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 .     

First order properties of random posets

Let =(n,p) be a binary relation on the set [n]={1, 2, ..., n} such that (i,i) for every i and (i,j) with probability p, independently for each pair i,j [n], where i<j. Define as the transitive closure of and denote poset ([n], ) by R(n, p). We show that for any constant p probability of each first order property of R(n, p) converges as n .     

Arithmetic and other properties of certain Delphic semigroups. I

Summary Some aspects of Delphic semigroups in general — in particular, the idea of an hereditary subsemigroup, which has many uses in connexion with Delphic semigroups — are first treated. After that, attention is directed to the arithmetic of ⁺, the semigroup of positive renewal sequences. In a Delphic semigroup the aboriginal elements are the simples and the members of I ₀: a class of simples of ⁺ is constructed and the simples are shown to be residual. I ₀ is explicitly identified, and this leads to a canonical factorization of ⁺. The properties of division in ⁺ are discussed.     

Hyperaffine planes over hyperfields

Summary In this paper we solve the problem of finding suitable conditions for an hyperaffine plane (,) under which a planar hyperfield K exists such that (,) is isomorphic to the hyperaffine plane (K²,').Research partially supported by C.N.R.-G.N.S.A.G.A.     

On Random Sets Connected to the Partial Records of Poisson Point Process

Random intervals are constructed from partial records in a Poisson point process in ]0,[×]0,[. These are used to cover partially [0,[; the purpose of this work is to study the random set that is left uncovered. We show that enjoys the regenerative property and identify its distribution in terms of the characteristics of the Poisson point process. As an application we show that is almost surely a fractal set and we calculate its dimension.     

Présentations des groupes de Fischer (I)

This work gives a method for constructing presentation (X, ) for a group generated by a class of 3-transpositions (named here a Fischer class) in which {ie275-1}(G/Z(G)) is simple and non-abelian. The generating set X is contained in D; is the set of relations (xy) ^m(x,y)=1, x and y in X, where m(x, y) is the order of the product xy in G, m(x, y) 3; is a set of relations between elements of X. The group G is constructed as a quotient of the Coxeter group (X, ).These results are applied to symplectic and orthogonal groups over {ie275-2}. Applications to other groups, in particular to the sporadic Fischer groups, will follow later.     

A dissipative transformation with a trivial tail-algebra

Summary In [1], an example was given of a measure-preserving dissipative transformation T in a -finite measure space (X, , ), such that T is conservative in the measure space (X, , ) where . Here we shall show that for this transformation we actually have R ={ØX}[].     

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   

Bruck nets,codes, and characters of loops

Numerous computational examples suggest that if _{k-1 k} are (k- 1)- and k-nets of order n, then rank _{p k} - rank _{p k-1} n - k + 1 for any prime p dividing n at most once. We conjecture that this inequality always holds. Using characters of loops, we verify the conjecture in case k = 3, proving in fact that if p ^e n, then rank _{p 3} 3n - 2 - e, where equality holds if and only if the loop G coordinatizing ₃ has a normal subloop K such that G/K is an elementary abelian group of order p ^e . Furthermore if n is squarefree, then rank _p = 3n - 3 for every prime p ¦ n, if and only if ₃ is cyclic (i.e., ₃ is coordinated by a cyclic group of order n).The validity of our conjectured lower bound would imply that any projective plane of squarefree order, or of order n 2 mod 4, is in fact desarguesian of prime order.     

Strong rightD-domains

An integral domainR is called rightD-domain if its lattice of all right ideals is distributive. In § 2 a sufficient condition for an integral domainR is given such thatR is a rightD-domain if and only ifR is a leftD-domain. For example each integral domain which is algebraic over its center satisfies this criterion. Furthermore, a rightD-domain is called strong if its lattice of all fractional right ideals is distributive. Examples of strong rightD-domains are given in §4. Each overring of a strong rightD-domain is also a strong rightD-domain whereas arbitrary rightD-domains may have overrings which are no rightD-domains. Section 3 is mainly concerned with the set * of all left invertible fractional right ideals and the mapping :**,II _l ^–1 whereI _l ^–1 denotes the left inverse ofI. For example, equivalent conditions are given for * to be a sublattice of and it is shown that is bijective if and only if (IJ)=(I)+(J) holds for allI,J*. Finally, §5 deals with (right)D-domains which are algebraic over their centersC. It is proved thatR is invariant if and only ifC is a commutative Prüfer domain andR the integral closure ofC inQ(R).     

Algorithmic aspects of intersection graphs and representation hypergraphs

Let be a family of sets. The intersection graph of is obtained by representing each set in by a vertex and connecting two vertices by an edge if and only if their corresponding sets intersect. Of primary interest are those classes of intersection graphs of families of sets having some specific topological or other structure. The grandfather of all intersection graphs is the class of interval graphs, that is, the intersection graphs of intervals on a line.The scope of research that has been going on in this general area extends from the mathematical and algorithmic properties of intersection graphs, to their generalizations and graph parameters motivated by them. In addition, many real-world applications involve the solution of problems on such graphs.In this paper a number of topics in algorithmic combinatorics which involve intersection graphs and their representative families of sets are presented. Recent applications to computer science are also discussed. The intention of this presentation is to provide an understanding of the main research directions which have been investigated and to suggest possible new directions of research.     

The Bruhat decomposition,Tits system and Iwahori ring for the monoid of matrices over a finite field

Let G=GL _n (F _q ) be the finite general linear group and let M=M _n (F _q ) be the monoid of all n×n matrices over F _q . Let B be a Borel subgroup of G, let W be the subgroup of permutation matrices, and let W be the monoid of all zero-one matrices which have at most one non-zero entry in each row and each column. The monoid plays the same role for M that the Weyl group W does for G. In particular there is a length function on which extends the length function on W and a C-algebra H _C (M, B) which includes Iwahori's Hecke algebra H _C (G, B) and shares many of its properties.For Jacques Tits on his sixtieth birthday     

Commutativity of operators on the lattice of existence varieties

In a previous paper the authors introduced seven complete congruences on the lattice _ev(I of e-varieties of regular semigroups of the form _P :U _P VPU=PV, whereP is drawn from a small set of e-varieties: left zero, right zero, rectangular bands, groups, left groups, right groups and completely simple semigroups. Four new complete congruences are introduced here of the form _P :U _P VPU=PV, whereP is one of the following classes of regular semigroups: left monoids, right monoids, monoids, idempotent generated semigroups. For each complete congruence on _ev(I) and eachU_ev(I), the -class ofU is an interval [U ,U ] so that there is associated with each such congruence an idempotent operatorUU on _ev(I). This paper establishes numerous results concerning the commutativity of operators of this form.This work was supported in part by NSERC Grant 4044.     

Discrepancy and approximations for bounded VC-dimension

Let (X, ) be a set system on ann-point setX. For a two-coloring onX, itsdiscrepancy is defined as the maximum number by which the occurrences of the two colors differ in any set in . We show that if for anym-point subset the number of distinct subsets induced by onY is bounded byO(m ^d) for a fixed integerd, then there is a coloring with discrepancy bounded byO(n ^1/2–1/2d(logn)^1+1/2d ). Also if any subcollection ofm sets of partitions the points into at mostO(m ^d) classes, then there is a coloring with discrepancy at mostO(n ^1/2–1/2dlogn). These bounds imply improved upper bounds on the size of -approximations for (X, ). All the bounds are tight up to polylogarithmic factors in the worst case. Our results allow to generalize several results of Beck bounding the discrepancy in certain geometric settings to the case when the discrepancy is taken relative to an arbitrary measure.Work of J.M. and E.W. was partially supported by the ESPRIT II Basic Research Actions Program of the EC under contract no. 3075 (project ALCOM). L.W. acknowledges support from the Deutsche Forschungsgemeinschaft under grant We 1265/1-3, Schwerpunktprogramm Datenstrukturen und effiziente Algorithmen.     

Fine Densities for Excessive Measures and the Revuz Correspondence

Suppose that U is the resolvent of a Borel right process on a Lusin space X. If is a U-excessive measure on X then we show by analytical methods that for every U-excessive measure with the Radon–Nikodym derivative d/d possesses a finely continuous version. (Fitzsimmons and Fitzsimmons and Getoor gave a probabilistic approach for this result.) We extend essentially a technique initiated by Mokobodzki and deepened by Feyel. The result allows us to establish a Revuz type formula involving the fine versions, and to study the Revuz correspondence between the -finite measures charging no set that is both -polar and -negligible (U being the potential component of ) and the strongly supermedian kernels on X. This is an analytic version of a result of Azéma, Fitzsimmons and Dellacherie, Maisonneuve and Meyer, in terms of additive functionals or homogeneous random measures. Finally we give an application to the context of the semi-Dirichlet forms, covering a recent result of Fitzsimmons.     

The Terwilliger algebras of certain association schemes over the Galois rings of characteristic 4

In a cyclotomic scheme over a finite field, there are some relations between the irreducible modules of the Terwilliger algebra and the Jacobi sums over the field. These relations were investigated in [3]. In this paper, we replace the finite field by a commutative local ring which is called a Galois ring of characteristic 4. Hence we want to find similar relations between the irreducible modules of the Terwilliger algebra and the Jacobi sums over the local ring. Specifically, if we let be a Galois ring of characteristic 4,X a cyclotomic scheme over with classD and the Terwilliger algebra ofX, then we show that most of the irreducible -modules have standard forms; otherwise, certain relations of the Jacobi sums hold. When the classD is three, we can completely determine the irreducible -modules using Jacobi sums.     

On inertial wave and flow structure at low Rossby number

Flow disturbances produced by the slow relative motion of an impermeable body in a large, rapidly rotating vessel are studied as an asymptotic theory for an inviscid, incompressible fluid at a small Rossby number (u _c /L1). The axial distance between the vessel wallsH is assumed to be so much larger than the body scaleL that the reduced height H/L is of unit order or greater. This flow admits a columnar structure near the body and an outer nonlinear structure of the inertial-wave type far above the forcing region, at distances along the rotation-axis (z=0(L/)). The inner boundary condition for the outer problem is provided by transferring the impermeable surface condition through the columnar structure; the outer solution in turn determines the vorticity and the solution of the columnar inner region.For thin obstacles or shallow ground topography (1), the outer solution is governed by an equation comparable to a linear one for inertial waves. The linear solutions obtained for steady transverse motion in an infinite domain () shows that, in this case, surface (perturbation) velocities are orthogonal to those in an irrotational (non-rotating) flow over the same ground topography. In the far field, i.e., ( z/L l) disturbances are confined mainly behind a wedge-shaped caustic front downstream of the rotation axis , wherein their amplitudes are comparable to , in general accord with Lighthill's result from group-velocity consideration. The field behind the caustic supports, however, lee waves extending far downstream with diminishing strength. Their wave lengths belong to orderLz/x, and therefore these waves should appear to be densely packed in the wedge-shaped region. The question on tilting of the Taylor column is delineated; the structure of the caustic zone and its upstream flow behavior are also analyzed.

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Zusammenfassung Die Strömung um einen Körper, der sich langsam in einem grossen, rotierenden Behälter bewegt, wird untersucht mit einer asymptotischen Theorie für eine reibungsfreie, inkompressible Flüssigkeit bei einer kleinen Rossby-Zahl, d.h. u _c /L1. Der axiale AbstandH zwischen den Gefässwänden wird als soviel grösser als die Körperabmessung angenommen, dass für die reduzierte Höhe =H/L 0(1) gilt. Diese Strömung erlaubt eine säulenähnliche Struktur (Taylor column) in Körpernähe und eine äussere, nichtlineare Struktur vom Wellentyp für Höhenz=0(L/) über dem Köper. Die innere Randbedingung für das äussere Problem wird erhalten, indem die Bedingung an der Wand durch die Säule verschoben wird. Die äussere Lösung bestimmt ihrerseits die Rotation und damit die Lösung im inneren, säulenartigen Bereich.Wenn Körper oder Bodenform flach sind (1), so wird die Aussenlösung durch eine Gleichung bestimmt, welche vergleichbar ist mit einer linearen Gleichung für Trägheitswellen. Lineare Lösungen werden für gleichförmige Bewegung von der Achse weg in einem unbegrenzten Bereich ) gegeben. Sie zeigen, dass die Oberflächenstromlinien orthogonal zu den Stromlinien sind, die bei rotationsfreier (nichtdrehender) Strömung über der gleichen Bodenform entstehen. Im Fernfeld ( z/L l) sind Störungen hauptsächlich auf ein keilförmiges Gebiet stromabwärts von der Rotationsachse ( ) begrenzt. In diesem Gebiet geht ihre Amplitude mit , in Uebereinstimmung mit Lighthill's Resultat, das mit der Gruppengeschwindigkeit hergeleitet wurde. Im keiförmigen Gebiet bestehen drei Familien von Lee-Wellen, welche sich mit unveränderter Stärke weit stromabwärts erstrecken. Ihre Wellenlängen gehören zur OrdnungLx/z, weshalb die Wellen im keilförmigen Bereich dicht gepackt erscheinen. Die Frage der Neigung der Taylor Säule behandelt und die Struktur des Randes vom keilförmigen Bereich wird analysiert.

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This work is dedicated to Professor Nicholas Rott on the occasion of his sixtieth birthday.     

Global Optimization Algorithm for the Nonlinear Sum of Ratios Problem

This article presents a branch-and-bound algorithm for globally solving the nonlinear sum of ratios problem (P). The algorithm economizes the required computations by conducting the branch-and-bound search in ^p, rather than in ⁿ, where p is the number of ratios in the objective function of problem (P) and n is the number of decision variables in problem (P). To implement the algorithm, the main computations involve solving a sequence of convex programming problems for which standard algorithms are available.     

Some properties of lattices of radical classes of lattice-ordered groups

Aradical class of lattice-ordered groups (-groups) is a class closed under taking convex-subgroups, joins of convex-subgroups, and-isomorphic images. Imposing various other closure conditions leads to many specific types of radical classes (e.g., torsion classes). For several of these types, the complete latticeT of radical classes of that type has been studied, and such latticesT are our object of study here. We give the characteristic properties of closed-kernel radical mappings and polar kernel radical mappings. We prove in many instances thatT isrelatively polarized, that is, for any ], T with ] there exists a unique largest T such that = ], and often we are able to explicitly identify. By using these properties we characterize meet irreducibility in the latticeT of polar kernel radical classes.Presented by M. Henriksen.     

On cubature formulas on the basis of lattices

Q (.. , L). Q . P(S_r(2)) — 2 (S _r(2) (r — ). , M(P(S _r(m=sup{t(·)t(·)₁:t P(S _r(2)),t 0}. , /4+(1)M(P(S _r(2)))/r ²15/17+(1)(r+). (Q), Q L.