On the error of compound quadrature formulas forr-convex functions

LetC _m be a compound quadrature formula, i.e.C _m is obtained by dividing the interval of integration [a, b] intom subintervals of equal length, and applying the same quadrature formulaQ _n to every subinterval. LetR _m be the corresponding error functional. Iff ^(r) > 0 impliesR _m [f] > 0 (orR _m [f] < 0),=" then=" we=" say=">C _m is positive definite (or negative definite, respectively) of orderr. This is the case for most of the well-known quadrature formulas. The assumption thatf ^(r) > 0 may be weakened to the requirement that all divided differences of orderr off are non-negative. Thenf is calledr-convex. Now letC _m be positive definite or negative definite of orderr, and letf be continuous andr-convex. We prove the following direct and inverse theorems for the errorR _m [f], where , denotes the modulus of continuity of orderr:

Sphere orders

Ann-sphere order is a finite partially ordered set representable by containment ofn-spheres in Euclidean (n+1)-space. We present a sequence {P _i } of ordered sets such that eachP _i is ann-sphere order only forni; one consequence is that we are able to determine the dimension of a Euclidean space-time manifold from the finite suborders of its causality order.Research supported by ONR grant N00014 85-K-0769.     

On the existence of equilibrium states in local quantum field theory

It is shown that any local quantum field theory admits thermodynamical equilibrium states (KMS-states) for all positive temperatures provided it satisfies a nuclearity condition, proposed by Wichmann and one of the authors, which restricts the admissible number of local degrees of freedom.Dedicated to E. H. Wichmann on the occasion of his 60th birthday     

Achieving ‘Desirable and Feasible’ Change: An Application of Soft Systems Methodology

Rational intervention in human affairs, if it is to constitute not only action but also research, so that future interventions may be made more effective, needs a well-defined methodological framework. Soft systems methodology (S.S.M.) provides one such framework. S.S.M. is doubly systemic: it is itself a learning system, and within that system it uses systems models, models of human activity systems. It accepts that such models are not models of parts of the real world, only models of ways of perceiving the real world, that is to say, models relevant to debate about ‘reality’ (one man's ‘terrorism’ is another's ‘freedom fighting’).S.S.M. is here illustrated by means of an account of a systems study of change in the information function of a sophisticated science-based company. The study is described from the point of view of a professional analyst who was helping a team of three managers carry out the study. The course of the study is described, and a separate commentary relates its activity to the seven stages of S.S.M. The study involved three cycles round stages 2–3–4–5 of S.S.M., the stages in which models of relevant human activity systems are built and compared with the real world in order to construct a rich debate about changes whose introduction in the problem situation world be both (systemically) desirable and (culturally) feasible.     

A comparison of some hierarchal agglomerative clustering algorithms for structure—property correlation

Six different hierarchal clustering algorithms were used to cluster eleven sets of compounds for which associated property data were available. The effectiveness of the clustering in each case was assessed by inspection of the resulting tree diagram representing the classification and by the utility of the classification for molecular property prediction. The best predictions were generally obtained by using the algorithm proposed by Ward while the widely-used nearest neighbour algorithm performed very badly. Unstandardised data gave better results than standardised data although the difference was less when structurally related, rather than structurally disparate, groups of compounds were tested.     

The prism algorithm for two-electron integrals

A new approach to the evaluation of two-electron repulsion integrals over contracted Gaussian basis functions is developed. The new scheme encompasses 20 distinct, but interrelated, paths from simple shell-quartet parameters to the target integrals, and, for any given integral class, the path requiring the fewest floating-point operations (FLOPS ) is that used. Both theoretical (FLOP counting) and practical (CPU timing) measures indicate that the method represents a substantial improvement over the HGP algorithm.     

Zur Kenntnis von Natrium-hexafluoromanganat(III)

Sodium hexafluoromanganate(III) has been synthesized by heating equimolecular quantities of Na₂MnF₅ and NaF in argon atmosphere. The compound is monoclinic witha=5.56 (1) Å,b=5.84 (1) Å,c=8.10 (2) Å, =90.7 (2) andZ=2. It is a high spin complex with _eff and the deformation of the octahedra is evident from its IR-spectra. Two enantiotropic transitions (at 562 and 653°C) and the melting point at 800°C have been observed.

---

---

8. Mitt.: Mh. Chem.106, 483 (1975).     

Pyrolysis of homoadamant-3-ene

Pyrolysis of homoadamant-3-ene ($\text{5}$), generated from 1-adamantylcarbene ($\text{7}$), leads to the same three olefins ($\text{2}$, $\text{3}$, and $\text{4}$) that are produced from pyrolysis of 3-homoadamantyl acetate ($\text{1}$).     

Some estimates of theH ∞-uniform distribution

The sequences (y _n )=(a+blogn+ _n ) are uniformly distributed in the sense of the summation methodH . In the present paper the speed of convergence of this procedure is estimated for these, sequences and for some other sequences. For the sequences (a+blogn) theH -means converge considerably faster than logarithmic means.