We compute the string production rate at the end of inflation, using the string spectrum obtained in [M. Li, W. Song, Y. Song, Quantizing strings in de Sitter space, JHEP 0704 (2007) 042, hep-th/0701258] in a near-de Sitter space. Our result shows that highly excited strings are hardly produced, thus the simple slow-roll inflation alone does not offer a cosmic string production mechanism.

In the analysis of cluster data the regression coefficients are frequently assumed to be the same across all clusters. This hampers the ability to study the varying impacts of factors on each cluster. In this paper, a semiparametric model is introduced to account for varying impacts of factors over clusters by using cluster-level covariates. It achieves the parsimony of parametrization and allows the explorations of nonlinear interactions. The random effect in the semiparametric model accounts also for within cluster correlation. Local linear based estimation procedure is proposed for estimating functional coefficients, residual variance, and within cluster correlation matrix. The asymptotic properties of the proposed estimators are established and the method for constructing simultaneous confidence bands are proposed and studied. In addition, relevant hypothesis testing problems are addressed. Simulation studies are

We show how two topologically distinct spaces-the Kahler K3 x T^ 2 and the non-Kahler T^ 2 bundle over K3-can be smoothly connected in heterotic string theory. The transition occurs when the base K3 is deformed to the T^ 4/Z_2 orbifold limit. The orbifold theory can be mapped via duality to M-theory on K3 x K3 where the transition corresponds to an exchange of the two K3's.

We show that the logarithmic derivatives of the convolution heat kernels on a uni-modular Lie group are exponentially integrable. This result is then used to prove an “integrated” Harnack inequality for these heat kernels. It is shown that this integrated Harnack inequality is equivalent to a version of Wang’s Harnack inequality. (A key feature of all of these inequalities is that they are dimension independent.) Finally, we show these inequalities imply quasi-invariance properties of heat kernel measures for two classes of infinite dimensional “Lie” groups.

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