On a remarkable semigroup of homomorphisms with respect to free multiplicative convolution

Let M denote the space of Borel probability measures on the real line. For every nonnegative t we consider the transformation $\mathbb B_t : M \to M$ defined for any given element in M by taking succesively the the (1+t) power with respect to free additive convolution and then the 1/(1+t) power with respect to Boolean convolution of the given element. We show that the family of maps {\mathbb B_t|t\geq 0} is a semigroup with respect to the operation of composition and that, quite surprisingly, every $\mathbb B_t$ is a homomorphism for the operation of free multiplicative convolution. We prove that for t=1 the transformation $\mathbb B_1$ coincides with the canonical bijection $\mathbb B : M \to M_{inf-div}$ discovered by Bercovici and Pata in their study of the relations between infinite divisibility in free and in Boolean probability. Here M_{inf-div} stands for the set of probability distributions in M which are infinitely divisible with respect to free additive convolution. As a consequence, we have that $\mathbb B_t(\mu)$ is infinitely divisible with respect to free additive convolution for any for every $\mu$ in M and every t greater than or equal to one. On the other hand we put into evidence a relation between the transformations $\mathbb B_t$ and the free Brownian motion; indeed, Theorem 4 of the paper gives an interpretation of the transformations $\mathbb B_t$ as a way of re-casting the free Brownian motion, where the resulting process becomes multiplicative with respect to free multiplicative convolution, and always reaches infinite divisibility with respect to free additive convolution by the time t=1.Comment: 30 pages, minor changes; to appear in Indiana University Mathematics Journa

Convolution powers in the operator-valued framework

We consider the framework of an operator-valued noncommutative probability space over a unital C*-algebra B. We show how for a B-valued distribution \mu one can define convolution powers with respect to free additive convolution and with respect to Boolean convolution, where the exponent considered in the power is a suitably chosen linear map \eta from B to B, instead of being a non-negative real number. More precisely, the Boolean convolution power is defined whenever \eta is completely positive, while the free additive convolution power is defined whenever \eta - 1 is completely positive (where 1 stands for the identity map on B). In connection to these convolution powers we define an evolution semigroup related to the Boolean Bercovici-Pata bijection. We prove several properties of this semigroup, including its connection to the B-valued free Brownian motion. We also obtain two results on the operator-valued analytic function theory related to the free additive convolution powers with exponent \eta. One of the results concerns analytic subordination for B-valued Cauchy-Stieltjes transforms. The other gives a B-valued version of the inviscid Burgers equation, which is satisfied by the Cauchy-Stieltjes transform of a B-valued free Brownian motion.Comment: 33 pages, no figure

Cyclic Diamondoid Structures with Shared Vertices, Edges, or 6-membered Rings

Diamondoid structures with shared vertices, edges, or 6-membered rings can theoretically be curved into toroidal structures whose calculated energy provides information about steric strain. Diamondoid hydrocarbons sharing one vertex between two adamantane units are called [n]spiromantanes, where n indicates the number of adamantane units. When a pair of adamantane units shares one CC bond, the resulting assembly is called one-edge-[n]mantane, specifying (by letters in square brackets) which bonds are shared by the adamantane units. Two adjacent edges may be shared by a pair of adamantane units, and the assembly is called two-edge-[n]mantane, again specifying by letters in square brackets the shared bonds. Catamantanes or perimantanes sharing a 6-membered ring of carbon atoms may form larger rings in an assembly which is called [n]cyclomantane; in the case of catamantanes, the structure of the diamondoid is specified by codes of the dualists. Finally, nanotubes derived from hexagonal diamond, as well as corresponding toroidal structures, are discussed

Science, Technology, and Medicine have Progressed Immensely during the Last Five Centuries, yet Mankind Is Threatened by Self-Destruction

During the last 5900 years, creative human minds have dispelled false beliefs about our universe, as well as chemical, physical, and biomedical phenomena. Life expectancies in most inhabited parts of our world have increased appreciably. A brief survey of those who contributed to the progress of science, technology, and understanding is presented. Yet, primitive instincts and ambitions still dominate our world, and conflicts threaten to destroy life on this planet by way of nuclear weapons. Societies are ruled by politicians and militaries, for scientists are “on tap, but not on top”. The contrast between reason-based progress and instinct-based aggression is mind-boggling

Susceptibility of the QCD vacuum to CP-odd electromagnetic background fields

We investigate two flavor QCD in presence of CP-odd electromagnetic background fields and determine, by means of lattice QCD simulations, the induced effective theta term to the first order in the scalar product of E and B. We employ a rooted staggered discretization and study lattice spacings down to 0.1 fm and Goldstone pion masses around 480 MeV. In order to deal with a positive measure, we consider purely imaginary electric fields and real magnetic fields, then exploiting analytic continuation. Our results are relevant to a description of the effective pseudoscalar QED-QCD interactions.Comment: 5 pages, 4 figures, 1 table. New data and references added. Matches the published versio

Adaptive Hexapod Simulator Motion Based on Aircraft Stability

This paper determined the feasibility of an adaptive hexapod simulator motion algorithm based on aircraft roll stability. An experiment was conducted that used a transport aircraft model in the Vertical Motion Simulator at NASA Ames Research Center. Eighteen general aviation pilots flew a heading-capture task and a stall task consecutively under four motion configurations: baseline hexapod, adaptive hexapod, optimized hexapod, and full motion. The adaptive motion was more similar to the baseline hexapod motion in the heading-capture task when the aircraft was more stable, and more similar to the optimized hexapod motion in the stall task when the aircraft was more unstable. Pilot motion ratings and task performance in the heading-capture task under the adaptive hexapod motion were more similar to baseline hexapod motion compared to optimized hexapod motion. However, motion ratings and task performance in the stall task under the adaptive motion were not significantly more similar to the optimized hexapod motion compared to baseline hexapod motion. Motion ratings and overall task performance under optimized hexapod motion as opposed to baseline hexapod motion were always more similar to the full motion condition. This paper showed that adaptive motion based on aircraft stability is feasible and can be implemented in a straightforward way. More research is required to test the adaptive motion algorithm in different tasks

Discordance between cosmogenic nuclide concentrations in amalgamated sands and individual fluvial pebbles in an arid zone catchment

Based on cosmogenic 10Be and 26Al analyses in 15 individual detrital quartz pebbles (16–21 mm) and cosmogenic 10Be in amalgamated medium sand (0.25–0.50 mm), all collected from the outlet of the upper Gaub River catchment in Namibia, quartz pebbles yield a substantially lower average denudation rate than those yielded by the amalgamated sand sample. 10Be and 26Al concentrations in the 15 individual pebbles span nearly two orders of magnitude (0.22 ± 0.01 to 20.74 ± 0.52 × 10610Be atoms g−1 and 1.35 ± 0.09 to 72.76 ± 2.04 × 10626Al atoms g−1, respectively) and yield average denudation rates of ∼0.7 m Myr−1 (10Be) and ∼0.9 m Myr−1 (26Al). In contrast, the amalgamated sand yields an average 10Be concentration of 0.77 ± 0.03 × 106 atoms g−1, and an associated mean denudation rate of 9.6 ± 1.1 m Myr−1, an order of magnitude greater than the rates obtained for the amalgamated pebbles. The inconsistency between the 10Be and 26Al in the pebbles and the 10Be in the amalgamated sand is likely due to the combined effect of differential sediment sourcing and longer sediment transport times for the pebbles compared to the sand-sized grains. The amalgamated sands leaving the catchment are an aggregate of grains originating from all quartz-bearing rocks in all parts of the catchment. Thus, the cosmogenic nuclide inventories of these sands record the overall average lowering rate of the landscape. The pebbles originate from quartz vein outcrops throughout the catchment, and the episodic erosion of the latter means that the pebbles will have higher nuclide inventories than the surrounding bedrock and soil, and therefore also higher than the amalgamated sand grains. The order-of-magnitude grain size bias observed in the Gaub has important implications for using cosmogenic nuclide abundances in depositional surfaces because in arid environments, akin to our study catchment, pebble-sized clasts yield substantially underestimated palaeo-denudation rates. Our results highlight the importance of carefully considering geomorphology and grain size when interpreting cosmogenic nuclide data in depositional surfaces