A technique for constructing divisible difference sets

An (m, n, k, ₁,₂) divisible difference set in a groupG of ordermn relative to a subgroupN of ordern is ak-subsetD ofG such that the list {xy^–1:x, y D} contains exactly ₁ copies of each nonidentity element ofN and exactly ₂ copies of each element ofG N. It is called semi-regular ifk > ₁ and k²=mn₂. We develop a method for constructing a divisible difference set as a product of a difference set and a relative difference set or a difference set and a subset ofG which we call a relative divisible difference set. The method results in several parametrically new families of semi-regular divisible difference sets.     

Über die Normalteiler scharf zweifach und scharf dreifach transitiver Permutationsgruppen

A sharply 2-transitive (3-transitive) groupT can be described by means of a neardomainF (a KT-field(F,)). We show, thatT has a least nontrivial normal subgroupA (S(F,)), ifF is a nearfield or if CharF 2. In this case the nontrivial normal subgroups ofT correspond bijectively with all normal subgroupsF ^* (the multiplicative group ofF) containing a setD (D(Q)). IfF is a nearfield or ifF has a suitable central element, then the group S(F,) is simple.

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Herrn Prof. Dr. Dr. h. c. Helmut Karzel zum 70. Geburtstag gewidmet     

Almost Fredholm operators in von Neumann algebras

LetA be a von Neumann algebra,J be the ideal of compact operators relative toA and letF ₊ be the left-Fredholm class ofA. We call almost left-Fredholm the class = {A A: if P A is a projection and AP J then P J}. Then and the inclusion is proper unlessA is semifinite and has a non-large center. satisfies all of the algebraic properties ofF ₊ but it is generally not open. IfA is semifinite then A iff there are central projectionsG with G = I such that AG F₊(AG). Let :A A/J. Then the left almost essential spectrum ofA A, , coincides with the set of eigenvalues of (A)     

Über die Anzahl der Lösungen der diskreten Theodorsen-Gleichung

Summary Discretization of the Theodorsen integral equation (T) yields the discrete Theodorsen-equation (T _d ), a system of 2N nonlinear equations. A so-called -condition may be fulfilled. It is known that (T) has exactly one continuous solution. This solution gives the boundary correspondence of the normalized conformal map of the unit disc onto a given domainG. It is also known that (T _d ) has one and only one solution if <1 and at least one solution if 1. We show here that for every 1 and N\ {1} there is a domainG satisfying an -condition such that (T _d ) has an infinite number of solutions. Moreover, givenK>0 and any domainG that fulfills an -condition, we will construct a domainG ₁ in the neighbourhood ofG that fulfills a max (1, +K)-condition such that (T _d ) forG ₁ has an infinite number of solutions. The underlying idea of the construction of those domains allows also to give important new facts about iterative methods for the solution of (T _d ), even in the case <1.

     

The density condition in subspaces and quotients of Frechet spaces

The aim of this note is to investigate the topological structure (in particular the density condition) of subspaces and separated quotients of Fréchet spaces. Our main result is the following one: LetE be a Fréchet space which is neither Montel nor isomorphic to a closed subspace ofX × , withX a Banach space, also assume thatE can be written asFG withF andG infinite dimensional closed subspaces ofE not isomorphic to , thenE contains a closed subspace with basis and not satisfying the density condition. We also prove that every Köthe echelon space of orderp, 1<p<, which is not quasinormable has a separated quotient with basis which does not satisfy the density condition.     

Generation of alternating groups by pairs of conjugates

Considering the conjugacy classes of the alternating group of degreen, those classes that contain a pair of generators are in the majority. In fact, the proportion of such classes is 1 –(n), and(n) 0 asn .     

On automorphism groups of Cayley graphs

LetX _G,H denote the Cayley graph of a finite groupG with respect to a subsetH. It is well-known that its automorphism groupA(X_G,H) must contain the regular subgroupL _G corresponding to the set of left multiplications by elements ofG. This paper is concerned with minimizing the index [A(X_G,H)L_G] for givenG, in particular when this index is always greater than 1. IfG is abelian but not one of seven exceptional groups, then a Cayley graph ofG exists for which this index is at most 2. Nearly complete results for the generalized dicyclic groups are also obtained.     

A remark on generalized iterates

Letf be an invertible function on the real lineR, and letZ denote the set of integers. For eachx Z, letf ^|n| denote then'th iterate off. Clearlyf ^|m|(f ^|n|(x))=f ^|m+n|(x) for allm,nZ and allxR. LetG be any group of orderc, the cardinality of the continuum, which contains (an isomorphic copy of)Z as a normal subgroup. If for eachxR, the iteration trajectory (orbit) ofx is non-periodic, then there exists a set of invertible functionsF={F ^||:G} on the real line with the properties (i)F ^||(F ^||(x))=F ^|+| (x) for allxR and (ii)F ^|n|(x)=f ^|n|(x) for allnZ andxR. That is,f can be embedded in a set ofG-generalized iterates. In particular,f can be embedded in a set of complex generalized iterates.Dedicated to Professor Janos Aczél on his 60th birthday     

Decompositions of complete graphs into isomorphic bipartite subgraphs

Let |E(G)|= andf, a 1-1 mapping ofV(G) into {0,1,...,}. Thenf is called a -valuation ofG if the induced function given by , for alluvE(G) is 1-1. A -valuationf is called an -valuation ofG if there exists a nonnegative number such that for everyuvE(G) withf(u)<f(v),f(u)<f(v). Let denote the graph of then-dimensionalG-cube. ForG=K _{3, 3},K _{4, 4}, andP _k ,it is shown that for any positive integern, then-dimensionalG-cube has an -valuation. This gives rise to decompositions of some complete graphs into certain bipartite graphs.     

Clifford classes for isoclinic groups

Let G be a finite group, N a normal subgroup of G, and an irreducible character of G. Clifford Theory studies a whole collection of related irreducible characters of all the subgroups of G that contain N. The relationships among these characters as well as their Schur indices are controlled by the Clifford class c Clif(G/N, F) of with respect to N over some field F. This is an equivalence class of central simple G/N-algebras. Assume now that G/N is cyclic. One can obtain a new isoclinic group and character by multiplying each element of each coset of N in G by an appropriate power of a fixed root of unity . We show that there is a simple formula to calculate the Clifford class of in terms of c and . Hence, the Clifford class c controls not only the Schur index of the characters of all the subgroups of G that contain N, it also controls the Schur indices of the characters of the corresponding characters of the isoclinic groups When is a |G/N|-th root of 1, our formula shows that then When = i and |G/N| = 2, the implicit transformation on Clif(Z/2Z, F) yields a group homomorphism of the group structure introduced on the Brauer-Wall group of F to describe the Schur indices of all the irreducible characters of the double covers of the symmetric and alternating groups.Received: 17 August 2001     

Isolation theorem for forms corresponding to purely real algebraic fields

Let M be the complete module of a purely real algebraic field of degree n 3, let be a lattice in this module, and let F(X) be its form. We use to denote any lattice for which we have = , where is a nondiagonal matrix for which – I . With each lattice we can associate a factorizable formF (X) in a natural manner. We denote the complete set of forms corresponding to the set {} by {F (X)}. It is proved that for any > 0 there exists an > 0 such that for eachF (X) {F } we have |F (X₀₎| for some integer vector X₀ 0.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 185, pp. 5–12, 1990.In conclusion, the author would like to express his deep gratitude to B. F. Skubenko for stating the problem and for his constant attention.     

A lemma on the approximation of finite-dimensional*-algebras

One proves an analog of Glimm's lemma [Trans. Am. Math. Soc.,95, No. 2, 318–340 (1960)] by replacing the uniform norm by the trace norm A=tr(AA^*)^1/2; if the elements of a finite type factor satisfy (up to ) the relations which have to be satisfied by the matrix units of a finite-dimensional*-algebraA, then in their -neighborhoods there exist operators which satisfy exactly these relations and depends only on the algebraic type ofA and of.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 47, pp. 175–178, 1974.     

Generalized sentinels defined via least squares

We address the problem of monitoring a linear functional (c, x)_Eof an unknown vectorx of a Hilbert spaceE, the available data being the observationz, in a Hilbert spaceF, of a vectorAx depending linearly onx through some known operatorA(E; F). WhenE=E ₁×E ₂,c=(c ₁ 0), andA is injective and defined through the solution of a partial differential equation, Lions ([6]–[8]) introduced sentinelssF such that (s, Ax)_Fis sensitive to x₁ E ₁ but insensitive to x₂ E₂. In this paper we prove the existence, in the general case, of (i) a generalized sentinel (s, ) ×E, where F withF dense in 80, such that for anya priori guess x₀ ofx, we have s, Ax + (, x₀)_E=(c, x)_E, where x is the least-squares estimate ofx closest tox ₀, and (ii) a family of regularized sentinels (s _n , _n ) F×E which converge to (s, ). Generalized sentinels unify the least-squares approach (by construction !) and the sentinel approach (whenA is injective), and provide a general framework for the construction of sentinels with special sensitivity in the sense of Lions [8]).     

Strong solution for an elastic-plastic plate

Given a regular bounded open set R ²,, >0 andg L ^q () withq>2, we prove, under compatibility and safe load conditions ong, the existence of a minimizing pair for the functional, over closed setsK ² and functionsu C⁰( ) C²(/K); here ¦[Du]¦ denotes the jump ofDu acrossK and ¹ is the 1-dimensional Hausdorff measure.Dedicated to Enrico Magenes for his 70th birthday     

On the factor-thickness of regular graphs

For eachr-regular graphG, define a binary sequence(G) = ( ₁, ₂,..., _r-1) by _k = 0 ifG has ak-factor, and _k = 1 otherwise. A binary sequence = ( _i |i = 1, 2,...,r – 1) is said to be realizable if there exists anr-regular graphG such that(G) = . In this paper we characterize all binary sequences which are realizable.     

Complementing lattice-ordered groups: The finite-valued case

A complementedl-groupG is one in which to eacha G there exists ab G so that¦a¦ ¦b¦=0, while¦a¦ ¦b¦ is a unit ofG. IfG is anl-subgroup ofH, and the latter is complemented, then we say thatH is a complementation ofG ifG ^c , the convex hull ofG inH, is a strongly rigid extension ofG andG ^c is az-subgroup ofH. This article presents necessary and sufficient conditions for a finite-valuedl-subgroup to admit a complementation.Presented by M. Henriksen.     

Covering properties and Følner conditions for locally compact groups

Summary LetG be a locally compact group with left Haar measurem _G on the Borel sets IB(G) (generated by open subsets) and write |E|=m _G (E). Consider the following geometric conditions on the groupG.(FC If >0 and compact setKG are given, there is a compact setU with 0<|U|< and |x U U|/|U|< for allxK.(A) If >0 and compact setKG, which includes the unit, are given there is a compact setU with 0<|U|< and |K U U|/|U|<.HereA B=(A/B)(B/A) is the symmetric difference set; by regularity ofm _G it makes no difference if we allowU to be a Borel set. It is well known that (A)(FC) and it is known that validity of these conditions is intimately connected with amenability ofG: the existence of a left invariant mean on the spaceCB(G) of all continuous bounded functions. We show, for arbitrary locally compact groupsG, that (amenable)(FC)(A). The proof uses a covering property which may be of interest by itself: we show that every locally compact groupG satisfies.(C) For at least one setK, with int(K)Ø and compact, there is an indexed family {x J}G such that {Kx } is a covering forG whose covering index at each pointg (the number of J withgKx ) is uniformly bounded throughoutG.     

Quelques observations concernant les ensembles de Ditkin d'un groupe localement compact

We prove that every closed normal subgroupH of a locally compact amenable groupG is a Ditkin set with respect to the Herz-Figà-Talamanca algebraA _p (G) (p>1). Let be the canonical map ofG ontoG/H andF a closed subset ofG/H. We show thatF is a Ditkin set if and only if everyuA _p (G), which vanishes on ^–1, lies on the norm closure of the subspace ofA _p (G) generated by {u _h |hH, vA _p (G)C ₀₀(G)} whereu _h (x)=u(x h). As far as we know, this result seems to be new even forG abelian andp=2.     

Sur certaines algèbres associées à une mesure de Guelfand

Let be a Guelfand measure (cf. [A, B]) on a locally compact groupG DenoteL ₁ (G)=*L ₁(G)* the commutative Banach algebra associated to . We show thatL ₁ (G) is semi-simple and give a characterization of the closed ideals ofL ₁ (G). Using the -spherical Fourier transform, we characterize all linear bounded operators inL ₁ (G) which are invariants by -translations (i.e. such that ¹(( _x f) )=( _x ((f)) for eachxG andfL ₁ (G); where _x f(y)=f(xy); x,y G). WhenG is compact, we study the algebraL ₁ (G) and obtain results analogous to ones obtained for the commutative case: we show thatL ₁ (G) is regular, all closed sets of its Guelfand spectrum are sets of synthesis and establish theorems of harmonic synthesis for functions inL _p (G) (p=1,2 or +).

     

Compact operators whose singular numbers have powerlike asymptotics

For a compact operator in a Hilbert space, let s_n(A), n =1, 2,... be the singular numbers and let N(s; A) =card{n N:s_n(A)>, s>0. For 0

a _p and not on the individual elementAa, (H. Weyl's lemma); this allows us to write _p (a), _pp (a), a_p. One obtains certain results regarding the functionals _p, _p (and about the analogous functionals for the positive and negative eigenvalues in the casea=a ^*=A ^*:A a. In particular: I. Ifa ₁ a _2p, then. II.Let a ₁,a _{2 pP} ,.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matetmaticheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 126, pp. 21–30, 1983.