Geometric Mean Rule Research Articles

Geometric and harmonic means based priority dispatching rules for single machine scheduling problems

<span lang="ES">This work proposes two new prority dispatching rules (PDRs) for solving single machine scheduling problems. These rules are based on the geometric mean (GM) and harmonic mean (HM) of the processing time (PT) and the due date (DD) and they are referred to as GMPD and HMPD respectively. Performance of the proposed PDRs is evaluated on the basis of five measures/criteria i.e. Total Flow Time (TFT), Total Lateness (TL), Number of Late Jobs (TNL), Total Earliness (TE) and Number of Early Parts (TNE). It is found that GMPD performs better than other PDRs in achieving optimal values of multiple performance measures. Further, effect of variation in the weight assigned to PT and DD on the combined performance of TFT and TL is also examined which reveals that for deriving optimal values of TFT and TL, weighted harmonic mean (WHMPD) rule with a weight of 0.105 outperforms other PDRs. The weighted geometric mean (WGMPD) rule with a weight of 0.37 is found to be the next after WHMPD followed by the weighted PDT i.e. WPDT rule with a weight of 0.76.</span>

Adolescents show collective intelligence which can be driven by a geometric mean rule of thumb.

How effective groups are in making decisions is a long-standing question in studying human and animal behaviour. Despite the limited social and cognitive abilities of younger people, skills which are often required for collective intelligence, studies of group performance have been limited to adults. Using a simple task of estimating the number of sweets in jars, we show in two experiments that adolescents at least as young as 11 years old improve their estimation accuracy after a period of group discussion, demonstrating collective intelligence. Although this effect was robust to the overall distribution of initial estimates, when the task generated positively skewed estimates, the geometric mean of initial estimates gave the best fit to the data compared to other tested aggregation rules. A geometric mean heuristic in consensus decision making is also likely to apply to adults, as it provides a robust and well-performing rule for aggregating different opinions. The geometric mean rule is likely to be based on an intuitive logarithmic-like number representation, and our study suggests that this mental number scaling may be beneficial in collective decisions.

Excited state double proton transfer of a 1:1 7-azaindole:H 2O complex and the breakdown of the rule of the geometric mean: Variational transition state theory studies including multidimensional tunneling

The rule of the geometric mean in rates and kinetic isotope effects has long been used as a criterion for identifying the reaction mechanism, e.g., stepwise vs. concerted, for double proton transfer reactions. Potential energy surfaces of double proton transfers for excited state tautomerization in a 1:1 7-azaindole:H 2O complex were generated at the MRPT2//CASSCF(10,9)/6-31G(d,p) level. Variational transition state theory, including multidimensional tunneling approximation, was used to calculate rates and kinetic isotope effects. No intermediates were present along the reaction coordinate. Two protons in the excited state tautomerization were transferred concertedly, albeit asynchronously. Positions of the variational transition states depend very much on the isotopic substitution. The asynchronicity of two protons in flight breaks the underlying assumption of the rule of the geometric mean so that the relation, k HH/ k HD ≈ k HD/ k DD, is no longer valid in the excited state double proton transfer. Breakdown of the geometric mean rule does not necessarily entail that the reaction mechanism is stepwise; therefore this rule should be used very carefully as a criterion for identification of the mechanism.

Secondary Kinetics of Methanol Decomposition: Theoretical Rate Coefficients for 3CH2 + OH, 3CH2 + 3CH2, and 3CH2 + CH3

Direct variable reaction coordinate transition state theory (VRC-TST) rate coefficients are reported for the (3)CH(2) + OH, (3)CH(2) + (3)CH(2), and (3)CH(2) + CH(3) barrierless association reactions. The predicted rate coefficient for the (3)CH(2) + OH reaction (approximately 1.2 x 10(-10) cm(3) molecule(-1) s(-1) for 300-2500 K) is 4-5 times larger than previous estimates, indicating that this reaction may be an important sink for OH in many combustion systems. The predicted rate coefficients for the (3)CH(2) + CH(3) and (3)CH(2) + (3)CH(2) reactions are found to be in good agreement with the range of available experimental measurements. Product branching in the self-reaction of methylene is discussed, and the C(2)H(2) + 2H and C(2)H(2) + H2 products are predicted in a ratio of 4:1. The effect of the present set of rate coefficients on modeling the secondary kinetics of methanol decomposition is briefly considered. Finally, the present set of rate coefficients, along with previous VRC-TST determinations of the rate coefficients for the self-reactions of CH(3) and OH and for the CH(3) + OH reaction, are used to test the geometric mean rule for the CH(3), (3)CH(2), and OH fragments. The geometric mean rule is found to predict the cross-combination rate coefficients for the (3)CH(2) + OH and (3)CH(2) + CH(3) reactions to better than 20%, with a larger (up to 50%) error for the CH(3) + OH reaction.

Modeling gas sorption in amorphous Teflon through the non equilibrium thermodynamics for glassy polymers (NET-GP) approach

The solubility of C 2H 6 and C 2F 6 in two amorphous glassy fluoropolymers, Teflon® AF1600 and AF2400, has been modeled with the non equilibrium statistical associating fluid (NE-SAFT) and non equilibrium perturbed hard sphere chain (NE-PHSC) theories. Experimental swelling data during sorption are available to obtain a complete prediction of the solubility isotherms at 35°C. The solubility of C 2F 6 in both fluoropolymers is rather satisfactorily predicted by the models with no adjustable parameters, while that of C 2H 6 is overestimated, and a correction factor equal to about 0.90 must be applied on the binary energetic interactions estimated with the first approximation mixing rule. This result, combined with what previously obtained with the NE-lattice fluid (NE-LF) model, reveals inadequate representation of the hydrocarbon solubility in fluoropolymers by the geometric mean rule used to estimate the binary energetic interactions.

Predictive theory for the combination kinetics of two alkyl radicals

An ab initio transition state theory based procedure for accurately predicting the combination kinetics of two alkyl radicals is described. This procedure employs direct evaluations of the orientation dependent interaction energies at the CASPT2/cc-pvdz level within variable reaction coordinate transition state theory (VRC-TST). One-dimensional corrections to these energies are obtained from CAS+1+2/aug-cc-pvtz calculations for CH3 + CH3 along its combination reaction path. Direct CAS+1+2/aug-cc-pvtz calculations demonstrate that, at least for the purpose of predicting the kinetics, the corrected CASPT2/cc-pvdz potential energy surface is an accurate approximation to the CAS+1+2/aug-cc-pvtz surface. Furthermore, direct trajectory simulations, performed at the B3LYP/6-31G* level, indicate that there is little local recrossing of the optimal VRC transition state dividing surface. The corrected CASPT2/cc-pvdz potential is employed in obtaining direct VRC-TST kinetic predictions for the self and cross combinations of methyl, ethyl, iso-propyl, and tert-butyl radicals. Comparisons with experiment suggest that the present dynamically corrected VRC-TST approach provides quantitatively accurate predictions for the capture rate. Each additional methyl substituent adjacent to a radical site is found to reduce the rate coefficient by about a factor of two. In each instance, the rate coefficients are predicted to decrease quite substantially with increasing temperature, with the more sterically hindered reactants having a more rapid decrease. The simple geometric mean rule, relating the capture rate for the cross reaction to those for the self-reactions, is in remarkably good agreement with the more detailed predictions. With suitable generalizations the present approach should be applicable to a wide array of radical-radical combination reactions.

接着の極性効果に対する幾何平均則は正しいか?

接着の極性効果に対する幾何平均則は正しいか?

Kinetics of the CH2Cl + CH3 and CHCl2 + CH3 Radical−Radical Reactions

The CH2Cl + CH3 (1) and CHCl2 + CH3 (2) cross-radical reactions were studied by laser photolysis/photoionization mass spectroscopy. Overall rate constants were obtained in direct real-time experiments in the temperature region 301-800 K and bath gas (helium) density (6-12) x 10(16) atom cm(-3). The observed rate constant of reaction 1 can be represented by an Arrhenius expression k1 = 3.93 x 10(-11) exp(91 K/T) cm3 molecule(-1) s(-1) (+/-25%) or as an average temperature-independent value of k1= (4.8 +/- 0.7) x 10(-11) cm3 molecule(-1) s(-1). The rate constant of reaction 2 can be expressed as k2= 1.66 x 10(-11) exp(359 K/T) cm3 molecule(-1) s(-1) (+/-25%). C2H4 and C2H3Cl were detected as the primary products of reactions 1 and 2, respectively. The experimental values of the rate constant are in reasonable agreement with the prediction based on the "geometric mean rule." A separate experimental attempt to determine the rate constants of the high-temperature CH2Cl + O2 (10) and CHCl2 + O2 (11) reaction resulted in an upper limit of 1.2 x 10(-16) cm(3) molecule(-1) s(-1) for k10 and k11 at 800 K.

Effect of combining rules for cubic equations of state on the prediction of double retrograde vaporization

In this work, the phenomenon of double retrograde vaporization (DRV) is simulated using the Peng–Robinson equation of state with the classical mixing rules and several combining rules for the cross-energy and cross-co-volume parameters. The binary interaction parameters are set equal to zero in all cases, i.e., the calculations are entirely predictive. An interesting conclusion is that the predictions using the classical combining rules (geometric mean rule for a ij and arithmetic mean rule for b ij ) provide the best agreement with the experimental data for all the systems tested: methane + n-butane, methane + n-pentane, ethane + limonene, and ethane + linalool. Another interesting observation is that several combining rules for b ij , other than the arithmetic mean rule, predict the existence of three phases in equilibrium in a very narrow temperature range close to the critical temperature of methane in the methane + n-pentane system, even though, literature data indicates that n-hexane is the first n-alkane to present partial liquid phase immiscibility with methane.

Kinetics of the CCl3 + CH3 Radical−Radical Reaction

The CCl3 + CH3 (1) cross-radical reaction was studied by laser photolysis/photoionization mass spectroscopy. Overall rate constants were obtained in direct real-time experiments in the temperature region 306−800 K and bath gas (helium) density region (3−12) × 1016 atoms cm-3. The observed rate constant of reaction 1 is independent of temperature and equal to (2.05 ± 0.30) × 10-11 cm3 molecule-1 s-1. Formation of C2H2Cl2 and C2H3Cl2 was detected; however, only the C2H3Cl2 radical can be identified as a product of reaction 1 on the basis of the experimental information since the contribution of reaction 1 to the signal of C2H2Cl2 could not be separated from that of the reaction of CCl2 with CH3 (CCl2 being a minor product of photolysis of all precursors of CCl3 used). The experimental values of the rate constant are in reasonable agreement with the prediction based on the “geometric mean rule”. A separate experimental attempt to determine the rate constant of the high-temperature CCl3 + O2 (10) reaction res...

The H/(CH3)3SiH/C2H4Reaction System and the Addition of (CH3)3Si to Ethene. An End-Product Study

Me3Si and C2H5 radicals have been generated by the reaction of H atoms with mixtures of Me3SiH and C2H4 and the subsequent reactions studied by end-product analysis. The primary radicals undergo self- and cross-combination and disproportionation reactions. The Me3Si radical also adds to the C2H4, generating another radical, Me3SiCH2CH2. This radical in turn undergoes a variety of self- and cross-combination as well as disproportionation reactions with all the radicals present. A number of relative rate constants were extracted from the data. In particular the ratio k(H + Me3SiH)/k(H+C2H4) was found to be in good agreement with recent absolute determinations of these two rate constants. For the addition reaction we obtained the relative rate constant k(Me3Si + C2H4)/k1/2(2Me3Si) = (8.7 ± 0.2) × 10-10 cm3/2 s-1/2 at room temperature. From studies in the temperature range 300−470 K the following values of the Arrhenius parameters for the addition reaction were obtained: 11.6 ≤ Ea/kJ mol-1 ≤ 14.1 and 7.1 × 10-13 ≤ A/cm3 s-1 ≤ 1.9 × 10-12. In addition, it was found that alkyl and silyl radicals closely obey the geometric mean rule and that disproportionation reactions between alkyl and silyl radicals are of only minor importance. The reactivity of the silyl radical is much higher than that of the alkyl radical in adding to C2H4, and this can be explained in terms of polar effects.

Rate constant and RRKM product study for the reaction between CH3 and C2H3 at T = 298 K

The total rate constant k1 has been determined at P = 1 Torr nominal pressure (He) and at T = 298 K for the vinyl-methyl cross-radical reaction CH3 + C2H3 yields products. The measurements were performed in a discharge flow system coupled with collision-free sampling to a mass spectrometer operated at low electron energies. Vinyl and methyl radicals were generated by the reactions of F with C2H4 and CH4, respectively. The kinetic studies were performed by monitoring the decay of C2H3 with methyl in excess, 6 < |CH3|(sub 0)/|C2H3|(sub 0) < 21. The overall rate coefficient was determined to be k1(298 K) = (1.02 +/- 0.53)x10(exp -10) cubic cm/molecule/s with the quoted uncertainty representing total errors. Numerical modeling was required to correct for secondary vinyl consumption by reactions such as C2H3 + H and C2H3 + C2H3. The present result for k1 at T = 298 K is compared to two previous studies at high pressure (100-300 Torr He) and to a very recent study at low pressure (0.9-3.7 Torr He). Comparison is also made with the rate constant for the similar reaction CH3 + C2H5 and with a value for k1 estimated by the geometric mean rule employing values for k(CH3 + CH3) and k(C2H3 + C2H3). Qualitative product studies at T = 298 K and 200 K indicated formation of C3H6, C2H2, and C2H5 as products of the combination-stabilization, disproportionation, and combination-decomposition channels, respectively, of the CH3 + C2H3 reaction. We also observed the secondary C4H8 product of the subsequent reaction of C3H5 with excess CH3; this observation provides convincing evidence for the combination-decomposition channel yielding C3H5 + H. RRKM calculations with helium as the deactivator support the present and very recent experimental observations that allylic C-H bond rupture is an important path in the combination reaction. The pressure and temperature dependencies of the branching fractions are also predicted.

Global phase diagrams of model and real binary fluid mixtures Part II. Non-Lorentz–Berthelot mixtures of attractive hard spheres

The global phase diagram of a binary mixture of attractive hard spheres described by the Boublik–Mansoori–Carnahan–Starling–Leland equation of state with a van der Waals mean field attractive term is investigated in its dependence on the ratio of the hard-sphere diameters and the strength of the attraction. Two values of the energetic combining rule parameter k12, determining the deviation from the geometric mean rule, are considered. In addition to phenomena reported recently for k12=1, the shield region of quadruple points is investigated for k12=0.8, along with its associated azeotropic phenomena. For k12=1.15, the mixture exhibits a rich set of low-temperature azeotropic phenomena: cusps on the p–T projections of azeotropic lines, exchange of branches of azeotropic lines, zero-temperature limited azeotropy, and double critical azeotropic points. Some of these azeotropic phenomena are described here for the first time.

Segregation of molecules in binary solvent mixtures without H bonds. A quantitative treatment based on the theory of mobile order and disorder

This treatment applies to binary liquid mixtures of volume fractions φ1 and φ2, but only in the absence of H bonding or ionisation. An environmental layer is defined around the individual volume v1 of a given molecule 1. At a given instant, a fraction α11 of this layer contains atoms belonging to molecules of the same kind as 1 whereas the complementary fraction α12 contains atoms of molecules of kind 2. Due to the spontaneous displacements of v1 in the liquid, α11 fluctuates between the extreme values 1 and 0. However, after a long time t, the time fraction defined by the integral γ11≡(1/t)∫0tα11 dt no longer depends on the time and has the same value for all the molecules of the same kind. The four time fractions γ11 and γ12, γ22, γ21 (defined in a similar way) are characteristic of the equilibrium. They are directly related to the partition of the cohesive energy in 11, 22 and 12 interactions, ξ11=γ11φ1; ξ22=γ22φ2; ξ12=2γ12φ1=2γ21φ2. Random mixing occurs when γ11rand=γ21rand=φ1 and γ12rand=γ22rand=φ2. In this case (ξ12rand)2/(ξ11randξ22rand)=4. However, in most cases, homogeneous environments are preferred and (ξ12)2/(ξ11ξ22)=4K. The "‘environmental’' constant K is then smaller than unity. K is related to the molar volumes V1 and V2 and to the differences in standard Gibbs energies between the pure state and the state of infinite dilution in the other liquid. K can be estimated by the geometric mean rule. For liquids of similar polarities, K does not differ markedly from unity, but for mixtures of polar liquids and liquid alkanes, K is significantly smaller and an important segregation is predicted. Using the experimental solubilities of solid n-alkanes in pure solvents, the equations predict the solubilities in mixtures of the two solvents without any adapted parameter. The predicted values agree in this case, considerably better with the experimental ones than do those derived from the hypothesis of random mixing. The validity of time fractions thermodynamics has thus been experimentally verified.

Mechanistic study of the Me2SiH–Me3Si radical system

Di- and trimethylsilyl radicals, generated by the reaction of H atoms with di- and trimethylsilane, react to produce three main products: 1,1,2,2-tetramethyldisilane, pentamethyldisilane and hexamethyldisilane. These products are formed by both radical combination and radical disproportionation reactions. The disproportionation reactions form Me2Si which inserts into the Si–H bonds of the reactants. From a quantitative determination of the disilane products as a function of the reactant ratio, a value for the branching ratio of cross-disproportionation of di- and trimethylsilyl radicals relative to the branching ratio for the disproportionation of dimethylsilyl radicals can be extracted. Our results provide strong evidence that the ratio of the rate constants for hydrogen abstraction from di- and trimethylsilane by H atoms is larger than absolute rate measurements suggest. Analysis also shows that the geometric mean rule for cross-radical reaction is closely obeyed. Disproportionation reactions yielding silaethenes occur to a minor extent and are responsible for the formation of six trisilanes. Secondary reactions, mainly initiated by H-atom abstraction from tetra- and pentamethyldisilane by silyl radicals, also take place. The relative rate constants estimated for these reactions are in agreement with a previous determination.

Reverberation loss in shallow water

In this paper, reverberation loss is considered as a basic physical quantity to describe shallow water reverberation, and reverberation loss versus time was calculated by using three models: the ray-based model, the WKBZ mode models with account taken and not taken of the effects of complex eigenvalues. Numerical simulations show that the effects of complex eigenvalues cannot be neglected, and at medium ranges the reverberation loss from the ray-based model is consistent with that from the WKBZ mode model with the effects of complex eigenvalues taken into account. The experimental results shows that reverberation loss is not dependent on the bandwidth. Reverberation in shallow swater with a thermocline shows strong depth dependence, and the theoretical geometric mean rule of reverberation intensity was demonstrated by experiment.

Phase equilibria in ternary Lennard-Jones systems

Gibbs ensemble Monte Carlo simulations are used to test the ability of the van der Waals one-fluid theory to predict phase equilibria for ternary systems of particles interacting with Lennard-Jones potentials. Simulation results for asymmetric systems are compared to theoretical results for mixtures that obey the Lorentz-Berthelot combining rules. Good agreement is found between theory and simulation far form the plait point. The theory is found to overestimate the plait point, as expected from previous studies of binary systems. Liquid-liquid equilibria in a symmetric ternary system with energy parameters deviating from the geometric mean (Berthelot) rule are also investigated at three different pressures. A three-phase region is found at high pressures. The agreement between theory and simulation is excellent for the symmetric systems studied here.

Phase equilibria of binary Lennard-Jones mixtures: simulation and van der Waals l-fluid theory

Abstract The Gibbs ensemble simulation technique is used to investigate the ability of van der Waals 1-fluid theory to predict phase equilibria for binary Lennard-Jones mixtures. Simulation data for highly asymmetric mixtures with size and energy parameter ratios equal to 1.00, 0.5, 0.4, 0.35 and to 0.5, 0.33, 0.25 respectively are compared to theoretical results for cases in which unlike-pair interactions follow the Lorentz-Berthelot combining rules. Additional comparisons are made for vapour-liquid and liquid-liquid equilibria of mixtures with energy parameter deviating from the geometric mean (Berthelot) rule, and for vapour-liquid and gas-gas equilibria of mixtures with size parameter deviating from the arithmetic mean (Lorentz) rule. Good agreement was found between theory and simulation when the energy parameter deviates from the Berthelot combining rule. The agreement is less satisfactory when the size parameter deviates from the Lorentz rule, especially for the case of gas-gas equilibria.

Rate constants for some radical-radical cross-reactions and the geometric mean rule

ADVERTISEMENT RETURN TO ISSUEPREVArticleNEXTRate constants for some radical-radical cross-reactions and the geometric mean ruleLeslie J. Garland and Kyle D. BayesCite this: J. Phys. Chem. 1990, 94, 12, 4941–4945Publication Date (Print):June 1, 1990Publication History Published online1 May 2002Published inissue 1 June 1990https://doi.org/10.1021/j100375a034RIGHTS & PERMISSIONSArticle Views98Altmetric-Citations41LEARN ABOUT THESE METRICSArticle Views are the COUNTER-compliant sum of full text article downloads since November 2008 (both PDF and HTML) across all institutions and individuals. These metrics are regularly updated to reflect usage leading up to the last few days.Citations are the number of other articles citing this article, calculated by Crossref and updated daily. Find more information about Crossref citation counts.The Altmetric Attention Score is a quantitative measure of the attention that a research article has received online. Clicking on the donut icon will load a page at altmetric.com with additional details about the score and the social media presence for the given article. Find more information on the Altmetric Attention Score and how the score is calculated. Share Add toView InAdd Full Text with ReferenceAdd Description ExportRISCitationCitation and abstractCitation and referencesMore Options Share onFacebookTwitterWechatLinked InReddit PDF (660 KB) Get e-Alerts

The van der Waals equation of state around the van Laar point

The discovery of van Laar that the van der Waals equation of state for binary mixtures can be solved analytically at the double point on the critical line, provided one introduces the geometric-mean condition on the interaction parameters, is used to obtain the explicit expression for the critical line for this case. The critical line is expressed as a function of the density variables, and the origin is shifted to this double point: the van Laar point. By doing so it is possible to show that this double point is also a tricritical point. This is different from the lattice gas, where the double point is always a tricritical point. Small deviations from this point in parameter space induce very different phase diagrams. The influence of excursions from the van Laar point is expressed as a function of the state variables. Both the k factor (the deviation from the geometric-mean rule) and the ‘‘asymmetry’’ coefficient e (the deviation from the crossing point) are introduced. The results are given in the form of polynomials in local coordinates in density space. Conditions under which the double point is maintained are given and the differences between the Scott and van Konynenburg classes II, III, IV, and IV* are explained.