On invariants and scalar chiral correlation functions in N=1 superconformal field theories

A general expression for the four-point function with vanishing total R-charge of anti-chiral and chiral superfields in N=1 superconformal theories is given. It is obtained by applying the exponential of a simple universal nilpotent differential operator to an arbitrary function of two cross ratios. To achieve this the nilpotent superconformal invariants according to Park are focused. Several dependencies between these invariants are presented, so that eight nilpotent invariants and 27 monomials of these invariants of degree d>1 are left being linearly independent. It is analyzed, how terms within the four-point function of general scalar superfields cancel in order to fulfill the chiral restrictions.Comment: 11 pages; v2: minor changes, references adde

Johann Faulhaber and sums of powers

Early 17th-century mathematical publications of Johann Faulhaber contain some remarkable theorems, such as the fact that the $r$-fold summation of $1^m,2^m,...,n^m$ is a polynomial in $n(n+r)$ when $m$ is a positive odd number. The present paper explores a computation-based approach by which Faulhaber may well have discovered such results, and solves a 360-year-old riddle that Faulhaber presented to his readers. It also shows that similar results hold when we express the sums in terms of central factorial powers instead of ordinary powers. Faulhaber's coefficients can moreover be generalized to factorial powers of noninteger exponents, obtaining asymptotic series for $1^{\alpha}+2^{\alpha}+...+n^{\alpha}$ in powers of $n^{-1}(n+1)^{-1}$

Measuring questions: relevance and its relation to entropy

The Boolean lattice of logical statements induces the free distributive lattice of questions. Inclusion on this lattice is based on whether one question answers another. Generalizing the zeta function of the question lattice leads to a valuation called relevance or bearing, which is a measure of the degree to which one question answers another. Richard Cox conjectured that this degree can be expressed as a generalized entropy. With the assistance of yet another important result from Janos Aczel, I show that this is indeed the case, and that the resulting inquiry calculus is a natural generalization of information theory. This approach provides a new perspective on the Principle of Maximum Entropy.Comment: 8 pages, 1 figure. Presented to the MaxEnt 2004 meeting in Garching Germany. To be published in: R. Fischer, V. Dose (eds.), Bayesian Inference and Maximum Entropy Methods in Science and Engineering, Garching, Germany 2004, AIP Conference Proceedings, American Institute of Physics, Melville N