On a pair of functional equations of combinatorial interest

The equations of the title appear in the author's paper Chromatic Sums for Rooted Planar Triangulations, V: Special Equations. (Canadian Journal of Mathematics, 26 (1974), 893–907). They appear in that paper as Equations (24) and (25). They are simultaneous equations for two unknown functionsl andy ₂ of two variablesy ₁ andz. A parameter is involved. The main result is that for = 2 cos (2/n), wheren is a positive integer >1, the two equations can be reduced to a single equation (numbered (49)). Solutions of this are known forn <7. From such solutions we can expect to get information about the averaged chromatic polynomials of planar triangulations with a given number of triangles.The present work is basically an expository paper on the theory given in Chromatic Sums, V, but it includes some new results and many simplifications.     

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.     

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.

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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.

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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.     

The reorientation of Cd2+-vacancy and Pb2+-vacancy complexes in KCl

The relaxation of Pb²⁺-vacancy and Cd²⁺-vacancy dipoles in purified KCl crystals was studied using a double crystal dc polarization method which self-corrects relaxation effects due to spurious causes. In the radius range from Cd²⁺ to Ba²⁺ these results and most others support Dreyfus' model. The reciprocal relaxation times for each impurity are given by

$\text{τ}{}^{\mathrm{?1}}{}_{\mathrm{<mtext>Cd</mtext>}}\text{=2.06x10}{}^{12}\text{exp(?s0.64±0.01}\text{eV}\text{)}\text{kT}$

;

$\text{τ}{}^{\mathrm{?1}}{}_{\mathrm{<mtext>Pb</mtext>}}\text{=1.19x10}{}^{13}\text{exp(?0.69±0.01}\text{eV}\text{)}\text{kT}$

.It is also shown that the presence of H₂O or its products greatly perturb the relaxation times observed.     

Fingerprint scaling increases the probability of identifying molecules with similar activity in virtual screening calculations.

Results of systematic virtual screening calculations using a structural key-type fingerprint are reported for compounds belonging to 14 activity classes added to randomly selected synthetic molecules. For each class, a fingerprint profile was calculated to monitor the relative occupancy of fingerprint bit positions. Consensus bit patterns were determined consisting of all bits that were always set on in compounds belonging to a specific activity class. In virtual screening calculations, scale factors were applied to each consensus bit position in fingerprints of query molecules. This technique, called "fingerprint scaling", effectively increases the weight of consensus bit positions in fingerprint comparisons. Although overall prediction accuracy was satisfactory using unscaled calculations, scaling significantly increased the number of correct predictions but only slightly increased the rate of false positives. These observations suggest that fingerprint scaling is an attractive approach to increase the probability of identifying molecules with similar activity by virtual screening. It requires the availability of a series of related compounds and can be easily applied to any keyed fingerprint representation that associates bit positions with specific molecular features.