The Erdős-Mordell inequality is equivalent to non-positive curvature

We prove that, in plane absolute geometry, the Erdős-Mordell inequality is equivalent to the statement that the sum of the angles of a triangle is less than or equal to two right angles.      

Riesz transforms for a non-symmetric Ornstein-Uhlenbeck semigroup

Let (ℋ _t ) _t≥0 be the Ornstein-Uhlenbeck semigroup on ℝ ^d with covariance matrix I and drift matrix −λ(I+R), where λ>0 and R is a skew-adjoint matrix and denote by γ _∞ the invariant measure for (ℋ _t ) _t≥0. Semigroups of this form are the basic building blocks of Ornstein-Uhlenbeck semigroups which are normal on L ²(γ _∞). We investigate the weak type 1 estimate of the Riesz transforms for (ℋ _t ) _t≥0. We prove that if the matrix R generates a one-parameter group of periodic rotations then the first order Riesz transforms are of weak type 1 with respect to the invariant measure γ _∞. We also prove that the Riesz transforms of any order associated to a general Ornstein-Uhlenbeck semigroup are bounded on L ^p (γ _∞) if 1<p<∞. The authors have received support by the Italian MIUR-PRIN 2005 project “Harmonic Analysis” and by the EU IHP 2002-2006 project “HARP”.     

On the Elliptic Equation △u ＋ K（x）e^2u -0 with K（x） Positive Somewhere

This paper considers the existence problem of an elliptic equation.which is equivalent to the prescribing conformal Gaussian curvature problem on R2.An existence result is pried.In particular,K(x)is allowed to be unbounded above.     

Solvability of backward stochastic differential equations with quadratic growth

We prove the existence of the unique solution of a general backward stochastic differential equation with quadratic growth driven by martingales. A kind of comparison theorem is also proved.     

Skew-Gaussian random field

In this article, we introduce the concept of skewness to the Gaussian random field theory by defining a new two-dimensional non-Gaussian random field called skew-Gaussian random field. We derive the expected Euler characteristic of its excursion set. Moreover, an approximation to the size distribution of one connected component is derived. As an illustration, we present a simulation study to compare the approximation mechanism with the exact one.     

Error bounds for approximations of traces of products of truncated Toeplitz operators

The paper establishes error orders for integral limit approximations to the traces of products of truncated Toeplitz operators generated by integrable real symmetric functions defined on the real line. These approximations and the corresponding error bounds are of importance in the statistical analysis of continuous-time stationary processes (asymptotic distributions and large deviations of Toeplitz type quadratic functionals, estimation of the spectrum, etc.). The results improve the rates obtained by the authors (in an earlier paper). An explicit second-order asymptotic expansion is found for the trace of a product of two truncated Toeplitz operators generated by the spectral densities of continuous-time stationary fractional Riesz-Bessel motions. The order of magnitude of the second term in this expansion is shown to depend on the long-memory parameters of the processes. The second-order term provides a substantially better approximation to the original functional, as compared with the first-order approximation. M. Ginovyan research was partially supported by National Science Foundation Grant #DMS-0706786.     

Robustness of nonuniform exponential dichotomies in Banach spaces

We give conditions for the robustness of nonuniform exponential dichotomies in Banach spaces, in the sense that the existence of an exponential dichotomy for a given linear equation x^′=A(t)x persists under a sufficiently small linear perturbation. We also establish the continuous dependence with the perturbation of the constants in the notion of dichotomy and of the “angles” between the stable and unstable subspaces. Our proofs exhibit (implicitly) the exponential dichotomies of the perturbed equations in terms of fixed points of appropriate contractions. We emphasize that we do not need the notion of admissibility (of bounded nonlinear perturbations). We also obtain related robustness results in the case of nonuniform exponential contractions. In addition, we establish an appropriate version of robustness for nonautonomous dynamical systems with discrete time.     

Bootstrap approximation of tail dependence function

For estimating a rare event via the multivariate extreme value theory, the so-called tail dependence function has to be investigated (see [L. de Haan, J. de Ronde, Sea and wind: Multivariate extremes at work, Extremes 1 (1998) 7-45]). A simple, but effective estimator for the tail dependence function is the tail empirical distribution function, see [X. Huang, Statistics of Bivariate Extreme Values, Ph.D. Thesis, Tinbergen Institute Research Series, 1992] or [R. Schmidt, U. Stadtmüller, Nonparametric estimation of tail dependence, Scand. J. Stat. 33 (2006) 307-335]. In this paper, we first derive a bootstrap approximation for a tail dependence function with an approximation rate via the construction approach developed by [K. Chen, S.H. Lo, On a mapping approach to investigating the bootstrap accuracy, Probab. Theory Relat. Fields 107 (1997) 197-217], and then apply it to construct a confidence band for the tail dependence function. A simulation study is conducted to assess the accuracy of the bootstrap approach.     

Propagation characteristics of TM Gaussian beam

The propagation characteristics of TM polarized Gaussian beam, which is the rigorous solution of an eigenfunction problem for a confocal resonator, have been investigated using the nonparaxial vectorial moment theory of light beam propagation. The analytical expressions of the beam propagation factors are given by means of Fourier transform. Both the transversal second-order moment beam widths follow a simple hyperbolic variational law. For nonparaxial case, however, beam has different propagating features in the two transversal directions. As to paraxial case, its propagation approximately reduces to that of scalar Gaussian beam TEM₀₀ mode.