A planar calculus for infinite index subfactors

We develop an analog of Jones' planar calculus for II_1-factor bimodules with arbitrary left and right von Neumann dimension. We generalize to bimodules Burns' results on rotations and extremality for infinite index subfactors. These results are obtained without Jones' basic construction and the resulting Jones projections.Comment: 56 pages, many figure

Subfactors of index less than 5, part 1: the principal graph odometer

In this series of papers we show that there are exactly ten subfactors, other than $A_\infty$ subfactors, of index between 4 and 5. Previously this classification was known up to index $3+\sqrt{3}$. In the first paper we give an analogue of Haagerup's initial classification of subfactors of index less than $3+\sqrt{3}$, showing that any subfactor of index less than 5 must appear in one of a large list of families. These families will be considered separately in the three subsequent papers in this series.Comment: 36 pages (updated to reflect that the classification is now complete

Semi-regular masas of transfinite length

In 1965 Tauer produced a countably infinite family of semi-regular masas in the hyperfinite $\mathrm{II}_1$ factor, no pair of which are conjugate by an automorphism. This was achieved by iterating the process of passing to the algebra generated by the normalisers and, for each $n\in\mathbb N$, finding masas for which this procedure terminates at the $n$-th stage. Such masas are said to have length $n$. In this paper we consider a transfinite version of this idea, giving rise to a notion of ordinal valued length. We show that all countable ordinals arise as lengths of semi-regular masas in the hyperfinite $\mathrm{II}_1$ factor. Furthermore, building on work of Jones and Popa, we obtain all possible combinations of regular inclusions of irreducible subfactors in the normalising tower.Comment: 14 page

Fixed-Point Results on Complete G-Metric Spaces for Mappings Satisfying an Implicit relation of New Type

We prove general fixed-point theorems (generalizing some recent results) in a complete G-metric space.Доведено загальні теореми про нерухому точку у повних G-метричних просторах, що узагальнюють дєякі результати, отримані нещодавно

Ab initio Translationally Invariant Nonlocal One-body Densities from No-core Shell-model Theory

[Background:] It is well known that effective nuclear interactions are in general nonlocal. Thus if nuclear densities obtained from {\it ab initio} no-core-shell-model (NCSM) calculations are to be used in reaction calculations, translationally invariant nonlocal densities must be available. [Purpose:] Though it is standard to extract translationally invariant one-body local densities from NCSM calculations to calculate local nuclear observables like radii and transition amplitudes, the corresponding nonlocal one-body densities have not been considered so far. A major reason for this is that the procedure for removing the center-of-mass component from NCSM wavefunctions up to now has only been developed for local densities. [Results:] A formulation for removing center-of-mass contributions from nonlocal one-body densities obtained from NCSM and symmetry-adapted NCSM (SA-NCSM) calculations is derived, and applied to the ground state densities of $^4$He, $^6$Li, $^{12}$C, and $^{16}$O. The nonlocality is studied as a function of angular momentum components in momentum as well as coordinate space [Conclusions:] We find that the nonlocality for the ground state densities of the nuclei under consideration increases as a function of the angular momentum. The relative magnitude of those contributions decreases with increasing angular momentum. In general, the nonlocal structure of the one-body density matrices we studied is given by the shell structure of the nucleus, and can not be described with simple functional forms.Comment: 13 pages, 11 Figure

Measurements of greenhouse gases and related tracers at Bialystok tall tower station in Poland

Quasi-continuous, in-situ measurements of atmospheric CO2, O2/N2, CH4, CO, N2O, and SF6 have been performed since August 2005 at the tall tower station near Bialystok, in Eastern Poland, from five heights up to 300 m. Besides the in-situ measurements, flask samples are filled approximately weekly and measured at Max-Planck Institute for Biogeochemistry for the same species and, in addition, for H2, Ar/N2 and the stable isotopes 13C and 18O in CO2. The in-situ measurement system was built based on commercially available analysers: a LiCor 7000 for CO2, a Sable Systems "Oxzilla" FC-2 for O2, and an Agilent 6890 gas chromatograph for CH4, CO, N2O and SF6. The system was optimized to run continuously with very little maintenance and to fulfill the precision requirements of the CHIOTTO project. The O2/N2 measurements in particular required special attention in terms of technical setup and quality assurance. The evaluation of the performance after more than three years of operation gave overall satisfactory results, proving that this setup is suitable for long term remote operation with little maintenance. The precision achieved for all species is within or close to the project requirements. The comparison between the in-situ and flask sample results, used to verify the accuracy of the in-situ measurements, showed no significant difference for CO2, O2/N2, CH4 and N2O, and a very small difference for SF6. The same comparison however revealed a statistically significant difference for CO, of about 6.5 ppb, for which the cause could not be fully explained. From more than three years of data, the main features at Bialystok have been characterized in terms of variability, trends, and seasonal and diurnal variations. CO2 and O2/N2 show large short term variability, and large diurnal signals during the warm seasons, which attenuate with the increase of sampling height. The trends calculated from this dataset, over the period August 2005 to December 2008, are 2.02±0.46 ppm/year for CO2 and -23.2±2.5 per meg/year for O2/N2. CH4, CO and N2O show also higher variability at the lower sampling levels, which in the case of CO is strongly seasonal. Diurnal variations in CH4, CO and N2O mole fractions can be observed during the warm season, due to the periodicity of vertical mixing combined with the diurnal cycle of anthropogenic emissions. We calculated increase rates of 10.1±4.4 ppb/year for CH4, (-8.3)±5.3 ppb/year for CO and 0.67±0.08 ppb/year for N2O. SF6 shows only few events, and generally no vertical gradients, which suggests that there are no significant local sources. A weak SF6 seasonal cycle has been detected, which most probably is due to the seasonality of atmospheric circulation. SF6 increased during the time of our measurement at an average rate of 0.29±0.01 ppt/year

Constraints on non-thermal Dark Matter from Planck lensing extraction

Distortions of CMB temperature and polarization anisotropy maps caused by gravitational lensing, observable with high angular resolution and sensitivity, can be used to constrain the sterile neutrino mass, offering several advantages against the analysis based on the combination of CMB, LSS and Ly\alpha forest power spectra. As the gravitational lensing effect depends on the matter distribution, no assumption on light-to-mass bias is required. In addition, unlike the galaxy clustering and Ly\alpha forest power spectra, the projected gravitational potential power spectrum probes a larger range of angular scales, the non-linear corrections being required only at very small scales. Taking into account the changes in the time-temperature relation of the primordial plasma and the modification of the neutrino thermal potential, we compute the projected gravitational potential power spectrum and its correlation with the temperature in the presence of DM sterile neutrino. We show that the cosmological parameters are generally not biased when DM sterile neutrino is included. From this analysis we found a lower limit on DM sterile neutrino mass m_s >2.08 keV at 95% CL, consistent with the lower mass limit obtained from the combined analysis of CMB, SDSS 3D power spectrum and SDSS Ly\alpha forest power spectrum ($m_{\nu_s}>1.7$ keV). We conclude that although the information that can be obtained from lensing extraction is rather limited due to the high level of the lensing noise of Planck experiment, weak lensing of CMB offers a valuable alternative to constrain the dark matter sterile neutrino mass.Comment: 15 pages, 6 figure

Unbiased bases (Hadamards) for 6-level systems: Four ways from Fourier

In quantum mechanics some properties are maximally incompatible, such as the position and momentum of a particle or the vertical and horizontal projections of a 2-level spin. Given any definite state of one property the other property is completely random, or unbiased. For N-level systems, the 6-level ones are the smallest for which a tomographically efficient set of N+1 mutually unbiased bases (MUBs) has not been found. To facilitate the search, we numerically extend the classification of unbiased bases, or Hadamards, by incrementally adjusting relative phases in a standard basis. We consider the non-unitarity caused by small adjustments with a second order Taylor expansion, and choose incremental steps within the 4-dimensional nullspace of the curvature. In this way we prescribe a numerical integration of a 4-parameter set of Hadamards of order 6.Comment: 5 pages, 2 figure

Ab initio Folding Potentials for Nucleon-Nucleus Scattering based on NCSM One-Body Densities

Calculating microscopic optical potentials for elastic nucleon-nucleus scattering has already led to large body of work in the past. For folding first-order calculations the nucleon-nucleon (NN) interaction and the one-body density of the nucleus were taken as input to rigorous calculations in a spectator expansion of the multiple scattering series. Based on the Watson expansion of the multiple scattering series we employ a nonlocal translationally invariant nuclear density derived from a chiral next-to-next-to-leading order (NNLO) and the very same interaction for consistent full-folding calculation of the effective (optical) potential for nucleon-nucleus scattering for light nuclei. We calculate scattering observables, such as total, reaction, and differential cross sections as well as the analyzing power and the spin-rotation parameter, for elastic scattering of protons and neutrons from $^4$He, $^{6}$He, $^{12}$C, and $^{16}$O, in the energy regime between 100 and 200~MeV projectile kinetic energy, and compare to available data. Our calculations show that the effective nucleon-nucleus potential obtained from the first-order term in the spectator expansion of the multiple scattering expansion describes experiments very well to about 60 degrees in the center-of-mass frame, which coincides roughly with the validity of the NNLO chiral interaction used to calculate both the NN amplitudes and the one-body nuclear density.Comment: 10 pages, 14 figures, 1 tabl