A remark on Burnside's theorem on matrix algebras

We present a simple proof of Burnside's theorem using techniques from elementary linear algebra.     

Quadratic forms and Klauder's phenomenon: A remark on very singular perturbations

We give a general analysis of a class of pairs of positive self-adjoint operators A and B for which A + λB has a limit (in strong resolvent sense) as λ ↓ 0 which is an operator A_p ≠ A!     

Heawood's empire problem

Heawood's map color theorem is generalized to maps of empires on surfaces.     

A remark on Hilbert's matrix

We exhibit a Jacobi matrix T which has simple spectrum and integer entries, and 0 commutes with Hilbert's matrix. As an application we replace the computation of the eigenvectors of Hilbert's matrix (a very ill-conditioned problem) by the computation of the eigenvectors of T (a nicely stable numerical problem).     

A remark on Nehari's problem

An example of a strickly contractive Hankel matrix is given such that the central contractive extensionf _c of satisfies f _c =1. This way we answer a problem raised by Ciprian Foias.Partially supported by a Georgia State University Research Grant.     

On Woodall's interval problem

For each k > 1, a sequence of 4(k ? 1) ? [log₂ (k ? 1)] sets (layers) of disjoint intervals in $\text{R}$ is constructed, such that every interval overlaps at least one interval in each preceding layer and no point of $\text{R}$ belongs to more than k layers. This improves a construction of Woodall. The approach is by way of the related “saturation” or “scope” problem.     

Mersenne's solution for Galileo's problem of the rotating earth

More than twenty years before Huygens and Newton developed formulas for centrifugal acceleration, Mersenne contrived a statisfactory solution for Galileo's problem of the extrusion of bodies from the earth as a result of its daily rotation. Mersenne was able to overcome an error in Galileo's approach without the use either of an explicit notion of infinitesimals or of any clear concept of force. His solution depends on comparing the lengths of two lines, a technique that several historians have claimed to be inadequate for this problem.     

A remark on the capitulation problem

We determine explicitly an infinite family of imaginary cyclic number fields k, such that the 2-class group of k is elementary with arbitrary large 2-rank and capitulates in an unramified quadratic extension K. The infinitely many number fields k and K have the same Hilbert 2-class field and an infinite Hilbert 2-class field tower.     

A single-level control in Moran's problem with feedback

Moran's well-known model for a dam is generalized to the case when both the random entering flow and outflow may depend on the current magnitude of the supply of water.     

Sylvester's problem of the 15 schoolgirls

A solution is given to the problem: Can 15 schoolgirls walk out in five rows of three, seven times a week for a term of 13 weeks, in such a way that any two girls are in the same row just once in each week, and any three just once in the term.