arXiv subjects
Publications and source records attributed to James S. Wolper.
Agnes Mary Clerke (1842-1907) was an important contributor to the discussions of Mathematics in the Eleventh Edition of the Encyclopedia Britannica. The Eleventh Edition of the Encyclopedia Britannica had the audacity to "presume to define and delineate the entire (and thus complete) opus of all human knowledge." This is a survey of the Encyclopedia's treatment of Mathematics and an appreciation of her writings.
Information Theoretic analysis of the periods of a hyperelliptic curve provides more information about the well--known but abstract relationship between the branch points and the periods. Here one constructs a canonical homology basis for a hyperelliptic curve that shows that its periods must satisfy certain constraints and defines an open set in the Siegel upper half space that cannot contain any period matrices of hyperelliptic curves.
One interpretation of Torelli's Theorem, which asserts that a compact Riemann Surface $X$ of genus $g > 1$ is determined by the $g(g+1)/2$ entries of the period matrix, is that the period matrix is a message about $X$. Since this message depends on only $3g-3$ moduli, it is sparse, or at least approximately so, in the sense of information theory. Thus, methods from information theory may be useful in reconstructing the period matrix, and hence the Riemann surface, from a small subset of the periods. The results here show that, with high probability, any set of $3g-3$ periods form moduli for the surface.
The Information-Theoretic Schottky Problem treats the period matrix of a compact Riemann Surface as a compressible signal. In this case, the period matrix of a smooth plane curve is characterized by only 4 of its columns, a significant compression.
Quadrature rules estimate the value of an integral when the function is given by a table of values. Every binary string defines a quadrature rule by choosing which endpoint of each interval represents the interval. The standard rules, such as Simpson's Rule, correspond to strings of low Kolmogorov complexity, making it possible to define new quadrature rules with no smoothness assumptions, as well as in higher dimensions. Error results depend on concepts from compressed sensing. Good quadrature rules exist for "sparse" functions, which also satisfy an error--information duality principle.