Is the GW150914-GBM really associated with the GW150914?

Finding the electromagnetic (EM) counterpart is critically important for a gravitational wave event. Although many efforts have been made to search for the purported EM counterpart of GW150914, the first gravitational wave event detected by LIGO, only Fermi/GBM reported an excess above background (i.e. GW150914-GBM) at 0.4 s after the LIGO trigger time, that is possibly associated with this GW event (Connaughton et al. 2016). However, since there is no significant detection by the INTEGRAL/SPI-ACS around the time of GW150914-GBM, a great debate has been raised about whether GW150914-GBM is of astrophysical origin and associated with the GW150914 (Savchenko et al. 2016). In order to answer this question, we re-analyzed the GBM data with a straightforward but sophisticated method. We find that the excess of GW150914-GBM mostly comes from those detectors with bad viewing angles to the GW event, whereas the good viewing detectors see nothing significant beyond background fluctuation around the trigger time of GW150914. This finding suggests that GW150914-GBM is very unlikely associated with the GW150914. Given that GW150914-GBM is the only event found by GBM that is possibly associated with this GW event in a comprehensive search, we conclude that GBM did not detect any electromagnetic radiation from the GW150914.Comment: 4 pages, 2 figures, comments are very welcom

Modulus of continuity and Heinz-Schwarz type inequalities of solutions to biharmonic equations

For positive integers $n\geq2$ and $m\geq1$, suppose that function $f\in\mathcal{C}^{4}(\mathbb{B}^{n},\mathbb{R}^{m})$ satisfying the following: $(1)$ the inhomogeneous biharmonic equation $\Delta(\Delta f)=g$ ($g\in \mathcal{C}(\overline{\mathbb{B}^{n}},\mathbb{R}^{m})$) in $\mathbb{B}^{n}$, (2) the boundary conditions $f=\varphi_{1}$ $(\varphi_{1}\in \mathcal{C}(\mathbb{S}^{n-1},\mathbb{R}^{m}))$ on $\mathbb{S}^{n-1}$ and $\partial f/\partial\mathbf{n}=\varphi_{2}$ ( $\varphi_{2}\in \mathcal{C}(\mathbb{S}^{n-1},\mathbb{R}^{m})$) on $\mathbb{S}^{n-1}$, where $\partial /\partial\mathbf{n}$ stands for the inward normal derivative, $\mathbb{B}^{n}$ is the unit ball in $\mathbb{R}^{n}$ and $\mathbb{S}^{n-1}$ is the unit sphere of $\mathbb{B}^{n}$. First, we establish the representation formula of solutions to the above inhomogeneous biharmonic Dirichlet problem, and then discuss the Heinz-Schwarz type inequalities and the modulus of continuity of the solutions.Comment: 23 page

The Schwarz type Lemmas and the Landau type theorem of mappings satisfying Poisson's equations

For a given continuous function $g:~\Omega\rightarrow\mathbb{C}$, we establish some Schwarz type Lemmas for mappings $f$ in $\Omega$ satisfying the {\rm PDE}: $\Delta f=g$, where $\Omega$ is a subset of the complex plane $\mathbb{C}$. Then we apply these results to obtain a Landau type theorem, which is a partial answer to the open problem in \cite{CP}. %The obtained results areComment: 19 page

Some sharp Schwarz-Pick type estimates and their applications of harmonic and pluriharmonic functions

The purpose of this paper is to study the Schwarz-Pick type inequalities for harmonic or pluriharmonic functions. By analogy with the generalized Khavinson conjecture, we first give some sharp estimates of the norm of harmonic functions from the Euclidean unit ball in $\mathbb{R}^n$ into the unit ball of the real Minkowski space. Next, we give several sharp Schwarz-Pick type inequalities for pluriharmonic functions from the Euclidean unit ball in $\mathbb{C}^n$ or from the unit polydisc in $\mathbb{C}^n$ into the unit ball of the Minkowski space. Furthermore, we establish some sharp coefficient type Schwarz-Pick inequalities for pluriharmonic functions defined in the Minkowski space. Finally, we use the obtained Schwarz-Pick type inequalities to discuss the Lipschitz continuity, the Schwarz-Pick type lemmas of arbitrary order and the Bohr phenomenon of harmonic or pluriharmonic functions.Comment: 39 page

Lipschitz type, radial growth and Dirichlet type spaces on functions induced by certain elliptic operators

In this paper, we investigate properties of classes of functions related to certain elliptic operators. Firstly, we prove that a main result of Dyakonov (Acta Math. 178(1997), 143--167) on analytic functions can be extended to this more general setting. Secondly, we study the radial growth on these functions and the obtained results are generalizations of the corresponding results of Makarov (Proc. London Math. Soc. 51(1985), 369--384) and Korenblum (Bull. Amer. Math. Soc. 12(1985), 99--102). Finally, we discuss the Dirichlet type energy integrals on such classes of functions and their applications.Comment: 23 page

Recursive utility optimization with concave coefficients

This paper concerns the recursive utility maximization problem. We assume that the coefficients of the wealth equation and the recursive utility are concave. Then some interesting and important cases with nonlinear and nonsmooth coefficients satisfy our assumption. After given an equivalent backward formulation of our problem, we employ the Fenchel-Legendre transform and derive the corresponding variational formulation. By the convex duality method, the primal "sup-inf" problem is translated to a dual minimization problem and the saddle point of our problem is derived. Finally, we obtain the optimal terminal wealth. To illustrate our results, three cases for investors with ambiguity aversion are explicitly worked out under some special assumptions

Solvability of one kind of forward-backward stochastic difference equations

In this paper, we study the solvability problem for one kind of fully coupled forward-backward stochastic difference equations (FBS{\Delta}Es). With the help of the necessary and sufficient condition for the solvability of the linear FBS{\Delta}Es, under the monotone assumption, we obtain the existence and uniqueness theorem for the general nonlinear ones.Comment: 19 pages. arXiv admin note: text overlap with arXiv:1812.1128

A generalized Neyman-Pearson lemma for two sublinear operators

In this paper we extended the Neyman-Pearson lemma by replacing two probabilities into two sublinear operators and divide our problem into two cases to get the reminiscent form of the optimal solution as in the linear case if the optimal solution exists. We also studied the existence of the optimal solution.Comment: 17 page

A note on pricing of contingent claims under G-expectation

In this paper, we study the pricing of contingent claims under G-expectation. In order to accomodate volatility uncertainty, the price of the risky security is supposed to governed by a general linear stochastic differential equation (SDE) driven by G-Brownian motion. Utilizing the recently developed results of Backward SDE driven by G-Brownian motion, we obtain the superhedging and suberhedging prices of a given contingent claim. Explicit results in the Markovian case are also derived.Comment: 15 pages. arXiv admin note: substantial text overlap with arXiv:1212.5403, arXiv:1206.588

Lipschitz continuity of holomorphic mappings with respect to Bergman metric

n this paper, we establish the sharp estimate of the Lipschitz continuity with respect to the Bergman metric. The obtained results are the improvement and generalization of the corresponding results of Ghatage, Yan and Zheng (Proc. Amer. Math. Soc., 129: 2039-2044, 2000).Comment: 8 page