Well-posedness for the diffusive 3D Burgers equations with initial data in H1/2H^{1/2}

In this note we discuss the diffusive, vector-valued Burgers equations in a three-dimensional domain with periodic boundary conditions. We prove that given initial data in $H^{1/2}$ these equations admit a unique global solution that becomes classical immediately after the initial time. To prove local existence, we follow as closely as possible an argument giving local existence for the Navier--Stokes equations. The existence of global classical solutions is then a consequence of the maximum principle for the Burgers equations due to Kiselev and Ladyzhenskaya (1957). In several places we encounter difficulties that are not present in the corresponding analysis of the Navier--Stokes equations. These are essentially due to the absence of any of the cancellations afforded by incompressibility, and the lack of conservation of mass. Indeed, standard means of obtaining estimates in $L^2$ fail and we are forced to start with more regular data. Furthermore, we must control the total momentum and carefully check how it impacts on various standard estimates.Comment: 15 pages, to appear in "Recent Progress in the Theory of the Euler and Navier--Stokes Equations", eds. J.C. Robinson, J.L. Rodrigo, W. Sadowski and A. Vidal-L\'opez, Cambridge University Press, 201

Operator-Valued Frames for the Heisenberg Group

A classical result of Duffin and Schaeffer gives conditions under which a discrete collection of characters on $\mathbb{R}$, restricted to $E = (-1/2, 1/2)$, forms a Hilbert-space frame for $L^2(E)$. For the case of characters with period one, this is just the Poisson Summation Formula. Duffin and Schaeffer show that perturbations preserve the frame condition in this case. This paper gives analogous results for the real Heisenberg group $H_n$, where frames are replaced by operator-valued frames. The Selberg Trace Formula is used to show that perturbations of the orthogonal case continue to behave as operator-valued frames. This technique enables the construction of decompositions of elements of $L^2(E)$ for suitable subsets $E$ of $H_n$ in terms of representations of $H_n$

Design and Demonstration of a Miniature Lidar System for Rover Applications

Public awareness of harmful human environmental effects such as global warming has increased greatly in recent years and researchers have increased their efforts in gaining more knowledge about the Earth's atmosphere. Natural and man-made processes pose threats to the environment and human life, so knowledge of all atmospheric processes is necessary. Ozone and aerosols are important factors in many atmospheric processes and active remote sensing techniques provide a way to analyze their quantity and distribution. A compact ground-based lidar system for a robotic platform meant for atmospheric aerosol measurements was designed, tested, and evaluated. The system will eventually be deployed for ozone and aerosol measurements in Mars and lunar missions to improve our knowledge and understanding of atmospheres on Mars and the Moon. Atmospheric testing was performed to test the operability of the receiver system to acquire the lidar return signal from clouds and aerosols

Rhodora

Index v.51-75 (1976

Rhodora

v.52 (1950

The 22-Class Tower of Q(−5460)\mathbb{Q}(\sqrt{-5460})

The seminal papers in the field of root-discriminant bounds are those of Odlyzko and Martinet. Both papers include the question of whether the field $\mathbb{Q}(\sqrt{-5460})$ has finite or infinite $2$-class tower. This is a critical case that will either substantially lower the best known upper bound for lim inf of root-discriminants (if infinite) or else give a counter-example to what is often termed Martinet's conjecture or question (if finite). Using extensive computation and introducing some new techniques, we give strong evidence that the tower is in fact finite, establishing other properties of its Galois group en route

Immersion scanning thermal microscopy:probing nanoscale heat transport in liquid environments

While Scanning Thermal Microscopy (SThM) using locally heated nanoscale probes is known for its ability to map heat transport and thermal properties of materials and devices with micro and nanoscale resolution. Such studies in the liquid environments were perceived to be impossible due to dominating heat dissipation from the heated probe into the surrounding liquid that would also deteriorate spatial resolution. Here we show that contrary to the common belief, the heat generated by the SThM nanoscale probe remains localised within the well-defined nanoscale volume, and that the amount of local heat transfer to the sample is comparable to the one of the standard ambient environment in organic and inorganic liquids. Moreover, the presence of liquid provides highly stable thermal contact between the probe tip and the sample eliminating one of the major drawbacks of the ambient or vacuum SThM’s – variability of such contact. We show that such immersion SThM, or iSThM can effectively observe the semiconductor devices with the resolution of few tens of nanometres, providing new tool for exploring thermal effects of chemical reactions and biological processes with nanoscale resolution

Primordial Non-Gaussianity: Baryon Bias and Gravitational Collapse of Cosmic String Wakes

I compute the 3-D non-linear evolution of gas and dark matter fluids in the neighbourhood of cosmic string wakes which are formed at high redshift ($z\simeq 2240$) for a ``realistic'' scenario of wake formation. These wakes are the ones which stand out most prominently as cosmological sheets and are expected to play a dominant r\^ole in the cosmic string model of structure formation. Employing a high-resolution 3-D hydrodynamics code to evolve these wakes until the present day yields results for the baryon bias generated in the inner wake region. I find that today, wakes would be $1.5 h^{-1}$ Mpc thick and contain a 70% excess in the density of baryons over the dark matter density in their centre. However, high density peaks in the wake region do not inherit a baryon enhancement. I propose a mechanism for this erasure of the baryon excess in spherically collapsed objects based on the geometry change around the collapsing region. Further, I present heuristic arguments for the consequences of this work for large scale structure in the cosmic string model and conclude that the peculiarities of wake formation are unlikely to have significant import on the discrepancy between power spectrum predictions and observations in this model. If one invokes the nucleosynthesis bound on $\Omega_b$ this could be seen as strengthening the case against $\Omega_m=1$ or for low Hubble constants.Comment: 21 pages, 7 figures, 2 tables, prepared with the AASTeX package. Minor modifications, results unchanged. ApJ in press, scheduled for Vol. 50

Is the Pale Blue Dot unique? Optimized photometric bands for identifying Earth-like exoplanets

The next generation of ground and space-based telescopes will image habitable planets around nearby stars. A growing literature describes how to characterize such planets with spectroscopy, but less consideration has been given to the usefulness of planet colors. Here, we investigate whether potentially Earth-like exoplanets could be identified using UV-visible-to-NIR wavelength broadband photometry (350-1000 nm). Specifically, we calculate optimal photometric bins for identifying an exo-Earth and distinguishing it from uninhabitable planets including both Solar System objects and model exoplanets. The color of some hypothetical exoplanets - particularly icy terrestrial worlds with thick atmospheres - is similar to Earth's because of Rayleigh scattering in the blue region of the spectrum. Nevertheless, subtle features in Earth's reflectance spectrum appear to be unique. In particular, Earth's reflectance spectrum has a 'U-shape' unlike all our hypothetical, uninhabitable planets. This shape is partly biogenic because O2-rich, oxidizing air is transparent to sunlight, allowing prominent Rayleigh scattering, while ozone absorbs visible light, creating the bottom of the 'U'. Whether such uniqueness has practical utility depends on observational noise. If observations are photon limited or dominated by astrophysical sources (zodiacal light or imperfect starlight suppression), then the use of broadband visible wavelength photometry to identify Earth twins has little practical advantage over obtaining detailed spectra. However, if observations are dominated by dark current then optimized photometry could greatly assist preliminary characterization. We also calculate the optimal photometric bins for identifying extrasolar Archean Earths, and find that the Archean Earth is more difficult to unambiguously identify than a modern Earth twin.Comment: 10 figures, 38 page