A non-local problem for the Fokker-Planck equation related to the Becker-D\"{o}ring model

This paper concerns a Fokker-Planck equation on the positive real line modeling nucleation and growth of clusters. The main feature of the equation is the dependence of the driving vector field and boundary condition on a non-local order parameter related to the excess mass of the system. The first main result concerns the well-posedness and regularity of the Cauchy problem. The well-posedness is based on a fixed point argument, and the regularity on Schauder estimates. The first a priori estimates yield H\"older regularity of the non-local order parameter, which is improved by an iteration argument. The asymptotic behavior of solutions depends on some order parameter $\rho$ depending on the initial data. The system shows different behavior depending on a value $\rho_s>0$, determined from the potentials and diffusion coefficient. For $\rho \leq \rho_s$, there exists an equilibrium solution $c^{\text{eq}}_{(\rho)}$. If $\rho\le\rho_s$ the solution converges strongly to $c^{\text{eq}}_{(\rho)}$, while if $\rho > \rho_s$ the solution converges weakly to $c^{\text{eq}}_{(\rho_s)}$. The excess $\rho - \rho_s$ gets lost due to the formation of larger and larger clusters. In this regard, the model behaves similarly to the classical Becker-D\"oring equation. The system possesses a free energy, strictly decreasing along the evolution, which establishes the long time behavior. In the subcritical case $\rho<\rho_s$ the entropy method, based on suitable weighted logarithmic Sobolev inequalities and interpolation estimates, is used to obtain explicit convergence rates to the equilibrium solution. The close connection of the presented model and the Becker-D\"oring model is outlined by a family of discrete Fokker-Planck type equations interpolating between both of them. This family of models possesses a gradient flow structure, emphasizing their commonality.Comment: Minor revised version accepted for publication in Discrete & Continuous Dynamical Systems -

Quantitative Analysis of Disparities in Juvenile Delinquency Referrals to the Fairbanks North Star Borough, FY2005-06

Minority youths in the Fairbanks North Star Borough are referred to the Alaska Division of Juvenile Justice (DJJ) for delinquent behavior at rates much higher than white youths.This report describes disproportionate minority contact with the Alaska juvenile justice system for youths referred to the Fairbanks office of the Division of Juvenile Justice during fiscal years 2005 and 2006 (July 1, 2004, to June 30, 2006). Possible sources of disproportionate minority contact are subsequently narrowed by examining the impact of race and ethnicity, gender, type of referral, and geography. By developing a detailed understanding of the scope of disproportionate minority contact, we become much better prepared to identify its causes and to develop evidence-based solutions.Alaska Division of Juvenile Justice. National Institute of Justice. Grant No. 2005-IJ-CX-0013. Office of Juvenile Justice and Delinquency Prevention. Grant No. 2001-JF-FX-0005.Index of Tables and Figures / Acknowledgements / Executive Summary / Quantitative Analysis of Disparities in Referrals / Sample and Data / Analysis / Results / Summary and Conclusions / Appendix A - Technical Notes on Relative Rate Indices / Appendix B - Technical Notes on Fisher's Exact Test / Appendix C - Technical Notes on Relative EB Rate Indices / Appendix D - Map

Topological modular forms and conformal nets

We describe the role conformal nets, a mathematical model for conformal field theory, could play in a geometric definition of the generalized cohomology theory TMF of topological modular forms. Inspired by work of Segal and Stolz-Teichner, we speculate that bundles of boundary conditions for the net of free fermions will be the basic underlying objects representing TMF-cohomology classes. String structures, which are the fundamental orientations for TMF-cohomology, can be encoded by defects between free fermions, and we construct the bundle of fermionic boundary conditions for the TMF-Euler class of a string vector bundle. We conjecture that the free fermion net exhibits an algebraic periodicity corresponding to the 576-fold cohomological periodicity of TMF; using a homotopy-theoretic invariant of invertible conformal nets, we establish a lower bound of 24 on this periodicity of the free fermions

Sexual Assault Study: Differences by Victim's Alcohol Use

Poster originally presented to the Anchorage Police Department and the 2004 Alaska Summit on Violence Against Women.This issue of Anchorage Community Indicators Series 2, "Sexual Assault Study," describes the spatial patterning and geographical concentration of 282 sexual assaults reported to the Anchorage Police Department in 2001 by victim's alcohol use.This research was supported by Grant No. 2000-RH-CX-K039 awarded by the Bureau of Justice Statistics and by a UAA Faculty Development Grant to the second author.Data Sexual assault densities by victim's alcohol use (maps) / Alcohol use (by suspect) / Victim injuries / Age (of victim; of suspect) / Race (of victim; of suspect) / Location (pick-up location; assault location) / Time to report / Relationshi