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A phenomenologically successful dual amplitude with logarithmic trajectories is studied. The delta function imaginary parts of the amplitude are approximated by an “averaged” imaginary part. As expected, the resulting average s -channel discontinuity vanishes when α ( t ) = 0, −1, −2, … and also gives rise to a double spectral function with a correct hyperbolic boundary. It is argued that the dual amplitude has properties which are very similar to those expected from a composite particle model.

Mitosis occurs synchronously in up to 108 nuclei in the syncytial plasmodium ofPhysarum polycephalum. Any two phases of the mitotic cycle may be mixed by fusing plasmodial pairs. A topological property of the synchronized phase of the fused pair as a function of parental phases, the arc discontinuity, characterizes the underlying oscillator, and indicates mitosis is controlled by a moderate relaxation oscillator which rotates more rapidly near its singularity than its limit cycle. A model oscillator is briefly described.

Transitions from symmetry-adapted to symmetry-broken solutions of variational problems are classified according to the nature of the discontinuity involved. Two systems in which a second-order transition occurs are studied.

Using the results on the n -reggeon cut obtained by Gribov, Pomeranchuk and Ter-Martirosyan, we construct formal solutions for a two-particle scattering amplitude involving the exchange of a pomeron and multi-pomeron cuts.

The upper bound solutions for the axi-symmetric extrusions with two-stage die and curved die were studied. For the two-stage extrusion the radial flow was used as the admissible velocity field. The optimum shape of the surface of velocity discontinuity was given as a sphere having its center at a certain position on the extrusion axis. The extrusion pressure and the distribution of strain in the extruded billet were calculated based on the optimum velocity field. Discussions were made on the effects of die angle and extrusion ratio of each stage on the calculated values. The upper bound solution for curved die (sigmoidal die) was obtained adopting a flow function, where the shape of die section was expressed with Fourier series. The extrusion pressure and the distribution of strain in the extruded billet was calculated. The hydrostatic extrusion of copper was made and the extrusion pressure and the distribution of Vickers hardness in the extruded billet were measured. The experimental results showed good agreement with the calculated values.

A new treatment of the left-hand cut for N / D problems is presented whereby, instead of poles, the cut is parametrized by a Legendre expansion which fixes the discontinuity. The method is tested through the use of analytic forms, and is found to work very well in a wide variety of circumstances. An extension is made to situations where there are errors in the input, and resulting problems are discussed. The method is applied to elastic pion-pion scattering. Both the rho and f 0 mesons are produced dynamically, but trouble is encountered for the s-waves. The relation of this trouble to ambiguities found in other s-wave calculations is discussed.

The uniaxial motion of interfaces between regions deforming elastically and regions deforming plastically is considered. The governing constitutive, stress-rate/strain-rate equations in both elastic and plastic regions are taken to be non-linear. Discontinuity relations across such interfaces are established by the repeated differentiation of existing relations. The relations given by previous workers (especially R.J. Clifton, T.C.T. Ting, E.H. Lee and Th. von Karman) are discussed. The precise situations in which they hold are considered, and it is shown that some of these relations, while apparently derived for different situations, can, in certain circumstances, be shown to be equivalent. It has been shown that six essentially different types of motion can occur, and, when the constitutive equations are linear, each type of motion is unique. This result is extended to the non-linear situation, by means of an established local expansion procedure. For the case of a meeting interaction of stress waves carrying initially linear profiles, the previous (linear) analysis given by L.W. Morland and A.D. Cox fails to distinguish between certain types of motion. This motion is reconsidered and it is shown how non-linearity in the constitutive laws serves to determine uniquely the type of motion that takes place.

Proofs have been given that the Bethe-Peierls approximation solves exactly the Ising problem on a Cayley tree. For a tree with coordination number $\ensuremath{\gamma}g2$, the approximation predicts, among other things, a phase transition in zero field at ${T}_{c}=2J {{\mathrm{ln}[\frac{\ensuremath{\gamma}}{(\ensuremath{\gamma}\ensuremath{-}2)}]}}^{\ensuremath{-}1}$, with a discontinuity in the specific heat. On the other hand, the partition function in zero field can be calculated exactly and turns out to be analytic for all $T$. This paradox is analyzed and resolved. The transition occurring on a Cayley tree is found not to be of the type usually studied in thermodynamics.

We consider monochromatic electromagnetic scattering from a fixed configuration of an arbitrary number of separate scatteres which are immersed in a medium with constant electric and magnetic properties. Within the scatterers the electric and magnetic properties are assumed to vary smoothly. By considering the integral equations for the scattering we show that the total $T$ matrix for the configuration of the scatterers can be expressed in terms of the $T$ matrices for the individual scatterers in the same way as was previously found for a similar configuration of scatterers, each having a discontinuity in the electric and magnetic properties at the surface and constant electric and magnetic properties in its interior.

A modification to the coupled channel equations for many-body scattering is introduced, based on the evaluation of the discontinuity equation for the matrix of transition operators. The resulting (modified) equations are shown to be in a form which resolves some problems associated with continuum states in an analysis of rearrangement collisions using the eigenstate expansion method.

The Earth-ionosphere waveguide is considered to be a parallel plate waveguide whose upper and lower boundaries are characterized by constant impedance matrices. Simple orthogonal relations between the resulting modes are found and used to obtain an approximate formula for the mode conversion coefficients at a land-sea discontinuity. Some numerical results are given for propagation below an anisotropic ionosphere in which the QL approximation is valid. These results are similar to those for the propagation of TE , not TM , modes below an isotropic ionosphere.

The effect of an aligned magnetic field on the non-linear instability of Kelvin-Helmholtz flow of an inviscid incompressible fluid with perfect electrical conductivity is investigated in the presence of gravity and surface tension. The intensities of the unperturbed magnetic field are uniform in each side of the interface. The equations are derived for the time variation of the amplitude of the small disturbances with wavenumbers near the neutral stability. It is found that the effect of the magnetic field on the non-linear stability of the flow depends on its discontinuity, the density ratio and the value of surface tension. Detailed discussion is given in the case of small value of surface tension.

In the context of the multiperipheral model, we use an effective-range approximation and the formula for the discontinuity of the partial-wave amplitude across the AFS cut to construct a model amplitude that incorporates the reggeon-pomeron cut in addition to the simple Regge pole. We use this amplitude in an attempt to explain the structure of the high-energy differential cross section at small t, as observed at ISR. The model explains in addition the discrepancy between the values for the rho-intercept obtained from the differential cross section on the one hand and the difference of π±p total cross sections, at high energy, on the other. We can also understand the absence of such a discrepancy in the case of the omega trajectory.

A study is made of steady and non-steady flow through a carburettor. A model is presented in which the carburettor is represented by an adiabatic pressure loss at a discontinuity in the intake system of an engine. It is shown that this simple model gives excellent predictions of the pressure-time development in the intake system except at small throttle openings where there is some error in the peak amplitude. Mass flow and trapped mass predictions are excellent.

In this paper, following a brief review of previous contributions to the field of Parametric Acoustics, plane and spherical wave solutions of Burgers' equation are derived by the method of successive approximations for the difference‐frequency signal which results from nonlinear interaction in the medium of two different‐frequency primary waves radiated simultaneously by a finite‐amplitude source. These solutions are then matched asymptotically obtain a farfield solution which includes the influence of nonlinear interaction within the Fresnel and Fraunhofer zones of the primary waves, from which an expression for the “equivalent difference‐frequency source level” is derived. The analysis assumes that the primary‐wave amplitudes, which induce the nonlinear interaction are below their respective shock thresholds as defined by a criterion which relates the steepest slope of the waveform to a predetermined upper bound approximation of an ideal discontinuity. In order to determine the predictive reliability of the solutions, a parameter which characterizes the “equivalent difference‐frequency source level” is compared with empirical results obtained over a wide range of experimental conditions. Finally, a quasiplane three‐dimensional equation (which in one dimension assumes the form of Burgers' equation) is also analyzed in order to obtain a second‐order farfield directivity function for the difference‐frequency signal.

The spherical waves in an elastic-plastic, isotropically work-hardening medium generated by radial stress uniformly applied at a spherical cavity r=r0 are studied (r denoting radial distance). The radial stress and its time derivative at the cavity may be discontinuous at time t=t0. If the applied radial stress is continuous while its time derivative is not, the discontinuity at (r0, t0) propagates into r > r0 along the characteristics and/or the elastic-plastic boundaries. If the applied radial stress itself is discontinuous, the discontinuity may propagate into r > r0 in the form of a shock wave, or a centered simple-wave, or a combination of both. In any case, the solutions in the neighborhood of (r0, t0) are obtained for all possible combinations of discontinuous loadings applied at r=r0. This is a systematic study on the nature of the solution near (r0, t0) where the applied load is discontinuous. Solutions for special materials, such as linearly work-hardening or ideally-plastic ones, and for special applied loadings at the cavity obtained by other workers, in which the nature of the solutions near (r0, t0) are assumed a priori rather than determined, are compared with the results obtained here. Some of the solutions are found to be in error because of incorrect a priori assumptions.

Waves produced in a homogeneous elastic body by time-dependent displacements of all or part of the body surface are determined by the geometrical theory of diffraction. Waves associated with the rays of geometrical acoustics, and with certain diffracted rays, are treated. The latter arise from curves of discontinuity of the applied displacement and from curves separating the displaced part of the surface from the free part. The expression for each type of diffracted wave contains a diffraction coefficient. Certain diffraction coefficients are determined from appropriate canonical problems which are solved exactly by the conical field method. The theory is applied to the impulsive and time-harmonic angular displacements of a circular region on the otherwise stress free surface of a half-space. The results are shown to agree with and extend those found previously in these cases by other methods.

The pair distribution functions evaluated for the 23S state of the helium isoelectronic sequence from the Hart and Herzberg correlated wave functions and those corresponding to the Hartree-Fock approximation are used to determine the shape of the corresponding Coulomb holes. As a consequence of a discontinuity in the Hartree-Fock solution between He and Li+, the Coulomb hole has a different shape for He than for Li+ and the other isoelectronic ions.

Finite difference method for the numerical solution of electromagnetic waveguide discontinuity problems is presented. The method of boundary relaxation is applied, using finite difference techniques in the nonuniform section of the waveguide and using a modal representation of the field in the uniform sections of the waveguide. To illustrate the process some two-dimensional diffraction problems in an electromagnetic waveguide with rectangular cross section are solved.

A theory of the power spectrum of the Barkhausen noise is presented, accounting for the frequency behaviour of the experimental spectra in the whole range of analysis frequency between 1 Hz and 20kHz, when the magnetizing frequency is kept relatively low (below 0.1 Hz). The theory assumes the existence of a correlation between the number of clustered pulses in a large Barkhausen discontinuity and the time interval from such a discontinuity to the preceding one: the longer this time interval, the larger the number of clustered pulses. An experimental check of the dependence of the noise power spectrum on the magnetizing frequency in rings of Fe−Si 4% is given. It is shown that the theoretical results fit the experimental data without the introduction of any arbitrary constant.

It often happens that scalar-valued observables of biological interest are points not on the real line, but on the circle… for example phases of periodic events, and colors. Sometimes one such quantity depends upon two others in a symmetric way, for example, in determining a compromise phase after slime mold plasmodia are fused at different phases of the cell cycle, or in determining the color of a mixture of two colors. In such cases the experimental result cannot depend on the two inputs in an unreservedly continuous way: there must be a point of ambiguity or discontinuity. Experiments involving the cell cycle and glycolysis are examined, in which the discontinuity appears to take two different forms.

The conditional probability of finding a cavityG(r) devoid of molecular centers of hard sphere fluid is modified by making use of the discontinuity atr=a/2 (a is the diameter of a molecule). The new equation of state thus obtained is bounded by the Percus-Yevick compressibility and virial pressure equations of state, which may be the upper and lower bounds to the results of machine calculation.

This paper describes a functional method of deriving families of sum rules that express the full content of isospin conservation. In particular, it is shown that there seem to be no isospin relations involving inclusive cross sections beyond those that follow from Iz conservation alone.

A thin cylindrical elastic tube, reinforced on the outer surface by a two parameter family of inextensible cords, is deformed in such a way that the inner surface assumes a given shape. A previous solution for this problem, valid only when the curvature of the deformed inner surface is continuous everywhere, is extended to the more general case where the curvature possesses discontinuities. The method of matched asymptotic expansions is used to construct the solution in the neighborhood of a discontinuity, and to join it smoothly with the earlier solution, which is shown to remain valid away from the discontinuity. The deformation described here occurs, for example, when a reinforced elastic tube is deformed due to an enclosed rigid mold or mandrel and an applied external pressure.

From measurements made by the Heos 2 satellite a spread of bow shock locations is observed, oblique shocks (in the hydromagnetic sense) lying more earthward than perpendicular ones. This result is interpreted as a dynamic effect rather than as being due to different field-determined static equilibrium configurations. A simple hydromagnetic model for the impulsive interaction of the bow shock with a discontinuity in magnetic field direction is stated and discussed quantitatively. It is found that velocities of the order of 10 km/s can be induced thereby. The observed spread in bow shock positions as well as the correlation with magnetic field orientation can be explained by our model.

Eleven large directional discontinuities in the solar wind, simultaneously observed by Explorers 33, 35 and Heos 1 in the period Dec. 1968 to Jan. 1969, have been studied by the triangulation method. Testing the discontinuity normals deduced from the wind speed and timing parameters against those given by the magnetometer observations on different models suggests that 8 were tangential, 2 were rotational while 1 could not be identified. 3 small tangential discontinuities arriving within a period ≈ 35 min were also studied and as a result it is suggested that the solar wind remained stratified in separation planes over a distance ≈ 0.01 AU during that time. The implication of these results is briefly discussed.

By exploiting discontinuity properties of the maximal convolution it is possible to drastically reduce dimensionality in finite dynamic programs. In fact, we show how the search over the usual M-dimensional state space can be reduced to a one-dimensional search over an imbedded state space. The versatility of our approach is illustrated on a number of example problems.

We give a detailed treatment of the analytic continuation of the two-Reggeon-cut discontinuity formula from above the four-particle threshold in the $t$ channel down to negative $t$. We confirm the negative sign of the two-Pomeron-cut contribution to the total cross section. We show how the Mandelstam graphs can be used as a check on this result. We trace the negative sign to signature factors and use this to argue that multi-Pomeron cuts should contribute to the total cross section with alternating signs.

A general approach for deriving quasi-static equivalent circuits for discontinuities in microstrip lines is presented. The formulation is based upon Galerkin's method applied in the Fourier transform domain. Numerical results are presented for a number of different configurations and compared with data available from other sources.

This paper investigates the reflection and transmission of charged particles incident on a rotational discontinuity whose thickness is small compared to their Larmor radius. This is done for the full range of parameters likely to occur for rotational discontinuities in the solar wind. Usually the proportion of particles reflected is similar to that predicted by adiabatic theory, except when the magnetic field ratio is close to unity, or the bend in the fieldlines is large.

A distinction is made between a mobility edge, which is a quantum-mechanical phenomenon predicted for non-crystalline conductors, and a percolation edge which occurs in classical theory as a consequence of long-range fluctuations in the degree of disorder. The mobility edge is an energy, denoted by E c, at which the conductivity σ(E) at zero temperature of a degenerate electron gas with Fermi energy E jumps discontinuously from zero to a value σmin (the minimum metallic conductivity); for energies above a percolation edge E p there is no discontinuity and σ varies as (E-E p)1·6 For wells of random depth (the Anderson model) σmin is given by the author's formula (∼ 0·025 e2/ka), the constant however depending strongly on the Anderson localization parameter. When there are long-range fluctuations, as for instance in an inhomogeneous glass, then as soon as tunnelling is allowed there will still be a discontinuity in σ it will however be much smaller than the value given above and will occur at a val...

Part II of this paper examines some important cases in which the analysis of Part I can be simplified. The new conditions of a wave bifurcating and of a strong discontinuity being either non-reflective or transmission suppressive with respect to a wave are identified. An application is made to shallow water waves advancing over a step in the sea bed.

The stability and characteristics of Kelvin-Helmholtz waves in an atmosphere that may be saturated over some of its height are investigated analytically and numerically. It is shown that, if there is a temperature jump at the interface, the Wegener hypothesis, i.e., the assumption that stability boundary curves are loci of neutral waves travelling with phase velocity equal to the mean of the velocities of the regions above and below the discontinuity, is invalid. Instead, the stability boundary corresponds to the singular neutral modes with phase speed equal to the velocity in one or the other layer, depending on the sign of the temperature jump and the presence of saturation. Furthermore, the effect of saturation on the stability is found to be substantial. For the common case of a shallow saturated layer adjacent to the interface, the system is shown to behave essentially as if the temperature jump were smaller by an amount proportional to the mixing ratio and thickness. Finally, the validity o...

Product array processing, also called split beam correlation, is often the favored process for both detection and tracking because it has higher spatial resolution than sum and square processing and because it has been thought to yield a near zero mean output for either independent or isotropic noise inputs. Numerical evaluation, however, shows that when steered near endfire a product line array embedded in isotropic noise will yield a substantially nonzero mean, e.g., 15% correlation. This effect is studied in terms of both interelement correlation and directional response patterns for both split arrays and element pairs. It is shown to be a consequence of the discontinuity in the isotropic noise field when considered in the total interelement delay space that the array is capable of being steered to. It is thus seen to apply to product arrays of arbitrary configuration whenever the product steering delays are nearly equal to the maximum possible delays in the medium, e.g., near endfire for any element pairs. This effect is also shown to apply to ambient noise fields that are highly nonisotropic. Finally, it is shown that this effect can be greatly reduced by the addition of noise at each of the array elements, the noise being correlated in such a manner as to extend the isotropic noise field further into the delay space covered by the array.

This study expands further on an earlier study reported in this journal wherein the power spectrum and total power of a moving point source in a round, slug flow jet were calculated. In the present study three further aspects are reported on. Firstly the effect of non-axial lines of source convection is explored with the aid of a plane jet, line source model problem. Secondly, for centerline source convection in a round jet, the effect of a small (non-zero) shear layer thickness is studied. This procedure is first illustrated by an application to the classical problem of reflection of plane sound waves from a velocity discontinuity. Finally inferences regarding the peak angle in the radiation pattern are drawn from the study and shown to be in rough agreement with experimental data.

A method is developed for solving optimal-control problems having cost functional or system functions that are continuous but have discontinuous partial derivatives with respect to the state variables. The method is based on splitting the total time interval on which the problem is defined into segments where the trajectory either totally avoids the points of discontinuity or stays fixed at such a point. Examples are given which motivate consideration of such problems and illustrate the proposed method of solution.

It is shown that an arbitrary discontinuity of an arbitrary multi- Regge amplitude factorizes in a large class of models which encompasses the dual resonance model. It is argued that multiple discontinuities also factorize and that the factorization of discontinuities is expected to hold on very general grounds as a result of the singularity structure of multiparticle amplitudes. These results allow the full amplitude as well as its discontinuities to be written as a product of Regge propagators, single-, double- and triple-Regge vertices and their discontinuities. (24 refs).

The equations for the xerographic discharge of the potential are solved analytically by means of a flow line technique. The solution is for single‐carrier discharge corresponding to charge generation using strongly absorbed light. Exact analytical solutions are given for two cases in which both the photoinjection efficiency and the mobility are field dependent, including the Poole‐Frenkel effect. The surface potential V(t) and d V(t)/d t are calculated for specific cases, and the effect of variations in mobility, thickness, and injection rate are discussed. Under certain conditions the discharge can remain emission limited for all times. It is shown that a discontinuity in the conduction current exists at the transit time of the lead carriers, corresponding to the cusp in the constant‐potential case when the current is space‐charge limited.

This paper describes an LMFBR core design optimization procedure, where the sequential unconstrained minimization technique (SUMT) using the conjugate direction method for the unconstrained minimizer is adopted as the optimization algorithm. To minimize computing effort in estimating the equilibrium core condition, the procedure utilizes the regression functions of the burn up characteristics in a simplified method of calculation, which is called the “power balance method”. As a numerical example, the procedure was applied to calculations on a 1,000 MWe LMFBR core design, on which an evaluation is also made of the sensitivity shown by the results to changes in critical constraints. It is shown that the optimization algorithm based on SUMT is well applicable to core design optimization, provided careful attention is paid in treating the penalty term of the transformed performance index in order to obviate a discontinuity occurring in the locus of the optimized solution at the constraint boundary. It is als...

Apodization, a process in which the energy in a diffraction pattern is redistributed among the main lobes and the side lobes, has proved its usefulness in optical imaging systems and in linear sonar arrays. It can also make the numerical reconstruction of holograms more adaptable to varying requirements. To demonstrate, the hologram of a rectangular slot is first reconstructed numerically in the conventional manner. It is then reconstructed after apodizing the diffraction pattern contained in the hologram. One type of apodization led to a reconstructed image that accentuated the discontinuity at the edge of the slot. Another type gave an image with a concentration of energy at the center of the object, thus minimizing interference from nearby objects. It is expected that the method can be extended to other objects by expressing them as summations of rectangular slots.

A mechanism is proposed by which discontinuities in slope can propagate along an ideal fibr-ereinforced beam which is inextensible in the direction of its axis. The equations of motion of the beam are formulated, including the dynamical conditions which must be satisfied at the discontinuity. Constitutive equations for a rigid-plastic fibre-reinforced beam are established, and it is shown that slope discontinuities may propagate in a strain-hardening material, but are stationary in a perfectlyplastic beam. The theory is illustrated by its application to the problem of a beam moving in a direction normal to its axis brought to rest by striking a rigid stop at its mid-point. It is shown that in the subsequent motion slope discontinuities travel outwards from the centre of the beam. A complete explicit solution is obtained for the case of a beam with linear strain-hardening.

We determine the ωpp coupling constant by representing the discontinuity across the unphysical cut in the pp forward dispersion relation by one-meson and two-pion-exchange terms. The real part of the forward pp amplitude is computed from pp phase shifts below 460 MeV lab. kinetic energy. For the ωpp coupling constant we find values g ω 2 4π≈10 ± 1.2 .

It is shown that so far as pure advection is concerned, entities like density, density discontinuity (front) and humidity mixing ratio move with the wind at the level considered. Any disparity between actual movement and streamline movement must be put down to development. A pressure-field at a given level is advected in a direction and with a speed given by the average wind at will levels above that level. Similarly the difference in pressure between two fixed levels is advected by the average wind between those levels. Any disparity between actual movement and that determined by the average wind must be attributed to development. The advection of contours and thickness lines can be inferred from advection of isobars and pressure-differences in the absence of development. Winds are normally available up to pressure levels of 60 mb. in the British radiosonde ascents, so that over 90% of the atmosphere can be utilised in computing average winds. There is reason to believe that averages based on this data will be sufficiently accurate to give a satisfactory measure of advection of tropospheric phenomena. The average wind may also be taken as a measure of frontal advection, in the sence that the frontal through of pressure must be advected in the same way as other pressure fields when development is not of importance. A measure of upward motion might be obtained on some occasions from a consideration of the conflicting advections at different levels. The theory of the pressure-field being advected by the average wind at all levels accounts satisfactory for the fact that systems with “solid” wind currents do not move. An explanation of “thermal steering” is given.

It is demonstrated that the pure-Coulomb $t$ matrix satisfies a modified unitarity condition and that its discontinuity is not zero, as has been asserted elsewhere.

The calculation of the partftion function for a simple onedimensional order-disorder model is reduced to the solution of a certain functional equation. This equation is rigorously and it is shown that in the limit of indefinitely long-range interactions the model exhibits a finite discontinuity in the specific heat. (auth)

The problem of an unstable fluid overturning in a vertical tube filled with porous material is treated by an approximation of boundary-layer type. It is shown that the fluid can experience a pseudo-inertial effect, in which variations in density across the tube exhibit properties analogous to variations of momentum in an inertial flow. The mean fluid density and mean-square vertical velocity over a horizontal cross-section of the tube are related by a pair of hyperbolic equations, for which there exist two systems of characteristics. It is shown that changes in the mean density of the fluid can be propagated as discontinuities. For discontinuities of finite amplitude, two jump conditions are derived, one of which is found to involve an undetermined parameter λ. The theory is applied to the case of a vertical tube containing porous material saturated with water, which is attached at the top to a reservoir filled with an aqueous solution (an analogue of Taylor's (1954) experiment). The motion of a finite discontinuity which arises at the initial unstable interface is calculated by two approximate methods. These results compare satisfactorily with the data from three experiments, using tubes of circular cross-section, provided that the value of λ is about 0·75. If the theoretical interpretation is correct, it appears that convective flow ceases when the vertical density gradient is slightly less than the neutral value.

The shape of a jet of Newtonian liquid issuing from a capillary needle into air is considered. The results of two theoretical approaches are presented. One approach is a perturbation analysis about the final state of the jet and the other is a boundary-layer analysis near the point of jet formation. Comparison of the predictions with experimental jet shapes shows them to be in semi-quantitative agreement. Especially interesting is the presence of a ‘discontinuity’ in the empirical exponential decay rate of the jet radius occurring at a Reynolds number somewhere between 14 and 20 and the correspondence of this discontinuity with the peculiar behaviour in this range of the Reynolds number of the theoretical eigenvalue.

The nature of the scattering surfaces for thermal diffuse scattering (TDS) and the contribution of TDS to the intensity measured during a scan through a Bragg reflexion are considered for a model in which all phonons are assumed to have the same velocity in the crystal. For neutrons which are faster than this sound velocity the contribution is independent of the neutron velocity and can be calculated using the formulae derived for X-ray scattering. However, for neutrons which are slower than the sound velocity the contribution is a function of the neutron velocity and will be less than that given by the X-ray formulae. The behaviour of the TDS contribution as a function of the neutron velocity is considered for a type scan for which an analytical evaluation is possible and these considerations are extended to the conventional ω and θ−2θ types of scan. It is concluded that there is no discontinuity in the TDS contribution when the velocity of the neutrons is the same as that of the phonons, but that evaluation of a reliable correction is difficult for slower-than-sound neutrons for the conventional types of scan. These conclusions will also apply for a more realistic case for which the phonon velocities are not all identical.

In the study of inclusive reactions where more than one particle is observed an application of Mueller's unitarity technique shows that we must study a certain discontinuity of the 4 → 4 amplitude. We show, using two models, the ladder diagrams in ∅3 perturbation theory and Gribov's hybrid model that this discontinuity factorises in the multi-Regge limit. This has implications on the distribution of produced particles in inclusive experiments.

The method of elementary solutions recently extended to treat steady problems with kinetic models with velocity-dependent collision frequency is now extended to cover also time-dependent problems. The theory is somewhat different from the previous one holding for the Bhatnagar, Gross, and Krook model, since the continuous spectrum of space transients covers now a two-dimensional region of the complex plane. Consequently, in order to solve explicitly half-space problems we are not faced with one-dimensional singular integral equations but two-dimensional equations with integrable kernel. This circumstance does not make it possible to use the Muskhelishvili techniques which were used in previous papers. However, a connection of the present theory with the theory of generalized analytic functions is shown, and, on this basis, an analytic method of solving the relevant equations is constructed. The present theory would be easily applied to typical time-dependent problems, such as the Rayleigh problem, the oscillating wall, and the evolution of an initial discontinuity.