Hodge Theory on Metric Spaces

Hodge theory is a beautiful synthesis of geometry, topology, and analysis, which has been developed in the setting of Riemannian manifolds. On the other hand, spaces of images, which are important in the mathematical foundations of vision and pattern recognition, do not fit this framework. This motivates us to develop a version of Hodge theory on metric spaces with a probability measure. We believe that this constitutes a step towards understanding the geometry of vision. The appendix by Anthony Baker provides a separable, compact metric space with infinite dimensional \alpha-scale homology.Comment: appendix by Anthony W. Baker, 48 pages, AMS-LaTeX. v2: final version, to appear in Foundations of Computational Mathematics. Minor changes and addition

Single particle analysis with a 360/sup 0/ light scattering photometer

Light scattering by single spherical homogeneous particles in the diameter range 1 to 20 ..mu..m and relative refractive index 1.20 is measured. Particle size of narrowly dispersed populations is determined and a multi-modal dispersion of five components is completely analyzed. A 360/sup 0/ light scattering photometer for analysis of single particles has been designed and developed. A fluid stream containing single particles intersects a focused laser beam at the primary focal point of an ellipsoidal reflector ring. The light scattered at angles theta = 2.5/sup 0/ to 177.5/sup 0/ at phi = 0/sup 0/ and 180/sup 0/ is reflected onto a circular array of photodiodes. The ellipsoidal reflector is situated in a chamber filled with fluid matching that of the stream to minimize refracting and reflecting interfaces. The detector array consists of 60 photodiodes each subtending 3/sup 0/ in scattering angle on 6/sup 0/ centers around 360/sup 0/. 32 measurements on individual particles can be acquired at rates of 500 particles per second. The intensity and angular distribution of light scattered by spherical particles are indicative of size and relative refractive index. Calculations, using Lorenz--Mie theory, of differential scattering patterns integrated over angle corresponding to the detector geometry determined the instrument response to particle size. From this the expected resolution and experimental procedures are determined.Ultimately, the photometer will be utilized for identification and discrimination of biological cells based on the sensitivity of light scattering to size, shape, refractive index differences, internal granularity, and other internal morphology. This study has demonstrated the utility of the photometer and indicates potential for application to light scattering studies of biological cells