P-N vector

The MD trajectories of 1018 ps were computed at T = 303 K. The zero point (z = 0) is the centre of the bilayer it was calculated during the MD simulations as the middle point between the centres of P-N vectors of the two monolayers (the centre of the P-N vector is the middle of P-N distance). The ratio between the CPU times (MD/SCF) needed to obtain these results is in the order of 105... 

Fig. 6.13 Distribution of the angle between P-N vector and bilayer normal in pure DPPC membrane (solid line) and in membranes containing cholesterol 11 mol% (dotted line), 50 mol% structure A (dash dot line), and structure B (dashed line). When cosine is positive, the P—N vector points into the water layer.

In case of M= the vector p n) is uniquely determined if p(o) is fixed the value of p n) is always the same. But if M >1 then more different p n) vectors can occur with different probability of the occurence. 

It can be easily seen, that this way requires much less computations, than if the p n) are computed from the p n) vectors. 

We now consider the connection between the preceding equations and the theory of Aharonov et al. [18] [see Eqs. (51)-(60)]. The tempting similarity between the structures of Eqs. (56) and (90), hides a fundamental difference in the roles of the vector operator A in Eq. (56) and the vector potential a in Eq. (90). The fomrer is defined, in the adiabatic partitioning scheme, as a stiictly off-diagonal operator, with elements (m A n) = (m P n), thereby ensuring that (P — A) is diagonal. By contiast, the Mead-Truhlar vector potential a arises from the influence of nonzero diagonal elements, (n P /i) on the nuclear equation for v), an aspect of the problem not addressed by Arahonov et al. [18]. Suppose, however, that Eq. (56) was contracted between (n and n) v) in order to handle the adiabatic nuclear dynamics within the Aharonov scheme. The result becomes... 

To rationalize the name projection operator, we take a general vector f> in and operate on it with P n ... 

On the other hand, one can also regard the same matrix X as an ordered array of n vectors of dimension p, each of the form ... 

Fig. 29.5. Geometrical interpretation of an nxp matrix X as either a row-pattern of n points P" in p-dimensional column-space S (left panel) or as a column-pattern of p points / in n-dimensional row-space S" (right panel). The p vectors Uy form a basis of 5 and the n vectors v, form a basis of 5".

In the matrix-by-vector product one postmultiplies an nxp matrix X with a p vector y which results in an n vector z ... 

The vector-by-matrix product involves an n vector which premultiplies an nxp matrix Y to yield a p vector z ... 

The outer product results from premultiplying an n vector x with a p vector... 

The matrix-to-vector product can be interpreted geometrically as a projection of a pattern of points upon an axis. As we have seen in Section 29.4 on matrix products, if X is an nxp matrix and if v is a p vector then the product of X with v produces the n vector s ... 

In multiple linear regression (MLR) we are given an nxp matrix X and an n vector y. The problem is to find an unknown p vector b such that the product y of X with b is as close as possible to the original y using a least squares criterion ... 

The number of singular vectors r is at most equal to the smallest of the number of rows n or the number of columns p of the data table X. For the sake of simplicity we will assume here that p is smaller than n, which is most often the case with measurement tables. Hence, we can state here that r is at most equal to p or equivalently that rindependent measurements in X. Independent measurements are those that cannot be expressed as a linear combination or weighted sum of the other variables. 

We have seen above that the r columns of U represent r orthonormal vectors in row-space 5". Hence, the r columns of U can be regarded as a basis of an r-dimensional subspace 5 of 5". Similarly, the r columns of V can be regarded as a basis of an r-dimensional subspace S of column-space 5. We will refer to S as the factor space which is embedded in the dual spaces S" and SP. Note that r

factor-spaces will be more fully developed in the next section. 

In NIPALS one starts with an initial vector t with n arbitrarily chosen values (Fig. 31.12). In a first step, the matrix product of the transpose of the nxp table X with the n-vector t is formed, producing the p elements of vector w. Note that in the traditional NIPALS notation, w has a different meaning than that of a weighting vector which has been used in Section 31.3.6. In a second step, the elements of the p-vector w are normalized to unit sum of squares This prevents values from becoming too small or too large for the purpose of numerical computation. The... 

Let us consider the special class of problems where all state variables are measured and the parameters enter in a linear fashion into the governing differential equations. As usual, we assume that x is the n-dimemional vector of state variables and k is the p-dimemional vector of unknown parameters. The structure of the ODE mode is of the form... 

Roy, I., Ohulchanskyy, T.Y., Bharali, D.J., Pudavar, H.E., Mistretta, R.A., Kaur, N. and Prasad, P.N. (2005) Organically modified silica nanoparticles A non-viral vector for in-vivo gene delivery and expression. Proceedings of the National Academy of Sciences of the United States of America, 102, 11539-11544. 

Let us also define a matrix Bcolumn vector is obtained by stacking the vector b /nrn ri (p p) under the vector bj /nrn, f2 - (p p). These vectors, for different n, Qk, are then placed side-by-side thereby generating a square matrix B 7111" whose dimensions are the total number of LHSFs (channels) used. The coupled hyperradial equation satisfied by this matrix has the form... 

Like electrons, some nuclei also have spin. Protons and neutrons also have spins of Thus, if a nucleus consists of p protons and n neutrons its total spin will be a vector sum of p + n spins of Each isotope will have its own spin value, but the laws governing the vector addition of nuclear spins are not known, and at the moment, nuclear spins are known only in terms of some empirical rules. 

A random n-vector X has a mean vector fi and an n x n covariance matrix . a is the diagonal matrix with standard deviations as diagonal terms and p the correlation matrix. Find the correlation matrix of the reduced vector given by... 

Remember that e = D/E, where the displacement vector is given by D = E -F 4 rP (CGS units), E being the macroscopic electric held and P the macroscopic polarization. Considering a density of atoms N, the macroscopic polarizahon is P = N (p) = 7Vq (Eioc> (where the symbol ( ) indicates an average valne) and so D = E + dTriVa (Eioc>. Now assnming (Eioc) = E, we obtain ... 

Let A be an m x p matrix, abias be an m x 1 vector, B be an n x m matrix, and bbias be an n x 1 vector. Vectors x and q are then p x 1 and n x 1 vectors, respectively, and m becomes an arbitrarily determined number equal to the rows in A, the length of abias, and the columns in B. The total fitted parameters in Equation 9.6 is then m(p + n + l) + n as such, an upper limit for m) is set by the number of x, q data records available for parameter fitting. A corresponding lower limit for m is determined empirically That lower limit is reached when is so low that Equation 9.6 can no longer map x to q with the required degree of accuracy. 

In the case of N and p being vectors, as they are for momentum and velocity, the Reynolds transport theorem takes the primative form... 

In the language of V.2 the total master equation is decomposable. The state space of vectors n decomposes into sublattices between which no transition is possible. A separate master equation applies to each sublattice. According to V.3 in each sublattice the probability tends to a unique stationary solution. Any solution of the total master equation tends to that stationary solution on each sublattice, but the weights allotted to the several sublattices are fixed by the initial condition. If the initial condition is P(n, 0) = 6(h, n°) all sublattices other than that accessible from n° have zero weight, as expressed by (3.4). If one wants to describe an ensemble of vessels, with various rt°, the weights will be different and depend on the initial ensemble. In particular, (3.1) may occur for a suitably chosen ensemble but it does not describe the fluctuations of n as they occur in a single closed vessel in the course of time. 

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