Similarity, defined

Anuther concept that is extremely powerful when considering lattice structures is the fi i i/imca/ lattice. X-ray crystallographers use a reciprocal lattice defined by three vectors a, b and c in which a is perpendicular to b and c and is scaled so that the scalar juoduct of a and a equals 1. b and c are similarly defined. In three dimensions this leads to the following definitions ... 

The parameters rj and T2 are the vehicles by which the nature of the reactants enter the copolymer composition equation. We shall call these radical reactivity ratios, although similarly defined ratios also describe copolymerizations that involve ionic intermediates. There are several important things to note about radical reactivity ratios ... 

The sequence g(r-i)w+2i igrN-t-i , is similarly defined between Tp and Tr-n lattices, by successive additions of randomly selected edges linking vertices a dis-... 

In Chapter 4 we considered an acid to be a substance that produces an excess of H+ ions in water. A base was similarly defined to be a substance that forms excess OH- ions in water solution. This approach, first proposed by Svante Arrhenius in 1884, is a very practical one, but it has one disadvantage. It severely limits the number of reactions that qualify as acid-base. 

We consider a nuclear wave function describing collisions of type A + BC(n) AC(n ) + B, where n = vj, k are the vibrational v and rotational j quantum numbers of the reagents (with k the projection of j on the reagent velocity vector of the reagents), and n = v, f, k are similarly defined for the products. The wave function is expanded in the terms of the total angular momentum eigenfunctions t X) [63], and takes the form [57-61]... 

The second new program allows the user to compare the shapes of molecular weight distributions. For example, if we have the cumulative distribution of hydrodynamic volume for two polymers we can plot the hydrodynamic volume corresponding to the 10th percentile of the distribution for polymer A against the similarly defined hydrodynamic volume for polymer B. Such a plot, made for the entire distribution of both polymers, is called a "quantile... 

In a cosmology with significant cosmological constant, one can similarly define Qa = A/3/P. 

The symmetric sample correlation matrix R (or p) is similarly defined by its current element... 

Danish chemist Julius Thomsen, regarded the heat evolved in a chemical reaction as a measure of chemical affinity, on the grounds that heat represented the work done by chemical forces. Thomsen and Berthelot similarly defined affinity as the force that unites the component parts of a chemical compound and that must be overcome by an equivalent force, the quantity of which can be measured by heat evolved. 52... 

The symbols suggested by Jensen, based on Niggli s proposals, describe the local coordination by means of coordination number ratios. For instance, a formula AEm/n will indicate a binary compound where m is the coordination number (defined as the nearest-neighbour number (NNN)) of atoms E around A and n is the similarly defined coordination number of A around E. 

The extreme forms, but not the intermediate forms, of asymmetric similarity defined by Tversky (6) given in Eqs. 2.26 and 2.27 can be transformed into two symmetric measures by taking the maximum and minimum of the set cardinalities in the denominators of the two equations. The forms of these equations are obtained in analogy to those developed by Petke (33) for vectors and field-based functions (see Subheadings 2.3. and 2.4. for further details) ... 

Hence, the elastic modulus corresponds in principle to the force per square millimeter that is necessary to extend a rod by its own length. Materials with low elastic modulus experience a large extension at quite low stress (e.g., rubber, = 1 N/mm ). On the other hand, materials with high elastic modulus (e.g., polyoxymethylene, s 3500 N/mm ) are only slightly deformed under stress. Different kinds of elastic modulus are distinguished according to the nature of the stress applied. For tension, compression, and bending, one speaks of the intrinsic elastic modulus ( modulus). For shear stress (torsion), a torsion modulus (G modulus) can be similarly defined, whose relationship to the modulus is described in the literature. 

Harkins defines two terms, the work of cohesion We which is the work done when a bar of liquid of unit cross sectional area is pulled apart against the cohesive forces. It is thus numerically equal to twice the sur ce energy of the liquid. The work of adhesion Wa is similarly defined as the work required to pull apart a composite bar consisting of half of one liquid and half of the other, at the place of junction. 

The time dependent centroid Hamiltonian may be similarly defined as... 

Equivalent Weight and Volume an d Their Precision Indexes for Comparison of Explosives in Air. Data for mean peak pressures and positive impulses determine figures of merit which express performances of expls fired in air. Equivalent Weight (EW) Equivalent Volume (EV) are easily interpretable by ordnance designers. The EW of a new expl is the ratio of wt of a known expl to the wt of a new expl which gives. equiv power as measured by peak pressure or positive impulse. The EV is similarly defined... 

The reader should check that Definition 8.7 is satisfied and notice that those factors of 1 /2 are necessary. Thus Ti is a Lie algebra homomorphism. To see that it is an isomorphism, note that the matrices on the three right-hand sides of the defining equations for Ti form a basis of su(2). Similarly, defining 72 0Q 5o(3) by... 

AG is change in the free energy as the process proceeds. It is calculated from the sum of the free energies of all the reaction product species minus the sum of the free energies of all the reactant species. These quantities depend on the temperature, pressure, and concentrations of each of the species. The changes in the enthalpy, AH, and entropy, A5, are similarly defined as the difference between the product and reactant values. 

Electrolytic resistivity tthe reciprocal of conductivity is similarly defined as the electrical resistance of a unit cuhe of solution, h is expressed in the same units as electrical resistivity, i.e.. ohms times u unit of length. Most commonly we find ohm-cin (J2-cml and ohm-meter (fi-m) ... 

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