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2. Fonctions de partitions a` parite´ pe´riodique
- Author
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Lahouar, Houda
- Subjects
- *
SET theory , *NUMBER theory , *MATHEMATICAL functions , *MATHEMATICS , *ALGEBRA - Abstract
Let
N be the set of positive integers andA a subset ofN . Forn∈N , letp(A,n) denote the number of partitions ofn with parts inA . In the paper J. Number Theory 73 (1998) 292, Nicolas et al. proved that, given anyN∈N andB⊂{1,2,…,N} , there is a unique setA=A0(B,N) , such thatp(A,n) is even forn>N . Soon after, Ben Saı¨d and Nicolas (Acta Arith. 106 (2003) 183) consideredσ(A,n)=∑d∣n,d∈Ad , and proved that for allk≥0 , the sequence(σ(A,2kn) mod 2k+1)n≥1 is periodic onn . In this paper, we generalise the above works for any formal power seriesf inF2[z] withf(0)=1 , by constructing a setA such that the generating functionfA ofA is congruent tof modulo 2, and by showing that iff=P/Q , whereP andQ are inF2[z] withP(0)=Q(0)=1 , then for allk≥0 the sequence(σ(A,2kn) mod 2k+1)n≥1 is periodic onn . [Copyright &y& Elsevier]- Published
- 2003
- Full Text
- View/download PDF
3. Algèbres graduées avec symétries
- Author
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Battikh, Naoufel
- Subjects
- *
ALGEBRA , *MATHEMATICAL symmetry , *DIFFERENTIAL forms , *HOMOLOGY theory , *MATHEMATICS , *MATHEMATICAL analysis - Abstract
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]
- Published
- 2011
- Full Text
- View/download PDF
4. Algèbres et cogèbres de Gerstenhaber et cohomologie de Chevalley–Harrison
- Author
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Aloulou, Walid, Arnal, Didier, and Chatbouri, Ridha
- Subjects
- *
MATHEMATICS , *MATHEMATICAL analysis , *ALGEBRA , *LINEAR algebra - Abstract
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]
- Published
- 2009
- Full Text
- View/download PDF
5. Générateurs de l'anneau des entiers d'une extension cyclotomique
- Author
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Ranieri, Gabriele
- Subjects
- *
NUMBER theory , *ALGEBRA , *ALGEBRAIC number theory , *ARITHMETIC functions - Abstract
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]
- Published
- 2008
- Full Text
- View/download PDF
6. Vecteurs unimodulaires et systèmes générateurs
- Author
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Ducos, Lionel
- Subjects
- *
MULTILINEAR algebra , *NOETHERIAN rings , *COMMUTATIVE rings , *ALGEBRA - Abstract
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]
- Published
- 2006
- Full Text
- View/download PDF
7. Spectre premier de <f>Oq(Mn(k))</f> image canonique et se´paration normale
- Author
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Cauchon, Gérard
- Subjects
- *
MATRICES (Mathematics) , *ALGEBRA - Abstract
Given any commutative field
k, denoteR=Oq(Mn(k)) the coordinate ring of quantumn×n matrices overk and assumeq is a nonzero element ink which is not a root of unity. Recall thatR is generated byn2 variablesXi,α ((i,α)∈⟦1,n〉2) subject (only) to the following relations:If is anyx y z t 2×2 sub-matrix ofX=(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 theR k -algebra generated by the same variablesXi,α subject to the same relations, except relations (b) which are replaced by: (c)tx=xt; so that is just the algebra of regular functions on some quantum affine space of dimensionR n2 overk. The theory of “derivative elimination” defines a natural embeddingϕ :Spec(R)→Spec( and asserts that:R ) In this paper, we give the precise description of the set- The “canonical image”
ϕ(Spec(R)) is a union of strataSpecw( (in the sense of [Goodearl, Letzter, in: CMS Conf. Proc., Vol. 22 (1998) 39–58]), whereR )w describes some subsetW ofP(⟦1,n〉2) .- The sets
Specw(R):=ϕ−1(Specw( R ))(w∈W) define the Goodearl–LetzterH -stratification ofSpec(R) in the sense of [Goodearl, Letzter, Trans. Amer. Math. Soc. 352 (2000) 1381–1403].W and we compute its cardinality. Using that description and the derivative elimination algorithm, we can verify (Theorems 6.3.1, 6.3.2) thatH -Spec(R) has anH -normal separation (in the sense of [Goodearl, in: Lecture Notes in Pure and Appl. Math. 210 (2000) 205–237]), so thatSpec(R) has normal separation (in the sense of [Brown, Goodearl, Trans. Amer. Math. Soc. 348 (1996) 2465–2502]). This property was conjectured by K. Brown and K. Goodearl. SinceR is Auslander–Regular and Cohen–Macaulay, this implies (by [Goodearl, Lenagan, J. Pure Appl. Algebra 111 (1996) 123–142]) thatR is catenary and satisfies the Tauvel''s height formula. [Copyright &y& Elsevier]- The “canonical image”
- Published
- 2003
- Full Text
- View/download PDF
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