Perturbation solutions of the Carson–Cambi equation

The periodic differential equation (1+ε cos t)y&#x030B; + py = 0, hereby termed the Carson–Cambi equation, is the simplest second-order differential equation having a periodic coefficient associated with the second derivative. Provided |ε|<1, which is the case we examine, then the differential equation is a Hill's equation and thus possesses regions of stability and instability in the p–ε plane. Ordinary perturbation theory is employed to obtain the stable (periodic) solutions to ε³. Two-timing theory is employed to obtain solutions for values of k near the critical points k = ±$\text{1}\text{2}$, ±$\text{3}\text{2}$, ±$\text{5}\text{2}$. Three-timing is employed to extend the solution near k = ±$\text{1}\text{2}$. The solutions of the Carson–Cambi equation are compared with the solutions of the corresponding Mathieu equation.     

The scattering of X-rays

Present Status of the Problem.—The scattering of X-rays is one of the outstanding problems of electromagnetic radiation which has not been solved satisfactorily. All theories (based on classical electrodynamics) presented thus far do not explain either the diminution in the scattering coefficient, or the observed asymmetry in the scattering, or both. Among such theories we may mention:J. J. Thompson's Theory.—Assuming that the scattering is done by a point electron, and making use of certain additional hypotheses, Thomson showed that the scattering coefficient of any substance is given by

$\text{σ=}\text{8πNpe}{}^4\text{3m}{}^2\text{c}{}^4$

and that the intensity of the scattered radiation is given by

$\text{I}{}_{\mathrm{\theta }}\text{=I}\text{e}{}^4\text{(I+cos}{}^2\text{θ)}\text{2r}{}^2\text{m}{}^2\text{c}{}^4$

where N is the number of atoms per c.c., p the number of electrons per atom, e the electronic charge, m the electronic mass, c the velocity of light, I_θ the intensity of the scattered radiation at an angle θ between the incident beam and the radius vector joining the centre of the electron and the point P distant r from the electron, and I is the intensity of the incident beam. This theory explains neither the asymmetry nor the decrease in the coefficient of scattering.Schott's Theory.—Among other things, the assumption is here made that the atom consists of coaxal rings of electron. The electrons in each ring are spaced at equal intervals and revolve with a uniform angular velocity, which, however, may be different for different rings. This theory fails to explain the observed diminution in the scattering coefficient.Debye's Theory.—In its essentials, Debye's theory has the same merits and demerits as that of Schott. Debye assumes that all the electrons in an atom are arranged in a single ring, and that they are spaced at equal intervals. This theory (and also Schott's theory) explains the asymmetry and the “excess scattering,” but is altogether unable to explain the diminution in the scattering coefficient.Modification of the Classical Theory.—The present paper presents a discussion of the possibility of modifying the classical theory (that of J. J. Thomson) so as to account for the decrease in the scattering coefficient as well as the dissymmetry. By assuming that the electron is made up of a number of parts—for simplicity, of two parts—it has been found possible to account for the diminution in the scattering coefficient without, at the same time, explaining the observed asymmetry. To accomplish both objects is what was aimed at in the combination of the present work with that of Debye. In this research the goal has not been perfection between predicted and observed results, but rather to discuss some possible modifications of the classical theory and their consequences.     

Air flow separation at high speed

The resistance coefficient of a body moving in a fluid depends on Reynolds Number R, Mach Number M and the parameter $\text{gL}\text{U}{}^2$, which is customarily neglected in view of small weight of the air. Here L denotes a characteristic length; U denotes the body's speed of translation. The author points that dimensional deduction of this parameter does not limit it to the acceleration of gravity, and that the resistance coefficient is affected by the general acceleration to which the air is subjected. Evaluation of the acceleration of the air flowing about spheres puts this parameter in the form $\text{L}\text{R}$, where the characteristic length L is interpreted as the mean free molecular path. Large and small spheres were found to have widely different values of the pressure coefficient $\text{Δp}\text{q}$ for the same Reynolds Number or Mach Number. Here Δp denotes the difference in pressure between front stagnation point and the rear portion of the sphere, and q denotes the dynamic pressure. The plot of $\text{Δp}\text{q}$ against the parameter $\text{L}\text{R}$ removes this confusion. The low values of $\text{Δp}\text{q}$ are found to be associated with $\text{L}\text{R}$ below a certain critical value, and high values of $\text{Δp}\text{q}$ with $\text{L}\text{R}$ above the critical value, which apparently indicates the condition under which the flow separation takes place. Attention is called to the effect of air pressure on the separation as shown by the parameter $\text{L}\text{R}$, and its possible bearing on the drag in high altitude flying.     

Theory of nonlinear systems

This paper gives a general review of the Theory of Nonlinear Systems. In 1960, the author presented a paper “Theory of Nonlinear Control” at the First IFAC Congress at Moscow. Professor Norbert Wiener, who attended this Congress, drew attention to his work on the synthesis and analysis of nonlinear systems in terms of Hermitian polynomials in the Laguerre coefficients of the past of the input.Wiener's original idea was to use white noise as a probe on any nonlinear system. Applying this input to a Laguerre network gives u₁, u₂,…, u_s, and then to a Hermite polynomial generator gives V(α)'s. Applying the same input to the actual nonlinear system gives output c(t). Putting c(t) and V(α)'s through a product averaging device, we get $\text{c(t)V(α)}\text{=}\text{A}{}_{\mathrm{\alpha }}\text{2л}{}^{\mathrm{<mtext>s</mtext><mtext>2</mtext>}}$, where the upper bar denotes time average and A_α's can be considered as characteristic coefficients of the nonlinear system. A desired output z(itt) may replace c(itt) to get a new set of A_α's.The Volterra functional method suggested by Wiener in 1942 has been greatlydeveloped from 1955 to the present. The method involves a multi-dimensional convolution integral with multi- dimensional kernels. The associated multi-dimensional transforms are given by Y.H. Ku and A.A. Wolf (J. Franklin Inst., Vol. 281, pp. 9–26, 1966). Wiener extended the Volterra functionals by forming an orthogonal set of functionals known as G-functionals, using Gaussian white noise as input. Volterra kernels and Wiener kernels can be correlated and form the characteristic functions of nonlinear systems.From an extension of the linear system to the nonlinear system, the input-output crosscorrelation φ_xy can be shown to be equal to the convolution of system impulse response h₁ with the autocorrelation φ_xx. Using the white noise as input, where its power density spectrum is a constant, say, A, the crosscorrelation is given by φ_xy(σ) = Ah₁(σ), while the autocorrelation is φ_xx(τ) = Au(τ). This extension forms the basis of an optimum method for nonlinear system identification. Measurement of kernels can be made through proper circuitry.Parallel to the Volterra series and the Wiener series, another series based on Taylor-Cauchy transforms developed since 1959 are given for comparison. The Taylor-Cauchy transform method can be applied in the analysis of simultaneous nonlinear systems. It is noted that the Volterra functional method and the Taylor-Cauchy transform method give identical final results.A selected Bibliography is appended not only to include other aspects of nonlinear system theory but also to show the wide application of nonlinear system characterization and identification to problems in biology, ecology, physiology, cybernetics, control theory, socio- economic systems, etc.     

Initial conditioned solutions of a second-order nonlinear conservative differential equation with a periodically varying coefficient

The exact solution of the equation

$\text{d}{}^2\text{x}\text{dt}{}^2\text{+dx+d′f(wt)x}{}^3\text{=0,}$

where d, d' and w are positive constants, and ?(wt) is a rectangular periodic function of time is discussed. The equation describes approximately the transversal movement of a particle in an alternating gradient accelerator. The exact solution is obtained in the form of a composite recurrent relation containing five particular solutions. Each of these solutions corresponds to a specific well-defined area of the phase plane of the initial conditions. The dynamical behaviour and the stability of the movement are examined analytically.     

Stabilization of carrier-frequency servomechanisms. III. Methods of obtaining required carrier phase-shift

In a servomechanism using a two-phase alternating current control motor, a 90° difference is required in the phases of the carrier-frequency voltages applied to the fixed and control windings. This part describes and compares various methods of obtaining the phase difference.The question of the possibility of a phase-shifting proportional-derivative parallel “T” is answered in the negative, by the result that in any parallel “T” transfer characteristic, if the quadratic factor in the numerator is of the proportional-derivative form at the correct resonant frequency, the amount of phase shift which may be obtained from the remaining portion of the transfer characteristic is less than are tan $\text{(}\text{2}\text{n}\text{)}$, where n is twice the carrier frequency divided by notch width. Thus for values of n high enough to have an appreciable stabilizing effect, the maximum obtainable intrinsic phase shift is negligible.In order to obtain a large phase shift it is necessary to add either a series input or a load impedance to the parallel “T,” or to use a phase-shifting network preceding or following the parallel “T.” Formulae and design charts are given for determination of the values of the components of phase lag networks.The method of calculation of tolerance requirements on the components, in terms of allowable deviation from the correct phase, is illustrated by an example of a phase lag network used in conjunction with a bridge “T” proportional-derivative network.     

Angular region: a measure of underdamped behavior

The natural modes of an underdamped dynamical system are given by the characteristic numbers of the quadratic operator pencil

$\text{P(s)=s}{}^2\text{I+sB+A,}$

where the operator A depends on the dissipative and reactive elements of the system, while B depends solely on the reactive elements. The operator P(s) for every applied stimulus vector signal x must satisfy:

$\text{(Bx,x)}{}^2\text{<4(Ax,x).}$

A measure of underdamped behaviour is suggested by predetermining an angular region |φ| containing all natural modes of the system,

$\text{|}\text{tan}\text{φ|?}\text{[4(Ax,x)?(Bx,x)}{}^2\text{]}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{(Bx,x)}\text{.}$

When a comparison between positive operators A and B is available, say B²=KA, then

$\text{|}\text{tan}\text{φ|?}\text{√(4?K}{}^2\text{)}\text{K}\text{.}$

The paper is motivated by Duffin-Krein-Gohberg's earlier mathematical contributions.     

The representation of radiation reaction in wave mechanics

It is well known that the wave mechanical ψ equation leads to the conclusion that the centroid of the wave mechanical electron should move according to the classical electrodynamic equation of motion in which, however, the terms representing what is commonly called radiation reaction are absent. If v is the velocity of the electron, the classical rate of change of momentum is $\text{md}\text{dt}\text{{}\text{v}\text{(}\text{I}\text{?}\text{v}{}^2\text{c}{}^2\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{}}$. The equation of motion including radiation reaction terms may be regarded as obtainable by replacing this quantity by one obtained by operating upon it with the operator P^?1

$\text{P={I?}\text{α}{}_1\text{kd}\text{dt}\text{+}\text{α}{}_2\text{d}\text{dt}\text{(}\text{kd}\text{dt}\text{)?·}}{}^{\mathrm{?}}$

where α₁, α₂, etc., are constants and $\text{k = (}\text{I}\text{?}\text{v}{}^2\text{c}{}^2\text{)}{}^{\mathrm{<mtext>?1</mtext><mtext>2</mtext>}}$. The main purpose of the paper is to show that if there be any relativistically invariant ψ equation which leads to the classical equation of motion without radiation reaction terms, then by replacing the vector and scalar potentials U and ? in that equation by P(U) and P(?), a relativistically invariant equation of motion will be obtained including the radiation reaction terms, provided that the $\text{d}\text{dt}$ in P be now regarded as $\text{?}\text{?t}$ + u · grad, where u is the velocity of the wave mechanical density distribution at a point. The purpose is to use the power to produce the equation of motion as a criterion for suggesting the proper modification of the ψ equation to apply in those cases where, on the classical theory, the electron would suffer great acceleration, as in ionization by rapidly moving corpuscles.     

The concept of resilience with respect to indicating instruments

This paper deals particularly with those instruments of the index-and-scale and value-controlling types of the class of non-integrating instruments, as distinguished from integrating instruments and those used for comparison purposes strictly. Independent corroboration is adduced regarding the general characteristics of the hysteresis phenomena discussed in the author's earlier paper on the variance of measuring instruments.It is shown that the area of the hysteresis loop obtained on complete cyclic calibration is a measure of the energy dissipated in the operation of the instrument and that the smallness of the area of this loop, which may be used as a factor in a function exactly analogous to the resiliency in the case of other quasielastic bodies, is a measure of the excellence of the instrument as regards the reproducibility or invariance of its indications, so far as concerns mechanical sources of variation. The specific properties of the hysteresis loop are set down analytically and the physical nature and causes of the imperfect resilience of instruments are discussed in detail.Since the motion of an index or pointer through a displacement implies the existence of a motive force to bring about the deflection, in the presence of an equivalent reactive or restoring force opposing it, the essentials of a work diagram apparently always exist in the results of a properly perforined cyclic calibration. Indications are given of the methods to be followed in the process of reducing the results of the calibration to such form that the loop obtained correctly represents energy dissipation during a cycle, typifying an integral of the form

$\text{∫}\text{B}{}_{\mathrm{y}}\text{D}{}_{\mathrm{y}}\text{[Ø}{}_1\text{(Y)?Ø}{}_2\text{(Y)] dy}$

when B_y and D_y are the projections of the extremities B and D of the loop on the, axis of deflection or reading.The restrictions which surround the carrying out of cyclic calibrations in a manner calculated to obtain results of the character required are discussed, including the requirements of slow, aperiodic change of the variables, unreversed between the chosen extreme turning points, and accomplished in the absence of jarring or vibration. Attention is directed to the necessity of accustoming the instrument to the particular cycle over which it is to be calibrated, to the end of regularizing its performance.Both the form and area of the hysteresis loop should be observed, in order to arrive at regional as well as aggregate effects of the causes producing the variance. The amount and scope of the effects of vibratory treatment in modifying and diminishing the causes of lag are indicated.The possible causes of the lag known to exist in. the case of instruments using a surface of discontinuity between fluids as the indicating element are discussed, and it is shown that known phenomena perhaps hitherto unrecognized in their relation to instrument design and calibration may account for the variancy noted. Detailed experimental consideration of these factors is hoped for.Comparison of instrument performances on the basis of the hysteresis loss requires equivalence of the ranges of operation, or possibly reduction of the results on the basis of information not vet available in a form, capable of general application.The resiliency determination opens up a field for type-testing and selection of instruments on a basis quite discrete from that pertaining to the ordinary methods of calibration, in that the former permits selection between types of instruments rather than between individuals, making clearly discriminable the differences between the characteristics of given operating principles, designs of mechanical details, or qualities of workmanship. The more general methods of diminishing variance educible from the foregoing considerations are indicated.     

Report on the work of the Bartol Research Foundation, 1933–1934

Using the velocity analyzer of Zartman with improved technique the combined velocity spectrum of Bi atoms and Bi₂ molecules was obtained at 827°, 851°, 875°, 899°, 922°, 947° C. From the spectral distribution curves the relative abundance of Bi atoms and Bi₂ molecules in the beams at the above temperatures could be determined to 1 per cent. The vapor pressure curve of Bi was obtained experimentally by the method of effusion and the values so obtained were combined with the degree of dissociation of the vapor as computed from the beams to give the heat of dissociation. The heat of dissociation was computed from the data, assuming the pressure to be given by the temperature of the crucible T_c. In calculating the heat of dissociation, the equilibrium temperature was taken as that of the slit chamber T_s which was 24° above T_c. The results of these calculations plotted with log₁₀K_p as ordinates against $\text{1}\text{T}{}_{\mathrm{s}}$ give a straight line whose slope yields the value of the heat of dissociation as 77,100±1200 calories. The curves for the distribution of velocities observed and computed on the assumption of a given ratio of Bi atoms to Bi₂ molecules in the beam were compared in an attempt to test the law of distribution of velocities. On the high velocity side agreement in two curves was obtained within the limits of experimental accuracy. On the low velocity side important deviations were noted of such a sort that the observed curves below a velocity $\text{α}\text{2}$, (α is the most probable velocity) gave more molecules than the theory demanded. Other deviations were observed on some of the runs taken with a fourth slit in which a deficiency of molecules was observed between velocities of .75α and $\text{α}\text{2}$. This deviation was probably due to a warping of the fourth slit carriage due to heat. The nature of the variation at velocities less than $\text{α}\text{2}$ indicated the presence of molecules of greater mass than Bi₂ in the beam and at the lower temperatures a distinct peak corresponding to Bi₈ molecules was observed which were present to less than 2 per cent. The vapor pressure curve for Bi was determined by least square reduction of the observed points to be given by $\text{log}{}_{10}\text{P = ? 52.23 ×}\text{195.26}\text{T}\text{+ 8.56}$ between 1100° and 1220° abs. It lies very close to the extrapolated curve given in the International Critical Tables.     

Accurate approximate solutions to oscillatory problems with perturbing singular damping

In this paper we attempt to obtain approximate solutions of improved accuracy for a class of differential equations of the form

$\text{d}{}^2\text{y}\text{d}\text{x}{}^2\text{+εμ(x)}\text{d}\text{y}\text{d}\text{x}\text{+ω}{}^2{}_{\mathrm{c}}\text{y = 0}$

, where ε is a real parameter less than unity, ω_c is a positive real constant of order unity and μ(x) is a singular function of x in the region of interest. It does not appear to be possible to find a general analytic expression for the error estimate of the approximate solution. For the case μ(x) = x^?2, however, it is shown that the approximate solution is accurate to 0(ε²), as x → 0^? from negative values, by comparing it with the numerically integrated solution. For the same case, the approximate solution is orders of magnitude more accurate than Poincaré's first-order perturbation solution, which is accurate to $\text{0(}\text{ε}{}^2\text{ln}\text{|x|}\text{|x|}\text{)}$ as x → 0^?. This work arose in search of analytic solutions to a linearized form of the restricted three-body problem.     

An equation of state for a system with a single type of transformation

Based on theory of a previous paper, the writer has developed an equation of state for a system with a single type of transformation. This equation is of the form

$\text{h=A+Bv+Cp+Dpv?T(E+Fv+Gp+Hpv)}$

where h = ε + pv is the total heat, p the pressure, v the specific volume, T the temperature, and p, v, T are considered independent variables. A, B, C, etc., are constants for the system. The latent eat at constant (p, T) is given by

$\text{λp,T=(v}{}_2\text{?v}{}_1\text{)(}\text{?h}\text{?v}\text{)P,T= (v}{}_2\text{?v}{}_1\text{)[(B?TF)+p(D?TH)]}$

. These equations are checked with data on saturated and superheated ammonia, and the agreement is good to within a few tenths of a per cent. Also, checks with data on saturated and superheated steam show agreement within several per cent.     

The porous structure of paper in relation to drying and impregnation

It is shown that water adsorbed by paper cannot be completely removed by pumping while immersed in an impregnating compound which wets the paper.A theory which explains this effect is presented, according to which the equilibrium moisture content of paper pumped under these conditions does not correspond to the external pressure but to one $\text{2λ}\text{R}$ greater than this where λ is the surface tension of the impregnant and R the effective pore radius.The application of these findings and the theory to the preparation of experimental samples of impregnated paper of controlled moisture content; to the determination of moisture in impregnated paper; to the drying and impregnation of paper and to the investigation of the pore structure of solids is discussed.     

The reflection of hydrogen atoms from lithium fluoride

The paper describes the phenomena associated with the reflection of a sharply defined beam of hydrogen atoms from a crystal of LiF. Of primary interest is the fact that the atoms show interference effects in agreement with the wave mechanics theory and plane grating diffraction patterns are photographed. Evidence of the thermal agitation of the surface ions is obtained from the diffuse reflection with surrounds the specular beam.The Schrödinger wave equation for the motion of a free particle of mass m is

$\text{▽}{}^2\text{ψ ?}\text{4πmi}\text{h}\text{?ψ}\text{?t}\text{= 0}\text{(I)}$

. The solution of this equation corresponding to the kinetic energy $\text{mv}{}^2\text{2}$ is

where

$\text{v }\text{mv}{}^2\text{2}\text{and}\text{σ}\text{mv}\text{h}$

. The motion of such a particle should have the characteristics of a plane wave of frequency ν and wave-length $\text{λ =}\text{1}\text{σ}$. The experiments of various investigators¹ have shown the validity of the wave theory of the motion of the free electron and have given values of the wave-length in agreement with the theory.The free motion of atoms, ions and molecules should likewise have wave characteristics. In the case of the hydrogen atom, as the simplest example, the complete wave equation may be written in the form

$\text{I}\text{m}\text{▽}{}^2\text{x,y,zψ +}\text{I}\text{μ}\text{▽}{}^2\text{η,μζψ ?}\text{8π}{}^2\text{μ?ψ}\text{mh}{}^2\text{η}{}^2\text{+ μ}{}^2\text{+ ζ}{}^2$

$\text{?}\text{4πi}\text{h}\text{?ψ}\text{?t}\text{= 0,}\text{(3)}$

where x, y, z, are the coördinates of the center of mass of the atom and ξ, η, ζ the coördinates of the electron with respect to the center of mass. If m_? and m₊ are the masses of electron and proton, m and μ have the significance

$\text{m = m}{}_{\mathrm{?}}\text{+ m}{}_{\mathrm{+}}\text{and}\text{I}\text{μ}\text{=}\text{I}\text{m}{}_{\mathrm{?}}\text{+}\text{I}\text{m}{}_{\mathrm{+}}$

. Equation (3) is solved by

$\text{ψ = U}{}_1\text{(x,y,z) U}{}_2\text{(η, ν ζ) ?}{}^{\mathrm{<mtext>2\pi iEt</mtext><mtext>h</mtext>}}$

, where E may have a continuous set of values and represents the total energy. U₁ and U₂ must satisfy the equations

$\text{▽}{}_1{}^2\text{U}{}_1\text{+}\text{8π}{}^2\text{mβU}{}_1\text{h}{}^2\text{= 0,}\text{(4)}$

and

$\text{▽}{}_2{}^2\text{U}{}_2\text{+}\text{8π}{}^2\text{μ}\text{h}{}^2\text{(α ?}\text{μ?}\text{m}\text{η}{}^2\text{+ ν}{}^2\text{+ ζ}{}^2\text{)}\text{U}{}_2\text{= 0}\text{(5)}$

, where

$\text{α + β + E}$

.     

Eigenvalue localization for the tikhonova model of crystal growth

Matrix A with characteristic polynomial Q(z) is defined positive or negative Hurwitz according to whether Q(z) or Q(-z) is a Hurwitz polynomial. Leading principle sections of the Tikhonova growth matrix have associated characteristic polynomials P_n(-z) which satisfy the recursion

$\text{P}{}_{\mathrm{n+1}}\text{(z)=zP}{}_{\mathrm{n}}\text{(z)+}\text{1}\text{n(n+1)}\text{P}{}_{\mathrm{n-1}}\text{(z),P}{}_0\text{(z)=1,P}{}_1\text{(z)=1+z}$

That the Tikhonova growth matrix is negative Hurwitz is established through applying the Wall-Stieltjes theory of continued fraction expansions to show the P_n(-z) are Hurwitz polynomials. The Kayeya-Enestrom theorem and a procedure for refinement of the Gerschgorin estimate are used to obtain analytical bounds on spectral radii for the Tikhonova model, which provides estimates of maximal growth rates. The theory allows generalization to more complicated growth models.     

The detonation wave from solid explosives

It has been shown that the detonation wave from a solid explosive changes in velocity after leaving the explosive. In the case of some low-velocity explosives, e.g., about 2400 metres per second, the velocity of the detonation wave increases by approximately 100–200 metres per second. With other high-velocity explosives, 5000–6000 m./sec., the velocity may decrease to around 4400 metres in some cases.Careful measurements have shown that a zone of uniform velocity exists beyond the surface of the explosive, the dimensions of the zone depending on the diameter of the sphere of the explosive or the diameter of the cartridge detonated. In the case of a $\text{1}\text{1}\text{4}$-inch dynamite cartridge, the zone of uniform velocity extended for about ten inches, while, with a sphere of explosive four inches in diameter, the zone extended for about 25 inches.An explanation of the reason for the high velocity of the detonation wave in the gaseous medium is suggested, linking it up with either the velocity of sound at the temperatures prevailing or the kinetic velocity of the gaseous particles themselves, the latter explanation appearing to be more in accordance with the determined facts. Such an explanation implies the assumption of extremely high momentary temperatures at the wave front, in some cases of over 40,000° C. These extremely high temperatures are due partly to the heat of combustion of the explosive ingredients but more to the heating effect resulting from the enormous pressures developed at the moment of explosion, as previously applied by Berthelot and Dixon in the study of gaseous explosions. Adiabatic conditions are assumed in such calculations. With such an explanation, reasonable agreement between theory and fact is obtained for a number of explosives, and the discrepancies that exist are in cases where discrepancies might be expected.     

Mathematical models of information system use

This report presents the results from a study of mathematical models relating to the usage of information systems. For each of four models, the papers developed during the study provide three types of analyses: reviews of the literature relevant to the model, analytical studies, and tests of the models with data drawn from specific operational situations. (1) The Cobb-Douglas model: x₀ = ax₁^bx₂^(1?b).This classic production model, normally interpreted as applying to the relationship between production, labor, and capital, is applied to a number of information related contexts. These include specifically the performance of libraries, both public and academic, and the use of information resources by the nation's industry. The results confirm not only the utility of the Cobb-Douglas model in evaluation of the use of information resources, but demonstrate the extent to which those resources currently are being used at significantly less than optimum levels. (2) Mixture of Poissons:

$\text{χ}{}_0\text{=}\text{∑}\text{i=0}\text{n}\text{i}\text{∑}\text{j=0}\text{p}\text{n}{}_{\mathrm{j}}\text{e}{}^{\mathrm{mj}}\text{(m}{}_{\mathrm{j}}\text{)′/i!}$

where x₀ is the usage and (n_j,m_j),j = 0 to p, are the p + 1 components of the distribution. This model of heterogeneity is applied to the usage of library materials and of thesaurus terms. In each case, both the applicability and the analytical value of the model are demonstrated. (3) Inverse effects of distance: x = a e^?md if c(d) = rdx = ad^?m if c(d) = r log(d).These two models reflect different inverse effects of distance, the choice depending upon the cost of transportation. If the cost,c(d), is linear, the usage is inverse exponential; if logarithmic, the usage is inverse power. The literature that discusses the relationship between usage of facilities and the distance from them is reviewed. The models are tested with data from the usage of the Los Angeles Public Library, both Central Library and branches, based on a survey of 3662 users. (4) Weighted entropy:

$\text{S(x}{}_1\text{,x}{}_2\text{,...,x}{}_{\mathrm{n}}\text{)= -}\text{∑}\text{i=1}\text{n}\text{r(x}{}_{\mathrm{i}}\text{P(x}{}_{\mathrm{i}}\text{)}\text{log}\text{(p(x}{}_{\mathrm{i}}\text{)).}$

This generalization of the “entropy measure of information” is designed to accommodate the effects of “relevancy”, as measured by r(x), upon the performance of information retrieval systems. The relevant literature is reviewed and the application to retrieval systems is considered.     

Nuclear Resonance Flourescence Studies of Levels in 206Pb, 207Pb, 208Pb and 209Bi below about 5MeV

Capabilities of the resonance flourescence technique are discussed. Specific reference is given to the study of ground state dipole and electric quadrople transitions below about 5 MeV in ²⁰⁶Pb, ²⁰⁷Pb, ²⁰⁸Pb and ²⁰⁹Bi. It is suggested that many of the states observed in ²⁰⁷Pb and ²⁰⁹Bi arise from the weak-coupling of a p_{$\text{1}\text{2}$} neutron hole or an h_{$\text{9}\text{2}$} proton to collective levels of ²⁰⁸Pb.     

An experimental investigation of forced vibrations

The solution of the differential equation y″ + 2Ry′ + n²y = E cos pt is written in a new form which clearly exhibits many important facts thus far overlooked by theoretical and experimental investigators. Writing s = n ? p, and Δn = n ? √n² ? R², it is found: (a) When s ≠ Δn, there are “beats,” and the first “beat” maximum is greater than any later maximum while the first “beat” minimum is less than any later “beat” minimum. The “beat” frequency is $\text{(s ?}\text{Δ}\text{n)}\text{2π}$. (b) When n² ? p² = R², there are no “beats,” and the resultant amplitude grows monotonically from zero to the amplitude of the forced vibration, (c) At resonance, when n = p, we still have maxima which occur with a frequency $\text{Δ}\text{n}\text{2π}$ in a damped system. (d) The absence of “beats” is neither a sufficient nor a necessary condition for resonance in a damped system.In the experimental investigation the upper extremity of a simple pendulum was moved in simple harmonic motion and photographic records obtained of the motion of the pendulum bob. Different degrees of damping were used, ranging from very small to critical.The experimental results are in excellent agreement with theory.     

Frequency and impact of spelling errors in bibliographic data bases

Using a composite sample of over 3600 index terms drawn from 11 different machine-readable bibliographic data bases, estimates were made of the spelling error frequencies of each of these data bases, as well as the frequency of posting to misspelled terms. The terms studied included assigned index terms as well as some terms from titles and abstracts. The frequency of index term misspellings ranged from a high of almost 23% for one data base to a low of less than $\text{1}\text{2}\text{\%}$ for another data base. The frequency of posting to misspelled terms ranged from about one posting in 8000 citations for one data base, to about one posting in 160 citations in another data base. The impact of these error rates is discussed for the tape supplier, tape user and end user. Some suggestions are given regarding search strategry.