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An analytical solution to the attenuation of flux within an integrating sphere due to spherical integrating source coating, exit port escape, and atmospheric absorption is derived employing a geometric probability distribution of completed sphere transits. This is used to determine the mean number of completed sphere transits and its variance. Equations that provide the attenuation ratios for the three extinction mechanisms are derived using the energy balance and summation of refection methods. The mean length of a transit of the sphere and its variance are presented and used to derive expressions for the mean and variance of photon path lengths in the sphere. OCIS codes: 000.4430, 120.0280.

For a large class of repulsive interaction models, the Mayer cluster integrals can be transformed into a tridiagonal real symmetric matrix [R.sub.mn], whose elements converge to two constants with 1/[n.sup.2] correction. We find exact expressions in terms of these correction terms for the two critical exponents describing the density near the two singular termination points of the fluid phase. We apply the method to the hard-spheres model and find that the metastable fluid phase terminates at [[rho].sub.t] = 0.751[5]. The density near the transition is given by [[rho].sub.t]-[rho] ~ [([z.sub.t] - z).sup.[sigma]'], where the critical exponent is predicted to be [sigma]' = 0.0877[25]. Interestingly, the termination density is close to the observed glass transition; thus, the above critical behavior is expected to be associated with the onset of glassy behavior in hard spheres. glass transition | Mayer cluster integrals

Byline: A. Cappelli (1), F. Colomo (1) Abstract: We analyze the spherical model with frustration induced by an external gauge field. The case of the infinite-dimensional model has recently been reduced to a problem of q-deformed oscillators with q parametrizing the amount of frustration. We find a complete analytic solution of the model by using a convenient representation of the q-oscillator algebra, the q-Hermite polynomials. The low-temperature phase does not exhibit a glassy behavior. With respect to the usual unfrustrated spherical model, the effect of frustration is only quantitative. A glassy low-temperature phase is expected for the more complicated XY model whose study is in progress. Bibliography: 15 titles. Author Affiliation: (1) Universita di Firenze, Italy Article History: Registration Date: 06/10/2004

We obtain bounds for the cardinality of multiple packing of Euclidean sphere. At zero rate these bounds are tight. Index Terms - Euclidean sphere, multiple packing.

The gravitational transient fall velocity of a rigid sphere in an otherwise quiescent viscous fluid is studied through examining the physical reasonableness of simulated results and comparing the results to Moorman's 1955 free-fall experiments. The published sphere dynamic equations from low-to-moderate sphere Reynolds number are solved numerically by a fourth order predictor-corrector method and iterations on sphere velocity and acceleration. Among the published sphere dynamic expressions, Mei and Adrian's 1992 expression has the best agreement with Moorman's data. The relative importance of the steady and unsteady drags along the gravitational transient process is also discussed. It is found that neglecting the unsteady drag indeed simplifies the computational procedure, but the accuracy on the time-varying fall velocity is significantly compromised. The added mass and history terms are of great importance at early stages of gravitational transient falling for low sphere-to-fluid density ratio and low sphere Reynolds number.

The effects of non-Lambertian scattering of the interior wall of an integrating sphere are examined through a sphere simulation model. The model employs Monte Carlo techniques. A sphere used for measurement of directional-hemispherical reflectance is modeled. The simulation allows sphere wall scattering to vary from perfectly Lambertian to perfectly specular in steps. The results demonstrate that significant measurement error can result as the scattering deviates from the Lambertian ideal. The error is found to be a strong function of the wall reflectance value as well: it is minimized for reflectances approaching 1.0 and increases as the reflectance value decreases to the minimum value examined of 0.5. The magnitudes of the errors associated with non-Lambertian scattering are also shown to be relatively independent of the specific field of view of the detector used in the measurement.

An elastic-plastic impact model for spheres is introduced as the basis to study the normal impact of rough surfaces. Statistics is applied to arrive at the ensemble behavior of many unit events alluded above, allowing the investigation of surface roughness effects. Dissipation of kinetic energy increases such as surface roughness, material compliance, and impact velocity is increased. The rebound velocity is shown to be dependent on surface topography and material properties, in addition to impact velocity.

A simple tool called the Area Principle is introduced and used to present deservingly elegant proofs of both Ceva's and Menelaus' Theorems. Volume Principle is proved which is valid for the Euclidean plane, the hyperbolic plane and the two-dimensional.

The article discusses the vibrational behavior of spherical shells under thermal stress. Equations solved by Galerkin's method are analyzed.

The representation of the closest possible packing of spheres in most texts confuses closest packing per layer with closest packing overall. It can be shown that square packing of spheres, although less efficient per layer than hexagonal packing, is as efficient as hexagonal packing. Square packed spheres make up for per-layer inefficiency with between-layer efficiency.