An Interactive Production Planning System in the Chemical Industry

This paper offers an example of a practical production planning system as used routinely by BP Chemicals Ltd. It is based on the interactive control of a linear programming model embedded in a system for data collection and report production. The model solves a "product mix" type of problem with a multi-criteria objective function.     

Best approximation and de la Vallée-Poussin sums

For the class C_ε={f∈C_2π: E_n, n≤Z₊} where \(\left\{ {\varepsilon _n } \right\}_{n \in Z_ + } \) is a sequence of numbers tending monotonically to zero, we establish the following precise (in the sense of order) bounds for the error of approximation by de la Vallée-Poussin sums: (1) $$c_1 \sum\nolimits_{j = n}^{2\left( {n + l} \right)} {\frac{{\varepsilon _j }}{{l + j - n + 1}}} \leqslant \mathop {\sup }\limits_{f \in C_\varepsilon } \left\| {f - V_{n, l} \left( f \right)} \right\|_C \leqslant c_2 \sum\nolimits_{j = n}^{2\left( {n + l} \right)} {\frac{{\varepsilon _j }}{{l + j - n + 1}}} \left( {n \in N} \right)$$ , where c₁ and c₂ are constants which do not depend on n orl. This solves the problem posed by S. B. Stechkin at the Conference on Approximation Theory (Bonn, 1976) and permits a unified treatment of many earlier results obtained only for special classes C_ε of (differentiable) functions. The result (1) substantially refines the estimate (see [1]) (2) $$\left\| {V_{n, l} \left( f \right) - f} \right\|_C = O\left( {\log {n \mathord{\left/ {\vphantom {n {\left( {l + 1} \right) + 1}}} \right. \kern-\nulldelimiterspace} {\left( {l + 1} \right) + 1}}} \right) E_n \left[ f \right] \left( {n \to \infty } \right)$$ and includes as particular cases the estimates of approximations by Fejér sums (see [2]) and by Fourier sums (see [3]).     

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.     

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.     

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.     

Simultane Bestimmung von elektrophoretischer Beweglichkeit und Teilchengröße bei überlagerter Konvektion an Suspensionen und Emulsionen

With AWPS (Amplitude Weighted Phase Structuration), a new signal processing scheme is demonstrated for the simultaneous determination of zeta potentials and particle sizes. It allows the measurements of a small electrophoretic mobility in the presence of large particle diffusion and constant velocity, e. g. due to thermal convection. Laser light scattering techniques instead of the former methods determine electrophoretic velocity more objectively and precisely. The applicability of laser measurement techniques by analysis of the frequency spectrum is limited for particles ?50 nm or very low potentials, because of the broadening of the spectral peak by Brownian motion. In contrast to AWPS a separation of the various kinds of collective motion is not possible. The presented results demonstrate that this separation is of considerable significance in the acquisition of reliable values. Additionally the novel signal processing scheme allows a significant increase in sensitivity and therefore the application of an oscillating field (50–100 Hz) with a very small field strength. The system is feasible for particle sizes in the range of a few nm up to several μm. Its high resolution allows experiments with low fields or with small zeta potentials, even in the critical particles size range of a few nanometers.