23 results
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2. Supergénérix.
- Author
-
Poizat, Bruno
- Subjects
- *
STABILITY theory , *GROUP theory , *RANKING (Statistics) , *SET theory , *BOOLEAN functions , *MATHEMATICAL proofs - Abstract
Abstract: Given a subroup G of a stable (in the model-theoric sense) group Γ, in particular when Γ is a group of finite Morley rank, the traces on G of the definable subsets of Γ have a remarkable property: if the definable closure of G is connected, they are either supergeneric, or supergenerically complemented, in the sense of the definition given at the very beginning of this paper. An example of this situation is provided by the linear groups: for some n, G is a subgroup of , where K is a field that we may take algebraically closed; the definable sets in the sense of are its constructible subsets, i.e. the boolean combinations of a finite number of its Zariski closed subsets. For any group G, the supergeneric subsets of G form a filter of large sets, which, to my best knowledge, is defined here for the first time. This paper undertakes the study of supergenericity in a general context, with no hypotheses of a model-theoric nature, but with a special attention given to the very specific properties of genericity possessed by the definable subsets of a stable group. It can be read without any knowledge of Logic, provided that one is ready to skip the proofs of the theorems showing precisely that these definable sets have these properties. [Copyright &y& Elsevier]
- Published
- 2014
- Full Text
- View/download PDF
3. Algèbres de Hecke avec paramètres et représentations dʼun groupe p-adique classique : Préservation du spectre tempéré
- Author
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Heiermann, Volker
- Subjects
- *
IDENTITIES (Mathematics) , *ORTHOGONALIZATION , *SYMPLECTIC groups , *P-adic groups , *MATHEMATICAL forms , *SMOOTHNESS of functions , *REPRESENTATIONS of algebras - Abstract
Abstract: Let G be the identity component of an orthogonal or a symplectic p-adic group (not necessarily split) or an inner form of a general linear p-adic group. In a previous paper, it was shown that the Bernstein components of the category of smooth representations of G are equivalent to the category of right modules over some Hecke algebra with parameters, or more general over the semi-direct product of such an algebra with a finite group algebra. The aim of the present paper is to show that this equivalence preserves the tempered spectrum and the discrete series representations. [Copyright &y& Elsevier]
- Published
- 2012
- Full Text
- View/download PDF
4. Image des opérateurs dʼentrelacements normalisés et pôles des séries dʼEisenstein
- Author
-
Mœglin, C.
- Subjects
- *
EISENSTEIN series , *GROUP theory , *NUMBER theory , *OPERATOR theory , *IRREDUCIBLE polynomials , *HOMOLOGY theory , *REPRESENTATIONS of algebras , *HOLOMORPHIC functions , *AUTOMORPHIC functions - Abstract
Abstract: This paper is about the pole of some Eisenstein series for classical groups over a number field. In a previous paper, we have shown how to normalize intertwining operators in such a way that they are holomorphic for positive parameters. Here we show that the image of such operators is (in the interesting cases) either 0 or an irreducible representation. This enables us to compute explicitly the residue of the Eisenstein series obtained from square integrable cohomological representations. At the end of the paper we give necessary and sufficient conditions in terms of Arthurʼs data in order that a square integrable cohomological representation is cuspidal; the conditions are not totally satisfactory and we explain what we expect when Arthurʼs results will be fully available. [Copyright &y& Elsevier]
- Published
- 2011
- Full Text
- View/download PDF
5. 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
6. Récréations et mathématiques mondaines au XVIIIe siècle: le cas de Guyot.
- Author
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Belhoste, Bruno and Hazebrouck, Denise
- Abstract
In this paper, we study the most popular book of recreational mathematics published in the second half of the 18th century: The Nouvelles Récréations by Guyot. We indicate the motivations of the author, a simple postman, and the conditions which led him to write this book. We describe the spirit of the book and the public at which it aims. The success of the Nouvelles Récréations illustrates the rise of a science in polite society whose main goal is to amaze and amuse. Then, we examine the place of mathematics in this project and analyze the repertoire of problems and tricks. We focus on problems of combinatorics proposed by Guyot, like anagrams and card shuffles, which inspired some real mathematical work on the part of Monge and Gergonne. [ABSTRACT FROM AUTHOR]
- Published
- 2014
- Full Text
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7. Sous-groupes de et arbres.
- Author
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Bellaïche, Joël and Chenevier, Gaëtan
- Subjects
- *
SET theory , *TREE graphs , *GROUP theory , *FIXED point theory , *REPRESENTATION theory , *INVARIANTS (Mathematics) - Abstract
Abstract: In this paper, we first characterize the subsets of the Bruhat–Tits tree of , K a complete valued field, that are the sets of fixed points of a subgroup G of . When G is irreducible, the are the “strips” in the tree. We then evaluate the form of the strip , in particular in terms of algebraic invariants of the group G. We give two applications to representation theory, one about a new generalization of Ribet's lemma on extensions, the other about the trace-convergence of a sequence of representations. [Copyright &y& Elsevier]
- Published
- 2014
- Full Text
- View/download PDF
8. Existence de normes invariantes pour.
- Author
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De Ieso, Marco
- Subjects
- *
EXISTENCE theorems , *MATHEMATICAL symmetry , *LOGICAL prediction , *REPRESENTATION theory , *GROUP extensions (Mathematics) , *FINITE groups - Abstract
Abstract: In [8] Breuil and Schneider (2007) stated a conjecture on the equivalence between the existence of invariant norms on certain locally algebraic representations of and the existence of certain de Rham representations of , where F is a finite extension of . In this paper we prove that in the case and under some conditions, the existence of certain admissible filtrations implies the existence of invariant norms. [Copyright &y& Elsevier]
- Published
- 2013
- Full Text
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9. Isotropie dʼune forme bilinéaire dʼAlbert sur le corps de fonctions dʼune quadrique en caractéristique 2
- Author
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Laghribi, Ahmed
- Subjects
- *
BILINEAR forms , *QUADRICS , *ALGEBRAIC fields , *ISOTROPY subgroups , *CHARACTERISTIC functions , *MATHEMATICAL analysis - Abstract
Abstract: In this paper we give a complete answer to the isotropy of an Albert bilinear form over the function field of a quadric in characteristic 2. As a consequence, we complete the classification of nongood bilinear forms of height and degree 2 given in Laghribi and Rehmann (2009) . [Copyright &y& Elsevier]
- Published
- 2012
- Full Text
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10. 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
11. Sur les dimensions des anneaux gradués
- Author
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Ducos, Lionel
- Subjects
- *
GRADED rings , *KRULL rings , *COMMUTATIVE rings , *ITERATIVE methods (Mathematics) , *MONOIDS , *DIMENSIONAL analysis - Abstract
Abstract: In this paper, we investigate some basic classical results on the Krull dimension of -graded commutative rings by using the concept of iterated boundary ideals and iterated boundary monoids. After having pointed out the link between the Krull dimension (Kdim) and the boundary ideals and monoids, we characterize in the same way the graded dimension (dimgr). The main goal is to compare , and , where is a graded ring (noetherian or not). To do this, we establish various properties using the boundary ideals, in particular some typical boundary ideals inclusions. [ABSTRACT FROM AUTHOR]
- Published
- 2010
- Full Text
- View/download PDF
12. Bornes effectives des fonctions d'approximation des solutions formelles d'équations binomiales
- Author
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Rond, Guillaume
- Subjects
- *
ALGEBRAIC functions , *APPROXIMATION theory , *BINOMIAL equations , *ARTIN algebras , *EXPONENTIAL functions , *MATHEMATICAL analysis - Abstract
Abstract: The aim of this paper is to give an effective version of the Strong Artin Approximation Theorem for binomial equations. First we give an effective version of the Greenberg Approximation Theorem for polynomial equations, then using the Weierstrass Preparation Theorem, we apply this effective result to binomial equations. We prove that the Artin function of a system of binomial equations is bounded by a doubly exponential function in general and that it is bounded by an affine function if the order of the approximated solutions is bounded. [Copyright &y& Elsevier]
- Published
- 2010
- Full Text
- View/download PDF
13. Induction automorphe pour
- Author
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Henniart, Guy
- Subjects
- *
AUTOMORPHIC functions , *ISOMORPHISM (Mathematics) , *MODULES (Algebra) , *REPRESENTATIONS of groups (Algebra) , *WEIL group , *UNITARY operators , *P-adic fields , *GLOBAL analysis (Mathematics) - Abstract
Abstract: For or , isomorphism classes of irreducible -modules for are parametrized by n-dimensional representations of the Weil group of F. We can induce to a representation of , which has index 2 in . That gives a process of “automorphic induction” which to an irreducible -module τ for associates an irreducible -module for . In the present paper we show that if τ is unitary and generic then π is determined by τ, up to isomorphism, via a character identity entirely analogous to the character identity occurring in the automorphic induction process for p-adic fields. This completes the theory of automorphic induction for local and global representations of over number fields. [Copyright &y& Elsevier]
- Published
- 2010
- Full Text
- View/download PDF
14. Réflexivité d'une extension d'un opérateur sous-normal par un opérateur algébrique
- Author
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Benlarbi Delaï, M'hammed, El-Fallah, Omar, and Elhachimi, Khalid
- Subjects
- *
LINEAR operators , *HILBERT space , *INVARIANT subspaces , *POLYNOMIALS , *MATHEMATICAL decomposition , *SUBNORMAL operators , *MATHEMATICAL analysis - Abstract
Abstract: A bounded linear operator on a Hilbert space is said to be reflexive if the operators which leave invariant the invariant subspaces of T are wot-limits of polynomials in T. In this paper we give a necessary and sufficient condition for an extension of a subnormal operator by an algebraic one to be reflexive.We also give a formula for the reflexivity defect of such extensions. [Copyright &y& Elsevier]
- Published
- 2010
- Full Text
- View/download PDF
15. Sur la K-théorie du foncteur norme
- Author
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Karoubi, Max and Lambre, Thierry
- Subjects
- *
K-theory , *FUNCTOR theory , *RING theory , *MATHEMATICAL sequences , *MATHEMATICAL proofs , *GALOIS theory , *NUMBER theory - Abstract
Abstract: The K-theory of a functor may be viewed as a relative version of the K-theory of a ring. After its description, we prove a Mayer–Vietoris exact sequence in this framework. In the case of a Galois extension of a number field with rings of integers A, B respectively, this K-theory of the “norm functor” is an extension of a subgroup of the ideal class group of F by the Tate cohomology group . The Mayer–Vietoris exact sequence enables us to describe in a quite explicit way a quotient of the subgroup of the ideal class group , where N is the norm. We also prove a short exact sequence where is the group of semi-local units of F. Finally, we conclude this paper by applications of our methods to Number Theory. [Copyright &y& Elsevier]
- Published
- 2009
- Full Text
- View/download PDF
16. Torsion de groupes munis d'une donnée radicielle
- Author
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Hée, Jean-Yves
- Subjects
- *
FINITE groups , *CHEVALLEY groups , *LINEAR algebraic groups , *KAC-Moody algebras - Abstract
Abstract: We consider groups endowed with root data associated with non-necessarily finite root systems. We generalise to these groups the twisting methods of Chevalley groups initiated by Steinberg and Ree. The resulting theorem (proved in 1988) can be applied to Kac–Moody groups: see for instance two papers published by J. Ramagge in 1995 [J. Ramagge, On certain fixed point subgroups of affine Kac–Moody groups, J. Algebra 171 (2) (1995) 473–514; J. Ramagge, A realization of certain affine Kac–Moody groups of types II and III, J. Algebra 171 (3) (1995) 713–806]. [Copyright &y& Elsevier]
- Published
- 2008
- Full Text
- View/download PDF
17. 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
18. Formes homogènes de degré 3 et puissances p-ièmes
- Author
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Billerey, Nicolas
- Subjects
- *
DIOPHANTINE equations , *PRIME numbers , *ELLIPTIC curves , *NUMBER theory - Abstract
Abstract: In this paper, we are interested in diophantine equations of type where F is a separable homogeneous form of degree ⩾3 with integer coefficients, d a fixed integer ⩾1 and p a prime number ⩾7. As a consequence of the abc conjecture, if p is sufficiently large and is a nontrivial proper solution of the above equation, we have . In the case where F has degree 3, we associate to an elliptic curve defined over called the Frey curve or Hellegouarch–Frey curve. This allows us to deduce our conjecture from another one about elliptic curves attributed to G. Frey and B. Mazur (which is itself a consequence of the abc conjecture). We then applied our construction to the study of an explicit form. We give some results about the set of nontrivial proper solutions of the equation considered for several values of d. [Copyright &y& Elsevier]
- Published
- 2008
- Full Text
- View/download PDF
19. Cup i-produit sur les formes différentielles non commutatives et carrés de Steenrod
- Author
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Battikh, Naoufel
- Subjects
- *
DIFFERENTIAL geometry , *HOMOLOGY (Biology) , *MATHEMATICS , *MATHEMATICAL ability - Abstract
Abstract: In this paper, we define in the framework of noncommutative differential forms the notion of cup i-product introduced by Steenrod in [N.E. Steenrod, Product of cocycles and extensions of mappings, Ann. Math. 48 (2) (1947)]. This cup i-product permits to define an operator who will be a generalisation of the operator b defined in Hochschild and cyclic homology. With the operator we define in an explicitly way the Steenrod and Thomas–Pontrjagin squares. [Copyright &y& Elsevier]
- Published
- 2007
- Full Text
- View/download PDF
20. 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
21. Groupes satisfaisant une condition d'Engel
- Author
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Abdollahi, Alireza
- Subjects
- *
FINITE groups , *MODULES (Algebra) , *SOLVABLE groups , *GROUP theory - Abstract
Abstract: Let n be a positive integer. We say that a group G satisfies the condition , if every set of elements of G contains a pair such that , for some positive integer k. In this paper, we study finite groups G satisfying this condition. In particular, if G is a finitely generated soluble group, then , where is the hypercentre of G. [Copyright &y& Elsevier]
- Published
- 2005
- Full Text
- View/download PDF
22. De´finition abstraite d'un syste`me de racines dans le cas syme´trisable
- Author
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Bardy, Nicole
- Subjects
- *
AXIOMATIC set theory , *SET theory , *MATHEMATICS - Abstract
This paper presents an axiomatic description in a base free way of infinite root systems as those found in the study of Kac–Moody–Borcherds' algebras. In 1979, R. Moody [Adv. Math. 33 (1979) 144] posed the problem of finding such a description. In [Comm. Algebra 23 (1995) 4791], J.G. Bliss has already presented two answers to this question with his notions of “geometric root systems” and “rational root systems.” This new answer to Moody's question fits into the framework of the earlier axiomatic theory of the « syste`mes ge´ne´rateurs de racines » developed in [N. Bardy, Me´m. Soc. Math. Fr. (N.S.) 65 (1996)]. [Copyright &y& Elsevier]
- Published
- 2004
- Full Text
- View/download PDF
23. Spectre premier de <f>Oq(Mn(k))</f> image canonique et se´paration normale
- Author
-
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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