathcal{B}}$ and a conditional operator, which can be interpreted in this model. We prove completeness at type ${\mathcal{B}}^n\to{\mathcal{B}}$ for every n by an algebraic method