Lambda Function Research Articles

New procedures for calculating the mutual inductance of the system: filamentary circular coil-massive circular solenoid

New and fast procedures for calculating the mutual inductance between a thin circular filament coil and a massive circular solenoid in air are presented. The coils are axisymmetric. These results are expressed over the complete elliptical integrals of the first and second kind, Heumann's Lambda function, and one term that has to be solved numerically. The numerical integration of this term is based on some numerical integrations to get the satisfactory accuracy and computational cost. These new expressions are accurate and simple to apply for useful applications. Also, another comparative method based on an approximation of the massive solenoid by Maxwell's coils is given. The paper discusses the computational cost and the accuracy. Results obtained by the two approaches are in excellent agreement.

New and fast procedures for calculating the mutual inductance of coaxial circular coils (circular coil-disk coil)

This paper deals with two efficient approaches for determining the mutual inductance between thin circular coils and disk coils in air. The first approach gives new expressions for calculating the mutual inductance of treated configurations. These results are expressed over the complete elliptical integrals of the first and second kind, Heuman's Lambda function, and one term that must be solved numerically. Another approach is based on the filament method where conductors are approximated by the set of Maxwell's coils. The obtained expressions are expressed over the complete elliptical integrals of the first and second kind and permit also fast calculation of the mutual inductance for mentioned systems. These new expressions are accurate and simple for useful applications. All results obtained by the two approaches are in excellent agreement.

Transparency of the bovine corneal stroma at physiological hydration and its dependence on concentration of the ambient anion.

De-epithelialised and de-endothelialised bovine corneal stromas with a hydration of 3.2 equilibrated at 154 mM NaCl and buffered at pH 7.4 had their optical density (400-750 nm) measured. Stromas equilibrated against 10, 20, 30, 50 or 100 mM NaCl made isotonic to 154 mM NaCl by supplementing with sorbitol were progressively more transparent as NaCl increased. Hypertonic equilibration against 300, 600 or 1000 mM NaCl resulted in a progressive loss of transparency compared with 154 mM NaCl. Light scattering as a function of wavelength fitted a lambda(-3) function well for 10, 30, 50, 100 and 154 mM NaCl preparations between 450 and 650 nm, but not at higher wavelengths. However, hypertonic 300, 600 and 1000 mM NaCl preparations showed a lambda(-2) dependence in the 450-750 nm range. Experiments with 154 mM NaCl and either 0 or 300 mM sorbitol suggested that the changes in light scattering in hypertonic preparations are unlikely to be caused by osmotic alterations to the stromal keratocytes. Psychophysical studies of the optical transmission function of preparations indicated that corneal stromas dialysed against 154 mM NaCl had usable optical properties, but preparations dialysed against 10 mM NaCl were effectively unable to transmit an image. The results are related to the known increase of fixed negative charge in the corneal matrix when chloride ions are adsorbed onto the matrix. It is suggested that the ordering force between corneal collagen fibrils, generated in part by anion binding, may be crucial to the physiological functioning of the visual system.

Improvement in calculation of the self- and mutual inductance of thin-wall solenoids and disk coils

The self-inductance expressions given by Yu and Han (1987) for air-core circular coils with rectangular cross sections, thin solenoids, and disk coils can be solved only by the numerical integration methods. We propose as an alternative a combined analytic and numeric approach. The approach brings some improvement in the calculations of self-inductance of thin-wall solenoids and disk coils that can be encountered in superconducting magnetic energy storage (SMES) problems. We also give a method for the calculation of mutual inductance of disk coils and of thin-wall solenoids. The results are obtained in an analytical form over the complete elliptic integrals of the first and second kind and Heuman's Lambda function. It is important to mention that the kernels of these integrals are always continuous functions on intervals of integration, so singularities are avoided. The results enable one to calculate the self-inductance and the mutual inductance of any thin air-core coil precisely and fast. For practical applications, the results are so simple to use that we recommend them to avoid the problems of solving the singular cases.

The effect of grain size on fracture: Dislocation-free zone in the front of the finite crack tip

Dislocation pileups between a finite crack tip and a grain boundary with a dislocation-free zone is investigated by the method of continuous dislocation modeling. Analytic solutions of both the dislocation distributions inside the crack and plastic zone and the stress field in the space are obtained. The stress intensity factor at the crack tip and the stress concentration at the grain boundary are derived. When the applied stress exceeds τΛ0 [cos−1(c/a),k], the dislocations emitted from the crack tip pile up on the grain boundary. Note that Λ0 is the Heuman lambda function and τ is the lattice friction stress. a and c are the grain size and surface crack length, respectively. K is defined as (c/b)<rl>(a2−b2)/(a2−c2)<rlx,>, where (b−c) is the dislocation-free zone. The stress concentration at the grain boundary increases with increasing grain size but decreases with increasing dislocation-free zone. In contrast, if the applied stress is less than or equal to τΛ0 [cos−1(c/a),k], no stress concentration exists at the grain boundary. The stress intensity factor at the crack tip increases with increasing size of dislocation-free zone but decreases with increasing grain size. Our results are reduced to several special cases.

The effects of nonequilibrium ionization on the radiative losses of the solar corona

The emissivity of the ions of carbon and oxygen has been recalculated for a set of solar coronal loop models with a steady state siphon flow. The ion densities were calculated from the plasma velocities, temperatures, and densities of the models, and large departures from equilibrium were found. For purposes of comparison, the emissivity was calculated with and without the approximation of ionization equilibrium. Considerable differences in the radiative loss function Lambda(T) curve between equilibrium and nonequilibrium conditions were found. The nonequilibrium Lambda(T) function was then used to solve again the steady state flow equations of the loop models. The differences in the structure of these models with respect to the models calculated adopting the Lambda(T) curve in equilibrium are discussed.

The physiological basis of heterochromatic flicker photometry demonstrated in the ganglion cells of the macaque retina.

1. Heterochromatic flicker photometry is a way of measuring the spectral sensitivity of the human eye. Two lights of different colour are sinusoidally alternated at, typically, 10-20 Hz, and their relative intensities adjusted by the observer until the sensation of flicker is minimized. This technique has been used to define the human photopic luminosity, or V lambda, function on which photometry is based. 2. We have studied the responses of macaque retinal ganglion cells using this stimulus paradigm. The responses of the phasic ganglion cells go through a minimum at relative radiances very similar to that predicted from the V lambda function. At this point, defined as equal luminance, an abrupt change in response phase was observed. A small residual response at twice the flicker frequency was apparent under some conditions. 3. The spectral sensitivity of parafoveal phasic cells measured in this way corresponded very closely to that of human observers minimizing flicker on the same apparatus. 4. Minima in phasic cell activity were independent of flicker frequency, as is the case in the psychophysical task. 5. The response minima of phasic cells obey the laws of additivity and transitivity which are important characteristics of heterochromatic flicker photometry. 6. As the relative intensities of the lights were altered responses of tonic, spectrally opponent cells usually underwent a gradual phase change with vigorous responses at equal luminance. The responses of tonic cells treated individually or as a population could not be related to the V lambda function in any meaningful way. 7. We conclude that the phasic, magnocellular cell system of the primate visual pathway underlies performance in the psychophysical task of heterochromatic flicker photometry. It is likely that other tasks in which spectral sensitivity conforms to the V lambda function also rely on this cell system.

Wavelength discrimination in the turtle Pseudemys scripta elegans.

Wavelength discrimination was measured with two individuals using a behavioral training technique. The delta lambda function has three minima, indicating best discrimination at 400, 510 and at 570 nm (S2) or 600 nm (S1). In the range between 450 and 510 nm the ability to discriminate wavelengths was absent. This can be explained by the filter effect of the colored oil droplets, as indicated by a model computation. Color mixture experiments revealed that the high discrimination ability at 400 nm is based on an ultraviolet sensitive photoreceptor which is the fourth cone type involved in turtles' color vision. Spectral sensitivity in the ultraviolet region (measured for one turtle) is maximal at 370-380 nm.

Perception of blackness.

We sought to measure the mechanisms underlying the perception of blackness in the following way. A central spot (45') of fixed luminance was surrounded by a dark ring (7.5'), and surrounding all was an annular zone (30') of light. This stimulus was presented in Maxwellian view for 0.5 sec every 3 sec. The radiance of the annulus required to make the central area (spot and ring) appear uniformly black was measured for different wavelengths (440-660 nm) of the annulus. These measurements were made for test spots that were either broadband or of wavelength 480, 500, 580, or 660 nm. In all conditions the measured spectral efficiency of induced blackness matched the inverse of the V lambda function. Using the same stimulus, we have also measured increment-threshold functions. For a fixed luminance of the spot, the radiance of the surrounding annulus required to bring the central spot to threshold was measured. These increment-threshold functions do not match the V lambda or blackness functions. Our results show that induced blackness is inversely related to the luminous efficiency function and that the spectral efficiency of induced blackness is distinct from the increment-threshold function measured under these conditions. Furthermore, blackness appears to be independent of the wavelengths of the inducing annulus as well as of the central spot. Thus these results link induced blackness to the luminance pathway and argue against the involvement of the chromatic pathways in the perception of blackness.

Finite amplitude vibrations of a neo-Hookean oscillator

The problem of the finite amplitude, horizontal oscillatory motion of a mass attached to a neo-Hookean rubber spring and supported by a fixed, ideally smooth and rigid horizontal surface is solved exactly in terms of the Heuman lambda and beta functions. Therefore, the period of the oscillations may be computed from tables of values of the complete lambda function. It is proved that the ratio of the amplitude-dependent frequency of any finite amplitude motion to the constant frequency of the small amplitude vibration of the same oscillator depends only on the assigned initial data. Therefore, the ratio is universal for every neo-Hookean oscillator regardless of its stiffness or of its design parameters. Upper and lower bounds for this ratio also are provided. Moreover, it is proved for assigned initial data, that the normalized amplitude of the vibrations is invariable for every neo-Hookean oscillator, and all energy curves are reduced to a single constant energy trajectory determined by the initial data alone. The slingshot effect that occurs for a slender neo-Hookean rubber cord also is described, and all results are illustrated graphically.

Analysis of a crack-tip dual zone model

An analytic model is proposed for an opening mode of crack face displacement with crack-tip dual zones, i.e. an elastic core zone plus a plastic strip zone. A presence of such dual zones in the vicinity of the crack tip was experimentally observed in a recent study based on the generation of dislocations. A Papkovich-Neuber formulation of the resulting four-part mixed boundary value problem leads to a set of quadruple integral equations which are solved with an application of finite Hilbert transform technique. With conditions of boundedness on the stresses in the plastic strip zone, the results show an inverse square root of the distance type singularity at the base of the crack tip and a relaxation of stresses in the crack-tip elastic core zone is realized. The stress intensity factors and the crack-tip opening displacements are presented in exact forms involving elliptic integrals and Heuman's lambda function and are shown to depend upon the crack size, the applied loading and the crack-tip dual zone lengths. The analytic and graphical solutions are compared with the Dugdale model to which they reduce as a limiting case of vanishing elastic core zone.

Spectral efficiency of blackness induction.

The spectral efficiency of blackness induction was measured in three normal trichromatic observers and in one deuteranomalous observer. The psychophysical task was to adjust the radiance of a monochromatic 60-120' annulus until a 45' central broadband field just turned black and its contour became indiscriminable from a dark surrounding gap that separated it from the annulus. The reciprocal of the radiance required to induce blackness with annulus wavelengths between 420 and 680 nm was used to define a spectral-efficiency function for the blackness component of the achromatic process. For each observer, the shape of this blackness-sensitivity function agreed with the spectral-efficiency function based on heterochromatic flicker photometry when measured with the same 60-120' annulus. Both of these functions matched the Commission Internationale de l'Eclairage V lambda function except at short wavelengths. Ancillary measurements showed that the latter difference in sensitivity can be ascribed to nonuniformities of preretinal absorption, since the annular field excluded the central 60' of the fovea. Thus our evidence indicates that, at least to a good first approximation, induced blackness is inversely related to the spectral-luminosity function. These findings are consistent with a model that separates the achromatic and the chromatic pathways.

Weight functions and Stress Intensity Magnification factors for elliptical and semi‐elliptical cracks under variable normal stress – Part II

AbstractSurface cracks under peak stresses are investigated. The calculational procedure is based on the general form of the weight function for an elliptical crack embedded in an infinite solid. Two points on the contour of the ellipse are investigated. A new correction procedure for transfer from the embedded crack to surface crack configurations is presented, which is valid for all a/t‐values. Weight functions for both points have been found with the crack aspect ratio a/c as parameter. For the point at the end of the minor axis all weight functions for embedded cracks are describable by one equation only (using Heuman's lambda function). For various a/c‐ratios of the surface crack under different stress distributions the stress intensity magnification factors are given.

Weight functions and Stress Intensity Magnification factors for elliptical and semi‐elliptical cracks under variable normal stress – part I

AbstractSurface cracks under peak stresses are investigated. The calculational procedure is based on the general form of the weight function for an elliptical crack embedded in an infinite solid. Two points on the contour of the ellipse are investigated. The superposition method is used for transfer from the embedded crack to surface crack configurations. Weight functions for both points have been found with the crack aspect ratio a/c as parameter. For the point at the end of the minor axis all weight functions are describable by one equation only (Heuman's lambda function). For various a/c ratios of the surface crack under different stress distributions the stress intensity magnification factors are given.

Free vibrations of a loaded rubber string

The problem of the finite amplitude, free vertical oscillatory motion of a massattached to a neo-Hookean rubberlike string is solved exactly in terms of elementary functions and the Heuman lambda function, which is related to the elliptic integral of the third kind. Hence, the period of the oscillations for the various possible motions may be computed from tables of values of the complete lambda function. It is shown that the results differ significantly from those obtained elsewhere for a string having linearly elastic behavior; and all of the results are described graphically.

Chaotic boundary of a Hamiltonial map

The half-plane maps are a class of complex Hamiltonial maps whose invariant curves at most fixed irrational frequencies can be obtained as a convergent Taylor series expansion. For these maps the boundary between regular ordered motion on invariant curves and irregular chaotic motion is given by the radius of convergence of the series. The successive terms of the series oscillate wildly, due to the presence of small divisors. Methods are presented for taming the series, based on the conversion of the convergence exponent C = - ln α c into the integral of a continuous but nondifferentiable lambda function, whose graph whows a similarity structure on small scales. Self-similarity properties are illustrated for the chaotic boundary function α c, where v is the frequency.

The ral gene of phage lambda. I. Identification of a non-essential gene that modulates restriction and modification in E. coli.

Host controlled restriction in Escherichia coli can be relieved by pre-infecting restricting cells with modified lambda helper phages. This process, in which intact unmodified phage genomes are allowed to escape restriction attack, is mediated by a newly identified lambda function called ral. The ral gene has been located by deletion mapping between cIII and N. Efficient expression of the ral gene requires the product of the regulator gene N. Polyacrylamide gel analysis of the lambda proteins specified by the cIII-N region failed to reveal the product of the ral gene, but demonstrated that protein Ea10 is encoded by a gene located immediately to the left of ral. From these results the map order cIII-Ea10-ral-TL1-N was deduced. Ral specifically alleviates restriction in E. coli K and E. coli B, but does not affect restriction systems EcoRI, EcoRII and EcoP1. In addition, ral enhances the modification activity of the EcoK and EcoB restriction enzymes: we observed that efficient modification of progeny phages obtained by propagating unmodified lambda phages in r-m+ hosts, is dependent upon the presence of ral. We thus conclude that the ral gene product acts by modulating the restriction and modification activities of the type I restriction systems in E. coli, and the possible mechanisms will be discussed.

Simulation of the CIE spectral tristimulus value z( λ) in a glass filter

A number of new filter glasses for photometric use have recently been developed at the NPRL. The visual spectral transmittance curves have bandpass characteristics closely resembling the standard photometric functions x( lambda ), V'( lambda ) and V( lambda ) (Res and Kok 1974, Res et al. 1975, 1977). Further work was carried out to develop glasses simulating the CIE spectral tristimulus value z( lambda ). Silicate glass doped with CuO together with different amounts of CeO2, TiO2 and CoO proved to be most promising. The work resulted in a glass whose spectral transmittance, in conjunction with a 2 mm thick Schott GG 420 filter, yielded quite close simulations of the z( lambda ) function.

Cherenkov radiation in biaxial crystals—I

The radiation emitted by a uniformly moving point charge in a biaxial crystal is studied by a new method. Fourier integrals for the field are set up for the general case without imposing any restriction on the dielectric and magnetic properties of the crystal or on the direction of motion of the charge. In the case of motion along a principal dielectric axis the integrals are evaluated exactly and closed expressions for the radiated energy are obtained. For radiation of either kind these expressions involve Heuman's Lambda function and the complete elliptic integral of the first kind.

Design of elliptical and circular aperture pencil-beam antennas using the modified lambda functions

A procedure for the synthesis of pencil-beam radiation patterns having a given null half-beamwidth and a given sidelobe level by elliptical or circular aperture antennas is presented. The given pattern is first approximated by a modified lambda function; the lambda function is then synthesized by a finite line-source current distribution, which is mapped onto the elliptical aperture. Design curves and formulas are presented.