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Hamiltonians for the polaron of fixed total momentum are defined using momentum cutoffs. A renormalized Hamiltonian of fixed total momentum is defined in two space dimensions by proving the strong convergence of the resolvents of the cutoff Hamiltonians. The Hamiltonian for the physical polaron is defined as the direct integral of the fixed momentum Hamiltonians.

A physical model of an irrigation border (bay) is constructed and tested. Equations are developed which describe the profile of the water advancing down the border, the rates of advance, and recession after the inflow is cut off, in terms of inflow, infiltration, slope, and surface roughness. Where possible these equations are simplified to a form more suitable for use in design. An illustrative analysis is done to determine correct inflows and cutoff times needed to uniformly irrigate to a desired depth.

Analog filters employed in most averaged electroencephalic response (AER) recording systems impose some phase distortion on recorded AER. Phase distortion of the response results in a real time shift of peak latencies. The extent of this distortion, in general, is determined by the cutoff frequencies of the filter band in relation to the spectrum of the response. If the filter band employed is quite restrictive, the shift of latencies can be large. For purposes of threshold audiometry, filter-imposed phase shifts will not usually prevent simple response identification. However, if such time shifts in the data are not accounted for in interclinic or published communications, the response information could be misleading and confusing.

In this paper we give a first application of a general method whose mathematical aspects will be fully developed in a forthcoming article. We are concerned with strongly singular perturbative series. Here we shall restrict ourselves to the most general two‐body repulsive singular potential for which a regularization exists. Various extensions of this case are discussed in the conclusion. We show that, knowing only a finite number of regularized Born terms, it is possible to construct an upper bound to the exact phase shifts and that this upper bound is the best possible for the given regularization. The method uses the construction of the [N/N] Pade approximation indifferently on the regularized partial waves of the K or T matrix and exploits the fact that the approximate corresponding phase shifts have an absolute minimum as a function of the regularization parameter (cutoff). Three numerical examples are provided which show, even for very large phase shifts, an excellent convergence.

A theoretical study, backed by experimental verification, was undertaken to determine the modal characteristics of quadruple-ridged circular and square waveguides. Field lines for the first few important modes and cutoff frequencies were determined. It is shown that for square waveguides quadruple-ridge loading always decreases the TE/sub 10/-TE/sub 11/ bandwidth whereas for circular waveguides only a small amount of additional separation between the first two fundamental modes may be obtained over a limited parameter range. Symmetrical excitation will not excite the asymmetrical higher-order modes. This feature makes these waveguides acceptable as feeds for wide-band reflector antennas and for similar applications but raises a question mark regarding their use as radiators in wide-band phases arrays.

The effects of Ekman layers on generalized Eady waves (i.e., height-varying static stability and shear) are examined. The non-constancy of N(z) and uz modify the classical Eady results but do not introduce any new effects. Thus, a short-wave cutoff is always found for flows with double Ekman layers but never for flows with a single Ekman layer. By comparing the analytical solutions with a numerically simulated annulus wave, we are able to categorize the latter quite accurately.

The class of renormalization procedures with mass-independent counterterms is considered. It is shown to be nonempty. All these procedures, of which 't Hooft and Weinberg have given particular examples, lead to equivalent renormalization-group equations, which can be solved at arbitrary momenta, unlike the equations in the Gell-Mann-Low and Callan-Symanzik methods. Dimensional regularization is used throughout, as needed for non-Abelian gauge theories. It is shown how Weinberg's method using an ultraviolet cutoff extends to one using dimensional regularization. Finally, some calculations in quantum electrodynamics are exhibited.

The particle size distribution in the coma and tail of Comet Bennett has been determined by several methods, each sensitive to a particular size range. It is confirmed that a minimum value of the particle density, size, and radiation pressure efficiency function exists at about .00003 to .00010 g/sq cm. The existence of such a cutoff is probably due to the decreasing radiation pressure efficiency for particles smaller than the wavelength of the light being scattered. An exact determination of this cutoff may allow identification of the particle type.

Spherical wave weak-scattering propagation theory is employed to show that the resolution of multifrequency atmospheric probing techniques is limited to inhomogeneity scale sizes that are larger than a certain cutoff scale size lc. The magnitude of lc depends on the frequency separation and the path geometry. Inhomogeneities smaller than lc cannot be resolved and inevitably contribute to measurement errors. Propagation experiments performed at radio and acoustic wavelengths confirm the existence of the predicted cutoff and are in good agreement with the theory. An example is given to illustrate how this cutoff limits the ultimate accuracy of a radio-optical comparison measurement aimed at determining path-averaged water vapor variations.

The concept of canonical ratings is introduced in which each S describes all the visual sensations produced by signal and noise trials in the expected spatial and temporal location of the stimulus. After many practice sessions, the S assigns one and only one numerical rating to each visual sensation. These canonical ratings are determined by the S, not the E, and are a, bbreviations for verbal descriptions of subjectively distinct visual sensations. The data consisted of canonical ratings at absolute visual detection for dim visual stimuli (signal) and blank (noise) trials containing no light at all. The physical stimulus is discrete since it is made up of absorptions of quanta of light that result in isomerizations of rhodopsin molecules or thermal decompositions of rhodopsin which are discrete noise events that mimic the action of quantal absorptions. Under these conditions, it is known from the laws of physics that these quantum-like events labsorptions plus thermal decompositions) follow a Poisson distribution. Previously, it had been shown that the canonical ratings follow the same Poisson distributions that the quantum-like events do. It was also shown that the data for one S were consistent with the hypothesis that the rating on any trial was equal to the number of quantum-like events that had occurred and for two other Ss, either one less or two less than this number. A signal detection theory analysis of these canonical ratings is performed, resulting in ROC curves and estimates of d’. In addition, it is shown that the Poisson canonical rating distributions can be approximated by cutoff Gaussian distributions. Hence it is possible to use a probit analysis, which is computationally simple, to calculate the maximum likelihood solutions for all means, standard deviations, d’, and b, as well as the standard errors of all these estimates. The rating is shown to be a linear function of the internal decision variable. The internal criteria are all greater than the mean of the noise distribution and they are all separated by steps of equal size. The probit analysis may be used whenever all the individual rating distributions are Gaussian in order to obtain the maximum likelihood estimates and standard errors of all parameters for each Gaussian distribution. Thus, this analysis may be applied to rating experiments, other than the one described here.

By means of data obtained during 1969 and 1971 with an airborne neutron monitor the pressure, cutoff rigidity, and solar modulation dependence of the attenuation coefficient of the neutron monitor were investigated. Special attention was given to the experimental and statistical errors introduced by the measurements and analyses. The pressure, cutoff rigidity, and solar modulation dependence of the attenuation coefficient are discussed and compared with other results and finally incorporated into a single empirical equation. It is shown that the effect of stopping muons on the attenuation coefficient of the neutron monitor may be neglected in transforming to a vertical attenuation coefficient. It is also confirmed that the specific or partial attenuation coefficient is independent of the solar activity and is in good correspondence with results obtained from Monte Carlo calculations.

The significance of retardation effects in photon emissions and absorption is emphasized in the calculation of self-energy. It is explicitly demonstrated that inclusion of such effects leads to a finite answer for the shifts of atomic energy levels in a nonrelativistic theory without cutoff. Ambiguities that exist in the mass renormalization in the non-relativistic approach are pointed out. Such ambiguities vanish from the relativistic theory. Explicit calculation is carried out in the case of hydrogen by utilizing the Coulomb Green's function. The advantage of the present approach in calculating self-energy shift in high Z hydrogenic ions is suggested. (auth)

Corrugated horns involve a junction between the corrugated surface and a conducting groundplane. Proper horn design requires an understanding of the electromagnetic properties of the corrugated surface and this junction. An integral equation solution has been used to study the influence of corrugation density and tooth thickness on the power loss, surface current, and the scattering from a groundplane-corrugated surface junction.

Coaxial dielectric waveguides were investigated in order to ascertain their useful bandwidth. The operating bandwidth is defined as a region of dominant HE11 mode propagation only, where 93% or more of the electromagnetic energy is transported internally. Guide loss characteristics are highly dependent on the loss tangents of the several dielectrics. Coaxial dielectric waveguides have more linear phase characteristics than do their hollow metallic counterparts. Cutoff wave numbers for the four lowest of the high‐order modes have been found for a number of combinations of dielectrics.

It is shown that local analyticity and unitarity lead to logarithmic cutting off of particle momenta, when particles are detected in the binar inclusive process in the domain of large angles. (auth)

A theoretical determination is made for the susceptance of a narrow conducting strip inserted vertically into a waveguide. The theory is based upon a variational form for the susceptance. A suitable current distribution along the strip is obtained for the variational equation, and is found to be similar to that determined from analysis of backscattering by a cylindrical obstacle irradiated from an incident plane wave. Accurate theoretical results may be obtained using a sinusoidal current distribution having a phase constant of /spl pi//2d, where d is the strip depth. Experimental results agree closely with the theory in the dominant-mode range and also at frequencies below cutoff.

The 6 Li(p, pd) 4 He reaction was tested for dominance of a pole diagram by measurement of the Treiman-Yang distribution at an incident energy of 19 MeV. The resulting T-Y distributions were consistent with the isotropic pole predictions in the 0–60 MeV/ c region of deuteron cluster momentum in 6 Li. Absolute cross sections were obtained in the region of the test and were fitted with a plane wave impulse approximation and surface cutoff as prescribed by Shapiro. The reduced width for a deuteron in 6 Li obtained for a cutoff radius of 5.7 fm was θ d 2 = 0.23±0.09 corresponding to an effective deuteron cluster number N eff = 0.13±0.05.

The nuclear level spacings determined from neutron resonance experiments for nuclei with 20 ≦ A ≦ 148 and 181 ≦ A ≦ 209 are compared with spacings calculated for spherical nuclei with a microscopic theory which includes the nuclear pairing interaction. Single particle levels of Seeger et al. and Nilsson et al. are used in the calculations. The gross features of the experimental data due to nuclear shells are reproduced with the microscopic theory. In addition, the absolute agreement between experiment and theory is reasonable (67 % of the 151 cases examined agree to within a factor of 2) in view of uncertainties in the experimental data, the theoretical single particle levels and the pairing strength. Values of the spin cutoff parameter σ2(E), calculated with a microscopic theory, are included also for several doubly even nuclei and discussed in terms of nuclear shells.

Modal cutoff, dispersion, and attenuation characteristics for the dielectric coaxial waveguide (three interfaces) are presented for representative parameters. These properties change considerably as the guide geometry and dielectric properties are varied. Discussion of dielectric guide operation and comparison of a particular design with a metal waveguide is given.

The purpose of this paper is to provide a comprehensive study of the generalized impact formalism as applied to overlapping neutral lines, especially to the 4471.48 A HeI line emitted in a plasma. First, we will discuss the specific algebraic modifications needed to adapt the impact theory to partially degenerate lines. Secondly, we will review briefly the time-averaged dipole width and shift functions. The corresponding impact parameter averages and other properties are given and discussed at length. Thirdly, we will show that the various proposals made for the long range cutoff of the electron impact parameter do not lead to significant discrepancies in the final line shape. The short range cutoff is shown to be dependent on the static splitting and it is the smallest for the nondiagonal elements of the collision matrix. Moreover, the nonmarkovian corrections to the frequency-independent impact theory appear to be of only marginal importance for the 4471.48 A line in the line core, in contradistinction to the wings where more significant discrepancies are exhibited. These results strongly suggest that the non-diagonal part of the collision operator is dependent on the close perturbers collisions (weak collisions with the smaller impact parameter), so that the extension of the impact formalism to off diagonal matrix elements appears as an acceptable approximation, as long as the impact theory retains its validity. The impact formulation is seen to be self-consistent with respect to slow electron collisions which are automatically excluded when the static splitting is introduced in the collision operator. Finally, our profile calculations are compared with the previous ones and additional results with 2 × 1016 ⩽ Ne ⩽ 6 × 1016e-cm−3 and Te = 2 × 104 K are presented. The dip between the two peaks is seen to be sensitively dependent on the perturber temperature. This effect combined with ion dynamic shielding and a self-consistent choice of the short impact parameter cutoff could allow for an important enhancement of the line intensity between the two peaks, and also a significent reduction of the theory-experiment discrepancy.

Low-loss strip-loaded guides, consisting of 7059 glass film on fused quartz substrate with sputtered SiO 2 as the loading strip, have been investigated. The number of modes supported by the strip-loaded structures were determined experimentally and compared with the values predicted by the application of an equivalent index analysis. Agreement between theory and experiment is good in the case of the smaller number of modes which result from small loading, with the 7059 film thickness far away from cutoff.

The unnormalized doubly cutoff Schwinger functions converge as the ultraviolet cutoff is removed. The limits, the finite volume unnormalized Schwinger functions, are tempered distributions and areC∞ in the coupling constant. They have asymptotic expansions given by perturbation theory. For λ sufficiently small they can be normalized and then they are the moments of a measure onI′ℝ(IR3).

The experimental nuclear level densities and spin cutoff factors for 55 Mn, 56 Fe, 59 Co, 60 Ni, 62 Ni, 63 Cu and 65 Cu are compared with a microscopic theory which includes the nuclear pairing interaction. Single particle levels of Seeger and Nix give quite similar results in this A -region and both sets of single particle levels give results which agree well with experiment. The values of ρ ( U ) and σ 2 ( U ) depend on the structure of the single particle levels and the effects of this structure are illustrated and discussed.

We discuss the possibility of producing superheavy elements in the astrophysical r-process. Thus, we consider a detailed calculation of fission-barrier heights and neutron separation energies for heavy neutron-rich nuclei. The dependence of the neutron-induced fission cutoff of the r-process on the uncertainty in the nuclear models is discussed in some detail.

In-band generation of synchronous linear data signals eliminates the need for preliminary generation of a baseband signal and sharp cutoff filtering. However, the ratio of carrier frequency to pulsing rate must, in general, be an integer if the signal is to be produced by successive pulsing of a single fixed filter. As a consequence, a singlesideband signal of bandwidth W and transmission rate 2W must occupy the frequency band [ 2NW,(2N +1)W ], N an integer, when it is generated from a single fixed filter. A partial-response signal satisfying this condition is described in this paper. A further analysis demonstrates that a finite number of fixed filters can be used to produce a pulse train in any band where the ratio of carrier frequency to pulsing rate is a rational number. Since this relaxed condition allows any transmission band to be approximated to an arbitrary degree of accuracy, frequency translation can be avoided. A single transversal filter with periodically varied tap weights is shown to be a practical realization.

An analytical procedure for generation of a special type of critical monotonic magnitude characteristic, having maximal flatness at a given frequency ω0 beyond the origin, with arbitrary transmission zeros, is presented. Compared with the known filters with maximally flat magnitude at the origin and the same transmission zeros, the new filters show distinct improvement in the cutoff performance.

1. The series of values of feeler gauge radii re normalized on the basis of the recommendation RS 3951-73 for mechanical filtration of waviness should be changed to the following: 2.5, 8, 35, and 80 mm. 2. This series of re values makes it possible to filter out waviness without undergauging, and the amplitude-sensitive transmission ratio of the roughness will not be greater than 5% in the majority of cases. 3. The series of values of cutoff wavelengths λR between roughness and waviness, normalized on the basis of the recommendation RS 3981-73, should be supplemented by the value 8 mm. 4. Methods of filtering out waviness proposed in the recommendation RS 3951–73 are not equivalent. Mechanical filtration with the aid of feeler gauges of large radii is nonlinear and does not define the frequency (pitch) characteristic of the filtration process unambiguously. Mechanical filtration depends not only on the radius of the feeler gauge, but also on the height of the asperities and is determined by the dimensionless ratio T2/rH, so that electrical filtration techniques must be recommended in the first instance. 5. Relationships linking the characteristics of electrical filters and the radii re of the feeler gauges were found, i.e., relationships between electrical filtration and mechanical filtration were arrived at. 6. The results obtained on the connection between re and λR values and the height WZ of the wave (see data in Table 1) can be employed in selecting the parameters of electrical and mechanical waviness filtration in practice.

The spectral cues used for median‐plane localization are described by three experiments. First, the frequency spectrum necessary for localization is measured by noting the accuracy of subjects localizing low‐ and high‐pass‐filtered white noise. Second, several high‐pass, low‐pass, bandpass, and bandstop filters are associated with the subjective impression of direction by observing what directions are most frequently perceived by subjects localizing white noise colored by each filter. Third, the frequency responses of several artificial ears are measured for different angles of median‐plane sound incidence. Results show that sound spectra from 4 to 16 kHz are necessary for localization. Frontal cues are a 1‐octave notch, with a lower‐frequency cutoff between 4 and 10 kHz and increased energy above 13 kHz. The “above” cue is a 1/4‐octave peak between 7 and 9 kHz, with a high‐frequency cutoff at 10 kHz. The “behind” cue is a small peak from 10 to 12 kHz. Increases in frontal elevation are signaled by an increase in the lower cutoff frequency of the 1‐octave notch. This notch appears to be generated by time‐delayed reflections off the posterior concha wall interfering with sound directly entering the external auditory canal.

The energy spectrum of the atmospheric γ rays has been measured utilizing the multiple scattering of the electron-positron pairs in a 30-gap spark chamber. The spectrum at 3 mbar and 4.5-GV cutoff is Iγ = (7.9 ± 1.6) × 10−3E−1.45±0.20 γ(s g sr Mev)−1 for 50 < Eγ < 250 MeV and Iγ = (8.7 ± 1.8) × 10−1E−2.3±0.4 γ(s g sr MeV)−1 for 250 < Eγ < 1000 MeV. At 12 GV the intensity is reduced by the factor 0.57. The shape of the spectrum is steeper than that predicted by Beuermann, but the discrepancy can probably be accounted for by uncertainties in the pion production in the atmosphere and in the intensity of the return albedo. The sea level energy spectrum is also fitted by a power law and the angular distribution by a series expansion in cos (nθ).

We have studied the radiative corrections to the angular distribution and energy spectrum of {3 rays from polarized nuclei. Explicit formulas of the radiative corrections are given for the angle-dependent and angle-independent terms and for the asymmetry factor, under the assumption of the effective current-current interaction of the V -pA type for the {3 decay. The lowest order corrections to the first two are almost the same and depend on the cutoff. The asymmetry factor of the {3-ray angular distribution is scarcely affected by the radiative corrections. In particular, these corrections are independent of the cutoff for the Gamow­ Teller transition. It is also shown that the radiative corrections to the {3-ray spectrum are consistent with those given by Kinoshita and Sirlin, and Berman and Sirlin. Numerical values of the radiative corrections are tabulated for the {3 decays, 12B-')12C+e- + :v and 12N-')12C+e+ +II.

Application of a coupled-mode formalism to longitudinally magnetized ferrite phase shifters provides an explanation of the increase or decrease of insertion phase with increasing magnetization which is observed in different types of phase shifters. If the higher order mode is TM, the phase shift increases with magnetization while the reverse happens if the higher order mode is TE. The generalized telegraphists' equations are used to analyze the TEM phase shifter. The maximum phase shift that can be obtained is determined by the effective permeability of the ferrite. However, coupling to higher order cutoff modes reduces the phase shift significantly.

A comparison of methods for the numerical solution of the hollow waveguide problem is presented in tabular form. Another table lists waveguide shapes and their cutoff characteristics that have been presented in the literature. These tables and the bibliography afford an aid towards the selection of a method.

Nonlinear acoustic theory is used to show that the cutoff frequencies and the resonant frequencies of modes in acoustic waveguides of finite length depend upon the mode amplitude. The amplitude en, of mode n > 0, produced by a periodic piston motion of amplitude δn, is also determined. It is found that at and near a resonant frequency, en≏δn1/3 while at and near the cutoff frequency en≏δn1/2. Away from these frequencies en≏δn, as linear theory predicts. Thus the infinite amplitude, which linear theory yields at cutoff and resonance, is avoided. The results are obtained simply by using the result for the propagation constant as a function of mode amplitude given by J. B. Keller and M. H. Millman [J. Acoust. Soc. Am. 49, 329–333(1971)].

Two kinds of modified Lippmann‐Schwinger equations are derived for the case of long‐range potentials. The equations of the first kind are homogeneous and are a direct result of the fact that the standard Lippmann‐Schwinger equations do not hold when long‐range forces are present. The equations of the second kind depend on the existence of an operator Z such that W± = s‐lim exp (iHt)Z exp(−iH0t). A general recipe for constructing Z is given and its computation is carried through for the case of asymptotically Coulombic potentials. The resulting equations are used to compare the long‐range theory with the theory with a space cutoff (i.e., screened potential) in the limit in which that cutoff is being removed.

We give the results of a numerical calculation of nonlinear effects of surface-helical instability of an electron-hole plasma, based on the theory, developed earlier, for the case of equal mobilitiesμe =μh =μ. The amplitudes of the helical waves, the nonlinear diminution of the current, and the frequency shift for the mode ¦ m ¦ = 1, k = (4/3)1/2 is calculated as a function of the cutoff magnetic field B in the range 0

The propagation of TE/sub no/ modes in rectangular wave-guides that contain two dielectric slabs parallel to the narrow wall and extending over the full height of the guide is investigated. Waveguide and dielectric are assumed to be lossless and infinitely long. Apart from these restrictions, the dielectric slabs may have arbitrary thickness, position, and dielectric constant. The analysis is restricted to TE/sub no/ modes with the E-field parallel to the narrow guidewall. The guide containing only one dielectric slab is covered by this analysis. The even modes n = 2, 4, 6, . . . of the guide with two slabs correspond to the odd modes n' = n/2 = 1, 2, 3, . . . of the guide with one slab half the width of the guide with two slabs. For six relative dielectric constants (/spl epsilon/= 2.25, 4.00, 9.00, 12.25, 16.00, 25.00) the cutoff frequencies for TE 10, 20, 30, 40, 60 modes and the normalized propagation constants for TE 10 and TE 20 modes between their respective cutoff frequencies and a frequency slightly above the second- and fourth-order mode cutoff frequency for the empty guide, respectively, have been computed for a large range of slab thicknesses and slab positions. Selected results are presented graphically. These results are discussed. The parametric dependence of field distributions, of normalized characteristic impedances, of the ratio of cutoff frequencies (fractional bandwidth), and of the ratio of magnetic field components (ellipticity) are illustrated.

The perturbation-theoretic result for the neutron-electron interaction, as calculated by Salzman in the Chew no-recoil theory, is in serious disagreement with experiment. The main part of the experimental interaction (\ensuremath{\sim}4.2 kev) is accounted for by the Foldy term (4.08 kev). But the latter is not included in the no-recoil approximation, where one calculates only the electron interaction with the static pion cloud of the physical neutron. For this part of the interaction the no-recoil perturbation result is too large by a factor of at least twenty. We investigate here the possibility that higher order corrections drastically alter the pion cloud charge distribution. Using the techniques developed by Miyazawa, one can express the higher order corrections in terms of weighted integrals over the pion-nucleon interaction cross sections. It is found that the higher order effects indeed reduce the pion cloud contribution, but only by 20 percent. Because of certain ambiguities involved in treating the cutoff function, it cannot be stated that this represents the true content of Chew's theory; but it seems unlikely that any reasonable treatment of the cutoff could reduce the discrepancy much further.

The photoproduction and electroproduction of pions is treated relativistically. A new solution for the 3-3 amplitudes is obtained which includes recoil and is insensitive to a cutoff. Corrections to the ${E}_{0+}$ and ${M}_{1\ensuremath{-}}$ amplitudes are evaluated by using fixed-$t$ dispersion relations. Good results are obtained in the region around the 3-3 resonance up to $\ensuremath{\pi}\ensuremath{-}N$ center-of-mass energies about 1300 MeV and electron momentum transfer up to 3.8 Be${\mathrm{V}}^{2}$.

The frequency response of bipolar transistors is calculated using a model which assumes that the spread in the transit time of minority carriers flowing from emitter to collector is the major factor in determining the frequency cutoff. The total emitter-to-collector transit time is determined by the combined effects of drift and diffusion but the spread in transit time is determined exclusively by diffusion. The analysis is adapted to point-contact transistors and checks experimental measurements of frequency cutoff in four groups of point-contact transistors having different structure and material parameters.

Compressional-wave velocity and attenuation measurements have been made for Pennsylvania slate over a frequency range from 64 kHz to 15 MHz. All the internal friction values have been made for a direction parallel to the cleavage plane. The internal friction varies by a factor of 10 over this frequency range decreasing at the high frequencies and at the lower frequencies. The high-frequency measurements are consistent with the high-frequency component of dislocation damping with little pinning of the loops by impurity atoms. The drop-off at low frequencies with a leveling off at 10 kHz is consistent with a low-frequency dislocation component. The width of the dislocation distribution is less than exponential, which is in agreement with the cutoff of the long dislocation loops by the finite grain size.

The abrasive cutoff operation in which a high-speed disk is fed radially into a steel member is analyzed. The feed rate and the wheel speed are the chief variables of importance. Cutting forces and wheel wear rate are analyzed in terms of chip size, grain spacing, and the length of work in the direction of cut. While the highest wheel speed possible should be used, there is an optimum feed rate. When the feed rate is low, an excessive wheel temperature is encountered, and wheel wear rate will be high. At a high feed rate, individual chips are too large for the space available, and wheel wear will also be high. Best performance from the point of view of wheel wear rate is obtained using a hard, dense wheel of large grain size (20 or coarser) operating at the highest wheel speed possible and with an intermediate feed rate.

Wind velocity measurements and smoke patterns are compared. The most frequent situation is a Gaussian wind velocity fluctuation distribution, which is in good agreement with the theoretical picture developed by C. F. Wandel and the author. Occasionally, under conditions usually classified as typical inversion situations, definite non-Gaussian distributions are observed. In such cases diffusion is slow, and a low-frequency cutoff in the Lagrangian power spectrum is proposed for the description of the phenomenon.

The degree of consistency of an oscillator model of a nucleus is examined by means of a type of Hartree–Fock calculation based on a simple form of internucleon potential valid at low energies. An effective mass equal to 0.757 times the mass of a free nucléon is used. The oscillator wave functions are found to be not far from self-consistent and the oscillator frequency derived is physically reasonable, but the bound on the binding energy is not good. It is also shown that the oscillator wave functions are a good approximation for the state functions of particles bound in a finite potential well having the shape of a cutoff oscillator so that the Hartree–Fock calculation can be used to prescribe a shell model potential.

Vibrational excitation cross sections of diatomic molecules have been calculated using the Born approximation by including the polarization interaction between the incident electron and the molecule. Because of the divergence of the polarization potential at the origin, a cutoff procedure must be used. Four different cutoff methods have been examined, and the excitation cross sections are not very sensitive to the choice of the cutoff parameters. For the nonpolar molecules, inclusion of the polarization term increases the cross sections by an order of magnitude or more over the previous calculations, while in the case of the CO molecule, the contribution from the polarization term is comparable to that from the monopole—dipole interaction.

Lifetimes of the high-frequency collective modes in simple argon-like classical liquids are estimated using a variational technique due to Zwanzig. It is assumed that the decay of these modes is due to the inhomogeneities in the system, giving rise to the mode-mode interactions. An expression for the lifetimes of both the longitudinal and the transverse components of the collective modes is written in terms of a cutoff parameter q 0 ; this corresponds to the correlation length or the first peak in the structure factor S ( k ). From our estimates, these collective excitations have lifetimes of about 1.4 × 10 -13 s , which is consistent with recent analysis of the molecular-dynamics experiments.

In a series of balloon flights, at 4.6 bv geomagnetic cutoff rigidity (corresponding to the geomagnetic latitudes of about 46° in northern Italy and about 42° in southern New Mexico, United States), with a light-weight neutron detector, the flux of the cosmic ray neutrons in the atmosphere has been investigated in the energy range from thermal to about 20 Mev. The energy spectrum proposed by Newkirk has been found to be in agreement with the experimental results. Values of the integrated neutron flux at different atmospheric depths are given; by extrapolating to zero pressure, the leakage flux has been determined as (0.45±0.07) neutrons cm−2 sec−1.