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

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

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

Analysis of Mono-, Di- and Oligosaccharides by CE Using a Two-Stage Derivatization Method and LIF Detection.

A sensitive CE with LIF method has been developed for quantitative analysis of small carbohydrates. In this work, 17 carbohydrates including mono-, di- and oligosaccharides were simultaneously derivatized with 4-fluoro 7-nitrobenzo furazane (NBD-F) via a twostep reaction involving reductive amination with ammonia followed by condensation with NBD-F. Under the optimized derivatization conditions all carbo-hydrates were successfully derivatized within 2.5 h and separated within 15 min using borate buffer (90 mmol/L, pH 9.2). For sugar standards LODs were in the range of 49.7 to 243.6 nmol/L. Migration time and peak area reproducibility were better than RSD 0.1 and 3%, respectively. The method was applied to measure sugars in nanoliter volume samples of phloem sap obtained by stylectomy from wheat and to honeydew samples obtained from aphids feeding from wheat and willow