Approximation under an arc length constraint

Let f C[a, b]. LetP be a subset ofC[a, b], L b – a be a given real number. We say thatp P is a best approximation tof fromP, with arc length constraintL, ifA[p] _b ^a [1 + (p(x)) ²]dx L andp – f q – f for allq P withA[q] L. represents an arbitrary norm onC[a, b]. The constraintA[p] L might be interpreted physically as a materials constraint.In this paper we consider the questions of existence, uniqueness and characterization of constrained best approximations. In addition a bound, independent of degree, is found for the arc length of a best unconstrained Chebyshev polynomial approximation.The work of L. L. Keener is supported by the National Research Council of Canada Grant A8755.     

An additivity principle for Goldie rank

LetA be a noetherian ring. In generalA will not admit a classical Artinian ring of quotients. Yet a problem in enveloping algebras leads one to consider the possible embedding ofA in a prime ringB which is finitely generated as a left and a rightA module. Under certain additional technical assumptions, it is shown that the setS of regular elements ofA is regular inB and is an Ore set in bothA andB withS ⁻¹ A andS ⁻¹ B Artinian. This enables one to establish the following additivity principle for Goldie rank. Let {P ₁,P ₂, …P ₁} be the set of minimal primes ofA. Then under the above conditions it is shown that there exist positive integersz ₁,z ₂, …,z, such that , where rk denotes Goldie rank. This applies to the study of primitive ideals in the enveloping algebra of a complex semisimple Lie algebra. This paper was written while the authors were guests of the Institute for Advanced Studies, The Hebrew University of Jerusalem. The first author was on leave of absence from the Centre Nationale de la Recherche Scientifique, France.     

Spectrophonic humidity-trace detection in hydrogen and air

A spectrophone with low resonant frequency compatible with the repetition frequency of a pulsed discharge CO laser is described. The spectrophone is used for humidity detection in hydrogen and air. The simple and inexpensive system provides a detection capacity of 10–100ppb/ . The responsivity of the resonant spectrophone is interpreted taking the dependence of the acoustical quality factor on the geometry into account.     

Critical surface of the ising model with first-neighbor,second-neighbor,and four-spin interactions

Exact values are obtained for the slopesK ₁ ^c (0, 0)/K ₂,K ₁ ^c (0, 0)/K ₄ of the critical surface of paramagnetic-ferromagnetic transitionsK ₁ ^c (K ₂,K ₄) for the two-dimensional Ising model on a square lattice with first-neighbor, second-neighbor, and four-spin couplingsK ₁,K ₂, andK ₄, respectively. The results are obtained using universality arguments to relate the slopes to known spin-spin correlation functions forK ₂=K ₄=0. The equivalence of different expressions for the slopes in terms of correlation functions yields sum rules for the divergent part of certain sums over the second-neighbor and four-spin energy-energy correlation functions.     

The level scheme of133Sb

Nuclear spectroscopy studies of ₅₁ ¹³³ Sb₈₂ have been performed at the recoil fission product separators JOSEF and LOHENGRIN. A tentative level scheme has been established for this nucleus with one proton in addition to the doubly closed shell core¹³²Sn. Two isomeric states with half-lives of 3 and 16 μs have been found at excitation energies of >4300 keV and 4526 keV, respectively. Lower limits of 15/2 are deduced for the spins of these levels. The h_11/2 proton state is suggested at 2792 keV in agreement with the systematics of the lighter Z=51 isotopes and the heavier N=82 isotones. Several states have been observed between 4300 and 4600 keV which are probably due to the coupling of the single proton to core-excited states which have energies between 4000 and 4800 keV.     

Heuristische Verfahren zur Tourenplanung bei der Hausmüllentsorgung in ländlichen Regionen

Zusammenfassung Tourenprobleme bei der Hausmüllsammlung lassen sich in die Reihe der Lieferplanprobleme einordnen. Zur Lösung praktischer Größenordnungen werden hier einfache heuristische Verfahren (Prioritätsregeln) vorgestellt, weniger um der Vielzahl heuristischer Verfahren weitere hinzuzufügen als vielmehr dem Praktiker einen vielversprechenden Weg beim Lösen seiner Probleme aufzuzeigen.

---

Summary The discussed solid waste collection problem shows the properties of the so called truck dispatching problem. To solve problems for typical problem structure heuristic rules are pointed out in order to show the planner an efficient way of problem solving.

---

---

Zusammenfassung dreier Vorträge im 4. planungstechnischen Seminar Tourenplanung bei der Abfallbeseitigung am Institut für Siedlungswasserwirtschaft der Universität Karlsruhe am 1.10.1976.     

Passive states and KMS states for general quantum systems

We characterize equilibrium states of quantum systems by a condition of passivity suggested by the second principle of thermodynamics. Ground states and -KMS states for all inverse temperatures 0 are completely passive. We prove that these states are the only completely passive ones. For the special case of states describing pure phases, assuming the passivity we reproduce the results of Haag et al.     

Crystalline titanium dichloride—an active catalyst in ethylene polymerization. II. Polymer structure,polymerization variables,and scope

Polymerization of ethylene with ball-milled titanium dichloride leads to a completely linear polymer with terminal unsaturation corresponding to approximately one carbon–carbon double bond per molecule. Polymerization rate is first-order in both monomer and catalyst concentration at 140°C. Due to a thermal deactivation of the catalyst, the polymerization rate falls sharply with temperature above 180°C. Propylene and butene-1 will copolymerization with ethylene in this system, propylene more efficiently than butene-1. Evidence for copolymerization of trans-2-butene, but not of the cis-isomer or of isobutene, in trace concentrations is presented. Propylene is homopolymerized to a product low in isotactic content. The significance of the structural and (limited) kinetic data in terms of the mechanism of polymerization are discussed.     

Dipolmomente 1, 6-überbrückter [10]- Annulene

The dipole moments of 1, 6-bridged [10]-annulenes of type I (X = CH₂, O, NH) and of bromo derivatives with X = CH₂, O are in agreement with the previously established structure of these compounds if a partial moment of approximately 0,8 D is assigned to the non-planar π-electron system.