Une variante à la Coxeter du groupe de Steinberg

LetG _n ()be the semi-direct product of the symmetric groupS _n by the Steinberg groupSt _n ()of a ringWe first prove thatG _n ()has a Coxeter-type presentation. The canonical morphism St _n () GL _n ()extends to a group homo G_n() GL _n ()We next determine the kernel of for n = We also give an expression for the generator of the algebraic K group K ₂(Z)of the integers in terms of permutation matrices.     

Dimension Theory and Nonstable K1 of Quadratic Modules

Employing Bak's dimension theory, we investigate the nonstable quadratic K-group K _1,2n (A, ) = G _2n (A, )/E _2n (A, ), n 3, where G _2n (A, ) denotes the general quadratic group of rank n over a form ring (A, ) and E _2n (A, ) its elementary subgroup. Considering form rings as a category with dimension in the sense of Bak, we obtain a dimension filtration G _2n (A, ) G _2n ⁰(A, ) ; G _2n ¹(A, ) ... E _2n (A, ) of the general quadratic group G _2n (A, ) such that G _2n (A, )/G _2n ⁰(A, ) is Abelian, G _2n ⁰(A, ) G _2n ¹(A, ) ... is a descending central series, and G _2n ^d(A)(A, ) = E _2n (A, ) whenever d(A) = (Bass–Serre dimension of A) is finite. In particular K _1,2n (A, ) is solvable when d(A) < .     

On Series of Translates of Positive Functions

For , a discrete infinite set of nonnegative real numbers, and a nonnegative measurable function f: R R ⁺, consider . The sets naturally break into two types. Type 1 consists of such that either C = R almost everywhere or else C = Ø a.e., for every f. Type 2 consists of all the other . We introduce a notion of asymptotic density for and the complementary notion of asymptotic lacunarity. We demonstrate that is of type 2 if it is asymptotically lacunary or else is asymptotically dense and exhibits asymptotically large Q-independent sets. We also give some examples of sets of both types.     

Stabilization theorem for the Milnor K2-functor

Let be an associative ring. For every natural number n there is a canonical homomorphism _n: K_2,n()K₂(), where K₂ is the Milnor functor and K_2,n() the associated unstable K-group. Dennis and Vasershtein have proved that if n is larger than the stable rank of , _n is an epimorphism. It is proved in the article that if n – 1 is greater than the stable rank of , the homomorphism _n is an isomorphism.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 64, pp. 131–152, 1976.     

Estimates for the minimal width of polytopes inscribed in convex bodies

The paper deals with the problem of approximating point sets byn-point subsets with respect to the minimal widthw. Let, in particular, ^d denote the family of all convex bodies in Euclideand-space, letA ^d and letn be an integer greater thand. Then we ask for the greatest number = _n (A) such that everyA A contains a polytope withn vertices which has minimal width at least w(A). We give bounds for _n ( ^d ), for _n (2133; ^d ), and for _n (W ^d ), where 2133; ^d ,W ^d denote the families of centrally symmetric convex bodies and of bodies of constant width, respectively.Dedicated to Professor L. danzer on the occasion of his sixtieth birthdayResearch for this paper was conducted in the academic year 1986/87 while both authors were visiting the University of Washington, Seattle. P. Gritzmann was supported by the Alexander von Humboldt Foundation.     

On the representation type of one point extensions of tame concealed algebras

Letk be an algebraically closed field and a finite dimensionalk-algebra. Letq be the quadratic Tits form associated with . If is tame we show thatq is weakly semipositive. Let be a one-point extension of a tame concealed algebra, then is tame iffq is weakly semipositive.     

Contraction of Exterior Powers in Characteristic 2 and the Spin Module

Let K be a field of characteristic 2 and letV be a vector space of dimension 2m over K. Let f be a non-degenerate alternating bilinear form defined on V × V. The symplectic group Sp(2m, K) acts on the exterior powers ^k V for 0 k. 2m There is a contraction map defined on the exterior algebra , which commutes with the Sp(2m, K) action and satisfies ² = 0 and ( ^k V) ^k–1 V We prove that ( ^k V)= ker ^k–1 V except when k=m+2. In the exceptional case, ( ^m+2 V) has codimension 2^m in ker ^m V and we show that the quotient module ker ^m V/ ^m+2 V is a spin module for Sp(2m,K). When K is algebraically closed, we show that this spin module occurs with multiplicity 1 in ^m V and multiplicity 0 in all other components of V.     

Harmonic analysis on tori

We consider measurable subsets {ofR}ⁿ with 0<m()<, and we assume that has a spectral set . (In the special case when is also assumed open, may be obtained as the joint spectrum of a family of commuting self-adjoint operators {H _k: 1kn} in L ² () such that each H _k is an extension of i(/x _k) on C _c (), k=1, ..., n.)It is known that is a fundamental domain for a lattice if is itself a lattice. In this paper, we consider a class of examples where is not assumed to be a lattice. Instead is assumed to have a certain inhomogeneous form, and we prove a necessary and sufficient condition for to be a fundamental domain for some lattice in {ofR}ⁿ. We are thus able to decide the question, fundamental domain or not, by considering only properties of the spectrum . Our criterion is obtained as a corollary to a theorem concerning partitions of sets which have a spectrum of inhomogeneous form.Work supported in part by the NSF.Work supported in part by the NSRC, Denmark.     

Folgenräume als universelle Generatoren für Klassen lokalkonvexer Räume

Motivated by the known characterizations of equicontinuity in the dual of a Schwartz space, a nuclear space, or a strongly nuclear space,we introduce the concepts of a -sequence and of a ()-sequence in the dual of an arbitrary lcs [E,], and we investigate the corresponding topologies and ₍₎ on E of uniform convergence on these sequences. Here is a normal sequence space such that . Under favorable enough conditions on , including the nuclearity of its normal topology , [,] acts as a universal generator for those lcs [E,] which satisfy =. Under somewhat weaker assumptions on , [,₍₎] is a universal generator for the lcs [E,] with =₍₎. These results cover e.g. the cases of -nuclear spaces and of nuclear spaces known from the recent literature. As an application we show that every non-trivial ultrabornological lcs is representable as an inductive limit of isomorphic copies of [, ( , )], where is any nuclear power series space of infinite type with stable exponent sequence.     

Isomorphism of one-place functors Ext

Let be an associative ring with identity. One considers the category of left (unitary) -modules m and also the contravariant and the covariant functors Ext ¹ ( ,A) and Ext ¹ (A, ): M_z M. One proves the following results: (1) If the homomorphism of -modules A B induces an isomorphism Ext ¹ ( ,A)Ext ¹ ( ,B), then there exist injective -modules J₁ and J₂ such that AJ₁BJ₂. (2) Every functorial morphism Ext ¹ ( ,A)Ext ¹ ( ,B) induces a certain homomorphism of -modules AB. One also obtains a dual result.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 112, pp. 71–74, 1981.     

Symmetric balanced ternary designs with ϱ1 = 1 or 2

Summary We prove the following two non-existence theorems for symmetric balanced ternary designs. If ₁ = 1 and 0 (mod 4) then eitherV = + 1 or 4₂ – + 1 is a square and (4₂ – + 1) divides ² – 1. If ₁ = 2 thenV = ((m + 1)/2) ² + 2,K = (m ² + 7)/4 and = ((m – 1)/2)² + 1 wherem 3 (mod 4). An example belonging to the latter series withV = 18 is constructed.     

Almost-Periodic Multipliers

In this work we study the necessary and sufficient conditions for a generalized trigonometric series in order for it to be the series of a Stepanoff almost-periodic function fS ^q (R),1q<. We consider analogous conditions for functions belonging to D(,R). Finally, we characterize the multipliers of invariance of the (B ¹(),B ¹()) type.     

Subnormalizer of net subgroups in the general linear group over a ring

Let be a commutative ring in which the elements of the form ²–1, * generate the unit ideal and assume that a is any D-net of ideals of of order n. It is shown that the normalizerN() of the net subgroup G() (RZhMat, 1977, 2A280) coincides with its subnormalizer in GL(n, ). For noncommutative the corresponding result is obtained under the assumptions: 1) in the elements of the form — 1, where runs through all invertible elements of the center of , generate the unit ideal, and 2) the subgroup G() contains the group of block diagonal matrices with blocks of order 2.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 116, pp. 14–19, 1982.     

Surgery on the Novikov complex

Let M be a closed connected smooth manifold with dim M=n6, and : ₁(M) Z be an epimorphism. Denote by the group ring of ₁(M) and let ^– be its Novikov completion. Let D _* be a free-based finitely generated chain complex over ^– . Assume that D _ii=0 for i1 and in–1 and that D _* has the same simple homotopy type as the Novikov-completed simplicial chain complex of the universal covering M. Let N be an integer. We prove that D _* can be realized, up to the terms of ^– of degree N as the Novikov complex of a Morse map : M S ¹, belonging to . Applications to Arnold's conjectures and to the theory of fibering of M over S ¹ are given.     

The Exterior Cube L-Function for GL(6)

We construct a Rankin Selberg integral to represent the exterior cube L function L(,³,s) of an automorphic cuspidal module of GL₆( _F ) (where F is a number field). We determine the poles of this L function and find period conditions for the special value L(,³,1/2). We use the Siegal Weil formula. We also state an analogue of the Gross–Prasad conjecture concerning a criterion for the nonvanishing of L(,³,1/2).     

On extremal perturbations of semi-Fredholm operators

Given a bounded linear operatorA in an infinite dimensional Banach space and a compact subset of a connected component of its semi-Fredholm domain, we construct a finite rank operatorF such that –A+F is bounded below (or surjective) for each ,F ²=0 and rankF=max min{dimN(–A), codimR(–A)}, if ind(–A)0 (or ind(–A)0, respectively) for each .     

Perversité et variation

Let S be the spectrum of a strictly henselian discrete valuation ring with residue characteristic p and =/, where is a prime number p and is an integer 1. For a scheme X of finite type over S and smooth over S along the special fiber X _s outside a closed point x, we study the vanishing cycles complex R() and the tame variation , for in the tame inertia group I _t . In particular, we show that if X is regular, flat over S of relative dimension n1, and is a topological generator of I _t , then R ^q () _x =0 for qn and is an isomorphism. Mathematics Subject Classification (2000):14F20, 14D05, 14D06     

A family of noetherian rings with their finite length modules under control

We investigate the category mod of finite length modules over the ring =A _k , where is a V-ring, i.e. a ring for which every simple module is injective, k a subfield of its centre and A an elementary k-algebra. Each simple module E _j gives rise to a quasiprogenerator P _j = A E _j . By a result of K. Fuller, P _j induces a category equivalence from which we deduce that mod _j mod EndP _j . As a consequence we can(1) construct for each elementary k-algebra A over a finite field k a nonartinian noetherian ring such that modA mod(2) find twisted versions of algebras of wild representation type such that itself is of finite or tame representation type (in mod)(3) describe for certain rings the minimal almost split morphisms in mod and observe that almost all of these maps are not almost split in Mod.     

A note on embedding of classes H ω

Among others, we extend an interesting theorem of M. V. Medvedeva pertaining to the embedding relation H BV, where BV denotes the set of the functions of -bounded variation.     

Multipliers on the Set of Rademacher Series in Symmetric Spaces

Let E be a symmetric space on [0,1]. Let (,E) be the space of measurable functions f such that fg E for every almost everywhere convergent series g=b _n r _n E, where (r _n) are the Rademacher functions. It was shown that, for a broad class of spaces E, the space (,E) is not order isomorphic to a symmetric space, and we study the conditions under which such an isomorphism exists. We give conditions on E for (,E) to be order isomorphic to L . This includes some classes of Lorentz and Marcinkiewicz spaces. We also study the conditions under which (,E) is order isomorphic to a symmetric space that differs from L . The answer is positive for the Orlicz spaces E=L _q with _q(t)=exp|t|^q-1 and 0相似文献