Negative Quantities Research Articles

Negative Größen bei Diophant? Teil II

In this second part of “Negative Grosen bei Diophant?” we start, as announced, by giving 33 places where Diophantus uses negative quantities as intermediate results; they appear as differences a − b of positive rational numbers, the subtrahend b being bigger than the minuend a; they each represent the (negative) basis \((\pi\lambda\varepsilon\upsilon\rho\acute{\alpha})\) of a square number \((\tau\varepsilon\tau\rho\acute{\alpha}\gamma\omega\nu o \zeta)\), which is afterwards computed by the formula (a - b)2 = a2 + b2 - 2ab. Finally, we report how the topic “Diophantus and the negative numbers” has been dealt with by translators and commentators from Maximus Planudes onwards.

Loop Selective Direction Measurement for Distance Protection

Distance relays achieve selective tripping by measurement of all short circuit fault conditions inside set reaches. The direction of the fault, forward or reverse is commonly determined with a dedicated measurement to ensure selectivity under all conditions. For the direction decision (measurement) a number of alternatives are available. This paper describes a loop selective direction measurement and illustrates by means of a typical fault why this is superior to a non loop selective direction measurement such as that based on negative sequence quantities.

An analysis of the multi-product newsboy problem with a budget constraint

This is a sequel of an earlier paper entitled “Exact, approximate, and generic iterative models for the multi-product newsboy problem with budget constraint’’ (Abdel-Malek et al., 2004) that appeared in this journal. Motivated by Lau and Lau's (Eur. J. Oper. Res., 94 (1996) 29) observation where infeasible ordering quantities (negative) were obtained when applying existing methods, the extension here examines the solution space of the problem in order to provide the necessary insight into this phenomenon. The resulting insight shows that the solution space can be divided into three regions that are marked by two distinct thresholds. The first region is where the budget is large and the solution is the same as the unconstrained problem. The second region is where the budget is medium and the constraint is binding, however the newsboy can order all the products on the list. The third region is where the budget is very tight and if the non-negativity constraints are relaxed negative order quantities may be obtained, and therefore some products have to be deleted from the original list. We show how the values of the thresholds that divide the regions are computed and extend the previous methods, when necessary, to cover each of the three-solution's domains in order to determine the optimum order quantity for the various products. Numerical examples are given to illustrate the application of the developed procedures.

Equal emissions per capita over time – a proposal to combine responsibility and equity of rights for post‐2012 GHG emission entitlement allocation

AbstractAny effective future climate policy regime based on the Kyoto Protocol requires the determination of the concrete contribution regarding time and quantity for each party. Based on the two justice principles, responsibility and equity of rights, that form the basis for the so‐called Brazilian Proposal and Contraction and Convergence respectively, a new approach is developed: future emission rights are allocated on the basis of equal emissions per capita over time. By so doing not only are emissions per capita (EPC) taken into account during the allocation but also their evolution over time. This may result in negative quantities of emission rights for some parties due to their historical ‘burden’. On the other hand, parties with low EPC would be allocated large amounts of ‘fair air’. Even though this approach may currently lack political support by powerful parties, it offers another analytical reference point for the political bargaining process on future allocations. Copyright © 2004 John Wiley & Sons, Ltd and ERP Environment.

Equal Emissions per Capita over Time - A Proposal to Combine Responsibility and Equity of Rights

After Future climate policy regimes may be based on the Kyoto-Protocol or on other policy instruments such as carbon-taxes. Any effective regime based on the Protocol requires the determination of the concrete contributions by each Party. This involves namely the time of contribution and the quantification of the contribution itself. By now many proposals exist for the two issues, as for example thresholds like GDP per capita for the question of timing or emissions per capita for an allocation of emission rights. Based on the two justice principle responsibility and equity of rights that form the basis for the so-called Brazilian Proposal and Contraction & Convergence respectively, a new approach is developed: Future emission rights are allocated on the basis of equal emissions per capita over time. By so doing not only are emissions per capita (EPC) taken into account during the allocation but also their evolution over time. I show that nations with high EPC may even be allocated negative quantities of emissions right due to their historical burden. On the other hand, Parties with low EPC would be allocated large amounts of fair air which can increase the incentive to accept absolute emission targets. Even though this approach may currently lack political acceptability, it offers another analytical reference point for the political bargaining process on future allocations.

Nonideal mixing of dodecyltrimethylammonium halides and nonionic surfactant in adsorbed films and micelles

The ion–dipole interaction between dodecyltrimethylammonium cations and nonionic surfactant molecules in adsorbed films and micelles was investigated by concentrating on the difference in the degree of counterion binding by employing dodecyltrimethylammonium chloride (DTAC)–octyl methyl sulfoxide (OMS) and dodecyltrimethylammonium bromide (DTAB)–OMS mixtures. The phase diagrams of adsorption and micelle formation were constructed and then the nonideal mixing of different species of surfactants was demonstrated in terms of the excess Gibbs free energies of adsorption and micelle formation, and the surface excess areas. Furthermore the dependence of them on the counterion was clearly shown. All these results were found to support our previous view that the direct interaction between surfactant cation and the dipole of the hydrophilic part of a nonionic surfactant is essential in cationic-nonionic surfactant mixtures, i.e., the DTAC system with a lower counterion biding has more negative excess thermodynamic quantities than the DTAB system with a higher one.

Isothermal vapour-liquid equilibria data of methanol + propan-2-OL, methanol + 2-methyl propan-2-OL + 2-methylpropan-2-0L binary mixtures at 333.15k

The vapour-liquid equilibria (VLE) data of methanol + propan-2-ol + 2-methylpropan-2-ol binary mixtures are investigated at 333.15k, using gas chromatographic technique. For binary mixtures, the total pressure, the compositions of the components in the vapour phase are obtained from the instrument and these data are then used to calculate the second virial coefficients of pure components to account for the deviations of these mixtures from ideal solution behaviour. The excess Gibbs free energies of mixing for these binary systems are then determined. The y-x plots reveals that the mixtures methanol + propan-2-ol and methanol +2-methylpropan-2-ol deviate strongly from ideal solution behaviour. The excess Gibbs free energies (GE) of mixing for these binary systems are negative quantities, confirming negative deviations from ideality. Minimum GE value of 450 J.mol-1 is obtained for methanol + 2-methylpropan-2-ol mixtures at 0.45 mole fraction of methanol. Methanol + propan-2-ol mixtures exhibit a minimum GE value of 100 J mol-1 at 0.5 mole fraction of methanol. The least minimum GE value of 15 J. mol-1 is obtained for propan-2-ol +2-methyl propan-2-ol mixtures at 0.5 mole fraction of propan-2-ol. The consistency of these data is verified using Wilson's model for calculating GE values. The calculated change in Gibbs free energies (GEcalc) obtained were in good agreement with the experimentally determined GE values. KeyWords: VLE, Spectrometric, Deviation, Excess, Ideality. (Global Journal of Pure and Applied Sciences: 2002 8(4): 483-490)

The Transfer Dilemma

In this article we provide an overview of research on transfer, highlighting its main tenets. Then we look at interviews of two 5th-grade students learning about mathematical concepts regarding operations on positive and negative quantities. We attempt to focus on how their learning is influenced by their prior knowledge and experience. We take the position that transfer is a theory of learning and we attempt to show that it cannot provide a solid foundation for explaining such examples of learning.

Improved 5-step modeling of the Photosystem II S-state mechanism in cyanobacteria

We present a model of the S-state mechanism, as well as an improved eigenvalue analysis, that integrate into a coherent ensemble several features found since the S-state model was initially developed. These features include the presence of S-1, deactivations in the dark interval between flashes, and the change in the number of active PS II centers by photoinhibition or photoactivation. A new feature is the capacity to predict the steady-state distribution of S-states under conditions of steady photoinhibition or photoactivation. The improved eigenvalue analysis allowed the calculation of the initial S-state distribution. In addition, the model resolved 'true' photochemical misses from apparent misses due to deactivations in the dark interval between flashes. The model suggested that most of the misses that are commonly reported are due to deactivations, and not to an intrinsic inefficiency of the photochemical mechanism of PS II. Because models that allow double-hits encompassing the S2 to S3 transition often predict negative initial quantities of S2 in cyanobacteria, our proposed model specifically prohibited them. The model accounts for inhomogeneous misses and a steady-state distribution of the type (S2)≈(S1)>(S3)≈(S0). This 5-step model uses only 4 probabilities, and is therefore easy to handle. The use of this model is critical for the analysis of several cyanobacterial strains, as well as for any species that show non-negligible deactivations in the dark interval between flashes.

Thermodynamic and spectroscopic properties of 2-pyrrolidinones. 2. Dielectric properties of 2-pyrrolidinone in binary mixtures

Dielectric permittivities of 2-pyrrolidinone - acetone, -dimethyl sulfoxide,-2-propanol, -dichloromethane and -water systems were measured as a function of mole fraction over the whole composition range at 30 and 50°C. The excess dielectric permittivities are predominantly negative for all the mixtures and the excess molar polarizations are negative except for 2-pyrrolidinone - water and 2-pyrrolidinone - dimethyl sulfoxide mixtures. The large negative excess quantities are indicative of the strong specific interactions between the like and unlike components of the solution mixtures.

Surface Heat Transfer Fluctuations on a Turbine Rotor Blade Due to Upstream Shock Wave Passing

A theoretical model to explain observed rapid large-scale surface heat transfer rate fluctuations associated with the impingement of nozzle guide vane trailing edge shock waves on a transonic turbine rotor blade is described. Experiments were carried out in the Oxford Isentropic Light Piston Cascade Tunnel using an upstream rotating bar system to simulate the shock wave passing. High-frequency surface heat transfer and pressure measurements gave rapidly varying, large, transient signals, which schlieren photography showed to be associated with the impingement of passing shock waves on the surface. Heat transfer rates varying from three times the mean value to negative quantities were measured. A simple first-order perturbation analysis of the boundary layer equations shows that the transient adiabatic heating and cooling of the boundary layer by passing shock waves and rarefactions can give rise to high-temperature gradients near the surface. This in turn leads to large conductive heat transfer rate fluctuations. The application of this theory to measured fluctuating pressure signals gave predictions of fluctuating heat transfer rates that are in good agreement with those measured. It is felt that the underlying physical mechanisms for shock-induced heat transfer fluctuations have been identified. Further work will be necessary to confirm them in rotating experiments.

Peculiarity of high-Tc oxide superconductors

Abstract The following conclusions are arrived through analyzing the experiments: 1) The peculiarity of the high-Tc oxide superconductors BaPbBiO, LaSrCuO and YBaCuO is that their pseudo-Coulomb potentials μ∗ are negative quantities. 2) The superconductivity of these three oxides can be understood within the framework of the conventional BCS-Eliashberg theory if the said peculiarity is taken into account in the theory.

A Further Note on Sraffa's Negative Quantities of Allegedly Embodied Labour

A Further Note on Sraffa's Negative Quantities of Allegedly Embodied Labour

Crystal Symmetry: Theory of Colour Crystallography

This is a rather technical account of the groups which result when cyclic subgroups, expressed as permutations of colour–the simplest is the alternation of black and white or equivalently positive and negative quantities – are combined with the 230 space groups of traditional crystallography. The book seems to be confined to the two-colour case – the Shubnikov groups of which there are 1651.

Implementation of an incoherent optical image restoration method: limitations related to optical subtraction

Incoherent optical information processing is known to be far less sensitive to noise than coherent processing. However, it is restricted to real non-negative quantities. In cases like approximate deconvolution for image restoration, where positive and negative quantities are involved, the use of incoherent illumination is possible only at the expense of some additional sophistication needed to incorporate the negative parts; in particular, the restoration can be achieved by combining two incoherent convolutions and one subtraction. This paper describes an experiment of this type, with special emphasis on the practical implementation and on the difficulties related to the subtraction. Alignment and normalization of the two images to be subtracted are critical issues. If the subtraction is performed optically, the dynamic range of this operation is an additional problem.

Thermoelastic constants of plastic-impregnated fiberglas in a cross-wound shell

1. Theoretical calculations on layered Fiberglas material of cross-wound type have shown that there are winding angles and filler contents for which the effective thermal-expansion coefficient and Poisson's ratio become negative. 2. Theoretical calculations have been performed on the magnitudes and ranges of variation of these negative quantities. 3. The theoretical results have been confirmed by experiment for Fiberglas having parameters corresponding to negative values for the thermal-expansion coefficient.

A positioning system using three-valued codes

A significant simplification of the computation circuits for positioning systems is achieved by the use of three-valued codes. The use of the odd radix makes it possible to provide a control system symmetrical to positive and negative quantities which relieves the necessity to complement digital information. The paper discusses the construction of suitable cyclic codes and gives the logical design for the necessary conversion logic. Design proposals for the remainder of the control system are also corporated.

The Residual Error in the Canadian National Accounts

GRoss national product is estimated in the Canadian National Accounts in two substantially independent ways: by adding the various components of production (mainly incomes); and by adding the various components of final expenditure. As both aggregates measure the value of goods and services produced by Canadians in a given period they should be equal. However, all measurements are subject to some error so that there is a statistical discrepancy between the two totals. In this way the residual error item arises.' The procedure adopted in the tables in the National Accounts is to present one half the difference between the independently calculated aggregates as an addition to the smaller aggregate or as a deduction from the larger aggregate. This averaging procedure is regarded as preferable to showing the entire amount of the discrepancy on one side of the Accounts (as is the practice in most countries) for two reasons. First, rough appraisals of the relative accuracy of the product and expenditure sides of the Accounts suggest that they have approximately equal reliability and so should be given equal weight on the average. Secondly, the aggregate which generates the residual error is considerably larger than gross national product itself. This is so because the error-generating aggregate is obtained by summing the components of gross national product and gross national expenditure without regard to sign and then deducting certain items common to both totals (such as the imputed items) and this is considerably larger than gross national product or expenditure as both gross national product and expenditure contain very large negative quantities (such as imports and farm operating expenses). Consequently, comparing one half the basic discrepancy with gross national product gives roughly the same result as comparing the whole discrepancy with the larger aggregate.2 Clearly there will be some tendency for errors to cancel out so that a small residual error does not mean that components are free of error.

The Residual Error in the Canadian National Accounts

Gross national product is estimated in the Canadian National Accounts in two substantially independent ways: by adding the various components of production (mainly incomes); and by adding the various components of final expenditure. As both aggregates measure the value of goods and services produced by Canadians in a given period they should be equal. However, all measurements are subject to some error so that there is a statistical discrepancy between the two totals. In this way the residual error item arises.The procedure adopted in the tables in the National Accounts is to present one half the difference between the independently calculated aggregates as an addition to the smaller aggregate or as a deduction from the larger aggregate. This averaging procedure is regarded as preferable to showing the entire amount of the discrepancy on one side of the Accounts (as is the practice in most countries) for two reasons. First, rough appraisals of the relative accuracy of the product and expenditure sides of the Accounts suggest that they have approximately equal reliability and so should be given equal weight on the average. Secondly, the aggregate which generates the residual error is considerably larger than gross national product itself. This is so because the error-generating aggregate is obtained by summing the components of gross national product and gross national expenditure without regard to sign and then deducting certain items common to both totals (such as the imputed items) and this is considerably larger than gross national product or expenditure as both gross national product and expenditure contain very large negative quantities (such as imports and farm operating expenses). Consequently, comparing one half the basic discrepancy with gross national product gives roughly the same result as comparing the whole discrepancy with the larger aggregate.Clearly there will be some tendency for errors to cancel out so that a small residual error does not mean that components are free of error.

Vapor Pressure of Molecular Crystals. XIII. Vapor Pressure of α-Glycine Crystal. The Energy of Proton Transfer

Abstract Sublimation pressures of crystalline α-glycine were measured between 138 and 153°C by the effusion method. The pressure equation logP(mmHg)=13.666-(6816.8/T) fits well the experimental data. The heat, entropy and free energy (at 25°C) of sublimation were 31.18±0.49 kcal./mole, 49.33±1.17 cal./deg. mole and 16.47±0.60 kcal./mole, respectively. An energy cycle was constructed which correlates zwitterion and molecule in crystal and vapor states, and hydrated zwitterion and molecule in solutions. The observed heat of sublimation corresponds to the change in heat content in going from the zwitterioncrystal to gaseous molecules. The molecular configuration is more stable in vapor state. By using the experimental values of the heat of sublimation, the heat of solution of crystals into water and the temperature dependence of pKa in solution, with the help of the theoretical values of the lattice energy of the crystal, the following heat values were determined; the heat of proton jump in vapor, NH3+CH2COO− → NH2CH2COOH, −72 kcal./mole, the heat of proton jump in the crystal 7∼10 kcal./mole, hypothetical heat of sublimation of crystals composed of molecules 21∼24 kcal./mole, the heats of hydration of a zwitterion and a molecule −99.8 and −17.8 kcal./mole, respectively, the heat of solution of the hypothetical crystal to give a solution of molecular glycine 3∼6 kcal./mole, and the heat of self-neutralization in aqueous solution 9.7 kcal./mole. The negative quantities show that the processes are exothermic. It was shown that the proton transfer in a medium of large dielectric constant or in a strong crystalline field requires only a small amount of energy and its significance in chemical reactions and biochemical activities was suggested. Each of the derived heat values was examined by independent checks from different sources, except for the energies of proton transfer.