Given a domain Ω⊂ℂ^{$n$}, the Lempert function is a functional on the space $\text{Hol}(\mathbb{D},\Omega)$ of analytic disks with values in Ω, depending on a set of poles in Ω. We generalize its definition to the case where poles have multiplicities given by local indicators (in the sense of Rashkovskii) to obtain a function which still dominates the corresponding Green function, behaves relatively well under limits, and is monotonic with respect to the local indicators. In particular, this is an improvement over the previous generalization used by the same authors to find an example of a set of poles in the bidisk so that the (usual) Green and Lempert functions differ.

Integral simplicial volume is a homotopy invariant of oriented closed connected manifolds, defined as the minimal weighted number of singular simplices needed to represent the fundamental class with integral coefficients. We show that odd-dimensional spheres are the only manifolds with integral simplicial volume equal to 1. Consequently, we obtain an elementary proof that, in general, the integral simplicial volume of (triangulated) manifolds is not computable.

In this paper, we give a condition on the bicharacteristic which guarantees the global existence of the mild solution to the Boltzmann equation with an external force for the hard-sphere model and potentials with angular cutoff in infinite vacuum. This generalizes the previous results to the case when the force can have arbitrary strength. The constructive condition on the bicharacteristic is used to obtain the pointwise estimates on the collision operator so that the global existence comes from the contraction mapping theorem.

A self-similar measure on R^n is defined to be a probability measure satisfying \mu = \sum_{j=1}^N p_j\mu\circ S_j^{-1} + \sum_{j=1}^M q_j(\mu*\mu)\circ T_j^{-1}, where S_j x = \rho_j R_j x + b_j, T_j x = \eta_j Q_j x + c_j are contractive similarities, 0 < \rho_j < 1, 0 < \eta_j < 1/2, 0 < p_j < 1, 0 < q_j < 1; \sum_{j=1}^N p_j + \sum_{j=1}^M q_j = 1, R_j, Q_j are orthogonal matrix and \mu*\mu is the convolution of two measures. When M = 0, \mu is a linear self-similar measure, we establish the asymptotic behavior of averages of the derivative of the Fourier transform of \mu, such as \int_{|x|\le R} |(\frac{\partial}{\partial x})^\alpha \hat\mu(x)|^2 dx = O(R^{n-\beta}) for any order derivation of \hat\mu(x) as R\to\infty under certain additional hypotheses. When M > 0, \mu is a nonlinear self-similar measure, we get some results of L^p boundedness for maximal operators of l, from the pointwise asymptotic estimate of the Fourier transform of \mu made by Strichartz.

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