Ricci flow of unwarped and warped product manifolds

We analyse Ricci flow (normalised/un-normalised) of product manifolds --unwarped as well as warped, through a study of generic examples. First, we investigate such flows for the unwarped scenario with manifolds of the type $\mathbb S^n\times \mathbb S^m$, $\mathbb S^n\times \mathbb H^m$, $\mathbb H^m\times \mathbb H^n$ and also, similar multiple products. We are able to single out generic features such as singularity formation, isotropisation at particular values of the flow parameter and evolution characteristics. Subsequently, motivated by warped braneworlds and extra dimensions, we look at Ricci flows of warped spacetimes. Here, we are able to find analytic solutions for a special case by variable separation. For others we numerically solve the equations (for both the forward and backward flow) and draw certain useful inferences about the evolution of the warp factor, the scalar curvature as well the occurence of singularities at finite values of the flow parameter. We also investigate the dependence of the singularities of the flow on the inital conditions. We expect our results to be useful in any physical/mathematical context where such product manifolds may arise.Comment: 25 pages, 25 figures, some figures replace

A central limit theorem for Hilbert modular forms

For a prime ideal $\mathfrak{p}$ in a totally real number field $L$ with the adele ring $\mathbb{A}$, we study the distribution of angles $\theta_\pi(\mathfrak{p})$ coming from Satake parameters corresponding to unramified $\pi_\mathfrak{p}$ where $\pi_\mathfrak{p}$ comes from a global $\pi$ ranging over a certain finite set $\Pi_{\underline{k}}(\mathfrak{n})$ of cuspidal automorphic representations of GL$_2(\mathbb{A})$ with trivial central character. For such a representation $\pi$, it is known that the angles $\theta_\pi(\mathfrak{p})$ follow the Sato-Tate distribution. Fixing an interval $I\subseteq [0,\pi]$, we prove a central limit theorem for the number of angles $\theta_\pi(\mathfrak{p})$ that lie in $I$, as $\mathrm{N}(\mathfrak{p})\to\infty$. The result assumes $\mathfrak{n}$ to be a squarefree integral ideal, and that the components in the weight vector $\underline{k}$ grow suitably fast as a function of $x$.Comment: 12 page

De Quervain`s disease: evaluation by high resolution ultrasonography

Background: De Quervain's tenosynovitis is a stenosing tenosynovitis of the first extensor compartment of wrist and leads to wrist pain and impaired function of wrist and hand.  The aim of this study is to evaluate the role of high resolution ultrasonography in diagnosing suspected cases of de Quervain’s tenosynovitis and also to evaluate the role of high resolution ultrasonography in detecting the anatomical variants of the first extensor compartment which are predisposing conditions for de Quervain’s tenosynovitis.Methods: A prospective study of 15 consecutive cases who were referred with clinical diagnosis of de Quervain`s disease was done with ultrasonography in the department of Radio-diagnosis and findings were carefully analysed.Results: Thickened extensor retinaculum over the first extensor compartment was found in all the cases. Mean thickness of the thickened retinaculum is 1.65 mm. In 60% of cases multiple slips of APL tendon were found.Conclusions: From the study, we conclude that extensor retinaculum thickening is a common finding in de Quervain`s disease