Triple {\alpha} Resonances in the Decay of Hot Nuclear Systems *

The Efimov (Thomas) trimers in excited 12 C nuclei, for which no observation exists yet, are discussed by means of analyzing the experimental data of 70(64) Zn( 64 Ni) + 70(64) Zn( 64 Ni) reactions at beam energy of E/A=35MeV/nucleon. In heavy ion collisions, the {\alpha}s interact with each other and can form complex systems such as 8 Be and 12 C. For the 3{\alpha} systems, multi resonance processes give rise to excited levels of 12 C. The interaction between any two of the 3{\alpha} particles provides events with one, two or three 8 Be. Their interfering levels are clearly seen in the minimum relative energy distributions. Events of three couple {\alpha} relative energies consistent with the ground state of 8 Be are observed with the decrease of the instrumental error at the reconstructed 7.458 MeV excitation energy of 12C, which was suggested as the Efimov (Thomas) state.Comment: 5 pages,7figure

A critical regularity condition on the angular velocity of axially symmetric Navier-Stokes equations

Let $v$ be the velocity of Leray-Hopf solutions to the axially symmetric three-dimensional Navier-Stokes equations. It is shown that $v$ is regular if the angular velocity $v_\theta$ satisfies an integral condition which is critical under the standard scaling. This condition allows functions satisfying $$|v_\theta(x, t)| \le \frac{C}{r |\ln r|^{2+\epsilon}}, \quad r<1/2,$$ where $r$ is the distance from $x$ to the axis, $C$ and $\epsilon$ are any positive constants. Comparing with the critical a priori bound $$|v_\theta(x, t)| \le \frac{C}{r}, \qquad 0< r \le 1/2,$$our condition is off by the log factor $|\ln r|^{2+\epsilon}$ at worst. This is inspired by the recent interesting paper \cite{CFZ:1} where H. Chen, D. Y. Fang and T. Zhang establish, among other things, an almost critical regularity condition on the angular velocity. Previous regularity conditions are off by a factor $r^{-1}$. The proof is based on the new observation that, when viewed differently, all the vortex stretching terms in the 3 dimensional axially symmetric Navier-Stokes equations are critical instead of supercritical as commonly believed.Comment: 16 page

Some gradient estimates for the heat equation on domains and for an equation by Perelman

In the first part, we derive a sharp gradient estimate for the log of Dirichlet heat kernel and Poisson heat kernel on domains, and a sharpened local Li-Yau gradient estimate that matches the global one. In the second part, without explicit curvature assumptions, we prove a global upper bound for the fundamental solution of an equation introduced by G. Perelman, i.e. the heat equation of the conformal Laplacian under backward Ricci flow. Further, under nonnegative Ricci curvature assumption, we prove a qualitatively sharp, global Gaussian upper bound

Heat kernel bounds, ancient κ\kappa solutions and the Poincar\'e conjecture

We establish certain Gaussian type upper bound for the heat kernel of the conjugate heat equation associated with 3 dimensional ancient $\kappa$ solutions to the Ricci flow. As an application, using the $W$ entropy associated with the heat kernel, we give a different and shorter proof of Perelman's classification of backward limits of these ancient solutions. The current paper together with \cite{Z:2} and a different proof of universal noncollapsing due to Chen and Zhu \cite{ChZ:1} lead to a simplified proof of the Poincar\'e conjecture without using reduced distance and reduced volume.Comment: more references added, especially [CL]; some details added on p12-1

On the question of diameter bounds in Ricci flow

A question about Ricci flow is when the diameters of the manifold under the evolving metrics stay finite and bounded away from 0. Topping \cite{T:1} addresses the question with an upper bound that depends on the $L^{(n-1)/2}$ bound of the scalar curvature, volume and a local version of Perelman's $\nu$ invariant. Here $n$ is the dimension. His result is sharp when Perelman's F entropy is positive. In this note, we give a direct proof that for all compact manifolds, the diameter bound depends just on the $L^{(n-1)/2}$ bound of the scalar curvature, volume and the Sobolev constants (or positive Yamabe constant). This bound seems directly computable in large time for some Ricci flows. In addition, since the result in its most general form is independent of Ricci flow, further applications may be possible. A generally sharp lower bound for the diameters is also given, which depends only on the initial metric, time and $L^\infty$ bound of the scalar curvature. These results imply that, in finite time, the Ricci flow can neither turn the diameter to infinity nor zero, unless the scalar curvature blows up.Comment: Introduction to the note modified, reference and motivation added following suggestions by Professors Peter Topping and Mingliang Cai. A lower bound for the diameters is added. As a result, we now know that, in finite time, the Ricci flow can neither turn the diameter to infinity nor zero, unless the scalar curvature blows u

The solar abundance problem: the effect of the turbulent kinetic flux on the solar envelope model

Recent 3D-simulations have shown that the turbulent kinetic flux (TKF) is significant. We discuss the effects of TKF on the size of convection zone and find that the TKF may help to solve the solar abundance problem. The solar abundance problem is that, with new abundances, the solar convection zone depth, sound speed in the radiative interior, the helium abundance and density in the convective envelope are not in agreement with helioseismic inversions. We have done Monte Carlo simulations on solar convective envelope models with different profile of TKF to test the effects. The solar abundance problem is revealed in the standard solar convective envelope model with AGSS09 composition, which shows significant differences (\rm{\sim 10 %}) on density from the helioseicmic inversions, but the differences in the model with old composition GN93 is small (\rm{\sim 0.5 %}). In the testing models with different imposed TKF, it is found that the density profile is sensitive to the value of TKF at the base of convective envelope and insensitive to the structure of TKF in the convection zone. Required value of turbulent kinetic luminosity at the base is about \rm{-13%\sim-19%L_{\odot}}. Comparing with the 3D-simulations, this value is plausible. This study is for the solar convective envelope only. The evolutionary solar models with TKF are required for investigating its effects on the solar interior structure below the convection zone and the whole solar abundance problem, but the profile of TKF in the overshoot region is needed.Comment: 5 pages, 2 figures, accepted for publication in the ApJ Letter

Strong non-collapsing and uniform Sobolev inequalities for Ricci flow with surgeries

We prove a uniform Sobolev inequality for Ricci flow, which is independent of the number of surgeries. As an application, under less assumptions, a non-collapsing result stronger than Perelman's $\kappa$ non-collapsing with surgery is derived. The proof is shorter and seems more accessible. The result also improves some earlier ones where the Sobolev inequality depended on the number of surgeries

Confidence Intervals for Low-Dimensional Parameters in High-Dimensional Linear Models

The purpose of this paper is to propose methodologies for statistical inference of low-dimensional parameters with high-dimensional data. We focus on constructing confidence intervals for individual coefficients and linear combinations of several of them in a linear regression model, although our ideas are applicable in a much broad context. The theoretical results presented here provide sufficient conditions for the asymptotic normality of the proposed estimators along with a consistent estimator for their finite-dimensional covariance matrices. These sufficient conditions allow the number of variables to far exceed the sample size. The simulation results presented here demonstrate the accuracy of the coverage probability of the proposed confidence intervals, strongly supporting the theoretical results

Convective overshoot mixing in stellar interior models

The convective overshoot mixing plays an important role in stellar structure and evolution. However, the overshoot mixing is a long standing problem. The uncertainty of the overshoot mixing is one of the most uncertain factors in stellar physics. As it is well known, the convective and overshoot mixing is determined by the radial chemical component flux. In this paper, a local model of the radial chemical component flux is established based on the hydrodynamic equations and some model assumptions. The model is tested in stellar models. The main conclusions are as follows. (i) The local model shows that the convective and overshoot mixing could be regarded as a diffusion process, and the diffusion coefficient for different chemical element is the same. However, if the non-local terms, i.e., the turbulent convective transport of radial chemical component flux, are taken into account, the diffusion coefficient for each chemical element should be in general different. (ii) The diffusion coefficient of convective / overshoot mixing shows different behaviors in convection zone and in overshoot region because the characteristic length scale of the mixing is large in the convection zone and small in the overshoot region. The overshoot mixing should be regarded as a weak mixing process. (iii) The result of the diffusion coefficient of mixing is tested in stellar models. It is found that a single choice of our central mixing parameter leads to consistent results for a solar convective envelope model as well as for core convection models of stars with mass from 2M to 10M.Comment: 9 pages, 2 figures, accepted for publication in ApJ

A simple scheme to implement a nonlocal turbulent convection model for the convective overshoot mixing

The classical 'ballistic' overshoot models show some contradictions and are not consistence with numerical simulations and asteroseismic studies. Asteroseismic studies imply that overshoot is a weak mixing process. Diffusion model is suitable to deal with it. The form of diffusion coefficient in a diffusion model is crucial. Because the overshoot mixing is related to the convective heat transport (i.e., entropy mixing), there should be a similarity between them. A recent overshoot mixing model shows consistence between composition mixing and entropy mixing in overshoot region. A prerequisite to apply the model is to know the dissipation rate of turbulent kinetic energy. The dissipation rate can be worked out by solving turbulent convection models (TCMs). But it is difficult to apply TCMs because of some numerical problems and the enormous time cost. In order to find a convenient way, we have used the asymptotical solution and simplified the TCM to be a single linear equation for turbulent kinetic energy. This linear model is easy to be implemented in the calculations of stellar evolution with ignorable extra time cost. We have tested the linear model in stellar evolution, and have found that the linear model can well reproduce the turbulent kinetic energy profile of full TCM, as well as the diffusion coefficient, abundance profile and the stellar evolutionary tracks. We have also studied the effects of different values of the model parameters and have found that the effect due to the modification of temperature gradient in the overshoot region is slight.Comment: 20 pages, 10 figures, accepted for publication in Ap