We prove that every place P of an algebraic function field F |K of arbitrary characteristic admits local uniformization in a finite extension F of F. We show that F|F can be chosen to be Galois, after a finite purely inseparable extension of the ground field K. Instead of being Galois, the extension can also be chosen such that the induced extension FP |F P of the residue fields is purely insep...

We prove that every place P of an algebraic function field F |K of arbitrary characteristic admits local uniformization in a finite extension F of F . We show that F|F can be chosen to be Galois, after a finite purely inseparable extension of the ground field K. Instead of being Galois, the extension can also be chosen such that the induced extension FP |FP of the residue fields is purely insep...

In this paper, we study the Mordell-Weil group of an elliptic curve as a Galois module. We consider an elliptic curve E defined over a number field K whose Mordell-Weil rank over a Galois extension F is 1, 2 or 3. We show that E acquires a point (points) of infinite order over a field whose Galois group is one of Cn×Cm (n = 1, 2, 3, 4, 6, m = 1, 2), Dn×Cm (n = 2, 3, 4, 6, m = 1, 2), A4×Cm (m = ...

If a continued fraction K∞ n=1an/bn is known to converge but its limit is not easy to determine, it may be easier to use an extension of K∞ n=1an/bn to find the limit. By an extension of K ∞ n=1an/bn we mean a continued fraction K∞ n=1cn/dn whose odd or even part is K ∞ n=1an/bn. One can then possibly find the limit in one of three ways: (i) Prove the extension converges and find its limit; (ii...

This paper presents a new method, referred as Weight k − TSP , which generates simple and accurate decision rules that can be widely used for classifying gene expression data. The proposed method extends previous approaches: TSP and k−TSP algorithms by considering weight pairwise mRNA comparisons and percentage changes of gene expressions in different classes. Both rankings have been modified a...

Let ` be a prime number and let k be an algebraically closed field with char(k) 6= `. Let K/k be a function field of transcendence degree td(K/k) > 1 with absolute Galois group GK . Take G K = GK and for all i ≥ 1, let G (i+1) K = [G , G](G) ∞ . In other words, the groups {G K }i≥1 form the descending central `∞-series of GK . Then ΠK = G /G = Gal(K ′/K) is the Galois group of the maximal pro-`...

Let S be a compact surface with possibly non-empty boundary ∂S and let G be a graph. Let K be a subgraph of G embedded in S such that ∂S ⊆ K. An embedding extension of K to G is an embedding of G in S which coincides on K with the given embedding of K. Minimal obstructions for the existence of embedding extensions are classified in cases when S is the projective plane or the Möbius band (for se...

Let Gn be a double cover of either the alternating group An or the symmetric group Sn, and let Gn−1 be the corresponding double cover of An−1 or Sn−1. For every odd n ≥ 3 and every field k of characteristic 0, we prove that the following are equivalent: (i) there exists a generic extension for Gn−1 over k, (ii) there exists a generic extension for Gn over k. As a consequence, there exists a gen...

criticism of public extension services has resulted in various countries seeking alternative approaches for provision of these services. in all these approaches, extension personnel are the key factors to the success of the implementation. today, agricultural extension services in turkey bear the responsibility of the ministry of food, agriculture and livestock. a number of non-public extension...

Usually creative telescoping is used to derive recurrences for sums. In this article we show that the non-existence of a creative telescoping solution, and more generally, of a parameterized telescoping solution, proves algebraic independence of certain types of sums. Combining this fact with summation-theory shows transcendence of whole classes of sums. Parameterized telescoping Given f1(k), ....