Forward-backward eccentricity and participant-plane angle fluctuations and their influences on longitudinal dynamics of collective flow

We argue that the transverse shape of the fireball created in heavy ion collision is controlled by event-by-event fluctuations of the eccentricity vectors for the forward-going and backward-going wounded nucleons: $\vec{\epsilon}_n^{\mathrm{F}}\equiv \epsilon_n^{\mathrm{F}} e^{i n\Phi_n^{\mathrm{*F}}}$ and $\vec{\epsilon}_n^{\mathrm{B}}\equiv \epsilon_n^{\mathrm{B}} e^{i n\Phi_n^{\mathrm{*B}}}$. Due to the asymmetric energy deposition of each wounded nucleon along its direction of motion, the eccentricity vector of the produced fireball is expected to interpolate between $\vec{\epsilon}_n^{\mathrm{F}}$ and $\vec{\epsilon}_n^{\mathrm{B}}$ along the pseudorapidity, and hence exhibits sizable forward-backward(FB) asymmetry ($\epsilon_n^{\rm B}\neq\epsilon_n^{\rm F}$) and/or FB-twist ($\Phi_n^{\mathrm{*F}}\neq\Phi_n^{\mathrm{*B}}$). A transport model calculation shows that these initial state longitudinal fluctuations for $n=2$ and 3 survive the collective expansion, and result in similar FB asymmetry and/or a twist in the final state event-plane angles. These novel EbyE longitudinal flow fluctuations should be accessible at RHIC and the LHC using the event-shape selection technique proposed in earlier papers. If these effects are observed experimentally, it could improve our understanding of the initial state fluctuations, particle production and collective expansion dynamics of the heavy ion collision.Comment: 13 pages, 16 figure

Optimal Time Decay Rate for the Compressible Viscoelastic Equations in Critical Spaces

In this paper, we are concerned with the convergence rates of the global strong solution to constant equilibrium state for the compressible viscoelastic fluids in the whole space. We combine both analysis about Green's matrix method and energy estimate method to get optimal time decay rate in critical Besov space framework. Our result imply the optimal $L^{2}$-time decay rate and only need the initial data to be small in critical Besov space which have very low regularity compared with traditional Sobolev space.Comment: 20 page

Optimal Time Decay of Navier-Stokes Equations With Low Regularity Initial Data

In this paper, we study the optimal time decay rate of isentropic Navier-Stokes equations under the low regularity assumptions about initial data. In the previous works about optimal time decay rate, the initial data need to be small in $H^{[N/2]+2}(\mathbb{R}^{N})$. Our work combined negative Besov space estimates and the conventional energy estimates in Besov space framework which is developed by R. Danchin. Though our methods, we can get optimal time decay rate with initial data just small in $\dot{B}^{N/2-1, N/2+1} \cap \dot{B}^{N/2-1, N/2}$ and belong to some negative Besov space(need not to be small). Finally, combining the recent results in \cite{zhang2014} with our methods, we can only need the initial data to be small in homogeneous Besov space $\dot{B}^{N/2-2, N/2} \cap \dot{B}^{N/2-1}$ to get the optimal time decay rate in space $L^{2}$.Comment: arXiv admin note: text overlap with arXiv:1410.794

A note on characterizations of G-normal distribution

In this paper, we show that the G-normality of X and Y can be characterized according to the form of f such that the distribution of {\lambda}+f({\lambda})Y does not depend on {\lambda}, where Y is an independent copy of X and {\lambda} is in the domain of f. Without the condition that Y is identically distributed with X, we still have a similar argument

On LpL^p-Liouville property for smooth metric measure spaces

In this short paper we study $L_f^p$-Liouville property with $0<p<1$ for nonnegative $f$-subharmonic functions on a complete noncompact smooth metric measure space $(M,g,e^{-f}dv)$ with $\mathrm{Ric}_f^m$ bounded below for $0<m\leq\infty$. We prove a sharp $L_f^p$-Liouville theorem when $0<m<\infty$. We also prove an $L_f^p$-Liouville theorem when $\mathrm{Ric}_f\geq 0$ and $|f(x)|\leq \delta(n) \ln r(x)$.Comment: Preliminary version, all comments are welcome

Heat Kernel on Smooth Metric Measure Spaces with Nonnegative Curvature

We derive a local Gaussian upper bound for the $f$-heat kernel on complete smooth metric measure space $(M,g,e^{-f}dv)$ with nonnegative Bakry-\'{E}mery Ricci curvature, which generalizes the classic Li-Yau estimate. As applications, we obtain a sharp $L_f^1$-Liouville theorem for $f$-subharmonic functions and an $L_f^1$-uniqueness property for nonnegative solutions of the $f$-heat equation, assuming $f$ is of at most quadratic growth. In particular, any $L_f^1$-integrable $f$-subharmonic function on gradient shrinking or steady Ricci solitons must be constant. We also provide explicit $f$-heat kernel for Gaussian solitons.Comment: Revised version. Math. Annalen, to appea

Heat kernel on smooth metric measure spaces and applications

We derive a Harnack inequality for positive solutions of the $f$-heat equation and Gaussian upper and lower bounds for the $f$-heat kernel on complete smooth metric measure spaces $(M, g, e^{-f}dv)$ with Bakry-\'Emery Ricci curvature bounded below. The lower bound is sharp. The main argument is the De Giorgi-Nash-Moser theory. As applications, we prove an $L^1_f$-Liouville theorem for $f$-subharmonic functions and an $L^1_f$-uniqueness theorem for $f$-heat equations when $f$ has at most linear growth. We also obtain eigenvalues estimates and $f$-Green's function estimates for the $f$-Laplace operator.Comment: 30 page

Elucidating the event-by-event flow fluctuations in heavy-ion collisions via the event shape selection technique

The presence of large event-by-event flow fluctuations in heavy ion collisions at RHIC and the LHC provides an opportunity to study a broad class of flow observables. This paper explores the correlations among harmonic flow coefficients $v_n$ and their phases $\Phi_n$, and the rapidity fluctuation of $v_n$. The study is carried out usin Pb+Pb events generated by the AMPT model with fixed impact parameter. The overall ellipticity/triangularity of events is varied by selecting on the eccentricities $\epsilon_n$ or the magnitudes of the flow vector $q_n$ for n=2 and 3, respectively. The responses of the $v_n$, the event-plane correlations, and the rapidity fluctuations, to the change in $\epsilon_n$ and $q_n$ are then systematized. Strong positive correlations are observed among all even harmonics $v_2, v_4$, and $v_6$ (all increase with $q_2$), between $v_2$ and $v_5$ (both increase with $q_2$) and between $v_3$ and $v_5$ (both increase with $q_3$), consistent with the effects of nonlinear collective response. In contrast, an anti-correlation is observed between $v_2$ and $v_3$ similar to that seen between $\epsilon_2$ and $\epsilon_3$. These correlation patterns are found to be independent of whether selecting on $q_n$ or $\epsilon_n$, validating the ability of $q_n$ in selecting the initial geometry. A forward/backward asymmetry of $v_n(\eta)$ is observed for events selected on $q_n$ but not on $\epsilon_n$, reflecting dynamical fluctuations exposed by the $q_n$ selection. Many event-plane correlators show good agreement between $q_n$ and $\epsilon_n$ selections, suggesting that their variations with $q_n$ are controlled by the change of $\epsilon_n$ in the initial geometry. Hence these correlators may serve as promising observables for disentangling the fluctuations generated in various stages of the evolution of the matter created in heavy ion collisions.Comment: 14 pages, 20 figure

Studies on an inverse source problem for a space-time fractional diffusion equation by constructing a strong maximum principle

In this paper, we focus on a space-time fractional diffusion equation with the generalized Caputo's fractional derivative operator and a general space nonlocal operator (with the fractional Laplace operator as a special case). A weak Harnack's inequality has been established by using a special test function and some properties of the space nonlocal operator. Based on the weak Harnack's inequality, a strong maximum principle has been obtained which is an important characterization of fractional parabolic equations. With these tools, we establish a uniqueness result for an inverse source problem on the determination of the temporal component of the inhomogeneous term.Comment: 30 pages. arXiv admin note: text overlap with arXiv:1009.4852 by other author

Bayesian approach to inverse problems for functions with variable index Besov prior

We adopt Bayesian approach to consider the inverse problem of estimate a function from noisy observations. One important component of this approach is the prior measure. Total variation prior has been proved with no discretization invariant property, so Besov prior has been proposed recently. Different prior measures usually connect to different regularization terms. Variable index TV, variable index Besov regularization terms have been proposed in image analysis, however, there are no such prior measure in Bayesian theory. So in this paper, we propose a variable index Besov prior measure which is a Non-Guassian measure. Based on the variable index Besov prior measure, we build the Bayesian inverse theory. Then applying our theory to integer and fractional order backward diffusion problems. Although there are many researches about fractional order backward diffusion problems, we firstly apply Bayesian inverse theory to this problem which provide an opportunity to quantify the uncertainties for this problem.Comment: 31 pages. arXiv admin note: text overlap with arXiv:1302.6989 by other author