On the boundary convergence of solutions to the Hermite-Schr\"odinger equation

In the half-space $\mathbb{R}^d \times \mathbb{R}_+$, we consider the Hermite-Schr\"odinger equation $i\partial u/\partial t = - \Delta u + |x|^2 u$, with given boundary values on $\mathbb{R}^d$. We prove a formula that links the solution of this problem to that of the classical Schr\"odinger equation. It shows that mixed norm estimates for the Hermite-Schr\"odinger equation can be obtained immediately from those known in the classical case. In one space dimension, we deduce sharp pointwise convergence results at the boundary, by means of this link.Comment: 12 page

Risk reduction and real estate portfolio size

Despite a number of papers that discuss the advantages of increased size on risk levels in real estate portfolios there is remarkably little empirical evidence based on actual portfolios. The objective of this paper is to remedy this deficiency by examining the portfolio risk of a large sample of actual property data over the period 1981 to 1996. The results show that all that can be said is that portfolios of properties of a large size, on the average, tend to have lower risks than small sized portfolios. More importantly portfolios of a few properties can have very high or very low risk

Invalid Forensic Science Testimony and Wrongful Convictions

This is the first study to explore the forensic science testimony by prosecution experts in the trials of innocent persons, all convicted of serious crimes, who were later exonerated by post-conviction DNA testing. Trial transcripts were sought for all 156 exonerees identified as having trial testimony by forensic analysts, of which 137 were located and reviewed. These trials most commonly included testimony concerning serological analysis and microscopic hair comparison, but some included bite mark, shoe print, soil, fiber, and fingerprint comparisons, and several included DNA testing. This study found that in the bulk of these trials of innocent defendants - 82 cases or 60% - forensic analysts called by the prosecution provided invalid testimony at trial - that is, testimony with conclusions misstating empirical data or wholly unsupported by empirical data. This was not the testimony of a mere handful of analysts: this set of trials included invalid testimony by 72 forensic analysts called by the prosecution and employed by 52 laboratories, practices, or hospitals from 25 states. Unfortunately, the adversarial process largely failed to police this invalid testimony. Defense counsel rarely cross-examined analysts concerning invalid testimony and rarely obtained experts of their own. In the few cases in which invalid forensic science was challenged, judges seldom provided relief. This evidence supports efforts to create scientific oversight mechanisms for reviewing forensic testimony and to develop clear scientific standards for written reports and testimony. The scientific community can through an official government entity promulgate standards to ensure the valid presentation of forensic science in criminal cases and thus the integrity and fairness of the criminal process

Local Central Limit Theorem for Determinantal Point Processes

We prove a local central limit theorem (LCLT) for the number of points $N(J)$ in a region $J$ in $\mathbb R^d$ specified by a determinantal point process with an Hermitian kernel. The only assumption is that the variance of $N(J)$ tends to infinity as $|J| \to \infty$. This extends a previous result giving a weaker central limit theorem (CLT) for these systems. Our result relies on the fact that the Lee-Yang zeros of the generating function for $\{E(k;J)\}$ --- the probabilities of there being exactly $k$ points in $J$ --- all lie on the negative real $z$-axis. In particular, the result applies to the scaled bulk eigenvalue distribution for the Gaussian Unitary Ensemble (GUE) and that of the Ginibre ensemble. For the GUE we can also treat the properly scaled edge eigenvalue distribution. Using identities between gap probabilities, the LCLT can be extended to bulk eigenvalues of the Gaussian Symplectic Ensemble (GSE). A LCLT is also established for the probability density function of the $k$-th largest eigenvalue at the soft edge, and of the spacing between $k$-th neigbors in the bulk.Comment: 12 pages; claims relating to LCLT for Pfaffian point processes of version 1 withdrawn in version 2 and replaced by determinantal point processes; improved presentation version

Invalid Forensic Science Testimony and Wrongful Convictions

This is the first study to explore the forensic science testimony by prosecution experts in the trials of innocent persons, all convicted of serious crimes, who were later exonerated by post-conviction DNA testing. Trial transcripts were sought for all 156 exonerees identified as having trial testimony by forensic analysts, of which 137 were located and reviewed. These trials most commonly included testimony concerning serological analysis and microscopic hair comparison, but some included bite mark, shoe print, soil, fiber, and fingerprint comparisons, and several included DNA testing. This study found that in the bulk of these trials of innocent defendants - 82 cases or 60% - forensic analysts called by the prosecution provided invalid testimony at trial - that is, testimony with conclusions misstating empirical data or wholly unsupported by empirical data. This was not the testimony of a mere handful of analysts: this set of trials included invalid testimony by 72 forensic analysts called by the prosecution and employed by 52 laboratories, practices, or hospitals from 25 states. Unfortunately, the adversarial process largely failed to police this invalid testimony. Defense counsel rarely cross-examined analysts concerning invalid testimony and rarely obtained experts of their own. In the few cases in which invalid forensic science was challenged, judges seldom provided relief. This evidence supports efforts to create scientific oversight mechanisms for reviewing forensic testimony and to develop clear scientific standards for written reports and testimony. The scientific community can through an official government entity promulgate standards to ensure the valid presentation of forensic science in criminal cases and thus the integrity and fairness of the criminal process