A general Dixmier trace formula for the density of states on open manifolds

We give an abstract formulation of the Dixmier trace formula for the density of states. This recovers prior versions and allows us to provide a Dixmier trace formula for the density of states of second order elliptic differential operators on manifolds of bounded geometry satisfying a certain geometric condition. This formula gives a new perspective on Roe's index on open manifolds.Comment: 31 pages, no figure

Search for the Heliospheric Termination Shock (TS) and Heliosheath (HS)

Voyager 1 continues to measure the very distant Heliospheric Magnetic Field (HMF) beyond 95 AU at ~35 North latitude. The MAG instrument data covers more than a full 22 years solar magnetic cycle. The magnitude of the observed HMF is well described, on average, by Parker's Archimedean spiral structure if due account is made for time variations of the source field strength and solar wind velocity. The V1 magnetic field observations do not provide any evidence for a field increase associated with entry into a subsonic solar wind region, such as the heliosheath is expected to be, nor an exit from this regime. We see no evidence for crossing of the Termination Shock (TS) as has been reported at ~85 AU by the LECP instrument. Merged Interaction Regions are identified by an increased HMF and associated decreases in the flux of >70 MeV/nuc cosmic rays which are then followed by a flux recovery. This CR-B relationship has been identified in V1 data and studied since 1982 when V1 was at 11 AU. The variance of HMF, a direct measure of the energy**1/2 in the HMF fluctuations, shows no significant changes associated with the alleged TS crossings in 2002–2003. Thus, the absence of any HMF increase at the entry into the heliosheath appears not to be due to the onset of mesoscale turbulence as proposed by Fisk. The TS has yet to be directly observed in-situ by the V1 MAG experiment in data through 2003

Geometry of Financial Markets -- Towards Information Theory Model of Markets

Most of parameters used to describe states and dynamics of financial market depend on proportions of the appropriate variables rather than on their actual values. Therefore, projective geometry seems to be the correct language to describe the theater of financial activities. We suppose that the object of interest of agents, called here baskets, form a vector space over the reals. A portfolio is defined as an equivalence class of baskets containing assets in the same proportions. Therefore portfolios form a projective space. Cross ratios, being invariants of projective maps, form key structures in the proposed model. Quotation with respect to an asset X (i.e. in units of X) are given by linear maps. Among various types of metrics that have financial interpretation, the min-max metrics on the space of quotations can be introduced. This metrics has an interesting interpretation in terms of rates of return. It can be generalized so that to incorporate a new numerical parameter (called temperature) that describes agent's lack of knowledge about the state of the market. In a dual way, a metrics on the space of market quotation is defined. In addition, one can define an interesting metric structure on the space of portfolios/quotation that is invariant with respect to hyperbolic (Lorentz) symmetries of the space of portfolios. The introduced formalism opens new interesting and possibly fruitful fields of research.Comment: Talk given at the APFA5 Conference, Torino, 200

Nonlinear progressive wave equation for stratified atmospheres

The nonlinear progressive wave equation (NPE) [McDonald and Kuperman, J. Acoust. Soc. Am. 81, 1406–1417 (1987)] is expressed in a form to accommodate changes in the ambient atmospheric density, pressure, and sound speed as the time-stepping computational window moves along a path possibly traversing significant altitude differences (in pressure scale heights). The modification is accomplished by the addition of a stratification term related to that derived in the 1970s for linear range-stepping calculations and later adopted into Khokhlov-Zabolotskaya-Kuznetsov-type nonlinear models. The modified NPE is shown to preserve acoustic energy in a ray tube and yields analytic similarity solutions for vertically propagating N waves in isothermal and thermally stratified atmospheres