A control theorem for pp-adic automorphic forms and Teitelbaum's L\mathcal{L}-invariant

In this article, we describe an efficient method for computing Teitelbaum's $p$-adic $\mathcal{L}$-invariant. These invariants are realized as the eigenvalues of the $\mathcal{L}$-operator acting on a space of harmonic cocycles on the Bruhat-Tits tree $\mathcal{T}$, which is computable by the methods of Franc and Masdeu described in \cite{fm}. The main difficulty in computing the $\mathcal{L}$-operator is the efficient computation of the $p$-adic Coleman integrals in its definition. To solve this problem, we use overconvergent methods, first developed by Darmon, Greenberg, Pollack and Stevens. In order to make these methods applicable to our setting, we prove a control theorem for $p$-adic automorphic forms of arbitrary even weight. Moreover, we give computational evidence for relations between slopes of $\mathcal{L}$-invariants of different levels and weights for $p=2$.Comment: 26 page

Evolution of primordial planets in relation to the cosmological origin of life

We explore the conditions prevailing in primordial planets in the framework of the HGD cosmologies as discussed by Gibson and Schild. The initial stages of condensation of planet-mass H-4He gas clouds in trillion-planet clumps is set at 300,000 yr (0.3My) following the onset of plasma instabilities when ambient temperatures were >1000K. Eventual collapse of the planet-cloud into a solid structure takes place against the background of an expanding universe with declining ambient temperatures. Stars form from planet mergers within the clumps and die by supernovae on overeating of planets. For planets produced by stars, isothermal free fall collapse occurs initially via quasi equilibrium polytropes until opacity sets in due to molecule and dust formation. The contracting cooling cloud is a venue for molecule formation and the sequential condensation of solid particles, starting from mineral grains at high temperatures to ice particles at lower temperatures, water-ice becomes thermodynamically stable between 7 and 15 My after the initial onset of collapse, and contraction to form a solid icy core begins shortly thereafter. Primordial-clump-planets are separated by ~ 1000 AU, reflecting the high density of the universe at 30,000 yr. Exchanges of materials, organic molecules and evolving templates readily occur, providing optimal conditions for an initial origin of life in hot primordial gas planet water cores when adequately fertilized by stardust. The condensation of solid molecular hydrogen as an extended outer crust takes place much later in the collapse history of the protoplanet. When the object has shrunk to several times the radius of Jupiter, the hydrogen partial pressure exceeds the saturation vapour pressure of solid hydrogen at the ambient temperature and condensation occurs.Comment: 14 pages 7 figures SPIE Conference 7819 Instruments, Methods, and Missions for Astrobiology XIII Proceedings, Aug 3-5, 2010, San Diego, Ed. Richard B. Hoove

Why don't clumps of cirrus dust gravitationally collapse?

We consider the Herschel-Planck infrared observations of presumed condensations of interstellar material at a measured temperature of approximately 14 K (Juvela et al., 2012), the triple point temperature of hydrogen. The standard picture is challenged that the material is cirrus-like clouds of ceramic dust responsible for Halo extinction of cosmological sources (Finkbeiner, Davis, and Schlegel 1999). Why would such dust clouds not collapse gravitationally to a point on a gravitational free-fall time scale of $10^8$ years? Why do the particles not collide and stick together, as is fundamental to the theory of planet formation (Blum 2004; Blum and Wurm, 2008) in pre-solar accretion discs? Evidence from 3.3 $\mu$m and UIB emissions as well as ERE (extended red emission) data point to the dominance of PAH-type macromolecules for cirrus dust, but such fractal dust will not spin in the manner of rigid grains (Draine & Lazarian, 1998). IRAS dust clouds examined by Herschel-Planck are easily understood as dark matter Proto-Globular-star-Cluster (PGC) clumps of primordial gas planets, as predicted by Gibson (1996) and observed by Schild (1996).Comment: 8 pages, 2 figures, Conference FQMT'1

Scanning thermal profiler

Journal ArticleA new high-resolution profilometer has been demonstrated based upon a noncontacting nearfield thermal probe. The thermal probe consists of a thermocouple sensor with dimensions approaching 100 nm. Profiling is achieved by scanning the heated sensor above but close to the surface of a solid. The conduction of heat between tip and sample via the air provides a means for maintaining the sample spacing constant during the lateral scan. The large difference in thermal properties between air and solids makes the profiling technique essentially independent of the material properties of the solid. Noncontact profiling of resist and metal films has shown a lateral resolution of 100 nm and a depth solution of 3 nm. The basic theory of the new probe is described and the results presented

High resolution thermal microscopy

Journal ArticleA new high resolution thermal microscope has been demonstrated capable of imaging thermal fields with sub 1000 angstom resolution. It is based upon a non-contacting near field thermal probe. The thermal probe consists of a thermocouple sensor on the end of a tip with sub 1000 angstrom dimensions. The probe tip is scanned in close proximity to a solid or liquid surface and the local temperature is mapped with a resolution determined by the size of the tip. Material independent surface profiling has also been demonstrated with the thermal probe, providing a lateral resolution of approximately 300 angstroms. Temperature mapping and surface profiling results are presented on both electronic and biological materials

Optical ranging by wavelength multiplexed interferometry

Journal ArticleA new optical technique is described for measurement of absolute distance. The approach is based upon a wavelength multiplexed heterodyne interferometer with FM demodulation. By temporally multiplexing discrete wavelengths in a heterodyne interferometer, a complete elimination of interferometric range ambiguity can be achieved while maintaining the high range sensitivity and resolution of interferometry