Summation convention

The symbol M represents the masses of the nuclei in the molecule, which for simplicity are taken to be equal. The symbol is the Kionecker delta. The tensor notation is used in this section and the summation convention is assumed for all repeated indexes not placed in parentheses. In Eq. (91) the NACT appears (this being a matrix in the electronic Hilbert space, whose components are denoted by labels k, m, and a vector with respect to the b component of the nuclear coordinate R). It is given by an integral over the electron coordinates... 

The summation convention for double indices, for example, k in Eq. (113), is assumed, as before. However, we no longer make distinction between covariant and contravariant sets.) We set ourselves the task to find anti-Hermitean operators Xf, such that... 

Again, the summation convention is used, unless we state otherwise. As will appear below, the same strategy can be used upon tbe Dirac Lagrangean density to obtain the continuity equation and Hamilton-Jacobi equation in the modulus-phase representation. 

The momentum and continuity equations give rise to a 22 x 22 elemental stiffness matrix as is shown by Equation (3.31). In Equation (3.31) the subscripts I and / represent the nodes in the bi-quadratic element for velocity and K and L the four corner nodes of the corresponding bi-linear interpolation for the pressure. The weight functions. Nr and Mf, are bi-qiiadratic and bi-linear, respectively. The y th component of velocity at node J is shown as iPj. Summation convention on repeated indices is assumed. The discretization of the continuity and momentum equations is hence based on the U--V- P scheme in conjunction with a Taylor-Hood element to satisfy the BB condition. 

Aj = dj A/ (summation convention over the repeated index / is implied)... 

The range convention and the summation convention will be used. The range convention is any subscript that appears only once on one side of an expression takes on values 1, 2, and 3. The summation convention is any subscript that appears twice on one side of an expression is summed from 1 to 3. The repeated index is called the dummy index. 

Chirgwin, B. H., Phys. Rev. 107, 1013, Summation convention and the density matrix in quantum mechanics. ... 

This expression demonstrates use of the Einstein summation convention 6. The significance of r is made clear by examining a particle momentarily at rest in a Lorentz system. The components of the vector, transformed dx = (0, 0, 0, icdt ) and dr2 = —(1 /c dx dx = [dt )2. Thus dr is the time interval on a clock travelling with the particle and is therefore referred to as the interval of the particle s proper time or world time. The relationship between dr and an interval of time as measured in a given Lorentz system can be derived directly by expanding the equation... 

We assume the familiar summation convention often ascribed to Einstein, so the repeated jS is understood as running through the standard index set for pairs. 

The electrostatic interaction between two nonoverlapping charge distributions A and B, consisting of NA and NB atoms, respectively, and each represented by their atom-centered multipole moments, is given by (using the Einstein summation convention for the indices a, / , y) (Buckingham 1978)... 

Using Einstein s summation convention, the formula for the interaction of a charge-current distribution j ix) at a space-time point x = x, x, x, x ) = (ct,x) with the local electromagnetic four-potential A (x),... 

In this equation ut should be interpreted as the volumetric flux density (directional flow rate per unit total area). The indexes range from 1 to 3, and repetition of an index indicates summation over that index according to the conventional summation convention for Cartesian tensors. The term superficial velocity is often used, but it is in our opinion that it is misleading because n, is neither equal to the average velocity of the flow front nor to the local velocity in the pores. The permeability Kg is a positive definite tensor quantity and it can be determined both from unidirectional and radial flow experiments [20], Darcy s law has to be supplemented by a continuity equation to form a complete set of equations. In terms of the flux density this becomes ... 

Although we are concerned with linear algebra it will be convenient to use a notation slightly different from the usual matrix notation and to adopt the summation convention from tensor analysis. Thus asr will denote the element in the rth row and 5 th column of the matrix a = (a ). The product a fi of a with a matrix ft = (, ) is the matrix (otsrfi s), where summation over s = 1, 2,. .., s is implied. The range of a particular affix (i.e., s = 1,2, s) is given in the table of nomenclature. In this notation the order of any matrix is apparent and the use of partitioned matrices can be represented rather easily. 

Since the publication of some prolegomena to the rational analysis of systems of chemical reaction [1] other cognate work has come to light and some earlier statements have been made more precise and comprehensive. I would like first to advert to an earlier work previously overlooked and to mention some recent publications that partially fill some of the undeveloped areas noticed before. Secondly, I wish to extend the theorem on the uniqueness of equilibrium to a more general case and to establish the conditions for the consistency of the kinetic and equilibrium expressions ( 2, 3). Thirdly, the conception of a reaction mechanism is to be reformulated in a more general way and the metrical connection between the kinetics of the mechanism and those of the ostensible reactions clarified. The notation of the earlier paper ([1], hereinafter referred to as P) will be followed and augmented where necessary. In particular the reader is reminded that the range of each affix is carefully specified and the summation convention of tensor analysis is employed. 

For notational convenience, the components of r are here r1 r2, r3 rather than x y z. The Einstein summation convention, an implied summation over repeated indices, is used ineq. (11). 

Each response is the sum of terms in that row (observing the summation convention). The coefficients are tensors, and the rank n of T(n) is shown by the number of subscripts. 

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