MPEJ Volume 13, No. 1, 29 pp. Received: Jul 24, 2006. Revised: Oct 12, 2006. Accepted: Mar 28, 2007. G.Bouch and P.L.Bowers Evaluating Feynman Path Integrals via the Polynomial Path Family ABSTRACT: We present a difficult and detailed calculation of propagators for quadratic potentials using a filtration of the space of variational paths around the classical path by the dense set of polynomial paths filtered by degree. This presents evidence for a cohesive explanation of path integrals as limits of finite-dimensional integrations using a sequence of filtration measures determined for specific dense families of paths by the requirement that the limit yield the free particle propagator. http://www.maia.ub.es/mpej/Vol/13/1.ps http://www.maia.ub.es/mpej/Vol/13/1.pdf http://www.ma.utexas.edu/mpej/Vol/13/1.ps http://www.ma.utexas.edu/mpej/Vol/13/1.pdf http://mpej.unige.ch/mpej/Vol/13/1.ps http://mpej.unige.ch/mpej/Vol/13/1.pdf ------------------------------ MPEJ Volume 13, No. 2, 8 pp. ------------------------------ MPEJ Volume 13, No. 2, 8 pp. Received: Jul 11, 2006. Revised: Jan 28, 2007. Accepted: Mar 28, 2007. N.Goldstein Inertiality Implies the Lorentz Group ABSTRACT: In his seminal paper of 1905, Einstein derives the Lorentz group as being the coordinate transformations of Special Relativity, under the main assumption that all inertial frames are equivalent. In that paper, Einstein also assumes the coordinate transformations are linear. Since then, other investigators have weakened and varied the linearity assumption. In the present paper, we retain only the inertiality assumption, and do not even assume that the coordinate transformations are continuous. Linearity is deduced. Our result is described in the affine space, R(n+1), with coordinates x0,x1,...,xn. Using the notation t = x0 and y = (x1,...,xn), the slope of a line in R(n+1) is defined to be |\delta y/\delta t|, computed from any two points on the line. The slope is non-negative and possibly infinite. A line in R(n+1) is said to be time-like if the slope of the line is strictly less than 1. Since inertial frames agree on who is inertial, coordinate transformations must carry time-like lines to time-like lines. A bijection from R(n+1) to R(n+1) is said to be time-like if it maps any time-like line onto another time-like line. The bijection is not assumed to be continuous. This paper proves that a time-like bijection is continuous (in fact, affine linear). The bijection is said to be strictly time-like if both it and its inverse are time-like. It is elementary to deduce that the strictly time-like bijections form the group generated by the extended Poincare group and the dilations. http://www.maia.ub.es/mpej/Vol/13/2.ps http://www.maia.ub.es/mpej/Vol/13/2.pdf http://www.ma.utexas.edu/mpej/Vol/13/2.ps http://www.ma.utexas.edu/mpej/Vol/13/2.pdf http://mpej.unige.ch/mpej/Vol/13/2.ps http://mpej.unige.ch/mpej/Vol/13/2.pdf ------------------------------ MPEJ Volume 13, No. 3, 4 pp. ------------------------------ MPEJ Volume 13, No. 3, 4 pp. Received: Jun 16, 2006. Revised: Oct 6, 2006. Accepted: Dec 10, 2006. T. Jecko Erratum to "Non-trapping condition for semiclassical Schr\"odinger operators with matrix-valued potentials" ABSTRACT: http://www.maia.ub.es/mpej/Vol/13/3.ps http://www.maia.ub.es/mpej/Vol/13/3.pdf http://www.ma.utexas.edu/mpej/Vol/13/3.ps http://www.ma.utexas.edu/mpej/Vol/13/3.pdf http://mpej.unige.ch/mpej/Vol/13/3.ps http://mpej.unige.ch/mpej/Vol/13/3.pdf ------------------------------ MPEJ Volume 13, No. 4, 24 pp. ------------------------------ MPEJ Volume 13, No. 4, 24 pp. Received: May 4, 2006. Revised: Jan 16, 2007. Accepted: Jul 20, 2007. C.Fermanian Kammerer Normal forms for conical intersections in quantum chemistry ABSTRACT: In quantum chemistry, the dynamics of heavy molecules is described by Schr\"odinger equations with energy level crossings. These crossings generate energy transfers at leading order between the modes as was earlier noticed by Landau and Zener in the 30's. The mathematical analysis of these transfers relies on the use of normal forms. Here, we analyze the implications of Y. Colin de Verdi\`ere's recent result on normal forms in the context derived by G. Hagedorn for molecular propagation and extend these normal forms to codimension 5 crossings. http://www.maia.ub.es/mpej/Vol/13/4.ps http://www.maia.ub.es/mpej/Vol/13/4.pdf http://www.ma.utexas.edu/mpej/Vol/13/4.ps http://www.ma.utexas.edu/mpej/Vol/13/4.pdf http://mpej.unige.ch/mpej/Vol/13/4.ps http://mpej.unige.ch/mpej/Vol/13/4.pdf ------------------------------ MPEJ Volume 13, No. 5, 40 pp. ------------------------------ MPEJ Volume 13, No. 5, 40 pp. Received: Mar 1, 2007. Revised: Aug 14, 2007. Accepted: Sep 20, 2007. H.Schulz-Baldes Rotation numbers for Jacobi matrices with matrix entries ABSTRACT: A Jacobi matrix with matrix entries is a selfadjoint block tridiagonal matrix with positive definite blocks on the off-diagonals. A rotation number calculation for its eigenvalues is presented. This is a matricial generalization of the oscillation theorem for the discrete analogues of Sturm-Liouville operators. The three universality classes of time reversal invariance are dealt with by implementing the corresponding symmetries. For Jacobi matrices with random matrix entries, this leads to a formula for the integrated density of states which can be calculated perturbatively in the coupling constant of the randomness with an optimal control on the error terms. http://www.maia.ub.es/mpej/Vol/13/5.ps http://www.maia.ub.es/mpej/Vol/13/5.pdf http://www.ma.utexas.edu/mpej/Vol/13/5.ps http://www.ma.utexas.edu/mpej/Vol/13/5.pdf http://mpej.unige.ch/mpej/Vol/13/5.ps http://mpej.unige.ch/mpej/Vol/13/5.pdf