2-Galois groups and the Kaplansky radical

An accurate description of the Galois group GF(2) of the maximal Galois 2-extension of a field F may be given for fields F admitting a 2-henselian valuation ring. In this note we generalize this result by characterizing the fields for which GF(2) decomposes as a free pro-2 product F∗H where F is a free closed subgroup of GF(2) and H is the Galois group of a 2-henselian extension of F. The free product decomposition of GF(2) is equivalent to the existence of a valuation ring compatible with the Kaplansky radical of F. Fields with Kaplansky radical fulfilling prescribed conditions are constructed, as an application

Nanoscopic mechanical anisotropy in hydrogel surfaces

The bulk mechanical properties of soft materials have been studied widely, but it is unclear to what extent macroscopic behavior is reflected in nanomechanics. Using an atomic force microscopy (AFM) imaging method called force spectroscopy mapping (FSM), it is possible to map the nanoscopic spatial distribution of Young's modulus, i.e. “stiffness,” and determine if soft or stiff polymer domains exist to correlate nano- and macro-mechanics. Two model hydrogel systems typically used in cell culture and polymerized by a free radical polymerization process, i.e. poly (vinyl pyrrolidone) (PVP) and poly(acrylamide) (PAam) hydrogels, were found to have significantly different nanomechanical behavior despite relatively similar bulk stiffness and roughness. PVP gels contained a large number of soft and stiff nanodomains, and their size was inversely related to crosslinking density and changes in crosslinking efficiency within the hydrogel. In contrast, PAam gels displayed small nanodomains occuring at low frequency, indicating relatively uniform polymerization. Given the responsiveness of cells to changes in gel stiffness, inhomogeneities found in the PVP network indicate that careful nanomechanical characterization of polymer substrates is necessary to appreciate complex cell behavior

Cold atom gas at very high densities in an optical surface microtrap

An optical microtrap is realized on a dielectric surface by crossing a tightly focused laser beam with an horizontal evanescent-wave atom mirror. The nondissipative trap is loaded with $\sim$$10^5$ cesium atoms through elastic collisions from a cold reservoir provided by a large-volume optical surface trap. With an observed 300-fold local increase of the atomic number density approaching $10^{14}{\rm cm}^{-3}$, unprecedented conditions of cold atoms close to a surface are realized

Real-time and non-invasive measurements of cell mechanical behaviour with optical coherence phase microscopy

There is an unmet need in tissue engineering for non-invasive, label-free monitoring of cell mechanical behaviour in their physiological environment. Here, we describe a novel optical coherence phase microscopy (OCPM) set-up which can map relative cell mechanical behaviour in monolayers and 3D systems non-invasively, and in real-time. 3T3 and MCF-7 cells were investigated, with MCF-7 demonstrating an increased response to hydrostatic stimulus indicating MCF-7 being softer than 3T3, demonstrating the ability to provide qualitative data on cell mechanical behaviour. Quantitative measurements of 6% agarose beads have been taken with commercial Cell Scale Microsquisher® system demonstrating that their mechanical properties are in the same order of magnitude of cells, indicating that this is an appropriate test sample for the novel method desctibed

Undecidability in number theory

These lecture notes cover classical undecidability results in number theory, Hilbert's 10th problem and recent developments around it, also for rings other than the integers. It also contains a sketch of the authors result that the integers are universally definable in the rationals.Comment: 48 pages. arXiv admin note: text overlap with arXiv:1011.342

Totally Real Rigid Elements And Fπ-henselian Valuation Rings

[No abstract available]251136733697Arason, J.K., Elman, R., Jacob, B., Rigid Elements, Valuations, and Realization of Witt Rings (1987) J. of Algebra, 110, pp. 449-467Becker, E., Hereditarily-Pythagorean Fields and Orderings of Higher Level. (1978) Monografias de Matemática, 29. , Rio de Janeiro: IMPABerman, L., Quadratic Forms and Power Series Fields (1980) Pac. J. Math., 89, pp. 257-267Berman, L., Cordes, C.M., Ware, R., Quadratic Forms, Rigid Elements, and Formal Power Series Fields (1980) J. Algebra, 66, pp. 123-133Bröcker, L., Characterization of Fans and Hereditarily Pythagorean Fields (1976) Math. Zeitschr., 151, pp. 149-163Cordes, C.M., Ramsey Jr., J.R., Quadratic Forms over Fields with u = q/2 < +∞ (1978) Fund. Math., 99, pp. 1-10Endler, O., (1972) Valuation Theory, , Berlin, Heidelberg, New York: Springer-VerlagEngler, A.J., Totally Real Rigid Elements and Galois Theory, , in preparationElman, R., Lam, T.Y., Quadratic Forms and the u-Invariant I (1973) Math. Z., 131, pp. 283-304Jacob, B., Fans, Real Valuations, and Hereditarily-pythagorean Fields (1981) Pac. J. Math., 93, pp. 95-105Koenigsmann, J., From p-rigid Elements to Valuations (with a Galois-characterization of p-adic fields) (1995) J. Reine Angew. Math., 465, pp. 165-182Lam, T.Y., Orderings, Valuations and Quadratic Forms (1983) Conference Board of the Mathematical Science, (52). , Providence, RI: Amer. Math. SocSzymiczek, K., Quadratic Forms over Fields (1977) Dissertationes Math., 152, pp. 1-63Ware, R., Hasse Principles and the u-Invariant over Formally Real Fields (1976) Nagoya Math. J., 61, pp. 117-125Ware, R., Valuation Rings and Rigid Elements in Fields (1981) Canad. J. Math., 33, pp. 1338-1355Ware, R., Quadratic Forms and Pro 2-groups II: The Galois Group of the Pythagorean Closure of a Formally Real Field (1983) J. Pure Appl. Algebra, 30, pp. 95-107Zariski, O., Samuel, P., (1958) Commutative Algebra I, , New York: Van Nostran

Digging Holes In Algebraic Closures à La Artin. I

[No abstract available]265226327