Rank of the algebra

They are constructed from powers of the operators Xs and can be linear, quadratic, cubic,. Quite often a subscript is attached to C in order to indicate the order. For example, C2 denotes a quadratic invariant. The number of independent Casimir invariants of an algebra is called the rank of the algebra. It is easy to see, by using the commutation relation (2.3) that the operator... 

The situation is summarized in Table A.4, where the results for the exceptional algebras are also given. One may note that the number of integers that characterize the representations is also equal to the rank of the algebra. 

This is called a chain. Each subalgebra has one (or more) Casimir operator(s) C(G,) which commute with all the operators of that subalgebra. The Casimir operator is usually bilinear in the generators and the number of linearly independent Casimir operators is the rank of the algebra. In (59) the Casimir operator of the last subalgebra necessarily commutes with all the Casimir operators of the earlier subalgebras. The Hamiltonian is given as a linear combination of the Casimir operators for the chain of Eq. (59). 

Note that if desired, the Cartan subalgebra T may be interpreted as a proper subspace of transformations adp which corresponds to zero eigenvalue. The multiplicity of zero eigenvalue is equal to r, that is, to the rank of the algebra G of the dimension of the Cartan subalgebra (Fig. 14). 

Here m < 5, n = 8, p > 3. Choose D, V, i, k, and as the primary variables. By examining the 5x5 matrix associated with those variables, we can see that its determinant is not zero, so the rank of the matrix is m = 5 thus, p = 3. These variables are thus a possible basis set. The dimensions of the other three variables h, p, and Cp must be defined in terms of the primary variables. This can be done by inspection, although linear algebra can be used, too. 

We propose [15] a set of basis tensor algebra subroutines or btas. Tensors and tensor operators arise in many fields in the computational sciences, including computational quantum chemistry. The nomenclature BTAs(m,n), with m > n, where m and n are the respective ranks of the tensors, is proposed to establish a high level classification of tensor operations. The BTAS can be classified as follows -BTAS(1,0) BTAS(1,1)... 

Let us briefly recall a few of the basics of the algebra of tensors. An nth rank tensor in m-dimensional space is an object with n indices and rrf components. For a general tensor a distinction is made between contravari-ant (upper) indices and covariant (lower) indices. A tensor of rank mi + m2 may have mi contravariant indices and m2 covariant indices. The order of the indices is significant. Tensors can be classified according to whether they are... 

In what follows, we will suppose that the action of the group G in M and its projective actions in ft are regular and the orbits of these actions have the same dimension 5. This dimension is called the rank of the group G (or, alternatively, the rank of the Lie algebra g). Note that the condition rank G —. v is equivalent to the requirement that the relation... 

As a second step of the algorithm of symmetry reduction formulated above, we have to describe the optimal system of subalgebras of the algebra c(l, 3) of the rank 5 = 3. Indeed, the initial system has p = 4 independent variables. It has to be reduced to a system of differential equations in 4 — s = 1 independent variables, so that 5 = 3. 

Assertion 1. The list of subalgebras of the algebra p, 3) of the rank 3, defined within the action of the inner automorphism group of the algebra c(l,3), is exhausted by the following subalgebras ... 

Then, the critical points are characterized by two numbers, co and a, where to is the number of nonzero eigenvalues of H at the critical point (rank of the critical point) and a (signature) is the algebraic sum of the signs of the eigenvalues. Generally for molecules, the critical points are all of rank 3 then, four possible critical points may exist ... 

Constructions of basis vectors for the rows or columns of the coefficient matrix arise naturally in solving linear algebraic equations. The number, r, of row vectors in any basis for the rows is called the rank of the matrix, and is also the number of column vectors in any basis for the columns. 

The reaction network as represented by Equation (11.9) refers only to those among all possible reactions that may be regarded as independent. A set of reactions is said to be independent if no reaction from the set can be obtained by algebraic additions of other reactions (as such or in multiples thereof), and each member contains one new species exclusively. For any given set of reactions (including those that are not independent), the number of independent reactions is the rank of the reaction matrix (see Aris 1965 1969 Nauman 1987 Doraiswamy 2001). 

Before making the problem more complicated, we jga lore how to automate the preceding analysis by exploiting the stoichiometric matrix. If you are familiar with linear algebra, the issue of independence of reactions is obviously related to the rank of the stoichiometric matrix. Familiarity with these concepts, although helpful, is not required to follow the subsequent development. We now consider the stoichiometric matrix for the water gas shift reaction presented in Equation 2.7... 

The rank of the above matrix may be found by standard methods, for example by triangulation. After algebraic manipulations of rows and columns the final matrix is ... 

This is concluded generally by a theorem, of algebra which states that as many g s are independent as the rank of the matrix (a . ) of the coefficients of v, of the linear forms, gm> s, i.e.. 

Comments The elimination process can be done in many different ways and the surviving rows may contain different coefficients from ones obtained above, but the number of nonzero rows will be the same. This method comes from linear algebra and amounts to the determination of the rank of the matrix of the stoichiometric coefficients. 

In several practical applications of matrix methods, the rank of the matrix involved provides valuable information about the nature of the problem at hand. For example, in the solution of the system of linear algebraic equations by matrix methods, the number of independent solutions that can be found is directly related to the rank of the matrix involved. 

We use the Rank matrix to calculate the rank of the Dimension matrix. We need the rank of the Dimension matrix to determine the number of independent solutions that exist for our system of linear equations. From linear algebra, the rank of a matrix is the number of linearly independent rows, or columns, of a matrix [24]. In other words the rank of a matrix is the number of independent equations in a system of linear equations. Thus the number of variables in a system of linear equations, that is, the number of columns in the Dimension matrix minus the rank of the Dimension matrix equals the number of selectable unknowns [23]. Mathematically... 

The first condition, (4.31a), means that the matrices must be unitary. For matrices of rank 2, any unitary matrix can be expressed as a linear combination of the 2 x 2 unit matrix and the Pauli matrices. These four will not do, because even though the Pauli matrices anticommute, the unit matrix commutes with the Pauli matrices, and therefore the anticommutation requirement cannot be met. Thus, the rank of the a and P matrices must be greater than 2. Through some elementary matrix algebra, it may... 

The presence of conserved elements and conserved moieties cause linear dependence between the rows of the stoichiometric matrix p and decrease the rank of the stoichiometric matrix. In most cases, the number of species Ns is much less than the number of reaction steps N, that is, Ns < Wr. If the stoichiometric matrix p has N rows and Ns columns, and conserved properties are not present, then the rank of the stoichiometric matrix is usually Ns - If Nq conserved properties are present, then the rank of the stoichiometric matrix isN = Ns— Nq- In this case, the original system of ODEs can be replaced by a system of ODEs having N variables, since the other concentrations can be calculated from the computed concentrations using algebraic relations related to the conserved properties. 

It turns out the number of independent equations can also be found from the rank of the stoichiometric matrix, V/. Recall from linear algebra that the rank of a matrix is defined by the number of linearly independent rows in the matrix. It can be found using Gaussian elimination with partial pivoting or simply by using the rank(...) function in MATLAB. Once the rank is determined, we need to specify that number of independent... 

Internal Return Rate. Another rate criterion, the internal return rate (IRR) or discounted cash flow rate of return (DCERR), is a popular ranking criterion for profitabiUty. The IRR is the annual discounting rate that makes the algebraic sum of the discounted annual cash flows equal to zero or, more simply, it is the total return rate at the poiat of vanishing profitabiUty. This is determined iteratively. 

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