Exact solution of a coagulation equation with a product kernel in the multicomponent case

In this paper, we obtain the general solution for the continuous Smoluchowski equation in the multicomponent case with a product kernel as a series expansion. The solution of the problem involves the Laplace transform in several dimensions. We obtain a nonlinear partial differential equation (PDE) of the advective kind generalizing the one previously given by other authors for the mono-component case.As in its relative mono-component case, gelation is produced at some point, the conditions for its occurrence being the same as those for the mono-component case, though substituting a sum of derivatives by a derivative in the Laplace transform field. We demonstrate that for a multicomponent particle size distribution (PSD) of multiplicative form, it is sufficient for one of the marginal PSDs to generate instantaneous gelation for the occurrence of instantaneous gelation in the multicomponent PSD.The general solution is applied to several specific cases, a discrete case that recovers a previously known solution, and another two continuous cases which can be used to check numerical methods designed to directly solve the Smoluchowski equation in more general cases.We have compared the solutions for the multicomponent PSD for constant, additive and product kernels and we conjecture about the relation existing between the functional forms for the solutions both in the mono-component and the multicomponent case.Finally, we have analysed the shape of the solutions for multicomponent PSD for constant, additive and product kernels for very small masses of components, obtaining a qualitatively different behaviour for the product kernel. This has effects in the mixing state of the sol phase as time passes.     

Chiral symmetry breaking in an instantaneous approximation to Coulomb gauge QCD

We analyze the interplay between explicit and spontaneous chiral-symmetry breaking in Coulomb gauge QCD. Quark and pseudoscalar meson properties are investigated, using an instantaneous approximation to gluon exchange, with momentum-dependent coupling constants and current quark masses in agreement with the full QCD renormalization group equations. We show how a finite momentum-dependent constituent quark mass can be defined even for a confining interaction between the quarks, and derive an integral equation for this constituent mass from the renormalized Dyson-Schwinger equations. This equation is shown to be equivalent to a gap equation derived in a Bogoliubov-Valatin variational method from the model's hamiltonian. Including momentum-dependent current masses also ensures a finite value for the quark condensate. We report numerical results for a purely confining and for a Richardson potential for the Coulombic part of the quark-antiquark potential. Transverse gluons are included in the Breit approximation, neglecting retardation. As a confining Breit interaction leads to an infrared inconsistency in the model, and since there is mounting evidence for a dynamical gluon mass, such a mass is included. Numerical results for the constituent quark mass for one flavour, for different values of the current mass, are reported, together with the corresponding energy densities, quark condensates, pseudoscalar meson masses and pseudoscalar meson decay constants. The results are encouraging from a phenomenological point of view.     

Detection and discrimination of spectral peaks and notches at 1 and 8 kHz

The ability of subjects to detect and discriminate spectral peaks and notches in noise stimuli was determined for center frequencies fc of 1 and 8 kHz. The signals were delivered using an insert earphone designed to produce a flat frequency response at the eardrum for frequencies up to 14 kHz. In experiment I, subjects were required to distinguish a broadband reference noise with a flat spectrum from a noise with either a peak or a notch at fc. The threshold peak height or notch depth was determined as a function of bandwidth of the peak or notch (0.125, 0.25, or 0.5 times fc). Thresholds increased with decreasing bandwidth, particularly for the notches. In experiment II, subjects were required to detect an increase in the height of a spectral peak or a decrease in the depth of a notch as a function of bandwidth. Performance was worse for notches than for peaks, particularly at narrow bandwidths. For both experiments I and II, randomizing (roving) the overall level of the stimuli had little effect at 1 kHz, but tended to impair performance at 8 kHz, particularly for notches. Experiments III-VI measured thresholds for detecting changes in center frequency of sinusoids, bands of noise, and spectral peaks or notches in a broadband background. Thresholds were lowest for the sinusoids and highest for the peaks and notches. The width of the bands, peaks, or notches had only a small effect on thresholds. For the notches at 8 kHz, thresholds for detecting glides in center frequency were lower than thresholds for detecting a difference in center frequency between two steady sounds. Randomizing the overall level of the stimuli made frequency discrimination of the sinusoids worse, but had little or no effect for the noise stimuli. In all six experiments, performance was generally worse at 8 kHz than at 1 kHz. The results are discussed in terms of their implications for the detectability of spectral cues introduced by the pinnae.     

The experimental investigation of a glow discharge in a toroidal tube

A glow discharge in a toroidal tube with axial discharge current is investigated. The investigation was carried out for a discharge as a whole, not for electrodal zones and the positive column separately. The volt-ampere characteristics for different pressures and Paschen's curves for different gases were measured. The curves obtained are compared with analogous ones for a cylindrical tube. The conditions for the existence of a glow discharge in a toroidal tube and the processes passing there are discussed on the basis of the measured curves. The experiments were carried out in air, helium, argon and neon.     

Equations of Relativistic and Quantum Mechanics and Exact Solutions of Some Problems

Relativistic invariant equations are proposed for the action function and the wave function based on the invariance of the representation of the generalized momentum. The equations have solutions for any values of the interaction constant of a particle with a field, for example, in the problem of a hydrogen-like atom, when the atomic number of the nucleus Z > 137. Based on the parametric representation of the action, the expression for the canonical Lagrangian, the equations of motion and the expression for the force acting on the charge during motion in an external electromagnetic field are derived. The Dirac equation with the correct inclusion of the interaction for a particle in an external field is presented. In this form, the solutions of the equations are not limited by the value of the interaction constant. The solutions of the problem of charge motion in a constant electric field, problems for a particle in a potential well, and penetration of a particle through a potential barrier, as well as problem of a hydrogen atom are presented.     

Deflection of a 100-MeV positron beam by repeated reflections in thin crystals

The phenomenon of the deflection of a charged particle beam due to channeling in a bent crystal is thoroughly investigated and successfully applied for the extraction of the beam in high-energy accelerators, at the energies of about 10 GeV and higher. However, a big practical interest lies in the task of bending and extracting charged particles with energies below 1 GeV, for example, for the production of ultrastable beams of low emittance for medical and biological applications. That is why a novel crystal technique, namely thin straight crystal targets, is investigated in this article, using crystals as elements for extraction and collimation of the circulating beam in a ring accelerator. The advantages of reflection in straight crystals in comparison with bent crystal channeling consist in the small length of straight crystals along the beam that reduces the amount of nuclear interactions and improves the background. Experimental results were obtained for the bending of a 100 MeV positron beam with using five sequential straight crystals.     

Vowel errors in noise and in reverberation by hearing-impaired listeners

The effects of noise and reverberation on the identification of monophthongs and diphthongs were evaluated for ten subjects with moderate sensorineural hearing losses. Stimuli were 15 English vowels spoken in a /b-t/ context, in a carrier sentence. The original tape was recorded without reverberation, in a quiet condition. This test tape was degraded either by recording in a room with reverberation time of 1.2 s, or by adding a babble of 12 voices at a speech-to-noise ratio of 0 dB. Both types of degradation caused statistically significant reductions of mean identification scores as compared to the quiet condition. Although the mean identification scores for the noise and reverberant conditions were not significantly different, the patterns of errors for these two conditions were different. Errors for monophthongs in reverberation but not in noise seemed to be related to an overestimation of vowel duration, and there was a tendency to weight the formant frequencies differently in the reverberation and quiet conditions. Errors for monophthongs in noise seemed to be related to spectral proximity of formant frequencies for confused pairs. For the diphthongs in both noise and reverberation, there was a tendency to judge a diphthong as the beginning monophthong. This may have been due to temporal smearing in the reverberation condition, and to a higher masked threshold for changing compared to stationary formant frequencies in the noise condition.     

Intensity cross-correlation in light scattered by the rotational motion of macromolecules

Intensity cross-correlation in light scattered by the rotational motion of macromolecules is discussed in terms of pair- pair correlation. Expressions are given for a single pair for comparison with a rod, an equilateral triangle for comparison with a flat macromolecule, and a regular tetrahedron for comparison with other shapes. Anticorrelation occurs for the pair and triangle and correlation for the tetrahedron.     

Sound waves in a liquid with polydisperse vapor–gas bubbles

A mathematical model is presented for the propagation of plane, spherical, and cylindrical sound waves in a liquid containing polydisperse vapor–gas bubbles with allowance for phase transitions. A system of integro-differential equations is constructed to describe perturbed motion of a two-phase mixture, and a dispersion relation is derived. An expression for equilibrium sound velocity is obtained for a gas–liquid or vapor–liquid mixture. The theoretical results agree well with the known experimental data. The dispersion curves obtained for the phase velocity and the attenuation coefficient in a mixture of water with vapor–gas bubbles are compared for various values of vapor concentration in the bubbles and various bubble distributions in size. The evolution of pressure pulses of plane and cylindrical waves is demonstrated for different values of the initial vapor concentration in bubbles. The calculated frequency dependence of the phase sound velocity in a mixture of water with vapor bubbles is compared with experimental data.     

Numerical simulation of continuum models for fluid-fluid interface dynamics

This paper is concerned with numerical methods for two-phase incompressible flows assuming a sharp interface model for interfacial stresses. Standard continuum models for the fluid dynamics in the bulk phases, for mass transport of a solute between the phases and for surfactant transport on the interface are given. We review some recently developed finite element methods for the appropriate discretization of such models, e.?g., a pressure extended finite element (XFE) space which is suitable to represent the pressure jump, a space-time extended finite element discretization for the mass transport equation of a solute and a surface finite element method (SurFEM) for surfactant transport. Numerical experiments based on level set interface capturing and adaptive multilevel finite element discretization are presented for rising droplets with a clean interface model and a spherical droplet in a Poisseuille flow with a Boussinesq-Scriven interface model.     

Detecting the inseparability and distillability of continuous variable states in Fock space

The partial transposition (PT) operation is an efficient tool in detecting the inseparability of a mixed state. We give an explicit formula for the PT operation for the continuous variable states in Fock space. We then give the necessary and sufficient condition for the positivity of Gaussian operators. Based on this, a number of criteria on the inseparability and distillability for the multimode Gaussian states are naturally drawn. We finally give an explicit formula for the state in a subspace of a global Gaussian state. This formula, together with the known results for Gaussian states, gives the criteria for the inseparability and distillability in a subspace of the global Gaussian state.     

A new method for calculating the Zeeman split acceptor levels in cubic semiconductors

Within the effective-mass approximation we introduce a transformed hamiltonian for the acceptor problem in a homogeneous magnetic field. The new hamiltonian is expressed entirely in terms of spherical tensor operators. The reduced-matrixelement technique can thus be used in solving variationally the eigenvalue problem. In this way a simple non-perturbation numerical calculation of the Zeeman split acceptor states becomes for the first time feasible. As a preparatory test for the method we used the new hamiltonian in a variational “model” calculation with a very restricted basis set, to calculate the linear Zeeman parameters for some excited states of shallow acceptors in Ge and GaAs. The results can be compared with recent experimental data for these materials. We find already an excellent numerical agreement between the calculated and the measured linear Zeeman parameters for the final states of the D-transitions, and we obtain the correct order of magnitude for those of the C-line final states.     

Plane wave propagation in a uniaxial bianisotropic medium with an application to a polarization transformer

Uniaxial bianisotropic medium is a generalization of the well-studied bi-isotropic and chiral media. It is obtained, for example, when microscopic helices with parallel axes are positioned in a host dielectric in random locations. Plane wave propagation in such a medium is studied and a simple solution for the dispersion equation and for the eigenwaves are found. As a numerical example, polarization properties of a transverse wave propagating in a uniaxial bianisotropic medium is considered. The results give a simple possibility to construct a polarization transformer with a transversely uniaxial chiral medium for changing the polarization of a propagating plane wave.     

Brownian motion in periodic potentials; nonlinear response to an external force

The Brownian motion of particles in a periodic potential in response to a constant external force is investigated. By expanding the distribution function into Hermite-functions and into a Fourier-series, the Fokker-Planck-equation is transformed into a set of coupled equations for the expansion coefficients. These equations are solved by a continued fraction method for matrices. This continued fraction for the matrices converges for large, intermediate and even for very small damping constants. The mobility, the kinetic and potential energy for various damping constants and external forces are given for a cos-potential. The current-voltagecharacteristic of the Josephson tunneling junction is also shown.     

Calculation of short-range order in the method of cluster components

A scheme is proposed for calculating the short-range order in the method of cluster components for an n-component solid solution, taking into account an arbitrary number of spheres. The calculation of a single-, two-, and three-coordination spheres for a binary system and a calculation of a single sphere for a triple system are considered in detail. In the case of a binary system the possible type of property-composition dependence for a random distribution of the atoms in the case of one, two, and three coordination spheres are analyzed. The method is illustrated using a calculation of the concentration dependence of the elasticity constants of a KI_cBr_1-c solid solution.     

Measurement of the viscosity-density product using multiple reflections of ultrasonic shear horizontal waves

We have developed an on-line computer-controlled sensor, based on ultrasound reflection measurements, to determine the product of the viscosity and density of a liquid or slurry for Newtonian fluids and the shear impedance of the liquid for non-Newtonian fluids. A 14 MHz shear wave transducer is bonded to one side of a 45-90 degrees fused silica wedge and the base is in contract with the liquid. Twenty-eight echoes were observed due to the multiple reflections of an ultrasonic shear horizontal (SH) wave within the wedge. The fast Fourier transform of each echo was obtained for a liquid and for water, which serves as the calibration fluid, and the reflection coefficient at the solid-liquid interface was obtained. Data were obtained for 11 sugar water solutions ranging in concentration from 10% to 66% by weight. The viscosity values are shown to be in good agreement with those obtained independently using a laboratory viscometer. The data acquisition time is 14s and this can be reduced by judicious selection of the echoes for determining the reflection coefficient. The measurement of the density results in a determination of the viscosity for Newtonian fluids or the shear wave velocity for non-Newtonian fluids. The sensor can be deployed for process control in a pipeline, with the base of the wedge as part of the pipeline wall, or immersed in a tank.     

The applicability of a material-treatment laser pulse in non-destructive evaluations

A practical optodynamic study was performed to determine the usability of different lengths of laser pulses for the generation of ultrasonic transients in a solid material. The aim of the study was to evaluate the possibility of a dual use for a laser pulse-for laser material processing, on the one hand, and for the ultrasonic wave generation on the other-with both processes being combined on the same production line. The propagation of the laser-generated ultrasonic waves is evaluated by detecting and measuring with a PID-controlled stabilized interferometer. Thus, both systems provided the basic tools, the generation and detection of ultrasonic waves, for an ultrasonic, laser-based, non-destructive material evaluation. The ultrasonic transients generated by 'classical' nanosecond laser pulses were compared with the transients generated by industrial laser pulses with a duration of a few tenths of a microsecond. The experimental results are compared with the results of a time-of-flight analysis that also involved part of a mode-conversion analysis for both regimes in a layered material structure. The differences between the two waveforms were assessed in terms of their visibility, wavelength and resolution. The limit values were calculated and estimated for the laser-pulse parameters, when such pulses are intended for use in an ultrasonic, laser-based, non-destructive evaluation. The possibility of using an industrial marking laser for laser ultrasound generation is thus demonstrated.     

Low-dimensionality energy landscapes: Magnetic switching mechanisms and rates

In this paper we propose a new method for the study and visualization of dynamic processes in magnetic nanostructures, and for the accurate calculation of rates for such processes. The method is illustrated for the case of switching of a grain of an exchange-coupled recording medium, which switches through domain wall nucleation and motion, but is generalizable to other rate processes such as vortex formation and annihilation. The method involves calculating the most probable (lowest energy) switching path and projecting the motion onto that path. The motion is conveniently visualized in a two-dimensional (2D) projection parameterized by the dipole and quadrupole moment of the grain. The motion along that path can then be described by a Langevin equation, and its rate can be computed by the classic method of Kramers [4]. The rate can be evaluated numerically, or in an analytic approximation—interestingly, the analytic result for domain-wall switching is very similar to that obtained by Brown in 1963 for coherent switching, except for a factor proportional to the domain-wall volume. Thus in addition to its lower coercivity, an exchange-coupled medium has the additional advantage (over a uniform medium) of greater thermal stability, for a fixed energy barrier.     

Mathematical model of the direct current amplification mode in a channel of a microchannel plate

The boundary-value problem is solved for electric field in a channel of a microchannel plate for a preset value of current at the channel entrance. Analytic equations are determined for the distributions of field, potential, and current along the channel, as well as for the current gain and the current in the circuit of the source of constant voltage as a function of the input current.