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Consider a random sample of size $n$ taken from a continuous distribution $f(x)$. Let another random sample, independent of the first sample and also of size $n$, be drawn from the same population. Let $U^n_r$ be the random variable associated with the number of values in the second sample which exceed the $r$th smallest value in the first sample. Similarly let $V^n_s$ be the random variable associated with the number of values in the second sample which exceed the $s$th largest value in the first sample. Due to the fact that the $r$th smallest value in a sample of size $n$ is at the same time the $s$th largest value in the sample with $s = n - r + 1$, it follows that \begin{equation*}\tag{1} \mathrm{Pr} (U^n_r = x) \equiv \mathrm{Pr} (V^n_s = x),\end{equation*} $s = n - r + 1; \quad r = 1,2, \cdots, n;\quad x = 0,1,2, \cdots, n.$ The probability distribution of $U^n_r$ (and hence of $V^n_s)$ is given by: \begin{equation*}\tag{2} \mathrm{Pr} (U^n_r = x) = \binom{n-x+r-1}{r-1}\binom{n-r+x}{x}\binom/{2n}{n} = \frac{1}{2}P_{n-x+r-1,r-1} P_{n-r+x,x}/P_{2n,n},\end{equation*} $x = 0, 1, 2, \cdots, n.$ Formula (2) can be proved by combinatorial methods; details are omitted. An alternative formula, derived in another way [3], is \begin{equation*}\tag{2a} \mathrm{Pr}(U^n_r = x) = \frac{1}{2}\binom{n-1}{r-1}\binom{n}{x}/\binom{2n-1}{n-r_x} = \frac{1}{2} P_{n-1,r-1}P_{n,x}/P_{2n-1,n-r+x}.\end{equation*} In formulae (2) and (2a), $P_{n,x} = (\frac{1}{2})^n\binom{n}{x}$. Formulae in terms of $P_{n,x}$ are particularly convenient for hand computation, since one can use the extensive tables of the binomial probability distribution published by the National Bureau of Standards. If the values of $\mathrm{Pr} (U^n_r \leqq x), \text{for} x = 0, 1, 2, \cdots, n - 1, r = 1,2, \cdots, n$ are written (for fixed $n$) in matrix form, one notes certain useful symmetries, which can be expressed by the identities \begin{align*}\tag{3} \mathrm{Pr} (U^n_r &\leqq x) = \mathrm{Pr} (U^n_{x+1} \leqq r - 1), \\ \tag{4} \mathrm{Pr} (U^n_r &\leqq x) + \mathrm{Pr} (U^n_{n-r+1} \leqq n - x - 1) = 1. \\ \end{align*} If one takes $x = n - r$ in (4) and uses the relation (3), it is readily verified that \begin{equation*}\tag{5} \mathrm{Pr} (U^n_r \leqq n - r) = \frac{1}{2}.\end{equation*} Proofs of (3), (4), and (5) can be obtained by using the results of pages 257-258 of [3]. Because of these symmetries, the complete matrix (for any fixed $r$) can be constructed if one knows only the quantities, $\mathrm{Pr} (U^n_r \leqq x), r = 1(1)\lbrack n/2\rbrack, x = r - 1, r, r + 1, \cdots, n - r - 1.$ In Table 1 these values are given for $n = 2(1)15(5)20.$ To see how the complete matrix is obtained from Table 1, it is interesting to verify, using (3), (4), and (5), that the complete matrix, in the special case $n = 5,$ is given by Table 2. A somewhat different, but related, exceedance problem is to take two random samples of size $n$ from a continuous distribution $f(x).$ Let us for convenience attach the letter $x$ to one of the samples and the letter $y$ to the other sample. Further let $x_{r,n}$ and $y_{r,n}$ be respectively the $r$th smallest observations in each of the samples. Let us define $Z_{r,n} = \max (x_{r,n}, y_{r,n}).$ If $Z_{r,n} = x_{r,n}$, count the number of $y$'s which are $\geqq x_{r,n};$ if $z_{r,n} = y_{r,n}$ count the number of $x$'s which are $\geqq y_{r,n}.$ Denoting the number of exceedances as $W^n_r,$ it is readily seen from (1) that the probability distribution of $W^n_r$ is given by \begin{equation*}\tag{6} \mathrm{Pr}(W^n_r = x) = 2\binom{n-z+r-1}{r-1}\binom{n-r+x}{x}/\binom{2n}{n},\quad x = 0,1,2,\cdots, n - r.\end{equation*} It is evident from the definition that, \begin{equation*}\tag{7} \mathrm{Pr}(W^n_r \leqq x) = 1,\quad x \geqq n - r.\end{equation*} Clearly one can find the values of $\mathrm{Pr}(W^n_r \leqq x)$ by using Table 1. Thus, for example, in the special case $n = 5$ one obtains Table 3.

Contents: General characteristics of the research process; Desirable characteristics of research organizations; Recommendation on multiple sources of support; Allocation problem; The graphical model; Presentation of model in matrix form.

This chapter presents a cybernetic analysis and interpretation of the Marxian schemata of reproduction. The total product X , expressed in value units, may be defined as the sum of three components: X = c + ( v + m ). The first component c denotes the volume of outlays of the means of production needed to produce the amount X and the sum ( v + m ) is the outlay of direct labor. The chapter discusses a cybernetic analysis of the schema of expanded reproduction assuming that the national economy is divided into two divisions. It also discusses the matrix form of the balance equations of production allocation and outlays.

Given an interval matrix system with discrete- or continuous-time dynamics, the flow-invariance of an arbitrarily time-dependent rectangular set with respect to this system is introduced as a concept of geometric nature. The case of rectangular sets with exponential time-dependence is separately explored. If there exist flow-invariant rectangular sets approaching the state space origin for an infinite time horizon, then the interval matrix system exhibits two special types of asymptotic stability, which we have called componentwise asymptotic stability and componentwise exponential asymptotic stability. An interval matrix system is shown to be componentwise exponential asymptotically stable if and only if it is componentwise asymptotically stable. A necessary and sufficient condition for the componentwise asymptotic stability of an interval matrix system is derived in a matrix form. Brief comments focus on three classes of interval matrix systems for which the componentwise asymptotic stability is equivalent to the standard asymptotic stability (defined in terms of any consistent finite-dimensional norm).

Recently, a novel fast algorithm for 2-D N*N DCT (discrete cosine transform), where N=2/sup m/, was proposed. Only half the number of multiplications required for the conventional row-column approach are needed. However, the relationship between the input-output indices for the postaddition stage in the algorithm is seemingly very irregular. In the present work, the authors derive general and systematic expressions for the relation of the postaddition stage in the 2-D DCT algorithm by representing it in matrix form and developing a method for partitioning the matrices. The results show that the signal flow graph from input to output has a recursive structure where the structure for smaller N appears recursively for larger N. Hence, one can obtain an organized and regular structure for the input-output relation in the postaddition stage. >

We show that the possibility of storing more informations within a decreased volume is now opened. This is due mainly to the recording material we have used which is an improved polypeptide of dichromated gelatin. The input digital information is recorded and a diffractive digital image is reconstructed. The percentage of pixels error detection is less than 1% when decoded. We recorded a matrix form memory of n/spl times/m/spl times/p where n and m are rows and columns of the matrix and p the number of multiplexes. In this paper n=m=10 and p=20, for a 25 /spl mu/m thickness of the recorded medium. The diffraction efficiency uniformity is demonstrated.

A general method is presented for obtaining Clebsch‐Gordan coefficients, in a matrix form, of finite antiunitary groups, as a direct extension of a general method for unitary groups. It is shown that there is an essential difference as well as apparent similarities between two methods for unitary and antiunitary groups.

An iterative method for solving nonlinear eigenvalue problems in matrix form is presented. An example is given to illustrate the application of this method. This method may also be applied to linear eigenvalue problems.

B derivation of the "second-order modal design sensitivities" leads to results identical to the fir stand second-order terms obtained from Rayleigh's classical perturbation theory for variations in stiffness, but the second-order terms are in error for variations in mass. Rayleigh's methods are well known in the literature." His formulation is general and valid whether the eigenvalue problem is in matrix, differential, or integral equation form. Brandon's perturbation eigenvalue problem in matrix form is

Estimation of parameters is considered for several cases involving correlated errors. The cases include first and second order cumulative errors and a more general first order case. Estimators for first order cumulative errors are tabulated for five simple linear models. More general estimators for linear models are given in matrix form and it is demonstrated using maximum likelihood that many cases can be written simply in the form of differences. Expressions are also given that can be used to estimate the correlation coefficient and the error variance. Two examples are given to illustrate the new results.

The "shear" method of Humphries et al. (1981) is based on a path model intended to explain differences in form by multiple factors: one for size and one or more for shape differences. Its adaptation for "removing" the effects of a within-population size-factor from between-group morphometric analyses is presented in compact matrix form, simplified, and compared to the method of orthogonal projection proposed by Burnaby (1966). While the size- correction methods give similar results for most real data sets, Burnaby's procedure with k = 1 (i.e., using a single composite size variable) is recommended for this purpose owing to its geometrical and computational simplicity. An example based on artificial data demonstrates that sheared principal components are not necessarily uncorrelated with size. Path modeling of size and shape together is a different purpose than size-correction, and is better served by a different procedure. (Allometric growth; morphometrics; size; shape; principal components analysis.)

Thomas F Jordan 1986 Chichester: John Wiley xi + 259 pp price £25.50 ISBN 0 471 81751 1 To the majority of nonphysicists the prospect of attempting to unravel the mystical world of quantum mechanics (QM) is enough to prevent them from trying. In a book aimed specifically at students who are not specialising in physics, Jordan demonstrates that many of the fundamental properties of QM can be described and explained using simple matrix mechanics, as Heisenberg originally proposed.

The matrices in the X-ray intensity equation for one-dimensionally disordered structures are reduced by taking account of the symmetry character of the matrices.

SUMMARY An iterative procedure is presented for the estimation of constants and variance components in non-orthogonal data conforming to the mixed model. The procedure derives from Henderson's maximum likelihood technique for estimating fixed effects in the mixed model and method 3 for variance component estimation. This derivation is given, and relevant statements of method 3 are given in matrix form. An illustrative example is provided, and the application of the method to a large body of beef cattle data is described. In this application, the estimates of both

The effective rotational and translational stiffnesses of a set of isolators arbitrarily oriented with respect to the c.g. of an isolated rigid mass are developed by matrix techniques. Influence coefficients for determining isolator loadings due to arbitrary g loads are also presented in matrix form. Matrix equations of rigid‐body motion for a general isolation system are presented and conditions for decoupling translational and rotational motions are determined. The application of the matrix techniques developed to the design of vibration isolation systems is illustrated through several examples.

This paper adds the concept of interelement mutual coupling to a previously presented design technique for aerial arrays. With mutual coupling accounted for, one can realistically design electronically scanned arrays of wire elements that have both maximum gain and a number of independent, steerable nulls in the radiation pattern.