Morse-Bott functions on orthogonal groups

We make a detailed study of various (quadratic and linear) Morse-Bott trace functions on the orthogonal groups $O(n)$. We describe the critical loci of the quadratic trace function Tr$(AXBX^T)$ and determine their indices via perfect fillings of tables associated with the multiplicities of the eigenvalues of $A$ and $B$. We give a simplified treatment of T. Frankel's analysis of the linear trace function on $SO(n)$, as well as a combinatorial explanation of the relationship between the mod $2$ Betti numbers of $SO(n)$ and those of the Grassmannians $\mathbb{G}(2k,n)$ obtained from this analysis. We review the basic notions of Morse-Bott cohomology in a simple case where the set of critical points has two connected components. We then use these results to give a new Morse-theoretic computation of the mod $2$ Betti numbers of $SO(n)$.Comment: 28 page

The Case for Improving the Robinson Formulas

Auroral particle precipitation is the main source of ionization on the nightside, making it a critical factor in geospace physics. This magnetosphere‐ionosphere linkage directly contributes to, even controls, the nonlinear feedback within this coupled system. One study has dominated our understanding of this connection, presenting a pair of equations relating auroral particle precipitation to ionospheric Pedersen and Hall conductance, the famous Robinson formulas. This Commentary examines the history of the development and usage of the Robinson formulas and the recent studies exploring corrections and expansions to it. The conclusion is that more work needs to be done; the space physics research community should take up the task to develop improvements and enhancements to better quantify the connection of auroral precipitation to ionospheric conductance.Key PointsThe development and usage of the famous Robinson formulas, relating auroral precipitation to ionospheric conductance, is recountedRecent approaches to address shortcomings of these formulas are reviewed and discussed, noting shortcomings in some of these studiesThe space physics research community is urged to take on the action of improving the connection between precipitation and conductancePeer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/163473/2/jgra56025.pdfhttp://deepblue.lib.umich.edu/bitstream/2027.42/163473/1/jgra56025_am.pd

Pulses of Notch activation synchronise oscillating somite cells and entrain the zebrafish segmentation clock

Formation of somites, the rudiments of vertebrate body segments, is an oscillatory process governed by a gene-expression oscillator, the segmentation clock. This operates in each cell of the presomitic mesoderm (PSM), but the individual cells drift out of synchrony when Delta/Notch signalling fails, causing gross anatomical defects. We and others have suggested that this is because synchrony is maintained by pulses of Notch activation, delivered cyclically by each cell to its neighbours, that serve to adjust or reset the phase of the intracellular oscillator. This, however, has never been proved. Here, we provide direct experimental evidence, using zebrafish containing a heat-shock-driven transgene that lets us deliver artificial pulses of expression of the Notch ligand DeltaC. In DeltaC-defective embryos, in which endogenous Notch signalling fails, the artificial pulses restore synchrony, thereby rescuing somite formation. The spacing of segment boundaries produced by repetitive heat-shocking varies according to the time interval between one heat-shock and the next. The induced synchrony is manifest both morphologically and at the level of the oscillations of her1, a core component of the intracellular oscillator. Thus, entrainment of intracellular clocks by periodic activation of the Notch pathway is indeed the mechanism maintaining cell synchrony during somitogenesis

Brieskorn manifolds as contact branched covers of spheres

We show that Brieskorn manifolds with their standard contact structures are contact branched coverings of spheres. This covering maps a contact open book decomposition of the Brieskorn manifold onto a Milnor open book of the sphere.Comment: 8 pages, 1 figur

Trajectories of charged particles trapped in Earth's magnetic field

I outline the theory of relativistic charged-particle motion in the magnetosphere in a way suitable for undergraduate courses. I discuss particle and guiding center motion, derive the three adiabatic invariants associated with them, and present particle trajectories in a dipolar field. I provide twelve computational exercises that can be used as classroom assignments or for self-study. Two of the exercises, drift-shell bifurcation and Speiser orbits, are adapted from active magnetospheric research. The Python code provided in the supplement can be used to replicate the trajectories and can be easily extended for different field geometries.Comment: 10 pages, 7 figures. Submitted to American Journal of Physic

Corrections in the CHARMM36 parametrization of chloride interactions with proteins, lipids, and alkali cations, and extension to other halide anions

The nonpolarizable CHARMM force field is one of the most widely used energy functions for all-atom biomolecular simulations. Chloride is the only halide ion included in the latest version, CHARMM36m, and is used widely in simulation studies, often as an electrolyte ion but also as the biological substrate of transport proteins and enzymes. Here, we find that existing parameters systematically underestimate the interaction of C

Robust Rolling Horizon Optimisation Model for Offshore Wind Farm Installation Logistics

Our approach can be considered as both proactive and reactive, since uncertainty is considered both in creating the initial schedule and the schedule can be updated in real-time

Capital Fixity and Mobility in Response to the 2008-09 Crisis: Variegated Neoliberalism in Mexico and Turkey

The article examines the 2008-9 crisis responses in Mexico and Turkey as examples of variegated neoliberalism. The simultaneous interests of corporations and banks relative to the national fixing of capital and their mobility in the form of global investment heavily influenced each state authority’s policy responses to the crisis at the expense of the interests of the poor, workers, and peasantry. Rather than pitching this as either evidence of persistent national differentiation or some Keynesian state resurgence, we argue from a historical materialist geographical framework that the responses of capital and state authorities in Mexico and Turkey actively constitute and reconstitute the global parameters of market regulatory design and neoliberal class rule through each state’s distinct domestic policy formation and crisis management processes. While differing in specific content the form of Mexico and Turkey’s state responses to the crisis ensured continuity in their foregoing neoliberal strategies of development and capital accumulation, most notably in the continued oppression of workers. That is, the prevailing strategy of accumulation continues to be variegated neoliberalism