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The concept of ``equivalence,'' as it relates to comparing man and automaton, is explored and the notions of response equivalence and process equivalecne are contrasted. Some areas of psychological research are reviewed from the perspective of whether or not they concentrate on response or process identification. Finally, some questions concerning the description of biological automata are raised.

The phase corrected Stueckelberg and Landau-Zener formulae for the non-adiabatic transition probability in a two state problem are compared. It is shown that validity criteria on the derivation of the former imply precise equivalence between the two formulations for the linear curve crossing problem; but the Stueckelberg approach is seen to have greater flexibility. A proposed uniformization of the Stueckelberg form for use at low velocities is shown to be incorrect.

The three examples of ideal gas, square well, and harmonic oscillator particles illustrate the equivalence of the constant-energy and constant-temperature descriptions of statistical mechanics if the number of degrees of freedom is large. Significant differences between these descriptions appear, however, in the opposite extreme, i.e., the few-particle limit. The analysis presented here supports the conclusion that three degrees of freedom are necessary to localize a particle in the oscillator potential. This result is discussed in connection with the stability (finite rms particle displacement) of three-dimensional harmonic crystals.

The Fermi gas model of the nucleus at high temperatures can be used to define a ground state shell correction. We demonstrate analytically that the shell correction so obtained is the same as that obtained from the Strutinsky type method.

Differing methods of plotting have been used to describe the results of radial immunodiffusion (RID). Two, logarithm of antigen concentration (log c) vs. precipitin-ring diameter (d) (Fahey) and log c vs. area (A or d2) (Mancini), are linear while rings are enlarging. Another, c vs. A (or c vs. d2), becomes linear when enlargement ceases at equivalence (Mancini). Because of methodological inaccuracy, I could not experimentally determine whether log c = d or log c = d2 is appropriate; either one or both semilogarithmic graphs may be linear as precipitin circles enlarge. I could confirm the c = d2 linearity at equivalence. Because smaller circles cease growth first, intermediate readings produce plots that are partially curved and partially straight. Commercial RID plates produce linear and curved plots of log c vs. d and of c vs. d2. They often create too small a diameter range for great accuracy. I conclude that semilogarithmic plots are of questionable value. For greatest accuracy plates should be designed for measurement at equivalence. At this time, a straight-line graph of c = d2 will include a maximal diameter range.

Analysis of the stimulus composition of the three alternative forms of Benton's Visual Retention Test showed the three forms differed from each other in lateral distribution and vertical placement of peripheral details, in equality of chances for the elicitation of stabilizations, internal distortions, and mirror rotations. When 106 Ss were randomly assigned to perform one of the three forms, their mean “correct” and “error” scores were very close to each other. No significant influence of individual Ss on means was observed and the power of specific figures to elicit errors by individual Ss was spread evenly. However, the contribution of individual stimuli in each form to total error score was spread widely. Earlier findings as to differential frequency of the various error categories were confirmed for each form. Concentrations of certain errors in particular figures are discussed and tentative interpretations of these observations are advanced. In conclusion, certain changes are proposed so as to equate the three forms qualitatively and to increase the discriminative power of the instrument.

In a recent paper in this journal, Gilmer and Morgan attempted to prove that there is equivalence among certain forms of education subsidy programs-flat grants, foundation programs, and equalized apportionment formulas. This paper demonstrates that the proof presented by Gilmer and Morgan does not support the conclusion that these subsidy programs are equivalent.

Let Φ⊂F fields. With respect to the complex number plane we call the elements of Fpoints, the subsets Φm+b, m≠O,b∃F, lines and the bijektions z∃F→zm+b∃F (direct)similitudes. Two noncollinear point-triplets (a1,a2,a3) and (b1,b2,b3) are said to besimilar triangles if there exists a similitude, mapping a1 onto b1 for i=1,2,3. Therefore, similarity is an equivalence. relation on the set of all triangles. In this paper, we characterize these geometries axiomatically, starting from incidence structures with an abstract equivalence relation — called similarity — on the set of all triangles by imposing successively similarity-axioms for triangles.

SUMMARY Results in the theory of optimum regression designs are used to develop experimental designs for discriminating between several regression models. An extension of the Kiefer-Wolfowitz general equivalence theorem is used to establish properties of the procedures. A sequential data-dependent strategy is described and its properties investigated for specific examples. In conclusion, some difficulties in the Bayesian analysis of discrimination experiments are considered.

Two procedures are presented: one for diagnosis of single-gate failures and the other for deriving equivalence classes of single faults. Both procedures require use of a diagnostic table which lists gate sensitivities and primary output values for all input patterns in a set of tests which contains at most U tests, where U is an upper bound derived in the text. A gate is defined to be an n-input single-output combinational circuit (n ≥ 1) and a failure is any detectable transformation of the correct gate function. The information contained in a diagnostic table can be used to locate a faulty gate and describe its failed function. Equivalence classes are derived from a diagnostic table by first specifying the primary output values in accordance with a theorem given in the text and then deriving the corresponding gate functions. Examples are given to illustrate both the procedures.

In solving a mathematical problem where the input consists of data endowed with some precise property - as that of being limiting boundary values of holomorphic (and unitary) amplitudes - one can proceed in many (usually infinitely many) equivalent ways. It is important to notice that this equivalence is broken when noise appears under some form, either because of the lack of precision of the data one is provided with, or because of mere lack of knowledge concerning them on some regions. One is then faced with the meaningful question of finding among all tautological ways of solving the problem, that one which leads to results closest to the ideal (precise) ones. The present paper deals with the optimization in the above sense of the N D equations (started in a previous work). Its first part is concerned with the determination of allowed regions for the residua of the CDD poles, from conditions of consistency with a specified allowed error-channel. Its second part treats the question of optimization in the presence of inelasticities - first globally, with the Chew-Mandelstam and Froissart parametrizations - and finally generalizes the procedure to binary coupled channels (matrix N D equations). Apart from the mathematical aspect, it is the aim of this paper to provide the reader with workable formulas ready for applications.

It is proven that a large class of pionization Mueller Regge models are equivalent to the multiperipheral model. An orthogonal similarity transformation is found which allows one to transform in either direction between equivalent multiperipheral and Mueller models. Models with charge are discussed and the problem of constructing charge conserving Mueller models is solved.

A method of using molecular symmetry in MO calculations is described. The method consists of determining the point group of the molecule, and establishing equivalence tables for atoms equivalent under various applicable symmetry operations. Ways of setting up symmetry orbitals are outlined. The determination of the symmetry species of molecular orbitals, of configurations and states of molecules are described.

A simple test for the existence of an output feedback matrix K to achieve pole assignment in a multivariable system is described and a formula is given to calculate K when it exists. The method is illustrated by a numerical example and its equivalence with a previous method is also shown.

overloading the article, and other problems related with the development of programs of accelerated equivalent tests were not considered in the given paper. We must indicate only that the regimes for such tests are selected on the basis of the equality of the factors of safety (life) of the main components determining the operational capability of the article during operation according to a program taken as the reference and according to a program of accelerated equivalent tests. In this case one uses the known time relationships between the loads and operational capability of the structural components (usually the cumulative damage hypothesis). It is easy to show that sufficiently founded solutions can be obtained in this case only for a small number of structural elements. At the same time there is a pressing need for an objective evaluation of the operational capability also of those structural components for which the equivalence conditions are not fulfilled. Only in this case can accelerated equivalent tests be applied to a comprehensive evaluation of the operational capability and reliability of a structure. Ultimately this amounts to an evaluation of the reliability indices of the article, i.e., to finding the relation o~ = F @~

Summary Recently, social psychologists have set forth various hypotheses concerning the conditions that may affect the extent of attitude-behavior consistency. In this research we test the hypothesis, generally suggested by Fishbein, that attitude-behavior consistency is affected by the extent to which attitude and behavior are measured at approximately an equivalent level of generality. Behavior was measured at two levels of generality, and attitude was measured at five levels of generality. As the various attitudes were found to be moderately related, a multiple regression analysis was employed to estimate the independent effects and combined additive effects of the attitudes on each of the two levels of behavior. Briefly, the data supported the hypothesis, showing that both specific and general patterns of behavior were only affected by attitudes measured at an equivalent level of generality.

A new method is given for computing effective polycrystalline elastic constants from single‐crystal elastic coefficients. Agreement with observation is good. The method is based on the assumed equivalence of the lattice‐vibrational properties of single crystals and polycrystals of the same material; single‐crystal and polycrystal Debye temperatures are equated. Present predictions of polycrystal elastic moduli differ significantly from those of most other averaging methods by being lower than the familiar Voigt‐Reuss‐Hill results.

We investigate possible modifications of the nonforward Fubini-Dashen-Gell-Mann (FDG) sum rule in non-Abelian Compton scattering in the cases when the forward sum rule has 1) the form proposed by Barry, Gounaris and Sakurai, and 2) the usual form. The condition for the equivalence of the relations which arise on differentiating the FDG sum rule with respect toq2 andt, atq2=0,t=0, is also discussed. We suggest that our work provides a mechanism in which recent objections to the Cabibbo-Radicati sum rule can be overcome.

This paper provides a detailed survey and extension of certain properties of the stable regulator problem: determine \min\inf{u} \int\min{0}\max{\infty} x'Qx + u'u dt subject to \dot{x} = Fx + Gu; x(0) = x_{0}; \liminf{t \rightarrow \infty} x(t) = 0 where Q is not necessarily sign definite. First, equivalence conditions recently given by Willems for the existence of the minimum are extended to include statements in terms of the Hamiltonian matrix and spectral factorization. This provides a precise relation between the time-domain and frequency-domain solutions to the problems. Second, the inverse problem of whether a given feedback u = -Kx is optimal for some Q is easily resolved, as is the redundancy problem of distinct Q 1 and Q 2 , resulting in the same optimal control.

Approximate spectral representations are developed for the memory kernel which characterizes self-diffusion. These spectral representations are based upon approximate eigenfunctions constructed via the Rayleigh variational principle. A heuristic model is developed first in an effort to provide physical insight into the nature of the approximations employed, and then a number of specific trial functions are examined. These trial functions include sums of identical one- and two-particle functions as well as linear combinations of hydrodynamical variables. The results from these spectral representations indicate that the long-time behavior of the memory kernel (and thereby of the momentum autocorrelation function) is sensitive to the long-range effects of the interparticle potential. In addition, the equivalence of most of these spectral representations to specific low-order perturbation approximations is demonstrated.

In ‘Can and Might’ (this Journal, vol. 1, no. 1, September, 1971, pp. 87–92), Professor K. W. Rankin has presented three arguments that purport to refute the equivalence (E), ‘A (an action) is causally possible for P (a person) if and only if A is within P's power’. The first two arguments are attributed to Richard Taylor (Action and Purpose, Prentice-Hall, 1966, pp. 53–59), and the third is Professor Rankin's own. I will argue that none of these three arguments effectively refutes the above equivalence. My arguments are not to be construed as simply a rebuttal of Professor Rankin's paper since (a) he also appears to have some doubts about Taylor's arguments and (b) I refrain from comment on other aspects of his paper except these three arguments. I conclude with some general remarks on (E).

In his stability theorem on unfoldings of germs of C∞-functions G. Wassermann has proved the equivalence of several properties of an unfolding, e. g. “infinitesimally stable” and “universal”. This paper which is close to his work adds another equivalent property. Two examples show that this property can also be considered in the context of other stability theorems.

In the paper [1] I have given an example showing that the relation of equivalence of automata defined by Dr. R. Nowakowski1 is not a congruence with respect to the operation of coupling with retardation, In this paper I shall prove that a certain re? lation ? which seems to me to be inessential contraction of the relation of equivalence of automata defined by Nowakowski ? is the congruence with respect to the opera? tion of connecting and coupling with retardation2. Nowakowski's definition of equi? valence of automata is based on the notion of homomorphism3 of automata. Here I shall use a certain contraction of this notion. The contraction consists in putting identities for the functions b and c appearing in this definition. The relation of equi? valence of automata will be contracted to these automata in which the sets of states, inputs and outputs are the sets of sequences consisting of O's and l's, and the output function is defined on the set of states only. It is the only difference between the no? tion of equivalence of automata I shall use in this paper and the notion of equivalence of automata defined by Nowakowski. I give some lemmata which are necessary in proofs of basic theorems of this paper. Lemma 1. // the automaton is homomorphic to the automaton , then the automaton F?{0, 1}"\ {0, 1}% {0, l}Pl, ?j, Aj? w homomorphic to the automaton F ?{0, l} 2, {0, l}% {0, l}\ S2, A2?. Proof. Let 8J, A^, S2, A2 be the transit functions of the automata F ?{0, 1}% {0, l}\ {0, 1}% o\, Ai? and F?{0, lp, {0, 1}"?, {0, l}\ S2, A2? respectively. In view of the definition of coupling with retardation it is easy to prove the equalities: F?{o,i}"s{o,i}vo,i}'?,$i,x;? = = 5 1. F ?{0, 1}"., {0, 1}-., {0, i}*, s25 x2? = ; = .

Previous results have implied that pattern goodness, as defined by rotation/reflection equivalence set size, is a determinant of encoding time. A direct test of this implication was made via a discrete RT task using good and poor dot patterns in a two-stimulus discrimination. The following discrimination conditions were used: (1) good vs good; (2) poor vs poor; and (3) good vs poor. Overall RTs for Conditions 1 and 3 were equal, but each was significantly faster than Condition 2. In the first two conditions, there was no difference in RT within pairs and no consistent transitive ordering of the patterns. In Condition 3, however, the good pattern consistently produced the smaller RT. These results indicate that patterns of equal goodness are equally encodable, and the better the pattern, the faster it is encoded. In addition, there was evidence that pattern similarity affects RT.

The structure of two widely used methods, the Langevin method and the impedance-field method, for calculating device noise is considered. The equivalence of these two methods follows directly from identifying and comparing in detail the basic steps one follows in carrying each out.

Let (Ω,ℱ, P) be a probability space. Let R denote the set of real numbers and the set of all random variables defined on Ω. Throughout this work, random variables which differ only on a set of probability zero will be considered identical. EX represents, as usual, the expectation of .

The time adult Ss were allowed to explore stimuli was varied during intra- and cross-model equivalence matching involving vision and touch. Increasing time to explore either each standard, each comparison, or both standard and comparison from 4 to 16 sec significantly improved haptic intramodel matching. However, cross-modal matching, from either vision to touch or touch to vision, improved significantly only when time to explore each standard was increased. Videotape recordings of Ss’ hand movements revealed use of a greater variety of haptic scanning strategies by Ss in groups where increased exploration time enhanced accuracy. The difference in effects of exploration time on intra- compared to cross-model shape matching was discussed in terms of possible differences in requirements between the two tasks.

A method is presented for generating open-shell equivalence-restricted SCF orbitals in high-symmetry situations using Roothaan–Hartree–Fock programs which are adapted for lower symmetry.

The equivalence demonstrated by Cohen et al. between Delves' variational procedure and Dalgarno's perturbation method for calculating expectation values is shown to depend upon a specific choice of zero-order hamiltonian in the perturbation method. Theoretical considerations support the view that direct calculation of non-variational expectation values will yield results of a given accuracy more easily than either of the above methods, at least when only linear parameters are involved in the calculation.

Increasing interest has been expressed, particularly by social scientists, in the analysis of unbalanced data occurring in models not of full rank, i.e., linear models where the numbers of observations in the subclasses are unequal. The various discussions of this problem that have appeared in the psychological and educational literature differ depending upon the intent of the author(s). For example, some writers (Gourlay, 1955; Steinhorst & Miller, 1969; Tsao, 1942, 1946; Williams, 1972) have examined and at times compared different types of solutions, including approximate solutions, for the unbalanced case. Other writers (Overall & Spiegel, 1969; Rawlings, 1972) have focused upon the difficulty which arises in interpreting the results of unbalanced data analyses because the estimable functions involved in tests of hypotheses are not orthogonal. Though not specifically addressed to the unbalanced case but important in this regard is the work of Bottenberg and Ward (1960), Cohen (1968), Hurst (1970), and Jennings (1967) that demonstrates the equivalence of linear regression and the fixed-effects analysis of variance by the use of regression on dummy variables. This diversity in purpose, combined with the relative narrowness of the individual efforts, has resulted in a fragmented treatment of the problem of unbalanced data and in some cases confusion and controversy regarding methodology, e.g., the role of constraints in obtaining a solution for models not of full rank or the calculation of reductions in sums of squares (Overall & Spiegel, 1973; Rawlings, 1973; Smith, 1973). In response, at least one writer (Joe, 1971) has been prompted to call for a greater understanding and more detailed presentation of the models employed. Some of the more important issues encountered in the analysis of unbalanced data will be examined here within a single framework. These issues are broadly defined as (1) solutions to the normal equations for linear models where the incidence or design matrix is not of full column rank, (2) estimable functions of the parameters and tests of hypotheses, and (3) reductions in sums of squares and interpretation of results. Because of the increased availability of computer hardware and sophisticated least squares routines, the various approximate solutions for analyzing unbalanced data are not considered here.

SUMMARY An investigation into the effects of training in symbolic logic on the grouping strategies of 8-yr-old children is reported, Bruner's tests of equivalence (verbal and non-verbal) being used as criteria. Generally the training was found to be beneficial for the boys, but not for the girls, the training x sex interaction being significant for performance in the verbal test. Again, for both tests, more of the experimental group displayed above-average groupings in the functional and nominal categories, the difference from the controls being statistically significant. The implications of these findings are discussed.

Differential cross sections are derived from a multistate impact-parameter treatment of heavy-particle collisions. Various approximations are suggested and their relationship with previous expressions are discussed. The equivalence between the differential cross sections in the impact-parameter and wave versions of the Born approximation is established for elastic and inelastic scattering and is illustrated explicitly for the $1s$, $2s$, $2{p}_{0}$, and $2{p}_{\ifmmode\pm\else\textpm\fi{}1}$ excitations of atomic hydrogen by proton and hydrogen-atom impact.

Two combinatorial theorems relevant to Zimmermann's formulation of Lagrangian perturbation theory are proved in view of application to problems concerning symmetry breaking.

The mean values, standard deviations and statistical distributions are derived from measurements made on various parameters associated with the maxima of room frequency transmission responses. These statistics are presented together with a correlation analysis of relationships between the parameters. The results are discussed where they have bearings on the performance of assisted resonance channels. The most useful result was the indication of an equivalence between the average room peak response and the response of a simple filter.

A direct proof is given of the equivalence relation between the energy expressions ‘for dislocation loops in the basal plane of a hexagonal crystal and in an isotropic medium. The two sets of expressions can be used interchangeably by inserting the corresponding elastic constants.