Présentations des groupes de Fischer (II)

In this paper we give Coxeter presentation (X, ) for the three Fischer groupsG=Fi₂₂, Fi₂₃, Fi₂₄; we apply methods exposed in the first part. Each of these groups is generated by a class of 3-transpositions (named here a Fischer class) in which elements ofX are chosen. A subset of is the set of all the relations (xy) ^m(x,y)=1, wherex andy are inX and wherem(x,y) means the order ofxy inG. We obtainG as a specified quotient of the Coxeter group (X, ) with the appropriate diagram .     

Eine ErweiterungT(V′) des Ordinalzahlensystems 58-0158-0158-01(Λ0) von G. Jäger

Summary This paper gives a recursive generalization of a strong notation system of ordinals, which was devellopped by Jäger [3]. The generalized systemT(V) is based on a hierarchy of Veblen-functions for inaccessible ordinals. The definition ofT(V) assumes the existence of a weak Mahlo-ordinal. The wellordering ofT(V) is provable in a formal system of second order arithmetic with the axiom schema of ₂ ¹ -comprehension in a similar way, as it is proved in [6] for the weaker notation systemT(V).     

Numbers of Success-Runs of Specified Length Until Certain Stopping Time Rules and Generalized Binomial Distributions of Order k

A new distribution called a generalized binomial distribution of order k is defined and some properties are investigated. A class of enumeration schemes for success-runs of a specified length including non-overlapping and overlapping enumeration schemes is rigorously studied. For each nonnegative integer less than the specified length of the runs, an enumeration scheme called -overlapping way of counting is defined. Let k and be positive integers satisfying < k. Based on independent Bernoulli trials, it is shown that the number of (– 1)-overlapping occurrences of success-run of length k until the n-th overlapping occurrence of success-run of length follows the generalized binomial distribution of order (k–). In particular, the number of non-overlapping occurrences of success-run of length k until the n-th success follows the generalized binomial distribution of order (k– 1). The distribution remains unchanged essentially even if the underlying sequence is changed from the sequence of independent Bernoulli trials to a dependent sequence such as higher order Markov dependent trials. A practical example of the generalized binomial distribution of order k is also given.     

Estimates for a Group of Deviations in Generalized Hölder Metric

We present order relations for a group of deviations of a function f(·) H in terms of partial Fourier sums of this function in a generalized Hölder metric defined in a generalized Hölder space H _* H .     

Admissibility of unbiased tests for a composite hypothesis with a restricted alternative

This paper discusses -admissiblility and d-admissiblity which are important concepts in studying the performance of statistical tests for composite hypotheses. A sufficient condition for -admissibility is presented. When =1/m, the Nomakuchi-Sakata test, which is uniformly more powerful than the likelihood ratio test for hypotheses min (₁, ₁) = 0 versus min (₁, ₁) > 0, is generalized for a class of distributions in an exponential family, and its unbiasedness and -admissibility are shown. Finally, the case of 1/m is discussed in brief.     

On Negative Orbits of Finite Coxeter Groups

For a Coxeter group W, X a subset of W and a positive root, we define the negative orbit of under X to be {w · | w X} ^–, where ^– is the set of negative roots. Here we investigate the sizes of such sets as varies in the case when W is a finite Coxeter group and X is a conjugacy class of W.     

Superstable implicit Runge-Kutta methods for second order initial value problems

Using the well known properties of thes-stage implicit Runge-Kutta methods for first order differential equations, single step methods of arbitrary order can be obtained for the direct integration of the general second order initial value problemsy=f(x, y, y),y(x _o)=y _o,y(x _o)=y _o. These methods when applied to the test equationy+2y+ ² y=0, ,0, +>0, are superstable with the exception of a finite number of isolated values ofh. These methods can be successfully used for solving singular perturbation problems for which f/y and/or f/y are negative and large. Numerical results demonstrate the efficiency of these methods.     

On Non-Negative Elliptic-Type Differential Inequalities in ℝ n , Vanishing Almost Everywhere on Some Open Subset in ℝ n

It is proved in this article that any generalized solution of a sufficiently general class of elliptic-type differential inequalities in  ⁿ that is non-negative almost everywhere in  ⁿ and vanishes almost everywhere on an open set ⁿ is trivial in  ⁿ .     

On cyclic biquadratic fields related to a problem of Hasse

In this note we shall prove that there exist infinitely many cyclic biquadratic fieldsK whose integral bases are neither {1, , ², } nor {1, , , ³) for any numbers , inK. Next, we shall construct infinitely many cyclic biquadratic fieldsK which have the index 1, but still have not the integral basis {1, , ², ³) for every inK. Finally we shall give a class of biquadratic fields for a problem of Hasse concerning an integral basis.     

The Laplacian with Robin Boundary Conditions on Arbitrary Domains

Using a capacity approach, we prove in this article that it is always possible to define a realization of the Laplacian on L ²() with generalized Robin boundary conditions where is an arbitrary open subset of R ⁿ and is a Borel measure on the boundary of . This operator generates a sub-Markovian C ₀-semigroup on L ²(). If d=d where is a strictly positive bounded Borel measurable function defined on the boundary and the (n–1)-dimensional Hausdorff measure on , we show that the semigroup generated by the Laplacian with Robin boundary conditions has always Gaussian estimates with modified exponents. We also obtain that the spectrum of the Laplacian with Robin boundary conditions in L ^p () is independent of p[1,). Our approach constitutes an alternative way to Daners who considers the (n–1)-dimensional Hausdorff measure on the boundary. In particular, it allows us to construct a conterexample disproving Daners' closability conjecture.     

Sur les égalités du miroir et certaines formes faibles de la conjecture de Greenberg

Let p be an odd prime. For any CM number field K containing a primitive p^th-root of unity, class field theory and Kummer theory put together yield the well known reflection inequality ^+– between the plus and minus parts of the -invariant of K. Greenbergs conjecture asserts that ⁺ is always trivial. We study here a weak form of this conjecture, namely ⁺=^– if and only if ⁺=^–=0.     

On a generalized Eulerian distribution

The distribution with probability function p _k(n, , ) = A _{n, k}(, )/(+ )^[p], k = 0, 1, 2, ..., n, where the parameters and are positive real numbers, A _{n, k} (, ) is the generalized Eulerian number and ( + )^[n] = ( + )( + +1) ... ( + +n – 1), introduced and discussed by Janardan (1988, Ann. Inst. Statist. Math., 40, 439–450), is further studied. The probability generating function of the generalized Eulerian distribution is expressed by a generalized Eulerian polynomial which, when expanded suitably, provides the factorial moments in closed form in terms of non-central Stirling numbers. Further, it is shown that the generalized Eulerian distribution is unimodal and asymptotically normal.     

Darstellung einer Ordnung von Maßen durch Stoppzeiten

Summary Given a stochastic matrixP on the state spaceI an ordering for measures inI can be defined in the following way: iff(f)(f) for allf in a sufficiently rich subcone of the cone of positiveP-subharmonic functions. It is shown that, if, are probability measures with , then in theP-process (X _n)_n0 having as initial distribution there exists a stopping time such thatX is distributed according to. In addition, can be chosen in such a way, that for every positive subharmonicf with(f)< the submartingale (f(X _n))_n0 is uniformly integrable.     

On generalized gamma functions related to the Laurent coefficients of the Riemann zeta function

Summary We study a class of generalized gamma functions _k (z) which relate to the generalized Euler constants _k (basically the Laurent coefficients of(s)) as (z) does to the Euler constant. A new series expansion for _k is derived, and the constant term in the asymptotic expansion for log _k (z) is studied in detail. These and related constants are numerically computed for 1 k 15.     

Divisor Functions in the Ring of Gaussian Integers Weighted by the Generalized Klosterman Sum

Let _a,b,c () be the number of factorizations of a Gaussian number in the form = ₁ ^a ₂ ^b ₃ ^c , where a, b, and c are natural numbers. In the ring of Gaussian numbers, we construct an asymptotic formula for a summatory function of _a,b,c () weighted by the generalized Klosterman function.     

Variational problems in SBV and image segmentation

We show how it is possible to prove the existence of solutions of the Mumford-Shah image segmentation functional F(u,K) = _\K [u² + (u – g)²]dx + ^{n – 1}(K), u W ^1,2(\K), K closed in .We use a weak formulation of the minimum problem in a special class SBV() of functions of bounded variation. Moreover, we also deal with the regularity of minimizers and the approximation of F by elliptic functionals defined on Sobolev spaces. In this paper, we have collected the main results of Ambrosio and others.     

The core of a chain complete poset with no one-way infinite fence and no tower

Like dismantling for finite posets, a perfect sequence = P : of a chain complete posetP represents a canonical procedure to produce a coreP . It has been proved that if the posetP contains no infinite antichain then this coreP is a retract ofP andP has the fixed point property iffP has this property. In this paper the condition of having no infinite antichain is replaced by a weaker one. We show that the same conclusion holds under the assumption thatP does not contain a one-way infinite fence or a tower.Supported by a grant from The National Natural Science Foundation of China.     

On summability of subsequences of partial sums of trigonometric Fourier series

n- (n1) fL ^p ([–, ] ⁿ ),=1 = (L C) . , , f([–, ] ⁿ ).     

On contiguity of probability measures corresponding to semimartingales

, (ⁿ), - (P ⁿ ), P ⁿ (A ⁿ )>0P ⁿ (A ⁿ )0,n. [15] - , . , P ⁿ P ⁿ T ⁿ T ⁿ .     

The maximum size of a convex polygon in a restricted set of points in the plane

Assume we havek points in general position in the plane such that the ratio between the maximum distance of any pair of points to the minimum distance of any pair of points is at mostk, for some positive constant. We show that there exist at leastk ^1/4 of these points which are the vertices of a convex polygon, for some positive constant=(). On the other hand, we show that for every fixed>0, ifk>k(), then there is a set ofk points in the plane for which the above ratio is at most 4k, which does not contain a convex polygon of more thank ^1/3+ vertices.The work of the first author was supported in part by the Allon Fellowship, by the Bat Sheva de Rothschild Foundation, by the Fund for Basic Research administered by the Israel Academy of Sciences, and by the Center for Absorbtion in Science. Work by the second author was supported by the Technion V. P.R. Fund, Grant No. 100-0679. The third author's work was supported by the Natural Sciences and Engineering Research Council, Canada, and the joint project Combinatorial Optimization of the Natural Science and Engineering Research Council (NSERC), Canada, and the German Research Association (Deutsche Forschungsgemeinschaft, SFB 303).