Toroidal b-divisors and Monge–Ampère measures

We generalize the intersection theory of nef toric (Weil) b-divisors on smooth and complete toric varieties to the case of nef b-divisors on complete varieties which are toroidal with respect to a snc divisor. As a key ingredient we show the existence of a limit measure, supported on a balanced rational conical polyhedral space attached to the toroidal embedding, which arises as a limit of discrete measures defined via tropical intersection theory on the polyhedral space. We prove that the intersection theory of nef Cartier b-divisors can be extended continuously to nef toroidal Weil b-divisors and that their degree can be computed as an integral with respect to this limit measure. As an application, we show that a Hilbert–Samuel type formula holds for big and nef toroidal Weil b-divisors.

INTERMEDIATE ASSOUAD-LIKE DIMENSIONS FOR MEASURES

The upper and lower Assouad dimensions of a metric space are local variants of the box dimensions of the space and provide quantitative information about the ‘thickest’ and ‘thinnest’ parts of the set. Less extreme versions of these dimensions for sets have been introduced, including the upper and lower quasi-Assouad dimensions, [Formula: see text]-Assouad spectrum, and [Formula: see text]-dimensions. In this paper, we study the analogue of the upper and lower [Formula: see text]-dimensions for measures. We give general properties of such dimensions, as well as more specific results for self-similar measures satisfying various separation properties and discrete measures.

A Theory of Entanglement

This article presents the basis of a theory of entanglement. We begin with a classical theory of entangled discrete measures. Then, we treat quantum mechanics and discuss the statistics of bounded operators on a Hilbert space in terms of context coefficients. Finally, we combine both topics to develop a general theory of entanglement for quantum states. A measure of entanglement called the entanglement number is introduced. Although this number is related to entanglement robustness, its motivation is not the same and there are some differences. The present article only involves bipartite systems and we leave the study of multipartite systems for later work.Quanta 2020; 9: 7–15.

Motor axon excitability measures in the rat tail are the same awake or anaesthetized using sodium pentobarbital

BackgroundNerve excitability tests in sciatic motor axons are sensitive to anaesthetic choice. Results using ketamine/xylazine (KX) are different from those using sodium pentobarbital (SP). It is not clear which results are most similar to the awake condition, though results using SP appear more similar to human results.MethodsNerve excitability in tail motor axons was tested in 8 adult female rats with a chronic sacral spinal cord injury. These animals have no behavioural response to electrical stimulation of the tail and were tested awake and then anaesthetized using SP.ResultsThe nerve excitability test results in the awake condition were indistinguishable from the results when the same rats were anaesthetized with sodium pentobarbital. Summary plots of the test results overlap within the boundaries of the standard error and paired t-tests on the 42 discrete measures generated by nerve excitability testing yielded no significant differences (after Bonferroni correction for multiple comparisons).ConclusionsNerve excitability test results in rat motor axons are the same whether the animals are awake or anesthetized using sodium pentobarbital.