Let K be a field and f(X) be a separable polynomial in K[X]. The Galois group of f(X) over K permutes the roots of f(X) in a splitting field, and labeling the roots as r1, . . . , rn provides an embedding of the Galois group into Sn. We recall without proof two theorems about this embedding. Theorem 1.1. Let f(X) ∈ K[X] be a separable polynomial of degree n. (a) If f(X) is irreducible in K[X] t...

In this paper, a novel manifold learning algorithm for face recognition and gender classification ‐ orthogonal nearest neighbour feature line embedding (ONNFLE) ‐ is proposed. Three of the drawbacks of the nearest feature space embedding (NFSE) method are solved: the extrapolation/interpolation error, high computational load and non‐orthogonal eigenvector problems....

In this paper, we propose a texture representation framework to map local texture patches into a low-dimensional texture subspace. In natural texture images, textons are entangled with multiple factors, such as rotation, scaling, viewpoint variation, illumination change, and non-rigid surface deformation. Mapping local texture patches into a low-dimensional subspace can alleviate or eliminate t...

We describe the $$\nu$$ -lines of curvature on an embedding double torus into $$\mathbb{R}^{4}$$ , defined as complete intersection a small sphere with polynomial hypersurface in where is gradient such polynomial. Through this analysis, we present description foliation lines embedding, image stereographic projection $$\mathbb{R}^{3}$$ .

There has been growing interest in kernel methods for classification, clustering and dimension reduction. For example, kernel Fisher discriminant analysis, spectral clustering and kernel principal component analysis are widely used in statistical learning and data mining applications. The empirical success of the kernel method is generally attributed to nonlinear feature mapping induced by the ...

We classify normal supersingular K3 surfaces Y with total Milnor number 20 in characteristic p, where p is an odd prime that does not divide the discriminant of the Dynkin type of the rational double points on Y .

The existing nonlinear local methods for dimensionality reduction yield impressive results in data embedding and manifold visualization. However, they also open up the problem of how to define a unified projection from new data to the embedded subspace constructed by the training samples. Thinking globally and fitting locally, we present a new linear embedding approach, called Locally Embedded ...

Spectral methods have recently emerged as a powerful tool for dimensionality reduction and manifold learning. These methods use information contained in the eigenvectors of a data affinity (i.e., item-item similarity) matrix to reveal the low dimensional structure in the high dimensional data. The most popular manifold learning algorithms include Locally Linear Embedding, ISOMAP, and Laplacian ...