Making proofs without Modus Ponens: An introduction to the combinatorics and complexity of cut elimination

This paper is intended to provide an introduction to cut elimination which is accessible to a broad mathematical audience. Gentzen's cut elimination theorem is not as well known as it deserves to be, and it is tied to a lot of interesting mathematical structure. In particular we try to indicate some dynamical and combinatorial aspects of cut elimination, as well as its connections to complexity theory. We discuss two concrete examples where one can see the structure of short proofs with cuts, one concerning feasible numbers and the other concerning "bounded mean oscillation" from real analysis

The Hubbard transition and unsaturated hydrocarbons

We contrast a simple molecular orbital theory (Hückel) with a simple valence bond theory (Heisenberg–Dirac). We find for alternant systems in which both models have nondegenerate ground states that both models have ground states belonging to the most symmetrical irreducible representation of the molecular point group. We also find there exist nonalternant systems which have ground states with different irreducible representations. In these latter systems neither the Hückel nor the Heisenberg–Dirac model is sufficient to give a qualitative picture of the molecule. Instead a combined Hückel–Heisenberg–Dirac model (the Hubbard model) must be used. Finally we list some organic unsaturated hydrocarbons, whose ground state changes from one irreducible representation to another as the Hubbard parameters vary.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/71266/2/JCPSA6-90-5-2741-1.pd

Generalized interpolation and definability

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/32849/1/0000225.pd

Chi-Squared Portmanteau Statistics for Vector Autoregressive Models with Uncorrelated Errors

The portmanteau statistic based on the first m residual autocorrelations is used for testing the goodness-of-fit for vector autoregressive models with varying m. However, it is known that existing portmanteau statistics are approximately non chi-squared distributions in the presence of non-independent innovations. In this paper we propose a new portmanteau statistic that is asymptotically chi-squared even in the presence of non-independent innovations. We also study the joint probability of the multiple portmanteau statistics with different degrees of freedom. Monte Carlo experiments illustrate the finite sample performance for the proposed portmanteau test