This paper introduces a Projected Principal Component Analysis (Projected-PCA), which employees principal component analysis to the projected (smoothed) data matrix onto a given linear space spanned by covariates. When it applies to high-dimensional factor analysis, the projection removes noise components. We show that the unobserved latent factors can be more accurately estimated than the conventional PCA if the projection is genuine, or more precisely, when the factor loading matrices are related to the projected linear space. When the dimensionality is large, the factors can be estimated accurately even when the sample size is finite. We propose a flexible semi-parametric factor model, which decomposes the factor loading matrix into the component that can be explained by subject-specific covariates and the orthogonal residual component. The covariates effects on the factor loadings are further

Sflash had recently been broken by Dubois, Stern, Shamir, etc., using a differential attack on the public key. The C^{*-} signature schemes are hence no longer practical. In this paper, we will study the new attack from the point view of symmetry, then (1) present a simple concept (projection) to modify several multivariate schemes to resist the new attacks; (2) demonstrate with practical examples that this simple method could work well; and (3) show that the same discussion of attack-and-defence applies to other big-field multivariates. The speed of encryption schemes is not affected, and we can still have a big-field multivariate signatures resisting the new differential attacks with speeds comparable to Sflash.

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We discuss a general parametrization for vertices of quantum graphs and show, in particular, how the 's and 'couplings at an n-edge vertex can be approximated by means of n+ 1 couplings of the type provided the latter are properly scaled.

We derive asymptotic expansion for the spectrum of Hamiltonians with a strong attractive interaction supported by a smooth surface in R 3, either infinite and asymptotically planar, or compact and closed. Its second term is found to be determined by a Schrdinger type operator with an effective potential expressed in terms of the interaction support curvatures.