Vector algebra equality

F = objective function, g = algebraic inequality constraint vector, c = algebraic equality constraint vector,... 

In dealing with abstract vector or operator algebra, it is necessary to clarify the meaning of algebraic equality in equations such as... 

In order to bridge the gap between the discretized micro- and macro-worlds, averaging of the variables is necessary. Macroscopic variables in the N-S equation, are the density p and the momentum I, which are functions of the lattice space vector r and time t. The local density p is the summation of the average number of particles travelling along each of six (hexagonal) directions, with velocity c. Multiplication of the density p by the velocity vector u equals linear momentum (I = pu). Boolean algebra is applied for the expressions of the discretized variables density and momentum, respectively, as follows ... 

NO(7.5) = 0 after collision with Ar. The conditions necessary for this cancelation may be derived from classical mechanics using vector algebra. The velocity cancelation occurs when the velocity of the system s center-of-mass is equal in magnitude and opposite in direction to the recoil velocity of the scattered NOr.5 molecule in the center-of-mass frame of reference. That is, when = —Vcm, where the center-of-mass frame recoil velocity of... 

An interface, as in Figure 16-5, is defined by the kinematic property that fluid does not cross it. Hence, the velocity of the fluid normal to the interface must be equal to the velocity of the interface normal to itself. The velocity perpendicular to the surface, from vector algebra, is equal to -f (x,y,z,t)/V(fx ... 

The dot product on the left is composed by the rules of vector algebra. In other words, it equals the sum of the vector component products. Setting 0, = Cj / G = T - = 0 and using the tabulations of Table 2.3, we obtain... 

Exercise 8.9 Construct a Lie algebra representation (sii(2), V, p) with two highest weight vectors Vq and Vi such that their corresponding eigenvalues Ao Al (respectively) are not equal. 

Such a definition can, evidently, be extended to any number of routes. It is clear that if A(1), A(2), A<3) are routes of a given reaction, then any linear combination of these routes will also be a route of the reaction (i.e., will produce the cancellation of intermediates). Obviously, any number of such combinations can be formed. Speaking in terms of linear algebra, the reaction routes form a vector space. If, in a set of reaction routes, none can be represented as a linear combination of others, then the routes of this set are linearly independent. A set of linearly independent reaction routes such that any route of the reaction is a linear combination of these routes of the set will be called the basis of routes. It follows from the theorems of linear algebra that although the basis of routes can be chosen in different ways, the number of basis routes for a given reaction mechanism is determined uniquely, being the dimension of the space of the routes. Any set of routes is a basis if the routes of the set are linearly independent and if their number is equal to the dimension of the space of routes. 

Thus we consider only the commutation relation [F3, V+] = oJ+ and apply it to the basis vectors jm. After some algebra involving the substitution of Eqs. (40a), (40b), and (31), we obtain a linear relationship among several of the jm. Since these basis vectors are linearly independent we set their coefficients equal to zero and the result is a pair of difference equations which must be satisfied by the coefficients aj and cy... 

Any trajectory can end when p - I at a stationary point (SP), in which all the right-hand parts of equations (5.2) equal zero. In the case of the terminal model (2.8) all such SPs are those solutions of the non-linear set of the algebraic equations (4.13) which have a physical meaning. Inside m-simplex one can find no more than one SP, the location of which is determined by the solution of the linear equations (4.14). In addition to such an inner azeotrope of the m-simplex, azeotropes can also exist on its boundaries which are n-simplexes (2 S n m - 1). For each of these boundary azeotropes (m — n) components of vector X are equal to zero, so it is found to be an inner azeotrope in the system of the rest n monomers. Moreover, the equations (4.13) always have m solutions x( = 8is (where 8js is the Cronecker Delta-symbol which is equal to 1 when i = s and to 0 when i =(= s) corresponding to each of the homopolymers of the monomers Ms (s = 1,. ..,m). Such solutions together with all azeotropes both inside m-simplex and on its boundaries form a complete set of SPs of the dynamic system (5.2). 

An efficient way to treat such a system is to assemble all coefficients of the different terms of the mass-balance equations in a matrix and to apply methods of matrix algebra to solve the system for steady-state concentrations (level III) or for the concentrations as functions of time (level IV) [19]. We denote the matrix of coefficients (the fate matrix ) by S, the vector of concentrations in all boxes of the model by c, and the vector of all source terms by q. The set of mass-balance equations describing the temporal changes of the concentrations in all boxes then reads c = -S c + q. The steady-state solution is obtained by setting c equal to zero and solving for c. This leads to ss -1. j obtain the steady-state concentrations the emission vector has to be multiplied by the inverse of the matrix S. For the dynamic solutions of the system, the eigenvalues and eigenvectors of S have to be determined. 

In the language of linear algebra, N and b define vector spaces, and the dimension of a vector space corresponds to the number of linearly independent vectors, called basis vectors, that are needed to define the space. Then the multiplication in (7.4.2) can be interpreted as a transformation in which A maps a certain subspace of N into a subspace of b. In other words, only certain sets of mole numbers satisfy the elemental balances (7.4.2), and the possible sets of mole numbers depend on the chemical formulae for the species present in the system. That subspace of b, which is accessible to some N, is called the range of A the dimension of the range equals rank(A). According to (7.4.2), any basis vectors for the range automatically satisfy the elemental balances. For example, if we let N represent one particular basis vector for the range, then... 

A linear space of all left-invariant vector fields is a Lie algebra G with respect to the vector-field commutator. This Lie algebra is finite-dimensional, and its dimension is equal to the dimension of the group. It is called the Lie algebra of the Lie group 0. 

Here, p is mass density and yk th mass fraction, t is time and div the divergence operator v is local mass flow velocity (vector) and jk the it-th molecular diffusion flux vector, added to the term pykV representing the convection of particles Ck by the motion of a material element as a whole. So the instantaneous local change (increase) of the Ck-concentration (mass per unit volume) equals minus the amount that escapes from a volume element (the divergence term) plus the amount produced by chemical reactions. Physically, the balance makes sense if we know how the flux jk depends on the gradients (most simply by Pick s law), and how the rates of possible reactions depend on the local state of the element. If also the latter information is available then the balance takes the form of convective diffusion equation, possibly with chemical reactions. [If we have no information on the reaction rates, the w -terms can be eliminated from Eqs. (C.2) by an algebraic transformation in the same manner as in Chapter 4 indeed, it is sufficient to substitute for W, in (4.3.2) and to define the components of column vector n as follows from (C.2).] Observe finally that we have... 

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