On maximal immediate extensions of valued division algebras

We show an extension theorem for strictly contracting bilinear mappings into a spherically complete valued vector space and we apply this result to prove that every maximal valued division algebra having the same characteristic as its residue division algebra is spherically complete

The Hopf modules category and the Hopf equation

We study the Hopf equation which is equivalent to the pentagonal equation, from operator algebras. A FRT type theorem is given and new types of quantum groups are constructed. The key role is played now by the classical Hopf modules category. As an application, a five dimensional noncommutative noncocommutative bialgebra is given.Comment: 30 pages, Letax2e, Comm. Algebra in pres

An axiomatic approach to the non-linear theory of generalized functions and consistency of Laplace transforms

We offer an axiomatic definition of a differential algebra of generalized functions over an algebraically closed non-Archimedean field. This algebra is of Colombeau type in the sense that it contains a copy of the space of Schwartz distributions. We study the uniqueness of the objects we define and the consistency of our axioms. Next, we identify an inconsistency in the conventional Laplace transform theory. As an application we offer a free of contradictions alternative in the framework of our algebra of generalized functions. The article is aimed at mathematicians, physicists and engineers who are interested in the non-linear theory of generalized functions, but who are not necessarily familiar with the original Colombeau theory. We assume, however, some basic familiarity with the Schwartz theory of distributions.Comment: 23 page

Bohrification of operator algebras and quantum logic

Following Birkhoff and von Neumann, quantum logic has traditionally been based on the lattice of closed linear subspaces of some Hilbert space, or, more generally, on the lattice of projections in a von Neumann algebra A. Unfortunately, the logical interpretation of these lattices is impaired by their nondistributivity and by various other problems. We show that a possible resolution of these difficulties, suggested by the ideas of Bohr, emerges if instead of single projections one considers elementary propositions to be families of projections indexed by a partially ordered set C(A) of appropriate commutative subalgebras of A. In fact, to achieve both maximal generality and ease of use within topos theory, we assume that A is a so-called Rickart C*-algebra and that C(A) consists of all unital commutative Rickart C*-subalgebras of A. Such families of projections form a Heyting algebra in a natural way, so that the associated propositional logic is intuitionistic: distributivity is recovered at the expense of the law of the excluded middle. Subsequently, generalizing an earlier computation for n-by-n matrices, we prove that the Heyting algebra thus associated to A arises as a basis for the internal Gelfand spectrum (in the sense of Banaschewski-Mulvey) of the "Bohrification" of A, which is a commutative Rickart C*-algebra in the topos of functors from C(A) to the category of sets. We explain the relationship of this construction to partial Boolean algebras and Bruns-Lakser completions. Finally, we establish a connection between probability measure on the lattice of projections on a Hilbert space H and probability valuations on the internal Gelfand spectrum of A for A = B(H).Comment: 31 page

Zeros of the Möbius function of permutations

We show that if a permutation \pi contains two intervals of length 2, where one interval is an ascent and the other a descent, then the Möbius function \mu[1,\pi] of the interval [1,\pi] is zero. As a consequence, we prove that the proportion of permutations of length $\textit{n}$ with principal Möbius function equal to zero is asymptotically bounded below by (1\ -\ \sfrac{1}{e)^2} \geq 0.3995. This is the first result determining the value of \mu\left[1,\pi\right] for an asymptotically positive proportion of permutations \pi. We further establish other general conditions on a permutation \pi that ensure \mu\left[1,\pi\right]\ =\ 0, including the occurrence in \pi of any interval of the form \alpha\oplus\ 1\ \oplus\ \beta

Modification of Experimental Protocols for a Space Shuttle Flight and Applications for the Analysis of Cytoskeletal Structures During Fertilization, Cell Division , and Development in Sea Urchin Embryos

To explore the role of microgravity on cytoskeletal organization and skeletal calcium deposition during fertilization, cell division, and early development, the sea urchin was chosen as a model developmental system. Methods were developed to employ light, immunofluorescence, and electron microscopy on cultures being prepared for flight on the Space Shuttle. For analysis of microfilaments, microtubules, centrosomes, and calcium-requiring events, our standard laboratory protocols had to be modified substantially for experimentation on the Space Shuttle. All manipulations were carried out in a closed culture chamber containing 35 ml artificial sea water as a culture fluid. Unfertilized eggs stored for 24 hours in these chambers were fertilized with sperm diluted in sea water and fixed with concentrated fixatives for final fixation in formaldehyde, taxol, EGTA, and MgCl2(exp -6)H2O for 1 cell to 16 cell stages to preserve cytoskeletal structures for simultaneous analysis with light, immunofluorescence, and electron microscopy, and 1.5 percent glutaraldehyde and 0.4 percent formaldehyde for blastula and plueus stages. The fixed samples wre maintained in chambers without degradation for up to two weeks after which the specimens were processed and analyzed with routine methods. Since complex manipulations are not possible in the closed chambers, the fertilization coat was removed from fixation using 0.5 percent freshly prepared sodium thioglycolate solution at pH 10.0 which provided reliable immunofluorescence staining for microtubules. Sperm/egg fusion, mitosis, cytokinesis, and calcium deposition during spicule formatin in early embryogenesis were found to be without artificial alterations when compared to cells fixed fresh and processed with conventional methods

Facets of the Fully Mixed Nash Equilibrium Conjecture

In this work, we continue the study of the many facets of the Fully Mixed Nash Equilibrium Conjecture, henceforth abbreviated as the FMNE Conjecture, in selfish routing for the special case of n identical users over two (identical) parallel links. We introduce a new measure of Social Cost, defined to be the expectation of the square of the maximum congestion on a link; we call it Quadratic Maximum Social Cost. A Nash equilibrium is a stable state where no user can improve her (expected) latency by switching her mixed strategy; a worst-case Nash equilibrium is one that maximizes Quadratic Maximum Social Cost. In the fully mixed Nash equilibrium, allmixed strategies achieve full support. Formulated within this framework is yet another facet of the FMNE Conjecture, which states that the fully mixed Nash equilibrium is the worst-case Nash equilibrium. We present an extensive proof of the FMNE Conjecture; the proof employs a mixture of combinatorial arguments and ana-lytical estimations. Some of these analytical estimations are derived through some new bounds on generalized medians of the binomial distribution [22] we obtain, which are of independent interest.

Severe traumatic injury during long duration spaceflight: Light years beyond ATLS

Traumatic injury strikes unexpectedly among the healthiest members of the human population, and has been an inevitable companion of exploration throughout history. In space flight beyond the Earth's orbit, NASA considers trauma to be the highest level of concern regarding the probable incidence versus impact on mission and health. Because of limited resources, medical care will have to focus on the conditions most likely to occur, as well as those with the most significant impact on the crew and mission. Although the relative risk of disabling injuries is significantly higher than traumatic deaths on earth, either issue would have catastrophic implications during space flight. As a result this review focuses on serious life-threatening injuries during space flight as determined by a NASA consensus conference attended by experts in all aspects of injury and space flight