The structure and density of kk-product-free sets in the free semigroup

The free semigroup $\mathcal{F}$ over a finite alphabet $\mathcal{A}$ is the set of all finite words with letters from $\mathcal{A}$ equipped with the operation of concatenation. A subset $S$ of $\mathcal{F}$ is $k$-product-free if no element of $S$ can be obtained by concatenating $k$ words from $S$, and strongly $k$-product-free if no element of $S$ is a (non-trivial) concatenation of at most $k$ words from $S$. We prove that a $k$-product-free subset of $\mathcal{F}$ has upper Banach density at most $1/\rho(k)$, where $\rho(k) = \min\{\ell \colon \ell \nmid k - 1\}$. We also determine the structure of the extremal $k$-product-free subsets for all $k \notin \{3, 5, 7, 13\}$; a special case of this proves a conjecture of Leader, Letzter, Narayanan, and Walters. We further determine the structure of all strongly $k$-product-free sets with maximum density. Finally, we prove that $k$-product-free subsets of the free group have upper Banach density at most $1/\rho(k)$, which confirms a conjecture of Ortega, Ru\'{e}, and Serra.Comment: 31 pages, added density results for the free grou

Non-Homotopic Drawings of Multigraphs

A multigraph drawn in the plane is non-homotopic if no two edges connecting the same pair of vertices can be continuously deformed into each other without passing through a vertex, and is $k$-crossing if every pair of edges (self-)intersects at most $k$ times. We prove that the number of edges in an $n$-vertex non-homotopic $k$-crossing multigraph is at most $6^{13 n (k + 1)}$, which is a big improvement over previous upper bounds. We also study this problem in the setting of monotone drawings where every edge is an x-monotone curve. We show that the number of edges, $m$, in such a drawing is at most $2 \binom{2n}{k + 1}$ and the number of crossings is $\Omega\bigl(\frac{m^{2 + 1/k}}{n^{1 + 1/k}}\bigr)$. For fixed $k$ these bounds are both best possible up to a constant multiplicative factor.Comment: 19 page

Defective coloring of hypergraphs

We prove that the vertices of every (Formula presented.) -uniform hypergraph with maximum degree (Formula presented.) may be colored with (Formula presented.) colors such that each vertex is in at most (Formula presented.) monochromatic edges. This result, which is best possible up to the value of the constant (Formula presented.), generalizes the classical result of Erdős and Lovász who proved the (Formula presented.) case

Reconstructing a point set from a random subset of its pairwise distances

Let $V$ be a set of $n$ points on the real line. Suppose that each pairwise distance is known independently with probability $p$. How much of $V$ can be reconstructed up to isometry? We prove that $p = (\log n)/n$ is a sharp threshold for reconstructing all of $V$ which improves a result of Benjamini and Tzalik. This follows from a hitting time result for the random process where the pairwise distances are revealed one-by-one uniformly at random. We also show that $1/n$ is a weak threshold for reconstructing a linear proportion of $V$.Comment: 13 page

Flashes and Rainbows in Tournaments

Colour the edges of the complete graph with vertex set $\{1, 2, \dotsc, n\}$ with an arbitrary number of colours. What is the smallest integer $f(l,k)$ such that if $n > f(l,k)$ then there must exist a monotone monochromatic path of length $l$ or a monotone rainbow path of length $k$? Lefmann, R\"{o}dl, and Thomas conjectured in 1992 that $f(l, k) = l^{k - 1}$ and proved this for $l \ge (3 k)^{2 k}$. We prove the conjecture for $l \geq k^4 (\log k)^{1 + o(1)}$ and establish the general upper bound $f(l, k) \leq k (\log k)^{1 + o(1)} \cdot l^{k - 1}$. This reduces the gap between the best lower and upper bounds from exponential to polynomial in $k$. We also generalise some of these results to the tournament setting.Comment: 14 page

Mentoring In The Clinical Setting To Improve Student Decision-Making Competence

Introduction:  The physician-intern relationship can be difficult to develop. A new chiropractic intern in a teaching clinic undergoes a major transition from classroom to clinical practice and must learn to turn classroom knowledge into clinical application. The ability to start formulating clinical techniques and apply them on a patient is daunting. Developing a mentor relationship is difficult to do in a patient-care setting, but it can be done. Mentoring is a process of exchanging skills and values between 2 individuals with the goal of increasing the knowledge base and clinical skills of the intern. With this in mind, our group created short-duration small-group mentoring classes offered at different times on multiple days, permitting interns to have numerous opportunities to view procedure and ask questions about the topic of that day's presentation.Methods:  This project spanned a period of 3 months, during which time 6 clinicians were in charge of educating approximately 50 students. The mentorship model was developed so that in addition to the clinicians' regular duties of supervising patient care with interns, there would also be 3 15-minute sessions per week presenting a topic of a clinician's choice pertinent to the intern learning experience.Results:  209 evaluations were turned in, with 5 students not completing the evaluation. Students overwhelmingly believed that these sessions were beneficial to their learning and provided them with the opportunity to ask questions in regard to the topics.Conclusion:  Students agreed that these small group mentoring sessions provided them with more information than they previously had learned in the classroom. They thought that the sessions gave them enough information to be motivated to use the knowledge they learned from the session to make decisions on the topic when faced with a patient with a similar problem.  This is a small survey sample that will need further review and trials to determine if it will provide the necessary feedback to help improve small group presentations. It will also need to be spread to other clinical settings to determine if it is beneficial for this style of small groups to aid other learners and to evaluate its helpfulness to interns in putting clinical information and evaluations together for practice. Follow-up studies could also include evaluating students at a later date to determine if the students are using the information that they are learning from the sessions

The actor’s insight: Actors have comparable interoception but better metacognition than nonactors

Both accurately sensing our own bodily signals and knowing whether we have accurately sensed them may contribute to a successful emotional life, but there is little evidence on whether these physiological perceptual and metacognitive abilities systematically differ between people. Here, we examined whether actors, who receive substantial training in the production, awareness, and control of emotion, and nonactor controls differed in interoceptive ability (the perception of internal bodily signals) and/or metacognition about interoceptive accuracy (awareness of that perception), and explored potential sources of individual differences in and consequences of these abilities including correlational relationships with state and trait anxiety, proxies for acting ability, and the amount of acting training. Participants performed a heartbeat detection task in which they judged whether tones were played synchronously or delayed relative to their heartbeats, and then rated their metacognitive confidence in that judgment. Cardiac interoceptive accuracy and metacognitive awareness of interoceptive accuracy were independent, and while actors' and controls' interoceptive accuracy was not significantly different, actors had consistently superior metacognitive awareness of interoception. Exploratory analyses additionally suggest that this metacognitive ability may be correlated with measures of acting ability, but not the duration of acting training. Interoceptive accuracy and metacognitive insight into that accuracy appear to be separate abilities, and while actors may be no more accurate in reading their bodies, their metacognitive insight means they know better when they're accurate and when they're not. (PsycInfo Database Record (c) 2022 APA, all rights reserved)

RINGs, DUBs and Abnormal Brain Growth—Histone H2A Ubiquitination in Brain Development and Disease

During mammalian neurodevelopment, signaling pathways converge upon transcription factors (TFs) to establish appropriate gene expression programmes leading to the production of distinct neural and glial cell types. This process is partially regulated by the dynamic modulation of chromatin states by epigenetic systems, including the polycomb group (PcG) family of co-repressors. PcG proteins form multi-subunit assemblies that sub-divide into distinct, yet functionally related families. Polycomb repressive complexes 1 and 2 (PRC1 and 2) modify the chemical properties of chromatin by covalently modifying histone tails via H2A ubiquitination (H2AK119ub1) and H3 methylation, respectively. In contrast to the PRCs, the Polycomb repressive deubiquitinase (PR-DUB) complex removes H2AK119ub1 from chromatin through the action of the C-terminal hydrolase BAP1. Genetic screening has identified several PcG mutations that are causally associated with a range of congenital neuropathologies associated with both localised and/or systemic growth abnormalities. As PRC1 and PR-DUB hold opposing functions to control H2AK119ub1 levels across the genome, it is plausible that such neurodevelopmental disorders arise through a common mechanism. In this review, we will focus on advancements regarding the composition and opposing molecular functions of mammalian PRC1 and PR-DUB, and explore how their dysfunction contributes to the emergence of neurodevelopmental disorders

Can Rationing Through Inconvenience Be Ethical?

In this article, we provide a comprehensive analysis and a normative assessment of rationing through inconvenience as a form of rationing. By “rationing through inconvenience” in the health sphere, we refer to a non-financial burden (the inconvenience) that is either intended to cause or has the effect of causing patients or clinicians to choose an option for health‐related consumption that is preferred by the health system for its fairness, efficiency, or other distributive desiderata beyond assisting the immediate patient. We argue that under certain conditions, rationing through inconvenience may turn out to serve as a legitimate and, compared to direct rationing, even a preferable tool for rationing; we propose a research agenda to identify more precisely when that might be the case and when, alternatively, rationing through inconvenience remains ethically undesirable. After defining and illustrating rationing through inconvenience, we turn to its moral advantages and disadvantages over other rationing methods.We take it as a starting assumption that rationing, understood as scarce‐resource prioritization, is inevitable and, in a society that has goals beyond optimizing health care for individual patients—such as improving societal health care, education, or overall welfare—prudent and fair