Fix an integer n > 0. For a multivariate function defined on a (not necessarily rectangular) lattice, an extension is constructed to have, ∀k ≤ n, derivatives of total degree k that are bounded by the function’s tensor product divided differences of total degree k times a constant independent of the lattice and the function. The extension is locally constructed, can have any prescribed smoothne...

Fix a non-negative integer g and a positive integer I dividing 2g − 2. For any Henselian, discretely valued field K whose residue field is perfect and admits a degree I cyclic extension, we construct a curve C/K of genus g and index I. This is obtained via a systematic analysis of local points on arithmetic surfaces with semistable reduction. Applications are discussed to the corresponding prob...

Let G be a finite abelian group. A number field K is called a Hilbert-Speiser field of type G if for every tame G-Galois extension L/K, the ring of integers OL is free as an OKG-module. If OL is free over the associated order AL/K for every G-Galois extension L/K, then K is called a Leopoldt field of type G. It is well-known (and easy to see) that if K is Leopoldt of type G, then K is Hilbert-S...

Let G be a linear algebraic group defined over a field k. We prove that, under mild assumptions on k and G, there exists a finite k-subgroup S of G such that the natural map H(K,S) −→ H(K,G) is surjective for every field extension K/k. We give two applications of this result in the case where k an algebraically closed field of characteristic zero and K/k is finitely generated. First we show tha...

Let K be a field and v a valuation on K. We discuss some properties of extensions w of v to the field K(X1, . . . , Xn) of rational functions in n variables over K, for which the residue field of w has transcendence degree n over the residue field of v. 2000 Mathematics Subject Classification: 12J20, 12F20. 1.Introduction LetK be a field and v a valuation onK. Residual transcendental extensions...

Recently Burger et al. (2012) and Xie et al. (2012) proposed to use a denoising autoencoder (DAE) for denoising noisy images. They showed that a plain, deep DAE can denoise noisy images as well as the conventional methods such as BM3D and KSVD. Both of them approached image denoising by denoising small, image patches of a larger image and combining them to form a clean image. In this setting, i...

Clustering is one of the widely used data mining techniques for medical diagnosis. Clustering can be considered as the most important unsupervised learning technique. Most of the clustering methods group data based on distance and few methods cluster data based on similarity. The clustering algorithms classify gene expression data into clusters and the functionally related genes are grouped tog...

The fuzzy c-means algorithm is a soft version of the popular k-means clustering. As is well known, the k-means method begins with an initial set of randomly selected exemplars and iteratively refines this set so as to decrease the sum of squared errors. The k-centers clustering is moderately sensitive to the initial selection of centers, so it is usually rerun many times with different initiali...

We investigate the problem of existence a bounded extension to $C(K)$ $c_0(I)$-valued operator $T$ defined on subalgebra induced by continuous increasing surjection $\phi :K\to L$, where $K$ and $L$ are compact lines. Gener

Let p be a prime and suppose that K/F is a cyclic extension of degree p with group G. Let J be the FpG-module K/K of pth-power classes. In our previous paper we established precise conditions for J to contain an indecomposable direct summand of dimension not a power of p. At most one such summand exists, and its dimension must be p +1 for some 0 ≤ i < n. We show that for all primes p and all 0 ...