Sphere Of Unit Radius Research Articles

Sphere Theorem on the Stokes Equation for Three-Dimensional Viscous Flow

A theorem is presented which gives the perturbation pressure \(\tilde{p}({\mib x})\) and velocity \(\tilde{{\mib u}}({\mib x})\) directly when a sphere of unit radius r =1 with its center at the origin is introduced into an unlimited viscous fluid of viscosity µ obeying the Stokes equation, of which the pressure and the velocity are p ( x ) and u ( x ). They are \begin{aligned} \tilde{p}=-\left\{\mu \left(4r^{3}u_{r}^{\prime} +3 \int^{r}_{0}ru_{r} \,{\rm d}\,r\right) +(r^{2}-r^{4})p^{\prime}- \frac{1}{2} \int^{r}_{0} (3-r^{2})rp^{\prime} \,{\rm d}\,r \right\}^{\ast}, \label{eq:1} \end{aligned} and

An extremal problem of the approximation of integrals with fixed singularity in the class WrL2

Nikol'skii's general extremal problem in the class of functions W 0 r L 2 is solved for one class of integrals with fixed singularity. The result obtained is used for the approximate evaluation of singular integrals with a Cauchy kernel and integrals with fixed singularity, whose area of integration is a circle or sphere of unit radius.

Mathematical models for analysis and synthesis of spatial mechanisms—III Six-link spatial mechanisms

In this paper the decomposition method for kinematic synthesis and analysis is extended to all 6-link, single-loop spatial mechanisms. This is accomplished by solving sets of equations which can be used to describe mathematical models of all 6-link spatial mechanisms. These sets of equations are solved sequentially in an organised way by subdividing the sets of equations into groups of equations which describe the intersection of two planes with a unit radius sphere (computer programme PREB3), and the intersection of a plane and a unit radius sphere (computer programme PREB2). By way of examples mathematical models for RRRTRR, RRPRTR, RTRPRP, RRPPRC spatial mechanisms are described which are representative of the 130 distinct 6-link spatial mechanisms.

Elliptic polarization represented by the Carter and Smith charts

The graphical method using the Poincaré sphere for representing elliptically polarized light is based on the fact that the latitude and longitude of a point on the unit-radius sphere represent the ellipticity and the azimuth of the corresponding light ellipse. Stereographic projections of the unit-diameter Poincaré sphere either on a western plane tangent at the equator of the sphere or on an equatorial plane tangent to either of the poles lead to a representation of any elliptical polarization state on the so-called Carter charts, while charts derived from orthographic projections in the Poincaré sphere and consisting again of families of orthogonal circles traced inside either the equator or the principal meridian of the unit-radius Poincaré sphere yielding either ellipticity-azimuth or amplitude ratio-phase angle niveau lines constitute the Smith charts. Both types of charts consist of families of orthogonal circles. These families of circles, traced on either chart and corresponding to parametric families with different properties for the corresponding elliptically polarized light, yield an easy and accurate solution to the problem of graphical evaluation of the optical properties of the resulting polarization state when an elliptically polarized plane wave is passing through different optically active elements.

A Simplified Method of Quantitating Preferred Orientation

AbstractQualitative evaluation of preferred orientation in 10-rml Hastelloy N seamless tubing was carried out by obtaining conventional pole figures for several samples with the modified Schulz reflection technique. This established the texture as a (110)[112]. Quantitative determination of this texture was achieved for a large number of samples by a radial traverse of an azimuthally averaged (110) stereographic projection. This was accomplished with a Schulz goniometer by rotating the sample rapidly in its own plane while slowly varying the tilt angle ϕ. The first minimum observed in the resultant pattern approximates the pole density of the randomly oriented component of the sample. A scan obtained from a randomly oriented sample and suitably scaled to match pole density at these minima serves to distinguish the random from the oriented component of the sample. A ratio of the oriented to the random component as a function of the tilt angle ϕ is independent of the instrumental effects of defocusing and absorption. This ratio is proportional to the pole density of the oriented component since the pole density of the random component is a constant by definition. The same constant of proportionality yields a pole density of unity for the random component. Integrating the pole density for die random component over the surface of a reference sphere of unit radius gives the value 4π for the number of randomly oriented poles NR. The number of poles corresponding to the oriented component is obtained by evaluating the integralIn this equation, R(ϕ) is the ratio of the oriented to the random component, Φ is the tilt angle corresponding to the first minimum in the sample scan, and m is a multiplicity factor which compensates for integration over just one 110 face of each grain. Evaluation of this integral may be made by plotting it with the aid of a family of sine curves and measuring the area under the curve.The per cent of volume of the oriented component is then calculated by the relationshipThis method entails about 30min of an analyst's time and 40 to 120 min of machine time, depending on the grain size of the sample. Further information on the angular breadth of the texture may be obtained from the plot of the integrand. This technique is subject to the limitations that the material have a single texture and it must be possible tc prepare the specimens so that a suitably chosen pole figure contains a central concentration of poles which are delineated from other regions of pole concentration by a surrounding minimum in pole density. This is generally not too difficult in the case of sheet textures but requires that the specimen normal be parallel to ths fiber axis for a fiber texture.

A New Method of Analysing Dielectric Measurements

It is proposed that the Cole-Cole plot method of analysing dielectric data should be replaced by the use of a polarizability plot, in which the imaginary coordinate of the complex function, α(ω) = [epsilon(ω)-1]/[epsilon(ω) + 2], is plotted against its real coordinate. α(ω) is the complex polarizability of a dielectric sphere of unit radius. This quantity is chosen because, in a sphere, long-range dipole-dipole coupling vanishes and therefore α will be a good measure of the intrinsic properties of a substance. This method gives proper weighting to all polarization mechanisms and provides a ready means of comparing the dielectric behaviour of different substances. Because the α-plot is a bilinear transformation of the Cole-Cole plot, circular arcs are transformed into circular arcs and semicircles into semicircles. Experimental results for glycerol, an ethanol-water mixture and for n-propanol are analysed by the new method.

A wide-angle microwave radiator

For many years it has been known that, in principle, optical lenses could have been made from non-homogeneous glass. For example, if a glass sphere of unit radius were made with refractive index varying according to the relation.μ = √(2−r2)where μ = refractive index and r = radial distance from centre, it would act as a lens with the focus on the surface of the sphere.Such non-homogeneous glass cannot be produced in practice; however, a microwave analogue has been constructed using spaced conducting sheets to produce a region of refractive index varying in accordance with the above relation.This lens, being circularly symmetrical, is free from aberrations as the feed is moved around the circumference, and is therefore suitable for wide-angle scanning. It will operate only when the direction of polarization is such that the magnetic vector is normal to the conducting sheets.

The Pure Archimedean Polytopes in Six and Seven Dimensions

An Archimedean solid (in three dimensions) may be defined as a polyhedron whose faces are regular polygons of two or more kinds and whose vertices are all surrounded in the same way. For example, the “great rhombicosidodecahedron” is bounded by squares, hexagons and decagons, one of each occurring at each vertex. Thus any Archimedean solid is determined by the faces which meet at one vertex, and therefore by the shape and size of the “vertex figure,” which may be defined as follows. Suppose, for simplicity, that the length of each edge of the solid is unity. The further extremities of all the edges which meet at a particular vertex lie on a sphere of unit radius, and also on the circumscribing sphere of the solid, and therefore on a circle. These points form a polygon, called the “vertex figure,” whose sides correspond to the faces at a vertex and are of length 2 cos π/n for an n-gonal face. Thus the vertex figure of the great rhombicosi-dodecahedron is a scalene triangle of sides .