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In a factorial experiment involving two factors, one quantitative and one qualitative, one may simply treat both factors as qualitative and obtain an analysis of variance table by straightforward methods. The sum of squares due to the quantitative factor can then be partitioned into sums of squares due to the linear, quadratic, etc. orthogonal contrasts, while the sum of squares due to the qualitative factor can be partitioned into meaningful orthogonal contrasts. If, however, one of the levels of the quantitative factor is zero, the analysis is complicated by the fact that a zero amount of each level of the qualitative factor is the same treatment. It is clear, for example, that if one applies a zero amount of six different fertilizers to six experimental units, one treats each experimental unit in exactly the same way. The foregoing situation arises frequently in biological assay and detailed analyses have been derived under the assumption that the quantitative factor follows a particular response law for each level of the qualitative factor. A short summary of this type of analysis is given in Kempthorne [1, Sec. 18.8]. I have been unable to find in the literature any description of how to compute the analysis of variance table for the situation where one of the levels of the quantitative factor is zero. The computations for the analysis of variance table which is appropriate for experiments involving quantitative and qualitative factors including zero amounts are presented in the remainder of this paper. Suppose that a factorial experiment involves one quantitative factor, A, having a levels and one qualitative factor, B, having b levels. When none of the levels of the quantitative factor A is a zero amount, the usual analysis of variance table for a two-way classification with one observation per experimental unit applies. If yij denotes the yield of the experimental unit that receives level i of factor A and level j of factor B, the sum of squares due to factor B

Lipman Bers has pointed out to me that the principal result in my paper is too optimistically formulated. I prove, correctly, that a finitely generated group has only a finite number of linearly independent quadratic differentials, but I am not justified in concluding that the orbit space n2/r has only finitely many components. My mistake was to overlook the fact that a sphere with three punctures has a Poincare metric, but no quadratic differentials. The reasoning does not exclude the possibility of infinitely many components of this type, and it remains an open question whether this can actually occur. The mistake recurs in the proof of Theorem 6, but the assertion remains valid. Indeed, it has been shown by R. Accola (to appear) that in the presence of two invariant regions all other components of 0 are "atoms" in the sense that they are invariant only under the identity transformation. Since atoms cannot be triply punctured spheres the error does not influence the reasoning. On p. 418, line 13, read "restrictions" instead of "restriction," and on the same page, line 23, replace "0" by "o."

It is known [11 that a homogeneous polynomial equation in n variables of a given fixed degree r over a p-adic number field has a non-trivial solution in that field provided n exceeds some sufficiently large number which depends on r; say n ? c(r). Moreover, many examples exist showing that 0(r) > r2. It has been shown by Hasse [4] that for the quadratic case n ? 5 is sufficient. We here settle the cubic case and prove the following: THEOREM. If K is a complete field under a discrete non-archimedian valuation and has a finite residue class field, then every cubic homogeneous polynomial equation in n variables with coefficients in K, has a non-trivial solution in K, provided n ? 10. Recently, Demyanov [3] has proved this result under the additional hypothesis that the characteristic of the residue class field is not three, an assumption we have been able to avoid. Our proof was arrived at independently at the same time and is quite different from that of Demyanov, although both proofs make use of a crucial result of Chevalley [2].

GCF gross capital formation GNP gross national product N population r growth rate of GNP/N. After fitting this model to a cross section of 100 countries for 1966, they use the estimated coefficients to simulate a growth path of a typical economy. In this comment it will be shown that the simulation results depend critically on the quadratic sDecification of equation (1) .2 Although the estimation of equation (1) by Sommers and Suits gives satisfactory results, there is little empirical evidence for the declining range of the equation. This can easily be seen from the scatter diagram and the graph of the fitted equation (figure 1): GCF/GNP attains its maximum when GNP/N is $2,169. The sample, however, contains only 8 countries (out of 100) with a per capita income of more than $2,169. Except for the single case of the United States (GNP/N $3,763 and

Federer and Schlottfeldt discussed measurements of the heights of tobacco plants in an experiment on seven treatments arranged in eight randomized blocks. A fertility gradient within the blocks was suspected and they therefore calculated a quadratic covariance analysis on a measure of distance in this direction. For the study of a regression trend of higher degree, the computations are simplified by using standard orthogonal polynomial values from Fisher and Yates's Statistical Tables (2) based upon distance from the centre of the experiment, this modification involving no difference in principle. Table I reproduces the yields from (1) and also the covariates, xI to xs for the corresponding orthogonal polynomials. The analysis of squares and products up to the third degree is shown in Table II, which includes the quantities required for subsequent covariance adjustments and agrees with Tables III and IV of (1) with respect to xI , x2 and y. To estimate the regression coefficients in the cubic analysis, the following set of equations must be solved.

McCrimmon has defined a class of quadratic Jordan algebras of degree 3 obtained from a cubic form, a quadratic mapping and a base point. The structure of such an algebra containing no absolute zero-divisor is determined directly. A simple proof of Springer’s Theorem on isomorphism of reduced simple exceptional quadratic Jordan algebras is given.

An important issue in the financial literature concerns the conflict between the stochastic dominance (SD) and the mean-variance (EV) methods of choosing optimal portfolios of risky assets. Much of the recent theoretical and empirical work in portfolio analysis has been devoted to the extension and testing of the Markowitz two-moment model, in which it is assumed that either (a) decision makers have quadratic utility functions with negative second derivatives or (b) the probability functions are from some appropriate two-parameter family and the investor is risk averse.

where a,, 02, a3 , x4may be arbitrary marks in the GF [pf] such that o4 * 0. Hence, substitutions exist for which A((X) in the GF[p)n] is irreducible; the product of a linear factor and an irreducible cubic; the product of two distinct irreducible quadratics; the squ-are of an irreducible quadratic; the -product of an irreducible quadratic and two linear factors distinct or equal; finally, the product of four linear factors, some or all of which may be equal. Type I. If the characteristic determinant is irreducible, the substitution may be reduced to the canonical form

RECENT years have seen considerable progress toward an integration of the theory of consumer's choice with empirical demand analysis. Theory has been extended so as to bring dynamic adjustment and the effects of past expenditure decisions (primarily through the introduction of certain "state" variables) into its purview,1 while on the empirical side there now exist a number of studies whose demand functions respect to a letter the restrictions imposed by classical theory.2 Unlike the demand theorist, content to assume the existence of continuous partial derivatives through the second order and to specify the signs of first derivatives, the applied analyst must go considerably further and specify the actual analytical form of the utility function. Herein, however, lies one of the major obstacles to the continued progress in applied demand analysis, for the list of functions which are rich enough to incorporate the restrictions imposed by theory, but yet sufficiently simple to be estimated with the data and techniques at hand is not lengthy. Included in this list are: (1). The additive quadratic utility function used by Houthakker and Taylor (1970) and also by Phlips (1971); (2). The linear-expenditure system based on the Geary-Samuelson utility function employed by Stone and his associates (1954, 1965) and most recently by Phlips (1972); and (3). The "Rotterdam" system of demand functions developed by Barten and Theil.3 The present work was motivated initially by a desire to devise a better method of estimating the additive quadratic model (AQM) than that used by Houthakker and Taylor. In particular, H and T observed a tendency for the estimated marginal utility of income to decrease sharply at the very end of the sample period, and averred that this probably reflected a defect in the method of estimation (p. 230). However, once we began exploring this, it became clear that the defect was in the quadratic utility function itself. Accordingly, we then undertook a critical look at the appropriateness of the AQM as a tool for empirical research, and in so doing decided to do the same with the linear expenditure system (LES).

(1966). Quadratic Congruences with an Odd Number of Summands. The American Mathematical Monthly: Vol. 73, No. 2, pp. 138-143.

The main motivation of the analysis presented here is to investigate certain non-linear properties of the mechanism which may be supposed to generate a time series. I investigate specifically quadratic terms in the generating model. For a wide-sense stationary process the spectrum describes the linear mechanism. For a third moment stationary process the bi-spectrum describes the quadratic terms in the mechanism. I will discuss the derivation of the bi-spectrum. The interpretation of the bi-spectrum in terms of certain non-linear transformations will also be discussed. As I have chosen to compute bi-spectra by means of complexdemodulation, there will be a major digression into the computation and analysis of complex-demodulates. For background information concerning the estimation of spectra the reader is referred to Blackman and Tukey (1959). Analysis of certain non-linear systems is discussed in Wiener (1958), while the statistical theory of higher order spectra (polyspectra) is treated in Brillinger (1964). An extremely interesting application of the bispectrum is given in Hasselman et al. (1963).

Four common methods of analysing percentage (attribute) data were compared for the case of a 3 × 3 analysis without replication. Data with no interaction, linear‐ × ‐linear, linear‐ × ‐quadratic and quadratic‐ × ‐quadratic interactions were sequentially generated and the effects of interaction, and form of interaction, on the power of the F‐test, χ2 and Friedman's non‐parametric procedure were determined. Three‐way χ2 and Tukey's test for non‐additivity were compared for their power for finding interactions of various forms. In all cases the χ2 test proved superior.

Preface * Foreword * Introduction * 1. Examples * 2. The Linear Function * 3. The Function y=|x| * 4. The Quadratic Trinomial * 5. The Linear Fractional Function * 6. Power Functions * 7. Rational Functions * Problems for Independent Solution * Answers and Hints to Problems and Exercises Marked with the Sign +

Under the assumption that individuals are single-period maximizers of the expected utility of their future wealth, this essay extends the mean-variance security valuation model developed by Sharpe [10], Lintner [4, 5, and 6], and Mossin [7 and 8] to a general parameter-preference model, with and without the simplifications of homogeneous subjective probabilities and the existence of a risk-free security. Results with quadratic and cubic utility are developed as special cases.

In the statistical literature, Booth et al. (Ref. 1) and Hartley (Ref. 2) developed modifications of the well-known Gauss-Newton method of iterative solution and applied it to estimate a set of parameters for nonlinear regression problems by least squares. To ensure convergence of his method, Hartley restricts its applicability to problems that satisfy his assumptions. Herein, we describe a method that removes two of Hartley's three assumptions, thereby making it applicable to those problems for which his method fails to converge. The method we propose is based upon an algorithm due to Davidon (Ref. 3). It is an iterative descent method whose rate of convergence is quadratic in the limit. It was later modified by Fletcher and Powell (Ref. 4), and has been used for locating an unconstrained local minimum of a function of several variables. Fletcher and Powell's account of Davidon's method has been found to be very useful when first derivatives of the function are available. However, in realistic situations it frequently is practically impossible to calculate first derivatives. Therefore, the method we describe is a modification of the basic Fletcher and Powell method due to McGill (Ref. 5) and Taylor (Ref. 6) and includes the case for which the gradient is not given analytically. Our concern is with the application and introduction of the modified Davidon method (MDM) to nonlinear regression problems of the type discussed by Hartley. We have omitted all proofs concerning stability and rate of convergence of the method and refer the reader to the excellent paper of Fletcher and Powell for such proofs. Included is a description of the MDM, a description of ways of terminating the iterations, and an illustration of the method with the numerical example Hartley used in his paper. We show the results for both the case of analytic and approximate gradient. Also included are results obtained in applying both methods to a second example, which involves the estimation of six parameters for an exponential regression function. For this example, it is shown that the method described in this note converges when Hartley's method fails to converge.

plays an extremely important role, where kA is the adele ring of k, X is a nontrivial character of the additive group of kA, trivial on k, and dx is the canonical measure on k". For example, when f is a quadratic mapping, the sum Leekm g9f($) is closely related to the Eisenstein-Siegel series in the sense of Weil [12], and when m = 1 and f is arbitrary, the integral | f(9()dd is kA substantially the singular series in the sense of Ramanujan-Hardy-Littlewood [2, IX]. For an obvious reason, we propose to call the mapping p 9.Wfthe Gauss transform relative to f; when n = m and f is the identity, tqw is nothing but the Fourier transform ' of p. In case m = 1, it is also important to consider the integral

The trigonometric sums studied by Gauss [2], in connection with cyclotomy, were applied to quadratic forms by Dirichlet [1] in his determination of the class number of some binary forms, and these so-called "Gauss sums" have since become fundamental in studying certain questions in the arithmetic theory of quadratic forms. In [10], O'Meara used Gauss sums to classify local integral quadratic forms, in particular showing, as Theorem 4 of [10], that the conditions in Theorem 2 of [9], involving Hasse invariants, could be replaced by equality of Gauss sums. In the present paper, we shall develop a theory of Gauss sums for hermitian forms, and use this to classify hermitian forms over the integers of a local field, thus providing an alternate classification to that given in [4]. Our main result is the following. Let F be a local field of characteristic not 2, with an involution a-* a*; then equality of Gauss sums, together with the typeand " u "-invariants of [4], gives a classification for integral hermitian forms (with respect to the given involution) over F, under integral equivalence. Further, if F is non-dyadic, i. e., 2 is a unit, then the " u "-invariants are unnecessary; and if F is an unramified extension of the fixed subfield under the involution, then equality of Gauss sums alone suffices for the classification. The need to know when the Gauss sums of a particular lattice vanish will become evident as we proceed. This question was first considered by Minkowski ([8], pp. 50-58) for rational p-adic forms and Hiecke ([3], pp. 218-249) for unary forms over p-adic completions of algebraic number fields, later generalized by O'Meara ([10], pp. 695-698) to quadratic forms in any number of variables over any local field (of characteristic not 2). We shall study the same question for hermitian forms, obtaining in ? 7 results that are more precise than can be given in the quadratic case; as we observed in [4], the apparent complication of working with a pair of fields often yields simpler results in the hermitian case than the quadratic.

This, the problem of the least quadratic non residue, has often been investigated. The best result is due to Vinogradov, who proved that (1) n(k) = O(k1I(2Ve) where n(k) denotes the least positive quadratic non residue of the prime k. Also, several interesting results about n(k) have been obtained by A. Brauer through elementary methods. It has been conjectured that n(k) ko(E). This was first proved by Linnik on the assumption of the extended Riemann Hypothesis (E.R.H.). Recently, with the same hypothesis, S. Chowla and P. Erdds have improved Linnik's result to n(k) = O(exp ((log k)i+e)) In this paper I prove on the basis of the E.R.H. (2) n(k) = O((log k)2) S. Chow-la has proved that there exist infinitely many primes k where the first cl log k residues (mod k) are all quadratic residues. Hence, the upper bound on (2) cannot be improved beyond O(log k). The two results, the upper and lower bounds, are now not too far apart. The best possible bound for n(k) is probably not the bound in (2). One COuld(1 improve the bound in (2) by methods in this paper if one could derive any non trivial estimate in terms of k on

In a recent paper by Goodall (1961) criticisms are made of the method of analysis of pattern proposed by Greig-Smith (1952a). This method has been described fully elsewhere (Greig-Smith 1957, 1961b; Kershaw 1957) and need not be discussed in detail here. It depends on analysis of the variance of data, taken from grids or transects, into portions appropriate to blocks of increasing size. Peaks in the graph of mean square against block size are interpreted as corresponding to the scales of pattern present. Kershaw (1957) has used artificial 'communities' of known pattern to demonstrate the validity of the technique and Thompson (1955, 1958) has discussed the statistical aspects. Both Kershaw and Thompson have emphasized the importance of comparisons between analyses in assessing the significance of peaks (see also Greig-Smith 1961b). Goodall considers, we believe misguidedly, that the peaks can be explained entirely as chance fluctuations. The technique has already been used in a variety of situations and ecological conclusions have been drawn from the results (Agnew 1961; Anderson 1961a, b; Chadwick 1960; Cooper 1960; Greig-Smith 1952b, 1961a, c; Kershaw 1958, 1959, 1960; Kershaw & Tallis 1958; Phillips 1954). It therefore seems important to make clear our reasons for believing Goodall's criticisms to be invalid. Goodall has attempted to explain the form of the graphs in terms of a continued rise in variance with spacing between samples. (His use of 'spacing between samples' as opposed to 'block size' is unobjectionable but the latter term is not so misleading as he maintains. It is common practice in discussing analyses of variance to refer to 'variance within and between varieties', 'variance within and between treatments', 'variance within and between blocks', etc., although in fact it is the variance between mean values for constant sampling unit that are meant.) To test this he has applied a logarithmic transformation to both the variance and the 'spacing' and fitted a polynomial regression of log-variance on log-spacing. Of thirty-nine graphs prepared from data of his own or data made available to him he found twenty-three (59 %) to be fitted satisfactorily by a linear regression, six required a quadratic, six a cubic, two a quartic and two still had a significant amount of variation not accounted for after fitting a quartic regression. On testing twenty-two graphs from data of Kershaw (1957, 1958, 1959) and Kershaw & Tallis (1958) he found that only four (18 %) could be satisfactorily fitted by linear, four by quadratic, three by cubic and one by a quartic regression, with ten graphs still showing significant residual variation. Greig-Smith (1961a, b) has pointed out that the existence of an overall trend in representation of a species along the transects being sampled will necessarily result in high variance values at the larger block sizes which may mask scales of pattern present. The presence of such a trend may be tested by examining the regression of representation on position. (Its effect may be reduced sufficiently to expose the pattern present by deducting from the sums of squares at each block size a correction term for covariance

Best quadratic unbiased estimators (BQUE'S) of variance components from unbalanced data in the 1-way classification random model are derived under zero mean and normality assumptions. An estimator of the between-class variance is also suggested for the non-zero mean case. These estimators are functions of the ratio of the population variances, p = o/-c2. Numerical studies indicate that for badly unbalanced data and for values of p larger than 1 estimators of o-2 having variance less than that of the analysis of variance estimator can be obtained by substituting even a rather inaccurately predetermined value of p into the BQUE of oa.

Five procedures for discrimination with binary variables and small samples are discussed and evaluated. Two procedures are specific for binary variables and are based on first and second order approximations to multinomial probabilities. The third procedure, based on the full multinomial model, is completely general. The fourth and fifth procedures are the linear and quadratic discriminants. Evaluation is in terms of mean actual error and mean correlation between observed and true log likelihood ratios determined by Monte Carlo sampling. The concept of a “reversal” in log likelihood ratios is introduced to explain the results.

which is subject to the transformations of the total linear group C, modulo p (a prime number), a theory which has been developed only to the extent indicated in the summary below.t Examples of such invariantive functions have been constructed by A. Hurwitz, Miss Sanderson, Dickson, and the present author, and it has been proved that the totality of pure invariants of the type forms a finite system when m * 0 (mod p). A variety of construction methods have been invented and certain particular complete systems of seminvariants and invariants derived. The present writer was the first to publish a fundamental system of formal covariants, that of the binary cubic for the modulus 2. The developments which follow are devoted to processes for the generation of complete systems of covariants and, in particular, give an extension of the principle of the modular covariant scale which was employed in the paper, quoted as IV above, on the system of the cubic modulo 2. In Section 6 the theory of concomitant scales is applied in determining a fundamental system of covariants modulo 3 of the binary quadratic, an interesting system composed of 18 invariants and covariants.

so that the variables are separable, the solution being of the form 2Xi, where Xi is a function of xi alone. In 18932 he showed that when the quadratic differential form 2H 2dxi so determined is taken as the Riemannian metric of a space Vn the equations of the geodesics of V, admit n 1 independent quadratic first integrals other than the fundamental form. In ??1, 2 we show that when this condition is satisfied, the fundamental quadratic form is of the Sthckel type. In 1927 Robertson3 showed that for an equation of the form

In analytic geometry of a fairly elementary sort, we frequently encounter the problem of solving simultaneously a number of linear equations, and a single quadratic equation. In homogeneous form this amounts to solving n 1 linear equations, and one irreducible quadratic in n + 1 homogeneous variables, or, to use the jargon of projective geometry, of finding the intersections of a straight line and a hyperquadric in a space of n dimensions. We all know that the problem can be solved, generally we content ourselves with saying that it is easy, or with writing down the single irrationality that enters into the solution. There do arise cases, however, when we must actually put the work through. We must actually have the coordinates of the intersections of a straight line and a conic, or those of the limiting points of a set of coaxal spheres, or those of the lines that intersect four skew lines in our space. The work of putting through the solution is somewhat repellant, yet when we search in the textbooks we fail to find there any solution in symmetrical form. Such, at least, has been the present writer's experience.* The fact is that every solution must be either unsymmetrical or involve arbitrary quantities dragged in from the outside. Of the two alternatives the latter is certainly preferable, and in the present paper a solution is given which seems to be about as compact and symmetrical as the problem will allow. The final form will be found in formulke (14) and (15). Let a point in n space have n + 1 homogeneous coordinates

The article of record as published may be found at http://dx.doi.org/10.2307/2002231 Eastlack has published a method for the solution of quadratic equations by means of a calculating machine. His process is ex­tended here to the solution of cubic equations. In the ordinary manual operation of calculating machines, the use of the method of solving cubic equations presented here will not be found to be as convenient as the use of certain other methods, such as that of Newton. The method is described here, however, in the belief that it may find application in large scale, automatic computing machines (such as the IBM Sequence Controlled Calculator or the ENIAC) where a large number of operations is not objec­tionable, provided that the operations are repetitive and sufficiently simple. We limit our discussion to real roots. Eastlack's method of solving quadratic equations is first reviewed so that the extension of the method to cubic equations may be clearer.

Let K be a commutative field of characteristic zero and let K (i) be a quadratic adjunction field of K, where i is a zero of the polynomial x2 a, irreducible in K. If A is a matrix with elements in K (i), or more shortly a matrix over K(i), the matrix A* is defined to be the conjugate transposed of A, so that A* A'. In particular, if A is a matrix over K, A* -A', the transposed of A. Let (1) A ==rA +sB

SUMMARY A linear and a quadratic index are developed as selection criteria for quadratic models of total merit, these models being those in which total merit includes squares and cross-products as well as first-order powers of the traits involved. The linear index consists of weighted phenotypic deviations and the quadratic index consists of the same weighted phenotypic deviations plus the weighted squares and crossproducts of the phenotypic deviations. The weights are determined for both indices by minimizing the squared difference between the index and total merit, both expressed as deviations from expectations. The quadratic index is shown to be equivalent to a maximum-likelihood estimate of total merit. The use of the indices is illustrated by two examples.

The standard analysis for latin and Youden squares may need modification for change-over trials with dairy cows. It may be necessary to introduce parameters for residual effects of diets. Also the error terms for successive periods cannot usually be regarded as independently distributed. It is useful to represent the records for successive periods by a series of orthogonal polynomials. The linear polynomial has the highest variance, and the quadratic may be more variable than higher-order polynomials. The variability of the logarithm of milk yield, in four-period and five-period trials with the Colworth herd, has been analysed. Correlations between polynomials were not large enough to affect routine analyses seriously. Separate analyses were made for trials in which cows were fed for production and for those in which equalized feeding was used; and the ratios of variances of the polynomials did not vary significantly from trial to trial within either feeding method. Information on the treatment parameters can be obtained from each degree of polynomial. The standard least-squares analysis (here called Method 1) gives equal weight to each set of information. Patterson [1951] suggested a weighted analysis (Method 2) and Cox [1958] suggested discarding the information given by linear, and perhaps by quadratic, polynomials (Method 3). The precision of the estimates of treatment means, given by these methods, is compared. It depends on the ratios of variances of polynomials, and for the Colworth herd Method 2 is to be preferred. The weights should be based where possible on ratios obtained from the pooled results of past experiments: this may cause some bias in the estimated variance of treatment parameters, but it will be small. If some observations are missing, the analysis by Method 2 is complex. A suggestion for reducing the labour of analysis is made.