Spiraling of approximations and spherical averages of Siegel transforms

We consider the question of how approximations satisfying Dirichlet's theorem spiral around vectors in $\mathbb{R}^d$. We give pointwise almost everywhere results (using only the Birkhoff ergodic theorem on the space of lattices). In addition, we show that for $\textit{every}$ unimodular lattice, on average, the directions of approximates spiral in a uniformly distributed fashion on the $d-1$ dimensional unit sphere. For this second result, we adapt a very recent proof of Marklof and Str\"ombergsson \cite{MS3} to show a spherical average result for Siegel transforms on $\operatorname{SL}_{d+1}(\mathbb{R})/\operatorname{SL}_{d+1}(\mathbb{Z})$. Our techniques are elementary. Results like this date back to the work of Eskin-Margulis-Mozes \cite{EMM} and Kleinbock-Margulis \cite{KM} and have wide-ranging applications. We also explicitly construct examples in which the directions are not uniformly distributed.Comment: 20 pages, 1 figure. Noteworthy changes from the previous version: New title. New result added (Theorem 1.1). Strengthening of Theorem 1.

On the Boundary Entropy of One-dimensional Quantum Systems at Low Temperature

The boundary beta-function generates the renormalization group acting on the universality classes of one-dimensional quantum systems with boundary which are critical in the bulk but not critical at the boundary. We prove a gradient formula for the boundary beta-function, expressing it as the gradient of the boundary entropy s at fixed non-zero temperature. The gradient formula implies that s decreases under renormalization except at critical points (where it stays constant). At a critical point, the number exp(s) is the ``ground-state degeneracy,'' g, of Affleck and Ludwig, so we have proved their long-standing conjecture that g decreases under renormalization, from critical point to critical point. The gradient formula also implies that s decreases with temperature except at critical points, where it is independent of temperature. The boundary thermodynamic energy u then also decreases with temperature. It remains open whether the boundary entropy of a 1-d quantum system is always bounded below. If s is bounded below, then u is also bounded below.Comment: 12 pages, Latex, 1 eps-figure; v2: some expository material added, a slightly more condensed version of the paper is publihed in Phys. Rev. Let

Fast-Light in a Photorefractive Crystal for Gravitational Wave Detection

We demonstrate superluminal light propagation using two frequency multiplexed pump beams to produce a gain doublet in a photorefractive crystal of Ce:BaTiO3. The two gain lines are obtained by two-wave mixing between a probe field and two individual pump fields. The angular frequencies of the pumps are symmetrically tuned from the frequency of the probe. The frequency difference between the pumps corresponds to the separation of the two gain lines; as it increases, the crystal gradually converts from normal dispersion without detuning to an anomalously dispersive medium. The time advance is measured as 0.28 sec for a pulse propagating through a medium with a 2Hz gain separation, compared to the same pulse propagating through empty space. We also demonstrate directly anomalous dispersion profile using a modfied experimental configuration. Finally, we discuss how anomalous dispersion produced this way in a faster photorefractive crystal (such as SPS: Sn2P2S6) could be employed to enhance the sensitivity-bandwidth product of a LIGO type gravitational wave detector augmented by a White Light Cavity.Comment: 14 pages, 5 figure

A first-order Green's function approach to supersonic oscillatory flow: A mixed analytic and numeric treatment

A frequency domain Green's Function Method for unsteady supersonic potential flow around complex aircraft configurations is presented. The focus is on the supersonic range wherein the linear potential flow assumption is valid. In this range the effects of the nonlinear terms in the unsteady supersonic compressible velocity potential equation are negligible and therefore these terms will be omitted. The Green's function method is employed in order to convert the potential flow differential equation into an integral one. This integral equation is then discretized, through standard finite element technique, to yield a linear algebraic system of equations relating the unknown potential to its prescribed co-normalwash (boundary condition) on the surface of the aircraft. The arbitrary complex aircraft configuration (e.g., finite-thickness wing, wing-body-tail) is discretized into hyperboloidal (twisted quadrilateral) panels. The potential and co-normalwash are assumed to vary linearly within each panel. The long range goal is to develop a comprehensive theory for unsteady supersonic potential aerodynamic which is capable of yielding accurate results even in the low supersonic (i.e., high transonic) range

Generalizations of Weighted Trapezoidal Inequality for Monotonic Mappings and Its Applications

In this paper, we establish some generalizations of weighted trapezoid inequality for monotonic mappings, and give several applications for r − moment, the expectation of a continuous random variable and the Beta mapping

The relationships between illness perceptions, social support, coping on mood after first-time myocardial infarction

The aim of this study was to examine how illness perceptions, social support, and coping influence first-time MI patients' moods after six months post-MI. The second aim was to examine the roles of first-time MI couples' illness perceptions on their own moods. One hundred and twenty six first-time MI patients filled in the questionnaires during their hospitalisation, and 91 of them completed all three assessments during the first six months. Data from 42 first-time MI couples were also collected during the patients' hospitalisation. Thirty-five of the 42 MI couples completed all three assessments during the first six months. Therefore, information from 91 MI patients and 35 MI couples were used for longitudinal data analyses. The results indicated that when comparing with healthy people, first-time MI patients reported higher levels of depression, state anxiety and negative affect. Those held negative illness perceptions (e.g., worse illness consequences, longer recovery time, and worse symptom perception) tended to feel more depressed or anxious. Hierarchical regressions indicated that symptom perception and illness consequence perception were two important contributors of these MI patients' moods. In addition, some types of coping strategies and mood variables were also important. Data of MI couples indicated that spouses were also influenced by the MI event, and they reported higher levels of depression, state anxiety and negative affect. Those couples who both had negative illness perceptions of the MI tended to feel more negative. However, couples' moods did not significant correlate with each other. Hierarchical regression further showed only their own illness perceptions and moods significantly contributed to their own moods. Their partners' illness perceptions and moods did not contribute to their own moods. The finding suggested illness perceptions played important roles on MI patients' and spouses' moods. To improve moods, future studies should focus more on interventions of illness perceptions