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.     

Equivalence Bimodule Between Non-Commutative Tori

The non-commutative torus C ^*(ⁿ,) is realized as the C*-algebra of sections of a locally trivial C*-algebra bundle over S with fibres isomorphic to C ^*n/S, ₁) for a totally skew multiplier ₁ on ⁿ/S. D. Poguntke [9] proved that A is stably isomorphic to C(S) C(^*( Zⁿ/S, ₁) C(S) A M_kl( C) for a simple non-commutative torus A and an integer kl. It is well-known that a stable isomorphism of two separable C*-algebras is equivalent to the existence of equivalence bimodule between them. We construct an A-C(S) A-equivalence bimodule.     

Limits of balayage measures

LetA be a subset of a balayage space (X,W) and a measure onX. It is shown that for every sequence _n of measures such that lim_nn and lim_{n n} ^A = the limit measure is of the formf+[(1-f)]^A for some (unique) Borel function 0f1_Cb(A). Furthermore, conditions are given such that any such functionf occurs.     

Implicit elliptic boundary-value problems with discontinuous nonlinearities

Let be a bounded domain in ⁿ (n3) having a smooth boundary, let be an essentially bounded real-valued function defined on × ^h, and let be a continuous real-valued function defined on a given subset Y of Y ^h. In this paper, the existence of strong solutions u W ^2,p (, ^h) W _o ^1,p (n/2<p<+) to the implicit elliptic equation (–u)=(x,u), with u=(u₁, u₂, ..., u_h) and u=(u ₁, u ₂, ..., u _h), is established. The abstract framework where the problem is placed is that of set-valued analysis.     

A graded scale of parametric families of distributions,and parameter estimates based on the sample mean

Summary Denote by _k a class of familiesP={P} of distributions on the line R¹ depending on a general scalar parameter , being an interval of R¹, and such that the moments µ₁()=xdP ,...,µ_2k ()=x ^2k dP are finite, ₁ (), ..., _k (), _k+1 () ..., _k () exist and are continuous, with ₁ () 0, and _j +1 ()= ₁ () _j () +[₂() -₁()²] _j ()/ ₁ (), J=2, ..., k. Let ₁=¯x=x ₁ + ... +x _n/n, ₂=x ₁ ² + ... +x _n ²/n, ..., _k =(x ₁ ^k + ... +x _n ^k/n denote the sample moments constructed for a sample x₁, ..., x_n from a population with distribution Pg. We prove that the estimator of the parameter by the method of moments determined from the equation ₁= ₁() and depending on the observations x₁, ..., x_n only via the sample mean ¯x is asymptotically admissible (and optimal) in the class _k of the estimators determined by the estimator equations of the form ₀ () + ₁ () ₁ + ... + _k () _k =0 if and only ifP _k .The asymptotic admissibility (respectively, optimality) means that the variance of the limit, as n (normal) distribution of an estimator normalized in a standard way is less than the same characteristic for any estimator in the class under consideration for at least one 9 (respectively, for every ).The scales arise of classes _{1 2}... of parametric families and of classes _{1 2} ... of estimators related so that the asymptotic admissibility of an estimator by the method of moments in the class _k is equivalent to the membership of the familyP in the class _k .The intersection consists only of the families of distributions with densities of the form h(x) exp {C₀() + C₁() x } when for the latter the problem of moments is definite, that is, there is no other family with the same moments ₁ (), ₂ (), ...Such scales in the problem of estimating the location parameter were predicted by Linnik about 20 years ago and were constructed by the author in [1] (see also [2, 3]) in exact, not asymptotic, formulation.Translated from Problemy Ustoichivosti Stokhasticheskikh Modelei, pp. 41–47, 1981.     

Fractal Characteristics of Measures; An Approach Via Function Spaces

Let be a positive Radon measure in R ⁿ with compact support . Let Q _jm be cubes with side-length 2^-j+1 originating from the canonical tiling of R ⁿ where j\in N ₀ and m\in Z ⁿ. If \in R, 0 < p \le , 0 < q \le , then _pq is the mixed _q – _p -quasi-norm of the sequence 2 ^j (Q _jm ). Quantities of this type are considered in fractal geometry (multifractal formalism) and in the theory of the function spaces B ^s _pq (R ⁿ) and F ^s _pq (R ⁿ). In Theorem 1 we deal with the question when _pq is an equivalent quasi-norm in some of these spaces (-property). If || = 0, then _S consists of those points (t,s) in the t–s-diagram in Figure 1 for which belongs to B ^s _p (R ⁿ) with pt = 1. Theorem 2 deals with the interrelation of S and _pq . Some applications to truncated Riesz potentials, Bessel potentials and Fourier transforms of are given.     

Littlewood-paley operators on the generalized Lipschitz spaces

Littlewood-Paley operators defined on a new kind of generalized Lipschitz spaces ₀ ^,p are studied. It is proved that the image of a function under the action of these operators is either equal to infinity almost everywhere or is in ₀ ^,p , where –n<<1 and 1<p<.     

Twisted derivations and distinct sets of lines that cover the same pairs of points

A partial projective plane of ordern consists of lines andn ² +n + 1 points such that every line hasn+1 points and distinct lines meet in a unique point. Suppose that two essentially different partial projective planes and of ordern, n a perfect square, that are defined on the same set of points cover the same pairs of points. For sufficiently largen we show that this implies that and have at leastn(n+1) lines. This bound is sharp and there exist essentially two different types of examples meeting the bound.As an application, we can show that derived planes provide an example for a pair of projective planes of square order with as much structure as possible in common, that is, as many lines as possible in common. Furthermore, we present a new method (twisted derivations) to obtain planes from one another by replacing the same number of lines as in a derivation.     

On the lower Markov spectrum

C. Hightower found two infinite sequences of gaps in the Markov spectrum, ( _n , _n ) and ( _n , _n ) with _n and _n both Markov elements, converging to . This paper exhibits Markov elements _n ^* and _n ^* such that, for alln 1, ( _n ^* , _n ) and ( _{n n} ^* ) are gaps in the Markov spectrum. Other results include showing that, for alln 1, _n is completely isolated, while the other endpoints of the gaps are limit points in the Markov spectrum.     

On the product of sign vectors and unit vectors

For a fixed unit vectora=(a ₁,...,a _n )S ^n-1, consider the 2 ⁿ sign vectors=(₁,..., _n ){±1{ ⁿ and the corresponding scalar products^·a = _n ⁱ⁼¹ = _i a _i . The question that we address is: for how many of the sign vectors must^.a lie between–1 and 1. Besides the straightforward interpretation in terms of the sums ±a ₂ , this question has appealing reformulations using the language of probability theory or of geometry.The natural conjectures are that at least 1/2 the sign vectors yield |.a|1 and at least 3/8 of the sign vectors yield |.a|<1 (the latter excluding the case when |a _i |=1 for somei). These conjectured lower bounds are easily seen to be the best possible. Here we prove a lower bound of 3/8 for both versions of the problem, thus completely solving the version with strict inequality. The main part of the proof is cast in a more general probabilistic framework: it establishes a sharp lower bound of 3/8 for the probability that |X+Y|<1, whereX andY are independent random variables, each having a symmetric distribution with variance 1/2.We also consider an asymptotic version of the question, wheren along a sequence of instances of the problem satisfying ||a||0. Our result, best expressed in probabilistic terms, is that the distribution of .a converges to the standard normal distribution, and in particular the fraction of sign vectors yielding .a between –1 and 1 tends to 68%.This research was supported in part by the Institute for Mathematics and its Applications with funds provided by the National Science Foundation.     

On some formalized conservation results in arithmetic

I _n andB _n are well known fragments of first-order arithmetic with induction and collection for _n formulas respectively;I _n ⁰ andB _n ⁰ are their second-order counterparts. RCA₀ is the well known fragment of second-order arithmetic with recursive comprehension;WKL ₀ isRCA ₀ plus weak König's lemma. We first strengthen Harrington's conservation result by showing thatWKL ₀ +B _n ⁰ is ₁ ¹ -conservative overRCA ₀ +B _n ⁰ . Then we develop some model theory inWKL ₀ and illustrate the use of formalized model theory by giving a relatively simple proof of the fact thatI ₁ provesB _n+1 to be _n+2-conservative overI _n . Finally, we present a proof-theoretic proof of the stronger fact that the _n+2 conservation result is provable already inI ₀ + superexp. ThusI _n+1 proves 1-Con (B _n+1) andI ₀ +superexp proves Con (I _n )Con(B _n+1).The first author was partially supported by NSF Grant #DCR-860615     

The critical exponent of degenerate parabolic systems

The Cauchy problemu _t=u +v ^p ,v _t =v +u ^q is studied, wherex R ^N , 0 <t < and ,,p andq, are positive exponents. It is proved that ifp,q 1 and 1 <pq < 1 + 2 max(p + ,q + )/n then every nontrivial non-negative solution is not global in time; whereaspq > 1 + 2 max(p + , q + )/n then there exist both positive global solutions and non-global solutions. In addition, the decaying in time of solutions tou _t,=u inR ⁿ × (0, ), an equation which occurs naturally in our study of above systems, is studied and solutions with the fastest decaying in time are constructed.     

On the Hantzsche-Wendt manifold

We explicitly compute the outer automorphism group Out ₁ M of the fundamental group of the Hantzsche — Wendt manifoldM. It is an extension 1(₂)³Out₁ MS ₃₂1, but not the semidirect product (₂)³(S ₃₂) as claimed in [3] (see also [4]). As a consequence, we get a quick algebraic computation of the symmetry groups of the Borromean rings and the figure-8-knot.     

Some generalizations of the classical moment problem

The article is devoted to two generalizations of the classical power moment problem, namely: 1) instead of representing the moment sequence by ⁿ , a representation by polynomialsP _n (), ¹, connected with a Jacobi matrix, appears; 2) in the representation, instead of ⁿ , the expression ⁿ figures, where is a real generalized function (i.e., we investigate some infinite-dimensional moment problem).The work is partially supported by the DFG, Project 436 UKR 113/39/0 and by the CRDF, Project UM1-2090.     

Continuous Quotients for Lattice Actions on Compact Spaces

Let < SL _n ( ) be a subgroup of finite index, where n 5. Suppose acts continuously on a manifold M, where ₁(M) = ⁿ , preserving a measure that is positive on open sets. Further assume that the induced action on H ¹(M) is non-trivial. We show there exists a finite index subgroup < and a equivariant continuous map : M ⁿ that induces an isomorphism on fundamental group. We prove more general results providing continuous quotients in cases where ₁(M) surjects onto a finitely generated torsion free nilpotent group. We also give some new examples of manifolds with actions.     

An Inequality for Tail Probabilities of Martingales with Bounded Differences

Let M _n =X₁+...+X_n be a martingale with bounded differences X_m=M_m-M_m-1 such that {|X_m| _m}=1 with some nonnegative _m. Write ²= ₁ ² + ... + _n ² . We prove the inequalities {M _nx}c(1-(x/)), {M _n x} 1- c(1- (-x/)) with a constant . The result yields sharp inequalities in some models related to the measure concentration phenomena.     

Produktsätze für Verfahren zur Limitierung von Doppelfolgen

{p _mn } - ₀₀>0, (1, 1) (1.1) (1.2). {s _mn } J _p - ( bJ _p -lims _mn =), (1.3) 0<x,y<1 p _s (, )/p(x, y) x, y 1-. {r _mn } - , (1.5) 0<, <1. N _rp - , (1.6). , bJ _p -lims _mn = bJ _q -lim(N _rps) _mn =. J _p - . , .     

Global Attractivity of the Equilibrium of a Nonlinear Difference Equation

The authors consider the nonlinear difference equation x_n+1=x_n+x_n-kf(x_n-k),n=0,1,(0.1) where (0,1), k 0,1, and f C¹[[0,), [0,)] with f(x) < 0.They give sufficient conditions for the unique positive equilibrium of (0.1) to be a global attractor of all positive solutions. The results here are somewhat easier to apply than those of other authors. An application to a model of blood cell production is given.     

Arithmetic of the Modular Function j1,8

Since the genus of the modular curve X_1 (8) = _1 (8) * is zero, we find a field generator j _1,8(z) = ₃(2z)/₃(4z) (₃(z) := _n ein ^2z ) such that the function field over X ₁(8) is (j _1,8). We apply this modular function j _1,8 to the construction of some class fields over an imaginary quadratic field K, and compute the minimal polynomial of the singular value of the Hauptmodul N(j _1,8) of (j _1,8).     

Compact interval spaces in which all closed subsets are homeomorphic to clopen ones,I

A topological space X whose topology is the order topology of some linear ordering on X, is called an interval space. A space in which every closed subspace is homeomorphic to a clopen subspace, is called a CO space. We regard linear orderings as topological spaces, by equipping them with their order topology. If L and K are linear orderings, then L ^*, L+K, L·K denote respectively the reverse orderings of L, the ordered sum of L and K and the lexicographic order on L×K (so ·2=+ and 2·=). Ordinals are considered as linear orderings, and cardinals are initial ordinals. For cardinals , 0, let L(, )= + 1 + ^* . Main theorem. Let X be a compact interval space. Then X is a CO space if and only if X is homeomorphic to a space of the form + 1 + _i L( _i , _i ), where is any ordinal, n, for every ii, _i are regular cardinals and _{i i}, and if n>0, then max({ _i: i}) · . This first part is devoted to show the following result. Theorem: If X is a compact interval CO space, then X is a scattered space (that means that every subspace of X has an isolated point).Supported by the Université Claude-Bernard (Lyon-1), the Ben Gurion University of the Negev, and the C.N.R.S.: UPR 9016Supported by the City of Lyon