Shifted genus expanded W∞\cal{W}_{\infty} algebra and shifted Hurwtiz numbers

We construct the shifted genus expanded $\cal{W}_{\infty}$ algebra, which is isomorphic to the central subalgebra $\cal{A}_{\infty}$ of infinite symmetric group algebra and to the shifted Schur symmetrical function algebra $\Lambda^\ast$ defined by A. Y. Okounkov and G. I. Olshanskii. As an application, we get some differential equations for the generating functions of the shifted Hurwitz numbers, thus we can express the generating functions in terms of the shifted genus expanded cut-and-join operators.Comment: 16 pages, no figure, any comments welcome

Genus Expanded Cut-and-Join operators and generalized Hurwtiz numbers

To distinguish the contributions to the generalized Hurwitz number of the source Riemann surface with different genus, we define the genus expanded cut-and-join operators by observing carefully the symplectic surgery and the gluing formulas of the relative GW-invariants. As an application, we get some differential equations for the generating functions of the generalized Hurwitz numbers for the source Riemann surface with different genus, thus we can express the generating functions in terms of the genus expanded cut-and-join operators.Comment: Any comments welcome, revised versio

Estimating Mean and Covariance Structure with Reweighted Least Squares

Does Reweighted Least Squares (RLS) perform better in small samples than maximum likelihood (ML) for mean and covariance structure? ML statistics in covariance structure analysis are based on the asymptotic normality assumption; however, actual applications of structural equation modeling (SEM) in social and behavioral science research usually involve small samples. It has been found that chi-square tests often incorrectly over-reject the null hypothesis: Σ=Σ(θ), because when sample is small the sample covariance matrix becomes ill-conditioned and entails unstable estimates. In certain SEM models, the vector of parameter must contain both means, variances and covariances. Yet, whether RLS also works in mean and covariance structure remains unexamined. This research is an extended examination of reweighted least squares in mean and covariance structure. Specifically, we replace biased covariance matrix in traditional GLS function (Browne, 1974) with the unbiased sample covariance matrix that derives from ML estimation. Moreover, under the assumption of multivariate normality, a Monte Carlo simulation study was carried out to examine the statistical performance as compared with ML methods in different sample sizes. Based on empirical rejection frequencies and empirical averages of test statistic, this study shows that RLS performs much better than ML in mean and covariance structure models when sample sizes are small

The number of ramified covering of a Riemann surface by Riemann surface

Interpreting the number of ramified covering of a Riemann surface by Riemann surfaces as the relative Gromov-Witten invariants and applying a gluing formula, we derive a recursive formula for the number of ramified covering of a Riemann surface by Riemann surface with elementary branch points and prescribed ramification type over a special point.Comment: LaTex, 14 page

Evolution equations for abstract differential operators

AbstractWe study in this paper the wellposedness and regularity of solutions of evolution equations associated with abstract differential operators on a Banach space. The results can be applied to many partial differential equations on different function spaces