this text carries out some ideas about exact and p− exact sequences of transformationsemigroups. some theorems like the short five lemma (lemma 1.3 and lemma 2.3) are valid here as inexact sequences of r − modules.

At the bottom of page 1614, we are not precise in the definition of a Borel space. The condition should have read that there is a one-to-one measurable function with measurable inverse between (Ω,B) and (E,E), where E is a Borel subset of the reals and E is the Borel σ-field of subsets of E. After the remaining corrections below, our use of the term “Borel space” conforms with this definition. ...

The goal of this short paper is to give a slightly different perspective on the comparison between crystalline cohomology and de Rham cohomology. Most notably, we reprove Berthelot’s comparison result without using pd-stratifications, linearisations, and pd-differential operators. Crystalline cohomology is a p-adic cohomology theory for varieties in characteristic p created by Berthelot [Ber74]...

This is a correction to the proof of Theorem 2.1 of [1]. An error occurred in the " proof " of Lemma 2.2, which is false. The following should replace the proof of (c) (Sect. II, p. 821ff.), which will correct the proof of the theorem: The initial condition is satisfied by the summand in (2.3) coming from the identity permutation id. So what we have to show is σ =id C r

Let L be an Archimedean Riesz space with a weak order unit u. A sufficient condition under which Dedekind [σ-]completeness of the principal ideal Au can be lifted to L is given (Lemma). This yields a concise proof of two theorems of Luxemburg and Zaanen concerning projection properties of C(X)-spaces. Similar results are obtained for the Riesz spaces Bn(T ), n = 1, 2, . . . , of all functions o...

1 Overview In the last lecture we took a more in depth look at Chernoff Bounds and introduced subgaussian and subexponential variables. In this lecture we will continue talking about subgaussian variables and related random variables – subexponential and subgamma, and finally we will give a proof of famous Johnson-Lindenstrauss lemma using property of subgaussian/subgamma variables. Definition ...

and Applied Analysis 3 where ∗ denotes the corresponding symmetric terms, Σ 11 = Q 1 + τQ 2 − PC − CP − 2L 1 U 1 − 2M 1 U 4 , Σ 22 = − (1 − μ)Q 1 − 2L 1 U 2 , Σ 33 = Q 3 + τQ 4 − 2U 1 , Σ 44 = − (1 − μ)Q 3 − 2U 2 , Σ 55 = U 3 K (]) − 2U 4 , L 1 = diag (l− 1 l + 1 , . . . , l − n l + n ) , L 2 = diag (l− 1 + l + 1 , . . . , l − n + l + n ) , M 1 = diag (m− 1 m + 1 , . . . , m − n m + n ) , M 2 =...

The main aim of this paper is to deal with a fuzzy version of Farkas lemma involving trapezoidal fuzzy numbers. In turns to that the fuzzy linear programming and duality theory on these problems can be used to provide a constructive proof for Farkas lemma. Keywords Farkas Lemma, Fuzzy Linear Programming, Duality, Ranking Functions.