Random matrices, Virasoro algebras, and noncommutative KP

What is the connection of random matrices with integrable systems? Is this connection really useful? The answer to these questions leads to a new and unifying approach to the theory of random matrices. Introducing an appropriate time t-dependence in the probability distribution of the matrix ensemble, leads to vertex operator expressions for the n-point correlation functions (probabilities of n eigenvalues in infinitesimal intervals) and the corresponding Fredholm determinants (probabilities of no eigenvalue in a Borel subset E); the latter probability is a ratio of tau-functions for the KP-equation, whose numerator satisfy partial differential equations, which decouple into the sum of two parts: a Virasoro-like part depending on time only and a Vect(S^1)-part depending on the boundary points A_i of E. Upon setting t=0, and using the KP-hierarchy to eliminate t-derivatives, these PDE's lead to a hierarchy of non-linear PDE's, purely in terms of the A_i. These PDE's are nothing else but the KP hierarchy for which the t-partials, viewed as commuting operators, are replaced by non-commuting operators in the endpoints A_i of the E under consideration. When the boundary of E consists of one point and for the known kernels, one recovers the Painleve equations, found in prior work on the subject.Comment: 56 page

The Association of Polar Faculae with Polar Magnetic Patches Examined with Hinode Observations

The magnetic properties of the Sun's polar faculae are investigated with spectropolarimetric observations of the north polar region obtained by the Hinode satellite in 2007 September. Polar faculae are embedded in nearly all magnetic patches with fluxes greater than $10^{18}$ Mx, while magnetic patches without polar faculae dominate in the flux range below $10^{18}$ Mx. The faculae are considerably smaller than their parent patches, and single magnetic patches contain single or multiple faculae. The faculae in general have higher intrinsic magnetic field strengths than the surrounding regions within their parent patches. Less than 20% of the total magnetic flux contributed by the large (${\ge}10^{18}$ Mx) concentrations, which are known to be modulated by the solar cycle, is accounted for by the associated polar faculae.Comment: 14 pages, 10 figure

Clarke subgradients of stratifiable functions

We establish the following result: if the graph of a (nonsmooth) real-extended-valued function $f:\mathbb{R}^{n}\to \mathbb{R}\cup\{+\infty\}$ is closed and admits a Whitney stratification, then the norm of the gradient of $f$ at $x\in{dom}f$ relative to the stratum containing $x$ bounds from below all norms of Clarke subgradients of $f$ at $x$. As a consequence, we obtain some Morse-Sard type theorems as well as a nonsmooth Kurdyka-\L ojasiewicz inequality for functions definable in an arbitrary o-minimal structure

Self-Consistent MHD Modeling of a Coronal Mass Ejection, Coronal Dimming, and a Giant Cusp-Shaped Arcade Formation

We performed magnetohydrodynamic simulation of coronal mass ejections (CMEs) and associated giant arcade formations, and the results suggested new interpretations of observations of CMEs. We performed two cases of the simulation: with and without heat conduction. Comparing between the results of the two cases, we found that reconnection rate in the conductive case is a little higher than that in the adiabatic case and the temperature of the loop top is consistent with the theoretical value predicted by the Yokoyama-Shibata scaling law. The dynamical properties such as velocity and magnetic fields are similar in the two cases, whereas thermal properties such as temperature and density are very different.In both cases, slow shocks associated with magnetic reconnectionpropagate from the reconnection region along the magnetic field lines around the flux rope, and the shock fronts form spiral patterns. Just outside the slow shocks, the plasma density decreased a great deal. The soft X-ray images synthesized from the numerical results are compared with the soft X-ray images of a giant arcade observed with the Soft X-ray Telescope aboard {\it Yohkoh}, it is confirmed that the effect of heat conduction is significant for the detailed comparison between simulation and observation. The comparison between synthesized and observed soft X-ray images provides new interpretations of various features associated with CMEs and giant arcades.Comment: 39 pages, 18 figures. Accepted for publication in the Astrophysical Journal. The PDF file with high resplution figures can be downloaded from http://www.kwasan.kyoto-u.ac.jp/~shiota/study/ApJ62426.preprint.pdf

Mechanical properties of Pt monatomic chains

The mechanical properties of platinum monatomic chains were investigated by simultaneous measurement of an effective stiffness and the conductance using our newly developed mechanically controllable break junction (MCBJ) technique with a tuning fork as a force sensor. When stretching a monatomic contact (two-atom chain), the stiffness and conductance increases at the early stage of stretching and then decreases just before breaking, which is attributed to a transition of the chain configuration and bond weakening. A statistical analysis was made to investigate the mechanical properties of monatomic chains. The average stiffness shows minima at the peak positions of the length-histogram. From this result we conclude that the peaks in the length-histogram are a measure of the number of atoms in the chains, and that the chains break from a strained state. Additionally, we find that the smaller the initial stiffness of the chain is, the longer the chain becomes. This shows that softer chains can be stretched longer.Comment: 6 pages, 5 figure

The effect of thermal annealing on the properties of Al-AlOx-Al single electron tunneling transistors

The effect of thermal annealing on the properties of Al-AlOx-Al single electron tunneling transistors is reported. After treatment of the devices by annealing processes in forming gas atmosphere at different temperatures and for different times, distinct and reproducible changes of their resistance and capacitance values were found. According to the temperature regime, we observed different behaviors as regards the resistance changes, namely the tendency to decrease the resistance by annealing at T = 200 degree C, but to increase the resistance by annealing at T = 400 degree C. We attribute this behavior to changes in the aluminum oxide barriers of the tunnel junctions. The good reproducibility of these effects with respect to the changes observed allows the proper annealing treatment to be used for post-process tuning of tunnel junction parameters. Also, the influence of the annealing treatment on the noise properties of the transistors at low frequency was investigated. In no case did the noise figures in the 1/f-regime show significant changes.Comment: 6 pages, 7 eps-figure

Bispectral KP Solutions and Linearization of Calogero-Moser Particle Systems

A new construction using finite dimensional dual grassmannians is developed to study rational and soliton solutions of the KP hierarchy. In the rational case, properties of the tau function which are equivalent to bispectrality of the associated wave function are identified. In particular, it is shown that there exists a bound on the degree of all time variables in tau if and only if the wave function is rank one and bispectral. The action of the bispectral involution, beta, in the generic rational case is determined explicitly in terms of dual grassmannian parameters. Using the correspondence between rational solutions and particle systems, it is demonstrated that beta is a linearizing map of the Calogero-Moser particle system and is essentially the map sigma introduced by Airault, McKean and Moser in 1977.Comment: LaTeX, 24 page

Lectures on the Asymptotic Expansion of a Hermitian Matrix Integral

In these lectures three different methods of computing the asymptotic expansion of a Hermitian matrix integral is presented. The first one is a combinatorial method using Feynman diagrams. This leads us to the generating function of the reciprocal of the order of the automorphism group of a tiling of a Riemann surface. The second method is based on the classical analysis of orthogonal polynomials. A rigorous asymptotic method is established, and a special case of the matrix integral is computed in terms of the Riemann $\zeta$-function. The third method is derived from a formula for the $\tau$-function solution to the KP equations. This method leads us to a new class of solutions of the KP equations that are \emph{transcendental}, in the sense that they cannot be obtained by the celebrated Krichever construction and its generalizations based on algebraic geometry of vector bundles on Riemann surfaces. In each case a mathematically rigorous way of dealing with asymptotic series in an infinite number of variables is established

Structure and Dynamics of the 13/14 November 2012 Eclipse White-Light Corona

Continuing our series of observations of the motion and dynamics of the solar corona over the solar-activity cycle, we observed the corona from sites in Queensland, Australia, during the 13 (UT)/14 (local time) November 2012 total solar eclipse. The corona took the low-ellipticity shape typical of solar maximum (flattening index {\epsilon} = 0.01), showing a change from the composite coronal images that we had observed and analyzed in this journal and elsewhere for the 2006, 2008, 2009, and 2010 eclipses. After crossing the northeast Australian coast, the rest of the path of totality was over the ocean, so further totality was seen only by shipborne observers. Our results include measurements of velocities of a coronal mass ejection; during the 36 minutes of passage from the Queensland coast to a ship north of New Zealand, we find a speed of 413 km/s, and we analyze its dynamics. We discuss the shapes and positions of several types of coronal features seen on our higher-resolution composite Queensland images of the solar corona, including, many helmet streamers, very faint bright and dark loops at the base of helmet streamers, voids and radially oriented thin streamers. We compare our eclipse observations with a hairy-ball model of the magnetic field, confirming the validity of the prediction, and we relate the eclipse phenomenology seen with the near-simultaneous images from the Atmospheric Imaging Assembly on the NASA's Solar Dynamics Observatory (SDO/AIA), the Extreme Ultraviolet Imager on NASA's Solar Terrestrial Relations Observatory (STEREO/EUVI), ESA/ROB's PROBA2/SWAP, and NRL's LASCO on ESA's SOHO. For example, the southeastern CME is related to the solar flare whose origin we trace with a SWAP series of images.Comment: 33 pages, 21 figures in 32 part