A Generalisation of Dyson's Integration Theorem for Determinants

Dyson's integration theorem is widely used in the computation of eigenvalue correlation functions in Random Matrix Theory. Here we focus on the variant of the theorem for determinants, relevant for the unitary ensembles with Dyson index beta = 2. We derive a formula reducing the (n-k)-fold integral of an n x n determinant of a kernel of two sets of arbitrary functions to a determinant of size k x k. Our generalisation allows for sets of functions that are not orthogonal or bi-orthogonal with respect to the integration measure. In the special case of orthogonal functions Dyson's theorem is recovered

Ion implantation damage of silicon as observed by optical reflection spectroscopy in the 1 to 6 eV region

Optical reflection spectra of crystalline, sputtered, and ion implanted silicon specimens are presented. Characteristic aspects of the spectra of ion implanted specimens are related to lattice damage

Individual complex Dirac eigenvalue distributions from random matrix theory and lattice QCD at nonzero chemical potential

We analyze how individual eigenvalues of the QCD Dirac operator at nonzero chemical potential are distributed in the complex plane. Exact and approximate analytical results for such distributions are derived from non-Hermitian random matrix theory. When comparing these to lattice QCD spectra close to the origin, excellent agreement is found for zero and nonzero topology at several values of the chemical potential. Our analytical results are also applicable to other physical systems in the same symmetry class

HB 409, A Drastic Departure From Florida’s Traditional Stance On Will Execution Formalities

The baby boomer generation is aging, and many of the citizens that belong to this generation are retiring to Florida. Accordingly, Florida is expected to host one of the largest wealth transfers in history. And while the baby boomer population ages, our society is becoming more digitized. Things we traditionally did by pen and paper are now increasingly done by computer and keystroke, and wills are no exception. What was previously considered a document whose sacred nature could only be appreciated by the affixation of a handwritten signature at the bottom thereof, wills are now being drafted, signed, witnessed, and stored digitally. This Note analyzes Florida’s recently enacted legislation, HB 409, that authorizes electronic wills and the remote witnessing of such wills. The analysis proceeds against a backdrop defining the term “electronic will” and explaining how electronic wills diverge from what society has traditionally deemed a will. I begin by explaining the policy reasons behind statutory will act formalities and the four functions that are served by these traditional formalities. I also discuss the various positions that courts have taken when deciding whether to admit any purported will to probate. Next, I discuss the three categories of electronic wills and the shortcomings that each of these categories faces with respect to the “Four Functions.” After a brief discussion of how lawmakers and courts nationally and internationally have addressed the rise of electronic wills, this Note will turn the reader’s attention to Florida’s HB 409. This Note provides a summary of the legislation’s main provisions and an analysis of its specific “functional” shortcomings. After June 1, 2020, Florida courts should expect an influx of digitally signed and remotely witnessed electronic wills. Florida courts should also be aware of the entirely new grounds for will contests that HB 409 creates

Distributions of individual Dirac eigenvalues for QCD at non-zero chemical potential: RMT predictions and lattice results

For QCD at non-zero chemical potential $\mu$, the Dirac eigenvalues are scattered in the complex plane. We define a notion of ordering for individual eigenvalues in this case and derive the distributions of individual eigenvalues from random matrix theory (RMT). We distinguish two cases depending on the parameter $\alpha=\mu^2 F^2 V$, where $V$ is the volume and $F$ is the familiar low-energy constant of chiral perturbation theory. For small $\alpha$, we use a Fredholm determinant expansion and observe that already the first few terms give an excellent approximation. For large $\alpha$, all spectral correlations are rotationally invariant, and exact results can be derived. We compare the RMT predictions to lattice data and in both cases find excellent agreement in the topological sectors $\nu=0,1,2$