Non-negative

As the number of atoms in the asyimnetric unit increases, the solution of a structure by any of these phase-independent methods becomes more difficult, and by 1950 a PhD thesis could be based on a single crystal structure. At about that time, however, several groups observed that the fact that the electron density must be non-negative everywhere could be exploited to place restrictions on possible phases. The first use of this fact was by D Marker and J S Kasper [24], but their relations were special cases of more general relations introduced by J Karle and H Hauptman [25]. Denoting by A. the set of indices h., k., /., the Karle-Hauptman condition states that all matrices of the fonu... 

Beeause the level with this L and S quantum numbers eontains (2L+1)(2S+1) states with Ml and Ms quantum numbers running from -L to L and from -S to S, respeetively, one must remove from the original box this number of produet states. To do so, one simply erases from the box one entry with eaeh sueh Ml and Ms value. Aetually, sinee the box need only show those entries with non-negative Ml and Ms values, only these entries need be explieitly deleted. In the example, this amounts to deleting nine produet states with Ml, Ms values of 1,1 1,0 1,-1 0,1 0,0 0,-1 -1,1 -1,0 -1,-1. 

The eumulative exit age distribution is a non-negative, monotone non-deereasing funetion as shown in Figure 8-3. 

If solution gives negative flows or compositions, the non-negativity constraints can then be added end... 

These equations coupled with the non-negativity constraints form a linear program which can be modeled on LINGO as follows ... 

Studies on the electronic structure of carbon nanotube (CNT) is of much importance toward its efficient utilisation in electronic devices. It is well known that the early prediction of its peculiar electronic structure [1-3] right after the lijima s observation of multi-walled CNT (MWCNT) [4] seems to have actually triggered the subsequent and explosive series of experimental researches of CNT. In that prediction, alternative appearance of metallic and semiconductive nature in CNT depending on the combination of diameter and pitch or, more specifically, chiral vector of CNT expressed by two kinds of non-negative integers (a, b) as described later (see Fig. 1). 

The problem of Sec. 12 can be stated in this special situation as follows Let It be an arbitrary permutation group of degree 5 and /cj, /cj,. .., denote n non-negative integers whose sum is s. How many nonequivalent ways modulo H are there to place /Cj balls of the first, balls of the second,. .., k balls of the n-th color in 5 slots According to Sec. 16 the solution is established by introducing m cycle index of H and expanding the... 

The combinatorial interpretation (or the computations of Sec. 14 regardless of combinatorial considerations) implies the following useful result Substituting a power series with non-negative integer coefficients in the difference of the cycle indices of A, and we get a power series with non-negative integer coefficients. 

Since 0(0,0) 1 and since the last term on the right-hand side of the functional equation (2.22) has non-negative coefficients (cf. Sec. 

These statements are a consequence of the recursion relations obtained by identifying the coefficients of the power series expansion on the right- and left-hand side of the equation. For example, in (4.6), the coefficient of x" is (n > 1) on the left-hand side, and on the right-hand side a polynomial in R, . [cf. (2.56)], which implies the uniqueness. The coefficients of the polynomial mentioned are non-negative the term occurs, coming from x/, thus Rj > n-i statements that the coefficients are... 

Both F and G are polynomials in the unknowns Uq, Uy. U y. Obviously F has non-negative coefficients, hence, according to the induction assumption (4.11) we have... 

Some of the coefficients of G could be negative but G assumes nonnegative values for non-negative integers Uq,. (cf. final... 

In digital computation, numbers are represented by expansion in powers of a base (or radix) b, which is usually either 2 or 10, with coefficients that are non-negative integers <6. Thus... 

The ordinary euclidean length is such a norm, and, more generally, if Q is any positive definite matrix, then the non-negative square root of... 

Evidently this function is non-negative, and it assumes a local minimum of zero at any point x satisfying (2-23). If x0 is any approximation, and u is any vector, the function... 

We now have at our disposal the means for easily constructing examples of distribution functions. The simplest way to do this is to note that condition (iii) forces the derivative of Fx to be non-negative at all points where the derivative exists. We shall adopt the familiar... 

Conversely, if px(x) denotes any integrable, non-negative function satisfying... 

We conclude this section with examples of some particularly important probability density functions that will be used in later applications. In each of these examples, the reader should verify that the function px is a probability density function by showing that it is non-negative and has unit area. All of the integrals and sums involved are elementary except perhaps in the case of the gaussian distribution, for which the reader is referred to Cramer.7... 

The derivation can now be concluded by noting that since the numerator of Eq. (3-106) is the fraction of time something happens, it must always be non-negative. Consequently, the function p(xlaa 3), being the limit of non-negative quantities, must also be non-negative. 

We must now verify that the definition (3-219) does indeed generate a possible set of probability density functions. First of all, pXtfn, as defined by Eq. (3-219) is everywhere non-negative, and, moreover,... 

The functions defined by Eq. (3-231) are obviously non-negative and have unit area. One set of consistency conditions that must be met is of the form... 

A number of important properties of 8x(f) are now apparent. The fact that Rx(t) is a real, even69 function of r means that 8x(f) must also be real and even. Moreover, since the integral of 8x(f) over any band of frequencies, however narrow, is the power contained in those frequencies, this integral must be non-negative. The only way... 

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