Linear selections for the metric projection

A characterization is given of those proximinal subspaces of a normed linear space whose (set-valued) metric projections admit linear selections. This characterization is applied in each of the classical Banach spaces C₀(T) and L_p (1 ? p ? ∞), resulting in an intrinsic characterization of those one-dimensional subspaces whose metric projections admit linear selections.     

Interfacial long traveling waves in a two‐layer fluid with variable depth

Long wave propagation in a two‐layer fluid with variable depth is studied for specific bottom configurations, which allow waves to propagate over large distances. Such configurations are found within the linear shallow‐water theory and determined by a family of solutions of the second‐order ordinary differential equation (ODE) with three arbitrary constants. These solutions can be used to approximate the true bottom bathymetry. All such solutions represent smooth bottom profiles between two different singular points. The first singular point corresponds to the point where the two‐layer flow transforms into a uniform one. In the vicinity of this point nonlinear shallow‐water theory is used and the wave breaking criterion, which corresponds to the gradient catastrophe is found. The second bifurcation point corresponds to an infinite increase in water depth, which contradicts the shallow‐water assumption. This point is eliminated by matching the “nonreflecting” bottom profile with a flat bottom. The wave transformation at the matching point is described by the second‐order Fredholm equation and its approximated solution is then obtained. The results extend the theory of internal waves in inhomogeneous stratified fluids actively developed by Prof. Roger Grimshaw, to the new solutions types.     

Forbidden arithmetic progressions in permutations of subsets of the integers

Permutations of the positive integers avoiding arithmetic progressions of length 5 were constructed in Davis et al. (1977), implying the existence of permutations of the integers avoiding arithmetic progressions of length 7. We construct a permutation of the integers avoiding arithmetic progressions of length 6. We also prove a lower bound of $\frac{1}{2}$ on the lower density of subsets of positive integers that can be permuted to avoid arithmetic progressions of length 4, sharpening the lower bound of $\frac{1}{3}$ from LeSaulnier and Vijay (2011).     

Counting acyclic digraphs by sources and sinks

We count labeled acyclic digraphs according to the number sources, sinks, and edges.     

Equality and Freedom as Uncertainty in Groups

In this paper, I investigate a connection between a common characterisation of freedom and how uncertainty is managed in a Bayesian hierarchical model. To do this, I consider a distributed factorization of a group’s optimization of free energy, in which each agent is attempting to align with the group and with its own model. I show how this can lead to equilibria for groups, defined by the capacity of the model being used, essentially how many different datasets it can handle. In particular, I show that there is a “sweet spot” in the capacity of a normal model in each agent’s decentralized optimization, and that this “sweet spot” corresponds to minimal free energy for the group. At the sweet spot, an agent can predict what the group will do and the group is not surprised by the agent. However, there is an asymmetry. A higher capacity model for an agent makes it harder for the individual to learn, as there are more parameters. Simultaneously, a higher capacity model for the group, implemented as a higher capacity model for each member agent, makes it easier for a group to integrate a new member. To optimize for a group of agents then requires one to make a trade-off in capacity, as each individual agent seeks to decrease capacity, but there is pressure from the group to increase capacity of all members. This pressure exists because as individual agent’s capacities are reduced, so too are their abilities to model other agents, and thereby to establish pro-social behavioural patterns. I then consider a basic two-level (dual process) Bayesian model of social reasoning and a set of three parameters of capacity that are required to implement such a model. Considering these three capacities as dependent elements in a free energy minimization for a group leads to a “sweet surface” in a three-dimensional space defining the triplet of parameters that each agent must use should they hope to minimize free energy as a group. Finally, I relate these three parameters to three notions of freedom and equality in human social organization, and postulate a correspondence between freedom and model capacity. That is, models with higher capacity, have more freedom as they can interact with more datasets.     

Muscle spindles are concentrated in the superior vocalis subcompartment of the human thyroarytenoid muscle

It is hypothesized that different parts of the thyroarytenoid muscle (TA) are functionally specialized. Specifically, the TA is divided into a lateral muscularis compartment and a medial vocalis compartment. This study examined the distribution of muscle spindles throughout the human TA as an indicator of these functional differences. Histological cross-sections from the anterior, middle, and posterior regions of five human membranous vocal folds were examined for the number and location of muscle spindles. There was an average of 6.1 muscle spindles in sections from each region with no significant variation between the different regions (p < .05). However, in sections from all three regions, the muscle spindles were always found to be concentrated in the superior medial quadrant of the TA (mean 85.9%, p < .01). The inferior medial, superior lateral, and inferior lateral quadrants of the TA contained 11.96%, 2.17%, and 0%, respectively, of the total muscle spindles. Within the superior medial quadrant, most of the muscle spindles were localized in the most superficial part of the muscle.The results of this study demonstrate that the majority of TA muscle spindlesare concentrated in its superior medial quadrant, an area we have termed the superior vocalis subcompartment (SC. This finding suggests that the superior vocalis SC is functionally distinct from the remainder of the TA. It is hypothesized that tension in the superior vocalis SC can be controlled independently from the remainder of the TA, and this capability is used to effect the biomechanics of vocal fold vibration during phonation.     

Fluorescence Detection of the Ornamental Fish Cardinal Tetra (Paracheirodon axelrodi)

The fluorescence spectra of the tropical fish, Cardinal Tetra ( Paracheirodon axelrodi ), originating in the Amazon region of Brazil, were determined. These spectra were then treated using factor analysis, generating two contributing spectra and separating out the noise. Time-resolved fluorescence results indicated that the fluorescent system in the epidermis undergoes excited state reaction. Excited state proton transfer is suggested as being present. Both intentionally stressed and nonstressed individuals were used and some small differences were noted in the contributions of the two calculated contributing spectra to the experimental spectra, presumably as a function of stress. The results are compared with those obtained by the standard determination of cortisol level using the whole body extraction method and it is suggested that the method could be tested as an improved, nondestructive way to determine stress in this species, which is a necessary step in the development of best management practices of methods for storage and transport of the fish.