A Contour Integral Representation for the Dual Five-Point Function and a Symmetry of the Genus Four Surface in R6

The invention of the "dual resonance model" N-point functions BN motivated the development of current string theory. The simplest of these models, the four-point function B4, is the classical Euler Beta function. Many standard methods of complex analysis in a single variable have been applied to elucidate the properties of the Euler Beta function, leading, for example, to analytic continuation formulas such as the contour-integral representation obtained by Pochhammer in 1890. Here we explore the geometry underlying the dual five-point function B5, the simplest generalization of the Euler Beta function. Analyzing the B5 integrand leads to a polyhedral structure for the five-crosscap surface, embedded in RP5, that has 12 pentagonal faces and a symmetry group of order 120 in PGL(6). We find a Pochhammer-like representation for B5 that is a contour integral along a surface of genus five. The symmetric embedding of the five-crosscap surface in RP5 is doubly covered by a symmetric embedding of the surface of genus four in R6 that has a polyhedral structure with 24 pentagonal faces and a symmetry group of order 240 in O(6). The methods appear generalizable to all N, and the resulting structures seem to be related to associahedra in arbitrary dimensions.Comment: 43 pages and 44 figure

Eogenetic Karst from the Perspective of an Equivalent Porous Medium

The porosity of young limestones experiencing meteoric diagenesis in the vicinity of their deposition (eogenetic karst) is mainly a double porosity consisting of touching-vug channels and preferred passageways lacing through a matrix of interparticle porosity. In contrast, the porosity of limestones experiencing subaerial erosion following burial diagenesis and uplift (telogenetic karst) is mainly a double porosity consisting of conduits within a network of fractures. The stark contrast between these two kinds of karst is illustrated by their position on a graph showing the hydraulic characteristics of an equivalent porous medium consisting of straight, cylindrical tubes (n-D space, where n is porosity,D is the diameter of the tubes, and logn is plotted against logD). Studies of the hydrology of small carbonate islands show that large-scale, horizontal hydraulic conductivity (K) increases by orders of magnitude during the evolution of eogenetic karst. Earlier petrologic studies have shown there is little if any change in the total porosity of the limestone during eogenetic diagenesis. The limestone of eogenetic karst, therefore, tracks horizontally inn-D space. In contrast, the path from initial sedimentary material to telogenetic karst comprises a descent on the graph with reduction ofn during burial diagenesis, then a sideways shift with increasingD due to opening of fractures during uplift and exposure, and finally an increase inD andn during development of the conduits along the fractures. Eogenetic caves are mainly limited to boundaries between geologic units and hydrologic zones: stream caves at the contact between carbonates and underlying impermeable rocks (and collapse-origin caves derived therefrom); vertical caves along platform-margin fractures; epikarst; phreatic pockets (banana holes) along the water table; and flank margin caves that form as mixing chambers at the coastal freshwater-saltwater “interface”. In contrast, the caverns of telogenetic karst are part of a system of interconnected conduits that drain an entire region. The eogenetic caves of small carbonate islands are, for the most part, not significantly involved in the drainage of the island