Finite size corrections in random matrix theory and Odlyzko's data set for the Riemann zeros

Odlyzko has computed a data set listing more than $10^9$ successive Riemann zeros, starting at a zero number beyond $10^{23}$. The data set relates to random matrix theory since, according to the Montgomery-Odlyzko law, the statistical properties of the large Riemann zeros agree with the statistical properties of the eigenvalues of large random Hermitian matrices. Moreover, Keating and Snaith, and then Bogomolny and collaborators, have used $N \times N$ random unitary matrices to analyse deviations from this law. We contribute to this line of study in two ways. First, we point out that a natural process to apply to the data set is to thin it by deleting each member independently with some specified probability, and we proceed to compute empirical two-point correlation functions and nearest neighbour spacings in this setting. Second, we show how to characterise the order $1/N^2$ correction term to the spacing distribution for random unitary matrices in terms of a second order differential equation with coefficients that are Painlev\'e transcendents, and where the thinning parameter appears only in the boundary condition. This equation can be solved numerically using a power series method. Comparison with the Riemann zero data shows accurate agreement.Comment: 22 pages, 10 figures, Version 2 added some new references in bibliography, Version 3 corrected the scaling on the spacing distribution and some typo

Use of Drawings to Explore US Women's Perspectives on Why People Might Decline HIV Testing

The purpose of this research is to explore through drawings and verbal descriptions women's perspectives about reasons why persons might decline human immunodeficiency virus (HIV) testing. We asked 30 participants to draw a person that would NOT get tested for HIV and then explain drawings. Using qualitative content analysis, we extracted seven themes. We found apprehension about knowing the result of an HIV test to be the most commonly identified theme in women's explanations of those who would not get tested. This technique was well received and its use is extended to HIV issues

The CDC Revised Recommendations for HIV Testing: Reactions of Women Attending Community Health Clinics

The purpose of this study was to examine reactions to the Centers for Disease Control and Prevention revised recommendations for HIV testing by women attending community health clinics. A total of 30 women attending three community clinics completed semistructured individual interviews containing three questions about the recommendations. Thematic content analysis of responses was conducted. Results were that all agreed with the recommendation for universal testing. Most viewed opt-out screening as an acceptable approach to HIV testing. Many emphasized the importance of provision of explicit verbal informed consent. The majority strongly opposed the elimination of the requirement for pretest prevention counseling and spontaneously talked about the ongoing importance of posttest counseling. The conclusion was that there was strong support for universal testing of all persons 13 to 64 years old but scant support for the elimination of pretest prevention counseling. In general, respondents believed that verbal informed consent for testing as well as provision of HIV-related information before and after testing were crucial

A real quaternion spherical ensemble of random matrices

One can identify a tripartite classification of random matrix ensembles into geometrical universality classes corresponding to the plane, the sphere and the anti-sphere. The plane is identified with Ginibre-type (iid) matrices and the anti-sphere with truncations of unitary matrices. This paper focusses on an ensemble corresponding to the sphere: matrices of the form \bY= \bA^{-1} \bB, where \bA and \bB are independent $N\times N$ matrices with iid standard Gaussian real quaternion entries. By applying techniques similar to those used for the analogous complex and real spherical ensembles, the eigenvalue jpdf and correlation functions are calculated. This completes the exploration of spherical matrices using the traditional Dyson indices $\beta=1,2,4$. We find that the eigenvalue density (after stereographic projection onto the sphere) has a depletion of eigenvalues along a ring corresponding to the real axis, with reflective symmetry about this ring. However, in the limit of large matrix dimension, this eigenvalue density approaches that of the corresponding complex ensemble, a density which is uniform on the sphere. This result is in keeping with the spherical law (analogous to the circular law for iid matrices), which states that for matrices having the spherical structure \bY= \bA^{-1} \bB, where \bA and \bB are independent, iid matrices the (stereographically projected) eigenvalue density tends to uniformity on the sphere.Comment: 25 pages, 3 figures. Added another citation in version