Cohomogeneity one manifolds and selfmaps of nontrivial degree

We construct natural selfmaps of compact cohomgeneity one manifolds with finite Weyl group and compute their degrees and Lefschetz numbers. On manifolds with simple cohomology rings this yields in certain cases relations between the order of the Weyl group and the Euler characteristic of a principal orbit. We apply our construction to the compact Lie group SU(3) where we extend identity and transposition to an infinite family of selfmaps of every odd degree. The compositions of these selfmaps with the power maps realize all possible degrees of selfmaps of SU(3).Comment: v2, v3: minor improvement

Substantia nigra activity level predicts trial-to-trial adjustments in cognitive control

Effective adaptation to the demands of a changing environment requires flexible cognitive control. The medial and the lateral frontal cortices are involved in such control processes, putatively in close interplay with the BG. In particular, dopaminergic projections from the midbrain (i.e., from the substantia nigra [SN] and the ventral tegmental area) have been proposed to play a pivotal role in modulating the activity in these areas for cognitive control purposes. In that dopaminergic involvement has been strongly implicated in reinforcement learning, these ideas suggest functional links between reinforcement learning, where the outcome of actions shapes behavior over time, and cognitive control in a more general context, where no direct reward is involved. Here, we provide evidence from functional MRI in humans that activity in the SN predicts systematic subsequent trial-to-trial RT prolongations that are thought to reflect cognitive control in a stop-signal paradigm. In particular, variations in the activity level of the SN in one trial predicted the degree of RT prolongation on the subsequent trial, consistent with a modulating output signal from the SN being involved in enhancing cognitive control. This link between SN activity and subsequent behavioral adjustments lends support to theoretical accounts that propose dopaminergic control signals that shape behavior both in the presence and in the absence of direct reward. This SN-based modulatory mechanism is presumably mediated via a wider network that determines response speed in this task, including frontal and parietal control regions, along with the BG and the associated subthalamic nucleus

The general classical solution of the superparticle

The theory of vectors and spinors in 9+1 dimensional spacetime is introduced in a completely octonionic formalism based on an octonionic representation of the Clifford algebra \Cl(9,1). The general solution of the classical equations of motion of the CBS superparticle is given to all orders of the Grassmann hierarchy. A spinor and a vector are combined into a $3 \times 3$ Grassmann, octonionic, Jordan matrix in order to construct a superspace variable to describe the superparticle. The combined Lorentz and supersymmetry transformations of the fermionic and bosonic variables are expressed in terms of Jordan products.Comment: 11 pages, REVTe

Cutting the same fraction of several measures

We study some measure partition problems: Cut the same positive fraction of $d+1$ measures in $\mathbb R^d$ with a hyperplane or find a convex subset of $\mathbb R^d$ on which $d+1$ given measures have the same prescribed value. For both problems positive answers are given under some additional assumptions.Comment: 7 pages 2 figure

Classification of integrable Weingarten surfaces possessing an sl(2)-valued zero curvature representation

In this paper we classify Weingarten surfaces integrable in the sense of soliton theory. The criterion is that the associated Gauss equation possesses an sl(2)-valued zero curvature representation with a nonremovable parameter. Under certain restrictions on the jet order, the answer is given by a third order ordinary differential equation to govern the functional dependence of the principal curvatures. Employing the scaling and translation (offsetting) symmetry, we give a general solution of the governing equation in terms of elliptic integrals. We show that the instances when the elliptic integrals degenerate to elementary functions were known to nineteenth century geometers. Finally, we characterize the associated normal congruences

Constraint and gauge shocks in one-dimensional numerical relativity

We study how different types of blow-ups can occur in systems of hyperbolic evolution equations of the type found in general relativity. In particular, we discuss two independent criteria that can be used to determine when such blow-ups can be expected. One criteria is related with the so-called geometric blow-up leading to gradient catastrophes, while the other is based upon the ODE-mechanism leading to blow-ups within finite time. We show how both mechanisms work in the case of a simple one-dimensional wave equation with a dynamic wave speed and sources, and later explore how those blow-ups can appear in one-dimensional numerical relativity. In the latter case we recover the well known ``gauge shocks'' associated with Bona-Masso type slicing conditions. However, a crucial result of this study has been the identification of a second family of blow-ups associated with the way in which the constraints have been used to construct a hyperbolic formulation. We call these blow-ups ``constraint shocks'' and show that they are formulation specific, and that choices can be made to eliminate them or at least make them less severe.Comment: 19 pages, 8 figures and 1 table, revised version including several amendments suggested by the refere

Single ion implantation for single donor devices using Geiger mode detectors

Electronic devices that are designed to use the properties of single atoms such as donors or defects have become a reality with recent demonstrations of donor spectroscopy, single photon emission sources, and magnetic imaging using defect centers in diamond. Improving single ion detector sensitivity is linked to improving control over the straggle of the ion as well as providing more flexibility in lay-out integration with the active region of the single donor device construction zone by allowing ion sensing at potentially greater distances. Using a remotely located passively gated single ion Geiger mode avalanche diode (SIGMA) detector we have demonstrated 100% detection efficiency at a distance of >75 um from the center of the collecting junction. This detection efficiency is achieved with sensitivity to ~600 or fewer electron-hole pairs produced by the implanted ion. Ion detectors with this sensitivity and integrated with a thin dielectric, for example 5 nm gate oxide, using low energy Sb implantation would have an end of range straggle of <2.5 nm. Significant reduction in false count probability is achieved by modifying the ion beam set-up to allow for cryogenic operation of the SIGMA detector. Using a detection window of 230 ns at 1 Hz, the probability of a false count was measured as 1E-1 and 1E-4 for operation temperatures of 300K and 77K, respectively. Low temperature operation and reduced false, dark, counts are critical to achieving high confidence in single ion arrival. For the device performance in this work, the confidence is calculated as a probability of >98% for counting one and only one ion for a false count probability of 1E-4 at an average ion number per gated window of 0.015.Comment: 10 pages, 5 figures, submitted to Nanotechnolog