In this paper, we survey recent results on Gauss-Bonnet-Chern formulae and related issues for closed Riemannian manifolds with variable curvature. Among other things, we address the following problem: “if M is an oriented 2n-dimensional closed manifold with non-positive curvature, then is it true that its Euler number χ(M) satisfies the inequality (−1)χ(M) ≥ 0?” We will present some partial ans...

In order to improve estimation accuracy of nonliear system with linear measurement model, simplified gauss hermite filter based on sparse grid gauss hermite quadrature (SGHF) is proposed. Comparing to conventional Gauss-Hermite filter (GHF) based on tensor product gauss quadrature rule, simplified SGHF not only maintains GHF’s advantage of precission controllable, high estimation accuracy, but ...

Gauss mixtures are a popular class of models in statistics and statistical signal processing because they can provide good £ts to smooth densities, because they have a rich theory, and because the can be well estimated by existing algorithms such as the EM algorithm. We here extend an information theortic extremal property for source coding from Gaussian sources to Gauss mixtures using high rat...

In section 4.2 of the paper, we compared the distribution modeling of GOG to other distributions. Below, we describe the details of the compared methods. The Mean, Cov and Gauss are global distribution descriptors of pixel features within each region. The Cov-ofCov, Cov-of-Gauss and GOG are hierarchical distribution descriptors. The Cov-of-Cov uses covariance matrix in both patch and region mod...

We take the Gauss’ linking integral of two curves as a starting point to discuss the connection between the equation of continuity and the inhomogeneous Maxwell equations. Gauss’ formula has been discussed before, as being derivable from the line integral of a magnetic field generated by a steady current flowing through a loop. We argue that a purely geometrical result such as Gauss’ formula ca...

Finite Element problems are often solved using multigrid techniques. The most time consuming part of multigrid is the iterative smoother, such as Gauss-Seidel. To improve performance, iterative smoothers can exploit parallelism, intra-iteration data reuse, and inter-iteration data reuse. Current methods for parallelizing Gauss-Seidel on irregular grids, such as multi-coloring and ownercomputes ...

A classical theorem in complex algebraic geometry states that, for any smooth projective variety, the Gauss map is finite; in particular, a smooth variety and its Gauss image have the same dimension (with the obvious exception of a linear space). Furthermore, even when the variety is not smooth, Zak proved a lower bound on the dimension of its Gauss image in terms of the dimension of its singul...

In this paper, we introduce a kind of character sum which simultaneously generalizes the classical Gauss and Jacobi sums, and show that this “Gauss-Jacobi sum” also specializes to the Kloosterman sum in a particular case. Using the connection to the Kloosterman sums, we obtain in some special cases the upper bound (the “Weil bound”) of the absolute values of the Gauss-Jacobi sums. We also discu...