In response to a letter from Goldbach, Euler considered sums of the form where s and t are positive integers.As Euler discovered by a process of extrapolation (from s + f g l 3 ) ,

The lattice sums involved in the definition of Madelung's constant of an NaCI-type crystal lattice in two or three dimensions are investigated. The fundamental mathematical questions of convergence and uniqueness of the sum of these, not absolutely convergent, series are considered. It is shown that some of the simplest direct sum methods converge and some do not converge. In particular, the very common method of expressing Madelung's constant by a series obtained from expanding spheres does not converge. The concept of analytic continuation of a complex function to provide a basis for an unambiguous mathematical definition of Madelung's constant is introduced. By these means, the simple intuitive direct sum methods and the powerful integral transformation methods, which are based on theta function identities and the Mellin transform, are brought together. A brief analysis of a hexagonal lattice is also given.

We apply experimental-mathematical principles to analyze integralsThese are generalizations of a previous integral C n := C n,1 relevant to the Ising theory of solid-state physics [8]. We find representations of the C n,k in terms of Meijer G-functions and nested-Barnes integrals. Our investigations began by computing 500-digit numerical values of C n,k for all integers n, k where n ∈ [2, 12] and k ∈ [0, 25]. We found that some C n,k enjoy exact evaluations involving Dirichlet L-functions or the Riemann zeta function. In the process of analyzing hypergeometric representations, we found-experimentally and strikingly-that the C n,k almost certainly satisfy certain inter-indicial relations including discrete k-recursions. Using generating functions, differential theory, complex analysis, and Wilf-Zeilberger algorithms we are able to prove some central cases of these relations.

A systematic electron microscopic study was made of the structure of foveal cones of Macaca spp. Transverse sections of inner (IS) and outer segments (OS) were made in sequence, from the pigment epithelial zone (PEZ) to the outer limiting membrane (OLM). The smallest diameters of hundreds of cone sections were measured from electron micrographs with a Zeiss particle-size analyzer, and analyzed statistically. Some details are also included about Cebus photoreceptors. It is claimed in the literature that foveal cones are rod-like (cylindrical) and untapered. Our study shows the foveolar cone to be a tapered structure. There has been some confusion between the foveola, which is rod-free, and the fovea, which has a high concentration of cones, but is not rod-free. Within the fovea, as the ratio of cones to rods falls from infinity to 1, with distance from the central bouquet of cones, the cone center-to-center distances increase, the inner segment diameters increase, and the number of cones/sq mm decreases. The tapered calycal processes are more massive in M. irus than M. mulatta, and the lateral fins are better developed. Lateral fins are not present in the foveola. The cones are arranged in straight lines.

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