Hypergraphs do jump

We say that $\alpha\in [0,1)$ is a jump for an integer $r\geq 2$ if there exists $c(\alpha)>0$ such that for all $\epsilon >0$ and all $t\geq 1$ any $r$-graph with $n\geq n_0(\alpha,\epsilon,t)$ vertices and density at least $\alpha+\epsilon$ contains a subgraph on $t$ vertices of density at least $\alpha+c$. The Erd\H os--Stone--Simonovits theorem implies that for $r=2$ every $\alpha\in [0,1)$ is a jump. Erd\H os showed that for all $r\geq 3$, every $\alpha\in [0,r!/r^r)$ is a jump. Moreover he made his famous "jumping constant conjecture" that for all $r\geq 3$, every $\alpha \in [0,1)$ is a jump. Frankl and R\"odl disproved this conjecture by giving a sequence of values of non-jumps for all $r\geq 3$. We use Razborov's flag algebra method to show that jumps exist for $r=3$ in the interval $[2/9,1)$. These are the first examples of jumps for any $r\geq 3$ in the interval $[r!/r^r,1)$. To be precise we show that for $r=3$ every $\alpha \in [0.2299,0.2316)$ is a jump. We also give an improved upper bound for the Tur\'an density of $K_4^-=\{123,124,134\}$: $\pi(K_4^-)\leq 0.2871$. This in turn implies that for $r=3$ every $\alpha \in [0.2871,8/27)$ is a jump.Comment: 11 pages, 1 figure, 42 page appendix of C++ code. Revised version including new Corollary 2.3 thanks to an observation of Dhruv Mubay

A solution to the 2/3 conjecture

We prove a vertex domination conjecture of Erd\H os, Faudree, Gould, Gy\'arf\'as, Rousseau, and Schelp, that for every n-vertex complete graph with edges coloured using three colours there exists a set of at most three vertices which have at least 2n/3 neighbours in one of the colours. Our proof makes extensive use of the ideas presented in "A New Bound for the 2/3 Conjecture" by Kr\'al', Liu, Sereni, Whalen, and Yilma.Comment: 12 pages, 4 figures, 2 data files and proof checking code. Revised version to appear in SIAM Journal on Discrete Mathematic

Two Poems

Poetry by Preetinder Rahi

Hong-Ou-Mandel-like two-droplet correlations

We present a numerical study of two-droplet pair correlations for in-phase droplets walking on a vibrating bath. Two such walkers are launched towards a common origin. As they approach, their carrier waves may overlap and the droplets have a non-zero probability of forming a two-droplet bound state. The likelihood of such pairing is quantified by measuring the probability of finding the droplets in a bound state at late times. Three generic types of two-droplet correlations are observed: promenading, orbiting and chasing pair of walkers. For certain parameters, the droplets may become correlated for certain initial path differences and remain uncorrelated for others, while in other cases the droplets may never produce droplet pairs. These observations pave the way for further studies of strongly correlated many-droplet behaviors in the hydrodynamical quantum analogs of bouncing and walking droplets.Comment: 8 pages, 5 figure

Drumming To Communicate Emotion: Dual-Brain Imaging Informs An Intervention In A Carceral Setting

Drumming is an ancient nonverbal communication modality for expression of emotion. However, there has been limited exploration of its possible applications in clinical settings. Further, the underlying neural systems engaged during live communication with drumming have not been identified. We investigated the neural response to live, natural communication of emotion via drumming using a novel dual-brain neuroimaging paradigm to discover its unique neurophysiological mechanisms related to drum behavior and cross-brain coherence, and as compared to talking. We then investigated the application of a drumming intervention in an incarcerated, halfway house population to characterize intervention feasibility, elucidate the phenomenology of social and emotional effects of group drumming, and identify its possible benefits for treatment engagement and community reintegration. For neural response investigation, hemodynamic signals were acquired using whole-head functional near infrared spectroscopy. Dyads of 36 subjects participated in two conditions, drumming and talking, alternating between “sending” (drumming or talking to partner) and “receiving” (listening to partner) in response to emotionally salient images from the International Affective Picture System. Results indicated that increased frequency and amplitude of drum strikes was behaviorally correlated with higher arousal and lower valence measures, and neurally correlated with temporoparietal junction (TPJ) activation in the listener. Contrast comparisons of drumming greater than talking also revealed neural activity in right TPJ. For the interventional investigation, a group drumming program was implemented once a week for eight weeks for incarcerated participants in a halfway house. Twenty-eight participants were randomized to either the drum group or treatment as usual. Interviews and a focus group were conducted to assess the experienced benefits of the group drumming intervention, and halfway house retention rates were compared across groups. Retention rate was significantly higher in the drum group than in the treatment as usual group. Qualitative analysis elicited three themes: group drumming 1) functions therapeutically as a method of coping with difficulty, 2) offers opportunity for connection through building relationship and experiencing communion in a setting where isolation is the norm, and 3) provides an environment for personal growth, particularly toward re-humanization and self-empowerment. Neural findings suggest that emotional content communicated by drumming engages right TPJ mechanisms in an emotionally and behaviorally sensitive fashion; interventional findings suggest significant therapeutic potential in social and emotional domains that can have quantifiable impact on recovery process. Together, findings suggest that drumming may provide access to neural mechanisms with known sensitivity to social and emotional conditions that facilitates therapeutic aims. Informed by this research, drumming may provide novel, effective clinical approaches for treating social-emotional psychopathology

Teacher-Pupil Perceptions and their Interactive Behaviour in the Classroom

This study attempts to find out if there were significant differences between the perception of Democratic and Authoritarian teachers towards pupils who were academically good and weak and vice-versa and whether these differences affect their interactions in the classroom. The subjects of this study consisted of standard six teachers who were perceived by their pupils as Democratic and Authoritarian teachers respectively; and pupils who were perceived as academically good or weak by their teachers; from twenty schools in the Federal Territory, Malaysia. Both teachers and pupils were randomly selected. The grouping of teachers into Democratic and Authoritarian types was done by using questionnaires modified from statements prepared by Flanders. The questionnaires were answered by pupils. The perceived Democratic and Authoritarian teachers were interviewed using a prepared questionnaire on the perception of their pupils using the bi-polar construct of academically good and academically weak . Ten Democratic and ten Authoritarian teachers were chosen. Four hundred pupils were also chosen basing on their teachers 'perception of their academic performance Of these, two hundred were clas sified as academically good and two hundred as academically weak. The observations of the classroom interactions were done through time sampling using regular classroom sessions. The frequency of interactions between pupils and teachers was recorded in prepared recording sheets and computed. A modified Brophy and Good Dyadic Interaction Observation System was used as a guide for scoring. The interact ions were confined to initiatory verbal communication in the classroom. In the analysis of data, two approaches were utilised. The frequency count, and the t-test