Linear Function Of Distance Research Articles

Membrane potential and permeation of salts across bipolar membranes

This paper describes a theoretical consideration of membrane potential and permeabilities of uni-univalent salts in multi-lamellar ion exchange membranes. Model calculations are performed for bipolar membranes consisting of juxtaposed cation and anion exchange membranes. Donnan potential is predicted to be non-zero when the concentrations of external solution on the left- and right-hand sides of the bipolar membrane are equal, while this potential is estimated to be zero from TMS theory for single ion exchange membranes. Membrane potentials in both cases are, however, found to be zero. This is ascribed to the fact that the Donnan potential and the inter-Donnan potential, which is generated at the interfaces of anion and cation exchange layers in bipolar membranes, compensate each other. Concentration profiles and membrane potential profile in the bipolar membrane are calculated. The cation and anion concentration profiles are found to be approximately linear functions of distance from the surface of the membrane. The membrane potential profile, on the other hand, is convex. Permeability differences due to the direction of flux in bipolar membranes are also discussed in this study.

A General Theory of Shells and the Complementary Potentials

This theory incorporates the attributes which are essential to the approximation of shells by finite elements. It is limited only by one assumption: Displacement is a linear function of distance along the normal to a reference surface. Deformation is decomposed into rotation and strain; the rotation carries elements of the reference surface to the same surface in any subsequent state. Transverse-shear deformations accommodate simple elements. The theory is couched in the potential Pv and in the complementary potential Pc; these have the property, Pv + Pc= 0 for all admissible (equilibrated) states. The theory is also cast in the complementary functional P¯c of stress and displacement, and the functional P¯v of displacement, strain and stress; P¯c and P¯v are akin to the functionals of Hellinger-Reissner and Hu-Washizu. These alternate functionals provide the means to develop various hybrid elements.

Multivariate statistical analysis of shotgun pellet dispersion

Shotgun pellet dispersions were measured by using a shotgun with the following parameters being varied: barrel length, discharge distance, choke constriction and shot size. To reduce the number of experiments the statistical technique of Greco-Latin squares was used. A full statistical analysis of the data carried out and possible correlations between dispersion and the parameters examined. No association was found for shot size or barrel length, and little for choke, distance being the only large correlation with dispersion. Dispersion is not a truly linear function of distance and it is probable that another factor has to be incorporated into any mathematical expression for dispersion of shot.

‘Sink’ effect in a shock-induced boundary layer on attenuation

Experiments on shock-wave attenuation in a porous tube have shown that attenuation is a function of distance from the origin and the rate of cross-mass flow per unit area. It is a linear function of distance if the cross-mass flow rate per unit area is held constant. In a tube with solid walls the relationship between attenuation, mass extraction rate per unit area and distance is similar to that in a proous tube. The ratio of the local attenuation to the local cross-mass flow, or mass extraction rate, per unit area is the same linear function of distance in both sets of experiments. The mechanism of attenuation is therefore also identical.

Some Converse Theorems in Newtonian Field Theory and Their Applications

Two converse theorems related to a family of homogeneous homothetic bodies and connected to the theory of the Newtonian field are proved. In both of them the function characterising the attraction is unknown and it is demonstrated that this function in the first theorem is given by the one characterising the Newtonian field and in the second theorem it is given by this latter function with the addition of a linear function of distance. A second unknown function appears in the second theorem and it is proved that it is a linear function of the volume of the bodies. Moreover, in both the theorems it is proved that the unknown shape of the bodies must be spherical. The conjecture is made that the two theorems are still true without the hypothesis that the unknown function characterising the attraction has a pole. The full significance of the two theorems is briefly illustrated by the application of the second theorem to the case of a finite homogeneous fluid, the behaviour of which is isotropic with respect to an element of it. Among other results, it is found that the shape of the fluid is necessarily spherical and the forces at a distance which are exerted among the elements of the fluid are expressed by Newton's Law of gravitation. AMS Classification 31B20, 31B99, 76A02, 76A99

Polar flow induced by prescribed injection in an infinite channel

The micropolar flow induced by injection varying (I) as the inverse square of a linear function of distance along an infinite channel and (II) decreasing exponentially with this distance is discussed. Flow picture preceding the fully developed pattern is analysed and micropolar effects are demonstrated by calculating centre section velocity and microrotation in the section of maximum developed value.

Density Estimation by Variable Area Transect

The frequency distribution of the distance from a randomly selected point to the rth nearest organism contained in a variable area transect is described. Estimators of density and variance are derived for both randomly distributed and aggregated populations. For randomly distributed populations exact confidence intervals are available. On an effort-precision criterion the method is more useful than other spatial distribution based density estimators. Under the condition where effort is a linear function of distance covered and numbers of organisms counted, the effort expended in achieving a desired level of precision is the same as that for quadrats; the variance of effort depends on the difficulty in counting and in covering the distance. An example using pismo clams (Tivela stultorum) is given: the density of clams was found to be 1 per 2.87 m'. The method was found to be superior to that of other distance-measures and quadrats. J. WILDL. MANAGE. 43(2):484-492 Two methods are commonly employed when measuring the density of plant communities and closed animal populations where the animals are relatively stationary: quadrats and distance-measures. Quadrats are simple to apply. The fact that means of plot samples provide an unbiased estimate of true mean density enhances the usefulness of quadrats. Unfortunately, problems arise in quadrat sampling. Quadrats of different size yield different estimates of density (GreigSmith 1964). Also, the degree of nonrandomness detected depends on plot size (Pielou 1957). Distance-measure methods, where the density is based on the spatial distribution of organisms, avoid these difficulties. The density estimate is based on the distance between some randomly chosen point and some rth closest organism and not on a fixed plot. Unfortunately, there are also problems in using distance-measures. The precision of the estimate is inverse to the product of the number of samples taken, n, and the number in each sample, r (Pollard 1971). In the field, an r beyond 3 or 4 is often impractical. This difficulty arises not from the measurement to the rth organism but in determining which is the rth. Increased precision is afforded only at the expense of increasing the number sample points. This has discouraged many investigators from using distance-

An approximate method for the solution of fick's law problems

The simplifying approximation that the concentration of an electroactive species is a linear function of distance into the solution from the electrode surface is used to derive current-time relationships for controlled potential electrolysis. Relationships for thinlayer chronoamperometry, chronoamperometry with a preceding homogeneous reversible reaction and linear-sweep voltammetry are compared with existing relationships to determine the range of applicability of the approximation. The approximation is found to yield qualitatively correct results. To obtain quantitatively or near-quantitatively correct results, it is necessary to incorporate more information about the electrochemical system into the derivation than simply the usual initial and boundary conditions.

The Development of Linear Distance Functions for Distribution Analysis

The general method of estimating the road mileage between points identified only by their co-ordinates is to proportionately inflate the straight-line distance between them. We present the traditional single-region model and extend this to a generalized multi-region format. The inflating parameters are derived and various statistical tests carried out to evaluate the models. It is shown that, on average, the multi-region approach gives better mileage estimates for all types of distribution analysis. For specific problems, which are discussed, the single-region model is shown to produce results which are unacceptable for the planning of distribution networks.

The Development of Linear Distance Functions for Distribution Analysis

The Development of Linear Distance Functions for Distribution Analysis

Estimation of the probability of success in petroleum exploration

A probabilistic model for oil exploration can be developed by assessing the conditional relationship between perceived geologic variables and the subsequent discovery of petroleum. Such a model includes two probabilistic components, the first reflecting the association between a geologic condition (structural closure, for example) and the occurrence of oil, and the second reflecting the uncertainty associated with the estimation of geologic variables in areas of limited control. Estimates of the conditional relationship between geologic variables and subsequent production can be found by analyzing the exploration history of a “training area” judged to be geologically similar to the exploration area. The geologic variables are assessed over the training area using an historical subset of the available data, whose density corresponds to the present control density in the exploration area. The success or failure of wells drilled in the training area subsequent to the time corresponding to the historical subset provides empirical estimates of the probability of success conditional upon geology. Uncertainty in perception of geological conditions may be estimated from the distribution of errors made in geologic assessment using the historical subset of control wells. These errors may be expressed as a linear function of distance from available control. Alternatively, the uncertainty may be found by calculating the semivariogram of the geologic variables used in the analysis: the two procedures will yield approximately equivalent results. The empirical probability functions may then be transferred to the exploration area and used to estimate the likelihood of success of specific exploration plays. These estimates will reflect both the conditional relationship between the geological variables used to guide exploration and the uncertainty resulting from lack of control. The technique is illustrated with case histories from the mid-Continent area of the U.S.A.

Slender-body theory for particles of arbitrary cross-section in Stokes flow

A rigid body whose length (2l) is large compared with its breadth (represented by R0) is straight but is otherwise of arbitrary shape. It is immersed in fluid whose undisturbed velocity, at the position of the body and relative to it, may be either uniform, corresponding to translational motion of the body, parallel or perpendicular to the body length, or a linear function of distance along the body length, corresponding to an ambient pure straining motion or to rotational motion of the body. Inertia forces are negligible. It is possible to represent the body approximately by a distribution of Stokeslets over a line enclosed by the body; and then the resultant force required to sustain translational motion, the net stresslet strength in a straining motion, and the resultant couple required to sustain rotational motion, can all be calculated. In the first approximation the Stokeslet strength density F(x) is independent of the body shape and is of order μUε, where U is a measure of the undisturbed velocity and ε = (log 2l/R0)−1. In higher approximations, F(x) depends on both the body cross-section and the way in which it varies along the length. From an investigation of the ‘inner’ flow field near one section of the body, and the condition that it should join smoothly with the ‘outer’ flow which is determined by the body as a whole, it is found that a given shape and size of the local cross-section is equivalent, in all cases of longitudinal relative motion, to a circle of certain radius, and, in all cases of transverse relative motion, to an ellipse of certain dimensions and orientation. The equivalent circle and the equivalent ellipse may be found from certain boundary-value problems for the harmonic and biharmonic equations respectively. The perimeter usually provides a better measure of the magnitude of the effect of a non-circular shape of a cross-section than its area. Explicit expressions for the various integral force parameters correct to the order of ε2 are presented, together with iterative relations which allow their determination to the order of any power of ε. For a body which is ‘longitudinally elliptic’ and has uniform cross-sectional shape, the force parameters are given explicitly to the order of any power of ε, and, for a cylindrical body, to the order of ε3.

Use of a New Photoelectric Device (Fotonic Sensor) for Vibration Measurements in the Ear

A new non-contact photoelectric device is described for displacement and vibration measurements. The device uses a bifurcated fiber-optic bundle as a probe. One part of the fiber-optic bundle is used as a light carrier and the other as a receiver of reflected light. The light source is a small incandescent light which is supplied by a dc current. The reflected light from the surface is transmitted through the receiver part of the bundle and sent through a sensitive photosensor. This signal, which is a direct linear function of distance to the surface, is amplified and displayed on an oscilloscope. The instrument is capable of resolving one micro-inch and the frequency responses go to the megacycle-per-second range. Direct calibration for any surface is achieved without contact and without any external precision-motion apparatus. The technique of calibration against various reflecting surfaces in the ear: tympanic membrane, stapedial footplate, round-window membrane, basilar membrane is described. The diameter of the probe used here is from 0.5 to 4 mm. Calibration for misalignment is also given.

S in Three European Earthquakes

Summary Readings have been made of records of three European earthquakes, with the object of testing the behaviour of S between 10° and 20°. In the earthquake of 1950 September 5, a linear formula fitted up to about 9°, but there appeared to be a jump of about 4 s at that distance. There were too few observations in the range in question on 1951 May 15 to give useful information. The Dalmatian earthquake of 1962 January 7, however, had 16 S readings between 10° and 20°. They showed no sign of a jump and were substantially earlier than the J. B. Table of 1940. It appears that a linear function of distance may fit the times of S in Europe up to 20°.

Analysis of a new crossed-field amplifier, the "EAL," or electromagnetic amplifying lens

A new form of crossed-field microwave and millimeter wave amplifier which combines an interaction process that results in axial gain with a quasi-optical handling of the RF input and output power is described. Because of its functional similarity to a microwave lens and the additional property of amplification, it is called the Electromagnetic Amplifying Lens. Its advantages are outlined, and its interaction mechanism considered in detail. It is shown that the voltage gain is a linear function of distance in the useful operating range. When the RF level is low, interference between forward- and backward-traveling waves on the waveguide results in loss of control of the space charge. This phenomenon limits the gain available with stable device operation; useful gain of at least 20 db is theoretically possible with a bandwidth of several per cent. The crossed-field interaction principle that is employed assures efficiencies of better than 50 per cent.

The Effect of Associated Salts on the Availability of Concentrated Superphosphate

Phosphorus uptake by oat plants grown in the greenhouse on 2 soils fertilized with pellets of concentrated superphosphate (CSP) alone or of mixtures of CSP and each of 5 salts was measured. With Hartsells fine sandy loam (pH 5.2) yields of P were in the order: CSP < CSP + KCl ≤ CSP + NH4Cl ≤ CSP + NH4NO3 < CSP + (NH4)2SO4 ≤ CSP + KNO3. With Mountview silt loam (pH 6.6) yields of P were in the order: CSP < CSP + KCl < CSP + (NH4)2SO4 ≤ CSP + NH4Cl ≤ CSP + NH4NO3 ≤ CSP + KNO3. After pellets of CSP, CSP + KNO3, and CSP + NH4NO3 had reacted with the moist soils for 3 weeks, samples of soil were taken at increments of distance from the center of the pellet residue. Both water-soluble P (1:15 soil-water extracts) and P extractable in 6N HCl were linear functions of distance of the sample from the center of the pellet residue. The constants in the linear equations depended upon the composition of the pellet. The summation, over the total volume of soil influenced by fertilizer, of the products of water-soluble P × volume of soil containing this concentration of water-soluble P was well correlated with the uptake of P by oats in the greenhouse experiment. Thus, the effect of the associated salts on the availability of the CSP was largely explained on the basis of measured chemical differences. Any effects on physiology of the plants were of secondary importance in this experiment.

Linearly Tapered Loaded Transmission Lines

Working formulas for the calculation of input impedances and attenuation are obtained for a transmission line in which the inductance and resistance per unit length are linear functions of distance, and in which the capacitance and leakance are constant. Generalized functions used in the formulas, which were developed in an earlier paper, are here expressed in terms of Bessel's functions of the initial constants and the rate of taper, a form more readily applied in some cases. For the tapered loaded submarine cable the formulas are further simplified through the use of the hyperbolic functions applicable to the smooth uniform line, with appropriate correction factors.

Sinusoidal Currents in Linearly Tapered Loaded Transmission Lines

Working formulas for the calculation of input impedances and attenuation are obtained for a transmission line in which the resistance and inductance per unit length are linear functions of distance, and in which the capacitance and leakance are constant. Generalized functions depending on the initial constants and the rate of taper are introduced, in such a way that the formulas and the functions are analogous to the usual formulas and hyperbolic functions. The latter are secured as special cases of the former. The theory is applicable to the tapered loaded submarine cable. Propagation in such a transmission line, as distinguished from that in the smooth uniform line, is characterized by the fact that the current attenuation is, in general, different from the voltage attenuation.