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This paper presents the research of conditions for introducing "algebra prior to letter". In this perspective we study the potential of certain categories of generalized situations. How can they promote algebraic thinking in elementary and junior high school classes before the introduction of algebraic symbolism? In this article we show the advantage of introducing a praxeological analysis to these situations by analyzing a situation of generalization in a 6th grade class in Quebec (12-year-old students). [ABSTRACT FROM AUTHOR]

The algebra of noncommutative differential forms has been defined by A. Connes in [4]. Using this algebra, M. Karoubi has defined cyclic homology and Hochschild homology groups (see [13]). These groups are related to the algebraic K-theory. The purpose of this paper is to provide the noncommutative differential forms algebra with the structure of Gerstenhaber--Voronov algebras. [ABSTRACT FROM AUTHOR]

Let N be the set of positive integers and A a subset of N. For n∈N, let p(A,n) denote the number of partitions of n with parts in A. In the paper J. Number Theory 73 (1998) 292, Nicolas et al. proved that, given any N∈N and B⊂{1,2,…,N}, there is a unique set A=A0(B,N), such that p(A,n) is even for n>N. Soon after, Ben Saı¨d and Nicolas (Acta Arith. 106 (2003) 183) considered σ(A,n)=∑d∣n,d∈Ad, and proved that for all k≥0, the sequence (σ(A,2kn) mod 2k+1)n≥1 is periodic on n. In this paper, we generalise the above works for any formal power series f in F2[z] with f(0)=1, by constructing a set A such that the generating function fA of A is congruent to f modulo 2, and by showing that if f=P/Q, where P and Q are in F2[z] with P(0)=Q(0)=1, then for all k≥0 the sequence (σ(A,2kn) mod 2k+1)n≥1 is periodic on n. [Copyright &y& Elsevier]

Abstract: In this paper we define the notion of “graded algebra with symmetries”. This notion is a generalization of the extended differential forms. We prove that for a graded algebra with symmetries T, we associate a subalgebra which generalizes the noncommutative differential forms. Using this algebra , we can define the Hochschild and cyclic homologies, cup i-products and the Steenrod squares. [Copyright &y& Elsevier]

In [Breuil et Schneider, J. reine angew. Math. 610: 149–180, 2007] is formulated a conjecture on the equivalence of the existence of invariant norms on certain locally algebraic representations of GL d+1( L) and the existence of certain de Rham representations of Gal(/ L), where L is a finite extension of ℚ p. In this paper, we prove the “easy” direction of the conjecture: the existence of invariant norms implies the existence of admissible filtrations. [ABSTRACT FROM AUTHOR]

Abstract: The fundamental example of Gerstenhaber algebra is the space of polyvector fields on , equipped with the wedge product and the Schouten bracket. In this paper, we explicitely describe what is the enveloping algebra of a Gerstenhaber algebra . This structure gives us a definition of the Chevalley–Harrison cohomology operator for . We finally show the nontriviality of a Chevalley–Harrison cohomology group for a natural Gerstenhaber subalgebra in . [Copyright &y& Elsevier]

Abstract: Let p be an odd prime and , where m is a positive integer. Let be a qth primitive root of 1 and be the ring of integers in . In [I. Gaál, L. Robertson, Power integral bases in prime-power cyclotomic fields, J. Number Theory 120 (2006) 372–384] I. Gaál and L. Robertson show that if , where is the class number of , then if is a generator of (in other words ) either α is equals to a conjugate of an integer translate of or is an odd integer. In this paper we show that we can remove the hypothesis over . In other words we show that if is a generator of then either α is a conjugate of an integer translate of or is an odd integer. [Copyright &y& Elsevier]

This paper is the continuation of a precedent one (cf. [2]). We decompose a subclass of semi-simples nuclear algebras as direct sum of simples ideals and we give a Wedderburn decomposition for these algebras. [ABSTRACT FROM AUTHOR]

Abstract: This paper shows a mean (using multilinear alternating forms) of getting unimodular vectors in a module over a commutative ring, without noetheriannity hypothesis. We show an elementary approach of Serre''s splitting off theorem, Bass''s stable range and cancellation theorems and Forster–Swan''s theorem. [Copyright &y& Elsevier]

Abstract: In this paper, we use p-adic families of automorphic forms for a unitary group in three variables, containing some non-tempered forms constructed by Rogawski, to prove some cases of the Bloch–Kato conjectures. [Copyright &y& Elsevier]

Given any commutative field k, denote R=Oq(Mn(k)) the coordinate ring of quantum n×n matrices over k and assume q is a nonzero element in k which is not a root of unity. Recall that R is generated by n2 variables Xi,α ((i,α)∈⟦1,n〉2) subject (only) to the following relations:If xyzt is any 2×2 sub-matrix of X=(Xi,α), then: (a) yx=q−1xy, zx=q−1xz, tz=q−1zt, ty=q−1yt, zy=yz; (b) tx=xt−(q−q−1)yz.Denote R the k-algebra generated by the same variables Xi,α subject to the same relations, except relations (b) which are replaced by: (c) tx=xt; so that R is just the algebra of regular functions on some quantum affine space of dimension n2 over k.The theory of “derivative elimination” defines a natural embedding ϕ :Spec(R)→Spec(R) and asserts that: