Characterizing normal crossing hypersurfaces

The objective of this article is to give an effective algebraic characterization of normal crossing hypersurfaces in complex manifolds. It is shown that a hypersurface has normal crossings if and only if it is a free divisor, has a radical Jacobian ideal and a smooth normalization. Using K. Saito's theory of free divisors, also a characterization in terms of logarithmic differential forms and vector fields is found and and finally another one in terms of the logarithmic residue using recent results of M. Granger and M. Schulze.Comment: v2: typos fixed, final version to appear in Math. Ann.; 24 pages, 2 figure

Likelihood Geometry

We study the critical points of monomial functions over an algebraic subset of the probability simplex. The number of critical points on the Zariski closure is a topological invariant of that embedded projective variety, known as its maximum likelihood degree. We present an introduction to this theory and its statistical motivations. Many favorite objects from combinatorial algebraic geometry are featured: toric varieties, A-discriminants, hyperplane arrangements, Grassmannians, and determinantal varieties. Several new results are included, especially on the likelihood correspondence and its bidegree. These notes were written for the second author's lectures at the CIME-CIRM summer course on Combinatorial Algebraic Geometry at Levico Terme in June 2013.Comment: 45 pages; minor changes and addition

A pragmatic effectiveness study of 10-session cognitive behavioural therapy (CBT-T) for eating disorders: Targeting barriers to treatment provision

Objective Ten‐session cognitive behavioural therapy (CBT‐T) for transdiagnostic eating disorders targets several barriers to treatment, including cost, therapist expertise, and lengthy wait lists. Method We used a case series design to investigate the effectiveness of CBT‐T delivered by trainee psychologists in a postgraduate training clinic. Participants were randomly allocated to commence treatment either immediately or after a 4‐week waitlist period. CBT‐T was delivered to 52 patients, by six different trainees under supervision. Measures of eating disorder cognitions and behaviours, quality of life, and general psychopathology were examined in completer and intention‐to‐treat analyses using multilevel modelling. Last observation carried forward was applied for abstinence, remission, and good outcome analyses to aid comparison with prior studies. Results Significant improvements, associated with medium to large effect sizes, were found for eating disorder cognitions, behaviours quality of life, and negative affect from baseline to posttreatment, and at 1‐ and 3‐month follow‐up. Attrition (38.5%) was comparable with other treatment studies. Conclusion Results provide evidence for the effectiveness of CBT‐T delivered by trainee psychologists for transdiagnostic eating disorder patients, thus tackling some important barriers for treatment. Longer follow‐up, randomised controlled trial designs, and moderator analyses will provide more robust evidence about which patients do best with a shorter therapy

Exact Expressions for the Critical Mach Numbers in the Two-Fluid Model of Cosmic-Ray Modified Shocks

The acceleration of relativistic particles due to repeated scattering across a shock wave remains the most attractive model for the production of energetic cosmic rays. This process has been analyzed extensively during the past two decades using the ``two-fluid'' model of diffusive shock acceleration. It is well known that 1, 2, or 3 distinct solutions for the flow structure can be found depending on the upstream parameters. The precise nature of the critical conditions delineating the number and character of shock transitions has remained unclear, mainly due to the inappropriate choice of parameters used in the determination of the upstream boundary conditions. We derive the exact critical conditions by reformulating the upstream boundary conditions in terms of two individual Mach numbers defined with respect to the cosmic-ray and gas sound speeds, respectively. The gas and cosmic-ray adiabatic indices are assumed to remain constant throughout the flow, although they may have arbitrary, independent values. Our results provide for the first time a complete, analytical classification of the parameter space of shock transitions in the two-fluid model. When multiple solutions are possible, we propose using the associated entropy distributions as a means for indentifying the most stable configuration.Comment: Accepted for publication in ApJ; corrected a few typos; added journal re