arXiv subjects
Publications and source records attributed to Joseph B. Keller.
It is shown that the first $n$ prime numbers $p_1,...,p_n$ determine the next one by the recursion equation $$ p_{n+1} =\lim\limits_{s\to +\infty} [\prod\limits^n_{k=1} (1-\frac{1}{p^s_k}) \sum\limits^\infty_{j=1} \frac{1}{j^s} -1]^{-1/s}. $$ The upper limit on the sum can be replaced by $2p_n -1$, and the result still holds.
A method is presented for evaluating authors on the basis of citations. It assigns to each author a citation score which depends upon the number of times he is cited, and upon the scores of the citers. The scores are found to be the components of an eigenvector of a normalized citation matrix. The same method can be applied to citation of journals by other journals, to evaluating teams in a league [1], etc.
Stirling's formula, the asymptotic expansion of $n!$ for $n$ large, or of $Γ(z)$ for $z\to \infty$, is derived directly from the recursion equation $Γ(z+1) =z Γ(s)$ and the normalization condition $Γ({1/2}) =\sqrtπ$.
The probability density of the resistance of a two dimensional rectangular network between two conducting plates is calculated. The nodes form an $M$ by $N$ lattice, and each edge has a random resistance. The Monte Carlo method is used.
It is shown how simple assumptions lead to piecewise linear behavior, which is observed in certain phase transitions.
The dimensions of sets of matrices of various types, with specified eigenvalue multiplicities, are determined. The dimensions of the sets of matrices with given Jordan form and with given singular value multiplicities are also found. Each corresponding codimension is the number of conditions which a matrix of the given type must satisfy in order to have the specified multiplicities.
A systematic derivation of a set of equations governing large scale atmospheric motions is presented. They are derived by introducing new variables scaled in terms of a small parameter.The solution is expanded in powers of this parameter.This leads to the hydrostatic pressure and geostrophic wind equations.The resulting set of equations is similar to that proposed by J. Charney.