Partially ordered permutation groups

In this note we discuss permutation groups (G, ) in which the set admits aG-invariant order. By aG-invariant partial order (G-partial order) we mean a partial order < of such that < implies g<g, for all and in andg inG. If the set admits aG-partial order which is a total order, then (G, ) is an O-permutation group (orderable permutation group).The main concern of this paper is the development of a foundation for partially ordered permutation groups analogous to the existing one for partially ordered groups, as found in Fuchs [2].     

Conditions for eigenvectors of a selfadjoint operator-function to form a basis

Consider a functionL() defined on an interval of the real axis whose values are linear bounded selfadjoint operators in a Hilbert spaceH. A point ₀ and a vector ₀ H( ₀ 0) are called eigenvalue and eigenvector ofL() ifL() ifL(₀) ₀ = 0. Supposing that the functionL() can be represented as an absolutely convergent Fourier integral, the interval is sufficiently small and the derivative will be positive at some point, it has been proved that all the eigenvectors of the operator-functionL() corresponding to the eigenvalues from the interval form an unconditional basis in the subspace spanned by them.     

Direct and inverse spectral problems for generalized strings

Let the functionQ be holomorphic in he upper half plane ⁺ and such that ImQ(z 0 and ImzQ(z) 0 ifz ⁺. A basic result of M.G. Krein states that these functionsQ are the principal Titchmarsh-Weyl coefficiens of a (regular or singular) stringS[L,m] with a (non-decreasing) mass distribution functionm on some interval [0,L) with a free left endpoint 0. This string corresponds to the eigenvalue problemdf + fdm = 0; f(0–) = 0. In this note we show that the set of functionsQ which are holomorphic in ⁺ and such that the kernel has negative squares of ⁺ and ImzQ(z) 0 ifz ⁺ is the principal Titchmarsh-Weyl coefficient of a generalized string, which is described by the eigenvalue problemdf +f dm + ² fdD = 0 on [0,L),f(0–) = 0. Here is the number of pointsx whereD increases or 0 >m(x + 0) –m(x – 0) –; outside of these pointsx the functionm is locally non-decreasing and the functionD is constant.To the memory of M.G. Krein with deep gratitude and affection.This author is supported by the Fonds zur Förderung der wissenschaftlichen Forschung of Austria, Project P 09832     

Topological hochschild homology of extensions of ?/p? by polynomials, and of ?[x]/(xn) and ?[x]/(xn?1)

Topological Hochschild homology is calculated for the rings /p[x]/(f(x)) (where p is prime and f(x) /p[x] any polynomial), [x]/(x ⁿ) and [x]/(x ⁿ–1). A spectral sequence argument is used for calculating the homology of the topological Hochschild homology spectrum, from which its stable homotopy structure can be read off since the spectrum is known for a priori reasons to be a restricted product of Eilenberg-MacLane spectra.     

On translations of double rays in graphs

Let be an infinite graph, let be a double ray in , and letd andd denote the distance functions in and in , respectively. One calls anaxis ifd(x,y)=d (x,y) and aquasi-axis if lim infd(x,y)/d (x,y)>0 asx, y range over the vertex set of andd (x,y). The present paper brings together in greater generality results of R. Halin concerning invariance of double rays under the action of translations (i.e., graph automorphisms all of whose vertex-orbits are infinite) and results of M. E. Watkins concerning existence of axes in locally finite graphs. It is shown that if is a translation whose directionD() is a thin end, then there exists an axis inD() andD(^–1) invariant under ^r for somer not exceeding the maximum number of disjoint rays inD().The thinness ofD() is necessary. Further results give necessary conditions and sufficient conditions for a translation to leave invariant a quasi-axis.     

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.     

Scaling limit of anticommuting self-adjoint operators and applications to Dirac operators

LetA andB be two anticommuting self-adjoint operators andV() be a symmetric operator in a Hilbert space, where >0 is a parameter. It is proven that, under some conditions forV(), the resolvents of A+² B±²|B|+V() converge as . Applications to the nonrelativistic-limit problem of Dirac operators and supersymmetry are discussed.This work is supported by the Grant-In-Aid 0560139 for science research from the Ministry of Education, Japan.     

On some algebras diagonalized by M-bases of ℓ2

We study reflexive algebrasA whose invariant lattices LatA are generated by M-bases of ². Examples are given whereA differs from ( being the rank one subalgebra ofA), and where together with the identity I is not strongly dense inA. For M-bases in a special class, we characterize the cases when they are strong, and also when the identity I is the ultraweak limit of a sequence of contractions in . We show that this holds provided that I is approximable by compact operators inA at any two points of ². We show that the spaceA+^* (where is the annihilator of ) is ultraweakly dense in (²), and characterize the M-bases in this class for which the sum is direct. We give a class of automorphisms ofA which are strongly continuous but not spatial.     

Compact perturbation of definite type spectra of self-adjoint quadratic operator pencils

Self-adjoint quadratic operator pencilsL()= ² A + B + C with a noninvertible leading operatorA are considered. In particular, a characterization of the spectral points of positive and of negative type ofL is given, and their behavior under a compact perturbation is studied. These results are applied to a pencil arising in magnetohydrodynamics.     

Nice decompositions of R n entirely into nice sets are mostly impossible

While there are several interesting examples of partitions of R ³ into elements which individually are geometrically nice — circles or segments — the partitions themselves fail to be nice, in the sense of forming continuous or upper semicontinuous decompositions. We show that this is no accident: R ³ has no continuous decomposition into circles, and no open subset of R ⁿ has an upper semicontinuous decomposition into convex compact nonsingleton sets.