Repeat index

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... 

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. 

In all expressions the Einstein repeated index summation convention is used. Xi, x2 and x3 will be taken to be synonymous with x, y and z so that o-n = axx etc. The parameter B will be temperature-dependent through an activation energy expression and can be related to microstructural parameters such as grain size, diffusion coefficients, etc., on a case-by-case basis depending on the mechanism of creep involved.1 In addition, the index will depend on the mechanism which is active. In the linear case, n = 1 and B is equal to 1/3t/ where 17 is the linear shear viscosity of the material. Stresses, strains, and material parameters for the fibers will be denoted with a subscript or superscript/, and those for the matrix with a subscript or superscript m. 

To illustrate how this works, let us suppose that there is only one operating variable, of which the right-hand side of (8) is always a unimodal function. Special as this may seem, it is the important case in Chapter 7. Let us denote df/dT by/r, dfidui by/,-, and use the summation convention on the repeated index i = I,. .. n. Then Eq. (8) may take one of three forms ... 

Tensors, from the same or different fields, can be combined by outer multiplication, denoted by juxtaposing indices with order preserved on the resultant tensor.33 It is possible that an index is present both in the covariant and contravariant index sets then with the repeated index summation convention, both are eliminated and a tensor of lower rank results. The elimination of pairs of indices is known as contraction, and outer multiplication followed by contraction is inner multiplication.33 In multiplication between tensors, contractions cannot take place entirely within one normal product (i.e., the generalized time-independent Wick theorem see Section IV) hence such tensors are called irreducible. 

We now show that the projection of the Schrodinger equation for the CCSDT wave function on the triply excited space [cf. Eq. (106)] can be written in terms of (at worst) products of unmodified rank 3 cluster coefficients and modified rank 2 integrals. Tensor notation with repeated index summation convention will be used, except, of course, for the permutation operator. 

According to the summation convention, we must sum over any repeated index over all possible values of that index. So the scalar product produces a scalar that is equal to A i Si -(- A2B2 + A3B3, whereas the vector product is a vector, the /th component of which is SijkAjBit (so, for example, the component in the 1 direction is A2B2 — A2B2), and the dyadic product is a second-order tensor with a typical component A, Bj (if we consider all possible combinations of i and j, there are clearly nine independent components). 

Here V and Gkm are the fluid velocity and the shear tensor components in the Cartesian coordinates X, X2, A3. The sum is taken over the repeated index m since the fluid is incompressible, it follows that the sum of the diagonal entries Gmm is zero. 

In both these equations there is no summation over the repeated index i. 

Here, as before, i,j= 1,2,3, where 1, 2, 3 stand for the axes x, y, z respectively, and the summation over repeating indexes is assumed. 

Here the summation is to be performed over repeating indexes. Substitution of (5.203) into (5.202) leads to... 

Here we use the summation rule we sum through the repeating indexes, e.g.. 

It represents the negative pressure at a site that promotes dilatation and can affect the critical threshold stress that governs the transition from elastic to plastic behavior. In the above equation, and in other eases that follow, a repeated index of subscripts implies summation over all eoordinates such as, e.g., in eq. (3.3). 

RBEE). It is necessary at this point to note that for the above equation and for all equations, Einstein s summation convention is in place which implies (if not explicitly defined otherwise) a summation over repeated index. In RBEE, C,y(A) = 5y if A e E, dyU if A is on a smooth boundary, and 0 if A V. The 2D first and second fundamental solutions of the Stokes flow, e.g., G(A, P) and //(A, P), that appear as kernels in the integrals in RBEE... 

Here the summation is performed over the repeating indexes. A is the transformation matrix with components Ay (ij= 1,2,3) and determinant det(A) = 1 the factor Ir denotes either the presence (tr=l) or the absence (tr = 0) of the time-reversal operation coupled to the space transformation Ay. For the case when the matrices A represent all the generating elements of the material point symmetry group (considered hereinafter) the identity = dY should be valid for nonzero components of the piezotensors. 

Here a, p = 1, 2, 3 and, following Einstein, the repeated index p means summation over p. 

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