An extension of abelian complexity, so called k-abelian complexity, has been considered recently in a number of articles. This paper considers two particular aspects of this extension: First, how much the complexity can increase when moving from a level k to the next one. Second, how much the complexity of a given word can fluctuate. For both questions we give optimal solutions.

The propositional ì-calculus is an extension of the modal system K with a least fixpoint operator. Kozen posed a question about completeness of the axiomatisation of the logic which is a small extension of the axiomatisation of the modal system K. It is shown that this axiomatisation is complete. §

We find a relationship between regular embeddings of G, an elementary abelian p-group of order p, into unipotent upper triangular matrices with entries in Fp and commutative dimension n degree 2 polynomial formal groups with nilpotent upper triangular structure matrices. We classify the latter when n = 3 up to linear isomorphism, and use that classification to determine the number of Hopf Galoi...

Let A be a Henselian discrete valuation ring with fractions K and with perfect residue field k of characteristic p > 0. Let G be a connected and reductive algebraic group over K, and let P be a parahoric group scheme over A with generic fiber P/K = G. The special fiber P/k is a linear algebraic group over k. If G splits over an unramified extension of K, we proved in some previous work that the...

— Let P (t) ∈ Q[t] be an irreducible quadratic polynomial and suppose that K is a quartic extension of Q containing the roots of P (t). Let NK/Q(x) be a full norm form for the extension K/Q. We show that the variety P (t) = NK/Q(x) 6= 0 satisfies the Hasse principle and weak approximation. The proof uses analytic methods.

RNA transcribed from the human epsilon globin gene was studied in K562 cells and human embryos of 5-10 weeks gestation. Using both primer extension and S1 analysis RNA molecules were found which extend 53 +/- 1 bp, 55 +/- 1 bp and 270 +/- 1 bp upstream from the first coding base and are colinear with the gene. The first pair of molecules represent transcripts initiating at the canonical cap sit...

dpH (G,EL) = dpH (G,Z/pZ)− dpH (G,Z/pZ) + dpH (G,Z/pZ). It is so crucial to find an upperbound for the p-rank dpH (G,EL) when Cl(L) is trivial. In this paper, we prove results about this rank in some special cases. More precisely, we compute this p-rank when L/K is an abelian unramified (also at infinity) p-extension whose Galois group can be generated by two elements. We also exhibit an explic...

‎In 1970‎, ‎Menegazzo [Gruppi nei quali ogni sottogruppo e intersezione di sottogruppi massimali‎, ‎ Atti Accad‎. ‎Naz‎. ‎Lincei Rend‎. ‎Cl‎. ‎Sci‎. ‎Fis‎. ‎Mat‎. ‎Natur. 48 (1970)‎, ‎559--562.] gave a complete description of the structure of soluble $IM$-groups‎, ‎i.e.‎, ‎groups in which every subgroup can be obtained as intersection of maximal subgroups‎. ‎A group $G$ is said to have the $FM$...

Working over a field k of characteristic zero, this paper studies algebraic actions SL2(k) on affine k-domains by defining and investigating fundamental pairs derivations. There are three main results: (1) The Structure Theorem for Fundamental Derivations (Theorem 3.4) describes the kernel derivation, together with its degree modules image ideals. (2) Classification 4.5) lists all normal SL2(k)...

The values of electronic components are always deviated, but the functions of the modern circuits are more and more precise, which makes the automatic fault diagnosis of analog circuits very complex and difficult. This paper presents an extension-neural-network-type-1ENN-1based method for fault diagnosis of analog circuits. This proposed method combines the extension theory and neural networks ...