The arithmetic-geometric-harmonic-mean and related matrix inequalities

AbstractRecently, Sagae and Tanabe defined a geometric mean of positive definite matrices and proved the harmonic-geometric-arithmetic-mean inequality. Here, we give a reversal of these results

The Hall instability of thin weakly-ionized stratified Keplerian disks

The stratification-driven Hall instability in a weakly ionized polytropic plasma is investigated in the local approximation within an equilibrium Keplerian disk of a small aspect ratio. The leading order of the asymptotic expansions in the aspect ratio is applied to both equilibrium as well as the perturbation problems. The equilibrium disk with an embedded purely toroidal magnetic field is found to be stable to radial, and unstable to vertical short-wave perturbations. The marginal stability surface is found in the space of the local Hall and inverse plasma beta parameters, as well as the free parameter of the model which is related to the total current through the disk. To estimate the minimal values of the equilibrium magnetic field that leads to instability, the latter is constructed as a sum of a current free magnetic field and the simplest approximation for magnetic field created by a distributed electric current.Comment: 13 pages, 7 figure

Symptoms predicting psychosocial impairment in bulimia nervosa

Purpose The current study aimed to determine which particular eating disorder (ED) symptoms and related features, such as BMI and psychological distress, uniquely predict impairment in bulimia nervosa (BN). Methods Two hundred and twenty-two adults with BN completed questionnaires assessing ED symptoms, general psychological distress, and psychosocial impairment. Regression analyses were used to determine predictors which account for variance in impairment. Results Four variables emerged as significant predictors of psychosocial impairment: concerns with eating; concerns with weight and shape; dietary restraint; and general psychological distress. Conclusions Findings support previous work highlighting the importance of weight and shape concerns in determining ED-related impairment. Other ED symptoms, notably dietary restraint and concerns with eating, were also significant predictors as was psychological distress. Results suggest that cognitive aspects of EDs, in addition to psychological distress, may be more important determinants of impairment than behavioural symptoms, such as binge eating or purging

Kinetic approaches to particle acceleration at cosmic ray modified shocks

Kinetic approaches provide an effective description of the process of particle acceleration at shock fronts and allow to take into account the dynamical reaction of the accelerated particles as well as the amplification of the turbulent magnetic field as due to streaming instability. The latter does in turn affect the maximum achievable momentum and thereby the acceleration process itself, in a chain of causality which is typical of non-linear systems. Here we provide a technical description of two of these kinetic approaches and show that they basically lead to the same conclusions. In particular we discuss the effects of shock modification on the spectral shape of the accelerated particles, on the maximum momentum, on the thermodynamic properties of the background fluid and on the escaping and advected fluxes of accelerated particles.Comment: 22 pages, 7 figures, accepted for publication in MNRA

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

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

Evolution of initially localized perturbations in stratified ionized disks

A detailed solution of an initial value problem of a vertically localized initial perturbation in rotating magnetized vertically stratified disk is presented. The appropriate linearized MHD equations are solved by employing the WKB approximation and the results are verified numerically. The eigenfrequencies as well as eigenfunctions are explicitly obtained. It is demonstrated that the initial perturbation remains confined within the disk. It is further shown that thin enough disks are stable but as their thickness grows increasing number of unstable modes participate in the solution of the initial value problem. However it is demonstrated that due to the localization of the initial perturbation the growth time of the instability is significantly longer than the calculated inverse growth rate of the individual unstable eigenfunctions.Comment: 10 pages, 5 figures. Accepted for publication in MNRA

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