Quasi-Invertibility in a Staircase Diagram

Invertibility in Sequent Calculi

The invertibility of the rules of a sequent calculus is important for guiding proof search and can be used in some formalised proofs of Cut admissibility. We present sufficient conditions for when a rule is invertible with respect to a calculus. We illustrate the conditions with examples. It must be noted we give purely syntactic criteria; no guarantees are given as to the suitability of the ru...

Geometry Ascending a Staircase

A large metal sculpture, consisting of four orbs, commissioned for the stairway atrium of the Fitzpatrick CIEMAS Engineering Building at Duke University embodies the important idea that Science, Technology, Engineering, Mathematics and Art are closely connected. Made from hundreds of pounds of powder-coated, laser-cut aluminum, with an underlying geometric design, it was assembled at a four-hou...

Invertibility in Infinite-dimensional Spaces

An interesting result of Doyle and Hocking states that a topological n-manifold is invertible if and only if it is a homeomorphic image of the n-sphere S. We shall prove that the sphere of any infinite-dimensional normed space is invertible. We shall also discuss the invertibility of other infinite-dimensional objects as well as an infinite-dimensional version of the Doyle-Hocking theorem.

On a Poisson Hyperbolic Staircase

We consider a process that starts at height y, stays there for a time X0 ∼ exp(y) when it drops to a level Z1 ∼ U(0, y). Thereafter it stays at level Zn for time Xn ∼ exp(Zn), then drops to a level Zn+1 ∼ U(0, Zn). We investigate properties of this process, as well as the Poisson hyperbolic process which is obtained by randomizing the starting point y of the above process. This process is assoc...

A Survey of Invertibility and Spectrum Preserving Linear Maps