Multiplication constant

Whereas most chemists focused their attention on speculation about atoms and the question of atomic weights, the constant multiplicity in compounds occupied an increasingly central role. The new concept of substitution, i.e., the replacement of one element by another in a compound, started to make a major impact on chemistry in the 1840s. It was probably Dumas, who in the 1830s at the request of his father-in-law (who was the director of the famous Royal Sevres porcelain factory) resolved an event that upset a royal dinner party at the Tuil-... 

The value of a determinant is not changed if to the elements of any row (or column) are added a constant multiple of the corresponding elements of any other row (or column). 

The proof of this theorem follows from theorem A A four-by-four matrix that commutes with the y commuted with their products and hence with an arbitrary matrix. However, the only matrices that commute with every matrix are constant multiples of the identity. Theorem B is valid only in four dimensions, i.e., when N = 4. In other words the irreducible representations of (9-254) are fourdimensional. 

The requirement that det 8 = 1, implies that Tr T = 0. Note that T is then uniquely determined by this requirement and Eq. (9-383). For assume that there were two such T s that satisfied Eq. (9-383). This difference would then commute with y , and, hence, by theorem A, their difference would be a constant multiple of the identity. But both of these T s can have trace zero only if this constant is equal to zero. This unique T is given by... 

We have derived previously [4, 5] that the following expression relates the noise on data to the noise of a constant multiple of that data ... 

Walters [24] examined the effect of chloride on the use of bromide and iodide solid state membrane electrodes, and he calculated selectivity constants. Multiple linear regression analysis was used to determine the concentrations of bromide, fluorine, and iodide in geothermal brines, and indicated high interferences at high salt concentrations. The standard curve method was preferred to the multiple standard addition method because of ... 

The length (norm) of the column vector v is the positive root Adv. A vector is normalized if its length is 1. Two vectors rj and rj of an n-dimensional set are said to be linearly independent of each other if one is not a constant multiple of the other, i.e., it is impossible to find a scalar c such that ri = crj. In simple words, this means that r - and Tj are not parallel. In general, m vectors constitute a set of linearly independent vectors if and only if the equation... 

Since by assumption income in (5.7) is a constant multiple of investment, it also follows that the rate of change of investment is equal to the rate of change of income ... 

The function Y0 at) so obtained is called Neumann s Bessel function of llio second kind of zero order. Obviously if we add to Yn f) a function which is a constant multiple of >/0(.t) the resulting function is also a solution of the differential equation... 

The argument at the end of the last section shows that the equation (41). 1) possesses solutions which tend to zero as J x -s- oo, if and only if the parameter X is of the form 1 -f- 2h where >i is a positive integer. When X is of this form the required solid ion of (40.1) is a constant multiple of the function lIfn[x) defined by the equation... 

The first of these equations is called the time-independent Schrodinger equation it is a so-called eigenvalue equation in which one is asked to find functions that yield a constant multiple of themselves when acted on by the Hamiltonian operator. Such functions are called eigenfunctions of H and the corresponding constants are called eigenvalues of H. 

We can apply Schur s lemma (Proposition 6.2) to see that for any function a e Z the linear transformation 7), is a constant multiple of Ta . Consider the linear transformation r o Tan Y By Proposition 6.1, this... 

The problem to be solved in this paragraph is to determine the rate of spread of the chromatogram under the following conditions. The gas and liquid phases flow in the annular space between two coaxial cylinders of radii ro and r2, the interface being a cylinder with the same axis and radius rx (0 r0 < r < r2). Both phases may be in motion with linear velocity a function of radial distance from the axis, r, and the solute diffuses in both phases with a diffusion coefficient which may also be a function of r. At equilibrium the concentration of solute in the liquid, c2, is a constant multiple of that in the gas, ci(c2 = acj) and at any instant the rate of transfer across the interface is proportional to the distance from equilibrium there, i.e. the value of (c2 - aci). The dispersion of the solute is due to three processes (i) the combined effect of diffusion and convection in the gas phase, (ii) the finite rate of transfer at the interface, (iii) the combined effect of diffusion and convection in the liquid phase. In what follows the equations will often be in sets of five, labelled (a),..., (e) the differential equations expression the three processes (i), (ii) (iii) above are always (b), (c) and (d), respectively equations (a) and (e) represent the condition that there is no flow over the boundaries at r = r0 and r = r2. 

Construct a temperature scale in which the freezing and boiling points of water are 100° and 400°, respectively, and the degree interval is a constant multiple of the Celsius degree interval. What is the absolute zero on this scale, and what is the melting point of sulfur (MP = 444.6°C) ... 

It is also possible to combine the supermolecule and continuum approaches by using specific solvent molecules to capture the short-range effects (i.e., those involving specific noncovalent interactions between solute and solvent) and a reaction field to treat longer range effects.33-35 Alternatively, structures along the gas phase reaction coordinate can be immersed in a box of hundreds (or more) of explicit solvent molecules that are treated using force field approaches.36,37 Each type of method - the SCRF, solvent box, and supermolecule approaches - tests the importance of particular features of the solvent on the reactivity of the solute dielectric constant, multiple specific classical electrostatic interactions, and specific local directional noncovalent interactions, respectively. 

It will be useful to have in mind another way of considering the problem a function on a coset space of G is essentially a function on G invariant under translation by the subgroup. When G is GL and H the upper triangular group, for instance, it is easy to compute that no nonconstant polynomial in the matrix entries is invariant under all translations by elements of H, and thus no affine coset space can exist. (What follows from (16.1) is that there are always semi-invariant functions, ones where each translate of/is a constant multiple of/) Our problem is to prove the existence of a large collection of invariant functions for normal subgroups. 

The same type of difficulty that is resolved by use of equation (35) for the partial-equilibrium approximation may also arise in connection with the steady-state approximation. For example, part of the sum of terms that contribute to the production rate of a primary species, to which the steady-state approximation is not applied, may be a constant multiple of cz . for an intermediary that is subject to the steady-state approximation, and the remaining terms in the production rate may be smaller than (U- even though (u. is small compared with. Under this condition, inaccurate results for the concentration history of the primary species will be obtained by use of the steady-state approximation for the intermediary unless a substitution... 

Equation 3-56 is a linear, homogeneous, second-order differential equation with constant coefficients. A fundamental theory of differential equations states (hat such an equation has two linearly independent solution functions, and its general solution is the linear combination of those two solution functions. A careful examination of the differential equation reveals that subtracting a constant multiple of the solution function 0 from its second derivative yields zero. Thus we conclude that the function 0 and its second derivative must be constant multiples of each other. The only functions whose derivatives are constant multiples of the functions themselves are the exponential functions (or a linear combination of exponential functions such as sine and cosine hyperbolic functions). Therefore, the solution functions of the differential equation above are the exponential functions e or or constant multiples of them. This can be verified by direct substitution. For example, the second derivative of e is and its substitution into Eq. 3-56... 

This condition is satisfied by the function but not by the other prospective solution function e" since it tends to infinity as x gets larger. Therefore, the general solution in this case will consist of a constant multiple of e" . The value of the constant multiple is determined from the requirement that at the fin base where x = 0 the value of 0 is Noting that = c = 1, the proper value of the constant is Oj, and the solution function we are looking for is 0 x) = This function satisfies the differential equation as well as the re-... 

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