Mori dream spaces

We explore the circle of ideas connecting finite generation of the Cox ring, Mori dream spaces and invariant theory.     

Linear and non‐linear simulations of feedback control in plane Poiseuille flow

This paper examines the performance of optimal linear quadratic state and output feedback controllers in stabilizing two‐dimensional perturbations in a plane Poiseuille flow. The synthesis of the controllers is based on a linearized model of the flow using a new set of interpolating polynomials in the wall‐normal direction, which automatically satisfy the homogeneous Dirichlet and Neumann boundary conditions at the walls and eliminate spurious eigenvalues. The controllers are implemented into a non‐linear Navier–Stokes solver, which is modified to compute the evolution of the flow perturbations. Two cases are examined, one with small initial disturbances that do not violate the linearity assumptions and the other with much larger disturbances that trigger the non‐linear convection terms. For the smallest disturbances, the solver accurately reproduced the results of the linear simulations of open‐ and closed‐loop systems. The simulations for the larger disturbances without control showed a rapid initial growth but the flow soon reached a saturated state in agreement with previous findings in the literature. The large initial growth is a consequence of the non‐normal nature of the system dynamics. The state feedback and output feedback controllers were able to reduce significantly the perturbation energy. For the larger disturbances, the energy calculated from the state variables is well below the energy evaluated by direct integration of the velocity field. This is probably due to the non‐linear terms transferring energy to harmonics of the considered wavenumber, which are not sensed by the linear controller. Copyright © 2008 John Wiley & Sons, Ltd.     

Boundedness of pluricanonical maps of varieties of general type

Using the techniques of [20] and [10], we prove that certain log forms may be lifted from a divisor to the ambient variety. As a consequence of this result, following [22], we show that: For any positive integer n there exists an integer r _n such that if X is a smooth projective variety of general type and dimension n, then is birational for all r≥r _n .     

Stochastic resonance in chaotic dynamics

It is suggested that chaotic dynamical systems characterized by intermittent jumps between two preferred regions of phase space display an enhanced sensitivity to weak periodic forcings through a stochastic resonance-like mechanism. This possibility is illustrated by the study of the residence time distribution in two examples of bimodal chaos: the periodically forced Duffing oscillator and a 1-dimensional map showing intermittent behavior.     

Time-Course of Alterations in the Endocannabinoid System after Viral-Mediated Overexpression of α-Synuclein in the Rat Brain

Since the discovery of α-synuclein as the major component in Lewy bodies, research into this protein in the context of Parkinson’s disease pathology has been exponential. Cannabinoids are being investigated as potential therapies for Parkinson’s disease from numerous aspects, but still little is known about the links between the cannabinoid system and the pathogenic α-synuclein protein; understanding these links will be necessary if cannabinoid therapies are to reach the clinic in the future. Therefore, the aim of this study was to investigate the time-course of alterations in components of the endocannabinoid system after viral-mediated α-synuclein overexpression in the rat brain. Rats were given unilateral intranigral injections of AAV-GFP or AAV-α-synuclein and sacrificed 4, 8 and 12 weeks later for qRT-PCR and liquid chromatography–mass spectrometry analyses of the endocannabinoid system, in addition to histological visualization of α-synuclein expression along the nigrostriatal pathway. As anticipated, intranigral delivery of AAV-α-synuclein induced widespread overexpression of human α-synuclein in the nigrostriatal pathway, both at the mRNA level and the protein level. However, despite this profound α-synuclein overexpression, we detected no differences in CB₁ or CB₂ receptor expression in the nigrostriatal pathway; however, interestingly, there was a reduction in the expression of neuroinflammatory markers. Furthermore, there was a reduction in the levels of the endocannabinoid 2-AG and the related lipid immune mediator OEA at week 12 post-surgery, indicating that α-synuclein overexpression triggers dysregulation of the endocannabinoid system. Although this research does show that the endocannabinoid system is impacted by α-synuclein, further research is necessary to more comprehensively understand the link between the cannabinoid system and the α-synuclein aspect of Parkinson’s disease pathology in order for cannabinoid-based therapies to be feasible for the treatment of this disease in the coming years.