اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدA refinement of Gunthers candle inequality
29
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
We analyze an upper bound on the curvature of a Riemannian manifold, using root-Ricci curvature, which is in between a sectional curvature bound and a Ricci curvature bound. (A special case of root-Ricci curvature was previously discovered by Osserman and Sarnak for a different but related purpose.) We prove that our root-Ricci bound implies Gunthers inequality on the candle function of a manifold, thus bringing that inequality closer in form to the complementary inequality due to Bishop.
قيم البحث
اقرأ أيضاً
The principal aim of this paper is to employ Bessel-type operators in proving the inequality begin{align*} int_0^pi dx , |f(x)|^2 geq dfrac{1}{4}int_0^pi dx , dfrac{|f(x)|^2}{sin^2 (x)}+dfrac{1}{4}int_0^pi dx , |f(x)|^2,quad fin H_0^1 ((0,pi)), end{a
lign*} where both constants $1/4$ appearing in the above inequality are optimal. In addition, this inequality is strict in the sense that equality holds if and only if $f equiv 0$. This inequality is derived with the help of the exactly solvable, strongly singular, Dirichlet-type Schr{o}dinger operator associated with the differential expression begin{align*} tau_s=-dfrac{d^2}{dx^2}+dfrac{s^2-(1/4)}{sin^2 (x)}, quad s in [0,infty), ; x in (0,pi). end{align*} The new inequality represents a refinement of Hardys classical inequality begin{align*} int_0^pi dx , |f(x)|^2 geq dfrac{1}{4}int_0^pi dx , dfrac{|f(x)|^2}{x^2}, quad fin H_0^1 ((0,pi)), end{align*} it also improves upon one of its well-known extensions in the form begin{align*} int_0^pi dx , |f(x)|^2 geq dfrac{1}{4}int_0^pi dx , dfrac{|f(x)|^2}{d_{(0,pi)}(x)^2}, quad fin H_0^1 ((0,pi)), end{align*} where $d_{(0,pi)}(x)$ represents the distance from $x in (0,pi)$ to the boundary ${0,pi}$ of $(0,pi)$.
The Crab Nebula is the only hard X-ray source in the sky that is both bright enough and steady enough to be easily used as a standard candle. As a result, it has been used as a normalization standard by most X-ray/gamma ray telescopes. Although small
-scale variations in the nebula are well-known, since the start of science operations of the Fermi Gamma-ray Burst Monitor (GBM) in August 2008, a ~ 7% (70 mcrab) decline has been observed in the overall Crab Nebula flux in the 15 - 50 keV band, measured with the Earth occultation technique. This decline is independently confirmed with three other instruments: the Swift Burst Alert Telescope (Swift/BAT), the Rossi X-ray Timing Explorer Proportional Counter Array (RXTE/PCA), and the INTErnational Gamma-Ray Astrophysics Laboratory Imager on Board INTEGRAL (IBIS). A similar decline is also observed in the ~3 - 15 keV data from the RXTE/PCA and INTEGRAL Joint European Monitor (JEM-X) and in the 50 - 100 keV band with GBM and INTEGRAL/IBIS. Observations from 100 to 500 keV with GBM suggest that the decline may be larger at higher energies. The pulsed flux measured with RXTE/PCA since 1999 is consistent with the pulsar spin-down, indicating that the observed changes are nebular. Correlated variations in the Crab Nebula flux on a ~3 year timescale are also seen independently with the PCA, BAT, and IBIS from 2005 to 2008, with a flux minimum in April 2007. As of August 2010, the current flux has declined below the 2007 minimum.
For a Poincare-Einstein manifold under certain restrictions, X. Chen, M. Lai and F. Wang proved a sharp inequality relating Yamabe invariants. We show that the inequality is true without any restriction.
Given a positive lower semi-continuous density $f$ on $mathbb{R}^2$ the weighted volume $V_f:=fmathscr{L}^2$ is defined on the $mathscr{L}^2$-measurable sets in $mathbb{R}^2$. The $f$-weighted perimeter of a set of finite perimeter $E$ in $mathbb{R}^
2$ is written $P_f(E)$. We study minimisers for the weighted isoperimetric problem [ I_f(v):=infBig{ P_f(E):Etext{ is a set of finite perimeter in }mathbb{R}^2text{ and }V_f(E)=vBig} ] for $v>0$. Suppose $f$ takes the form $f:mathbb{R}^2rightarrow(0,+infty);xmapsto e^{h(|x|)}$ where $h:[0,+infty)rightarrowmathbb{R}$ is a non-decreasing convex function. Let $v>0$ and $B$ a centred ball in $mathbb{R}^2$ with $V_f(B)=v$. We show that $B$ is a minimiser for the above variational problem and obtain a uniqueness result.
Numerical tools for constraints solving are a cornerstone to control verification problems. This is evident by the plethora of research that uses tools like linear and convex programming for the design of control systems. Nevertheless, the capability
of linear and convex programming is limited and is not adequate to reason about general nonlinear polynomials constraints that arise naturally in the design of nonlinear systems. This limitation calls for new solvers that are capable of utilizing the power of linear and convex programming to reason about general multivariate polynomials. In this paper, we propose PolyAR, a highly parallelizable solver for polynomial inequality constraints. PolyAR provides several key contributions. First, it uses convex relaxations of the problem to accelerate the process of finding a solution to the set of the non-convex multivariate polynomials. Second, it utilizes an iterative convex abstraction refinement process which aims to prune the search space and identify regions for which the convex relaxation fails to solve the problem. Third, it allows for a highly parallelizable usage of off-the-shelf solvers to analyze the regions in which the convex relaxation failed to provide solutions. We compared the scalability of PolyAR against Z3 8.9 and Yices 2.6 on control designing problems. Finally, we demonstrate the performance of PolyAR on designing switching signals for continuous-time linear switching systems.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
