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.     

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.     

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.     

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.     

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 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.     

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.     

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.     

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.     

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.     

Λ0-Nuclear Operators and Λ0-Nuclear Spaces in p-adic Analysis

For a Köthe sequence space, the classes of ₀-nuclear spaces and spaces with the ₀-property are introduced and studied and the relation between them is investigated. Also, we show that, for ₀c ₀, these classes of spaces are in general different from the corresponding ones for ₀=c ₀, which have been extensively studied in the non-archimedean literature (see, for example, [1]–[6]).     

Computational complexity of the integration problem for anisotropic classes

We determine the exact order of -complexity of the numerical integration problem for the anisotropic class W^r(I^d) and H^r(I^d) with respect to the worst case randomized methods and the average case deterministic methods. We prove this result by developing a decomposition technique of Borel measure on unit cube of d-dimensional Euclidean space. Moreover by the imbedding relationship between function classes we extend our results to the classes of functions W_p(I^d) and H_p(I^d). By the way we highlight some typical results and stress the importance of some open problems related to the complexity of numerical integration. Project supported by the fund of Personnel Division of Nankai University and the Program of One Hundred Distinguished Chinese Scientists of the Chinese Academy of Sciences.     

Implicitly representing arrangements of lines or segments

Anarrangement ofn lines (or line segments) in the plane is the partition of the plane defined by these objects. Such an arrangement consists ofO(n ²) regions, calledfaces. In this paper we study the problem of calculating and storing arrangementsimplicitly, using subquadratic space and preprocessing, so that, given any query pointp, we can calculate efficiently the face containingp. First, we consider the case of lines and show that with (n) space¹ and (n ^3/2) preprocessing time, we can answer face queries in (n)+O(K) time, whereK is the output size. (The query time is achieved with high probability.) In the process, we solve three interesting subproblems: (1) given a set ofn points, find a straight-edge spanning tree of these points such that any line intersects only a few edges of the tree, (2) given a simple polygonal path , form a data structure from which we can find the convex hull of any subpath of quickly, and (3) given a set of points, organize them so that the convex hull of their subset lying above a query line can be found quickly. Second, using random sampling, we give a tradeoff between increasing space and decreasing query time. Third, we extend our structure to report faces in an arrangement of line segments in (n ^1/3)+O(K) time, given(n ^4/3) space and (n ^5/3) preprocessing time. Lastly, we note that our techniques allow us to computem faces in an arrangement ofn lines in time (m ^2/3 n ^2/3+n), which is nearly optimal.The first author is pleased to acknowledge the support of Amoco Fnd. Fac. Dev. Comput. Sci. 1-6-44862 and National Science Foundation Grant CCR-8714565. Work on this paper by the fifth author has been supported by Office of Naval Research Grant N00014-87-K-0129, by National Science Foundation Grant NSF-DCR-83-20085, by grants from the Digital Equipment Corporation, and the IBM Corporation, and by a research grant from the NCRD—the Israeli National Council for Research and Development. The sixth author was supported in part by a National Science Foundation Graduate Fellowship. This work was begun while the non-DEC authors were visiting at the DEC Systems Research Center.     

Stability for Quadratic K1

The general quadratic group GQ _2n and its elementary subgroup EQ _2n are analogs in the theory of quadratic forms of the general linear group GL _n and its elementary subgroup E _n . This article proves that the stabilization map GQ _2n /EQ _2n GQ _2(n+1) /EQ _2(n+1) is an isomorphism whenever n +1 and S denotes the -stable rank of rings with anti-involution. As a corollary, a result is obtained which has been anticipated since the late 1960s: over rings of finite Bass–Serre dimension d, the stabilization map is an isomorphism whenever n d + 2.     

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   

Singular perturbed Markov chains and exact behaviors of simulated annealing processes

We study asymptotic properties of discrete and continuous time generalized simulated annealing processesX(·) by considering a class of singular perturbed Markov chains which are closely related to the large deviation of perturbed diffusion processes. Convergence ofX(t) in probability to a setS ₀ of desired states, e.g., the set of global minima, and in distribution to a probability concentrated onS ₀ are studied. The corresponding two critical constants denoted byd and withd are given explicitly. When the cooling schedule is of the formc/logt, X(t) converges weakly forc>0. Whether the weak limit depends onX(0) or concentrates onS ₀ is determined by the relation betweenc, d, and . Whenc>, the expression for the rate of convergence for each state is also derived.     

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     

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.     

Isomorphisms of Steinberg groups over commutative rings

LetA andR be commutative rings, andm andn be integers3. It is proved that, if :St _m (A)St _n (R) is an isomorphism, thenm=n. Whenn4, we have: (1) Every isomorphism :St _n(A)St _n(R) induces an isomorphism:E _n (A)E _n (R), and is uniquely determined by; (2) IfSt _n (A) St _n (R) thenK _2.n (A)K _2.n (R); (3) Every isomorphismE _n (A) E _n (R) can be lifted to an isomorphismSt _n(A)St _n(R); (4)St _n(A) St _n(R) if and only ifAR. For the casen=3, ifSt ₃(A) andSt ₃(R) are respectively central extensions ofE ₃(A) andE ₃ (R), then the above (1) and (2) hold.The Project supported by National Natural Science Foundation of China