Hughes equation

Resonance excitation of trapped ions is used extensively (see below) and is possible because in a quadrupole field, the trapped ions behave as simple harmonic oscillators (SHO s), which can be driven by suitable application of an RF field at a frequency related to the secular frequency of the oscillating ion. The approximate frequency of motion for such an oscillator is given by (see March and Hughes equation (3.2)) ... 

Development of weighted residual finite element schemes that can yield stable solutions for hyperbolic partial differential equations has been the subject of a considerable amount of research. The most successful outcome of these attempts is the development of the streamline upwinding technique by Brooks and Hughes (1982). The basic concept in the streamline upwinding is to modify the weighting function in the Galerkin scheme as... 

Brooks, A. N, and Hughes, T. J.R., 1982. Streamline-upwind/Petrov Galerldn formulations for convection dominated hows with particular emphasis on the incompressible Navier -Stokes equations. Cornput. Methods Appl Meek Eng. 32, 199-259. 

Wo shall introduce this method by examining the determination of sulfur in hydrocarbons as carried out by Hughes and Wilczewski.5 The fundamental relationship (by analogy with Equation 3-1) is... 

Mackay, D., Hughes, A.I. (1984) Three parameter equation describing the uptake of organic compounds by fish. Environ. Sci. Technol. 18, 439 144. 

The final rate expressions, which were used in the present work, were given by Hou and Hughes (2001). In these rate expressions all reaction rate and equilibrium constants were defined to be temperature-dependent through the Arrhenius and van t Hoff equations. The particular values for the activation energies, heats of adsorption, and pre-exponential constants are available in the original reference and were used in our work without alteration. 

The number of reflection intensities measured in a crystallographic experiment is large, and commonly exceeds the number of parameters to be determined. It was first realized by Hughes (1941) that such an overdetermination is ideally suited for the application of the least-squares methods of Gauss (see, e.g., Whittaker and Robinson 1967), in which an error function S, defined as the sum of the squares of discrepancies between observation and calculation, is minimized by adjustment of the parameters of the observational equations. As least-squares methods are computationally convenient, they have largely replaced Fourier techniques in crystal structure refinement. 

The presumption that family is the best place for children is part of the same, often untested, belief that kin relations are beneficial and that kin networks are stable and willingly supportive (Cramer and McDonald 1996). These expectations do not just operate on the social plane they are embedded deep in the framework of social welfare systems everywhere too (Kelley 1993). However, the addition of one or, often, more children to the household can have a major impact on its functioning. Family strain, conflict and instability can all be part of the equation where another family member assumes the care of children (Caliandro and Hughes 1998 Cramer and McDonald 1996). It might cause problems of overcrowding and resentment as already resident children are forced to live in cramped... 

The prototype of this sequence has been discovered by Brewis and Hughes as early as in 1965 in the carbonylation of 1,5-diene (Equation (26)), " though in this case the process is initiated not by oxidative addition of Pd(0) to C-X bond, but by the addition of hydridopalladium complex to double bond. ... 

Equations (1) and (4) or other variations of the 12-6 power law are often called the Lennard-Jones potential. The numerical values of the constants in the Lennard-Jones potential may be obtained from studies of the compressibility of condensed phases, the virial coefficients of gases, and by other methods. A summary of these methods and other expressions for the molecular interaction energy can be found in the book by Moelwyn-Hughes (1964). 

In 1933 the two still widely accepted mechanisms for nucleophilic displacement reactions were proposed by Hughes, Ingold, and Patel.4 They found that decomposition of quartenary ammonium salts, R4N+Y, to give R3N and RY exhibited two different kinds of kinetic behavior depending on the ammonium salt used. For example, when methyl alcohol was formed from trimethyl-n-decylammonium hydroxide (Equation 4.3), the rate of formation of methyl alcohol was found to be second-order, first-order each in trimethyl-n-decylam-monium cation and in hydroxide ion as in Equation 4.4. On the other hand, the rate of formation of diphenylmethanol from benzhydryltrimethylammonium... 

These predictions of effect of solvent polarity on reaction rates were first made by Hughes and Ingold in 1935. They searched the literature of direct displacement reactions and found that for charge types 1-3 the experimental facts agreed with their predictions. For example, the reaction of ethyl iodide with triethylamine (Equation 4.9) an SN2 displacement of type 2, does proceed more... 

It is only more recently that reactions of Charge type 4 have been known and studied, but the theory proposed by Hughes and Ingold predicted the results accurately here too, for the reaction of trimethylamine with trimethylsulfonium ion (Equation 4.12) proceeds more rapidly in nonpolar than in polar solvents.18... 

The first term, representing acid-"catalyzed" hydrolysis, is important in reactions of carboxylic acid esters but is relatively unimportant in loss of phosphate triesters and is totally absent for the halogenated alkanes and alkenes. Alkaline hydrolysis, the mechanism indicated by the third term in Equation (2), dominates degradation of pentachloroethane and 1,1,2,2-tetrachloroethane, even at pH 7. Carbon tetrachloride, TCA, 2,2-dichloropropane, and other "gem" haloalkanes hydrolyze only by the neutral mechanism (Fells and Molewyn-Hughes, 1958 Molewyn-Hughes, 1953). Monohaloalkanes show alkaline hydrolysis only in basic solutions as concentrated as 0.01-1.0 molar OH- (Mabey and Mill, 1978). In fact, the terms in Equation(2) can be even more complex both elimination and substitution pathways can operate, leading to different products, and a true unimolecular process can result from initial bond breaking in the reactant molecule. 

SUPG) developed by Brooks and Hughes [3]. Essentially, the finite element equations remain the same however, as shown here, modified shape functions are introduced on the upwind side of a nodal point. Hence, we have two interpolation, or shape, functions that define the temperature, or convected variable, distribution. One definition uses the conventional shape functions given by... 

A.N. Brooks and T.J.R. Hughes. Streamline upwind-Petrov-Garlerkin formulation for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations. Comp. Meth. Appl. Mech. Eng., 32 199, 1982. 

---