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LONG PATHS IN THE DISTANCE GRAPH OVER LARGE SUBSETS OF VECTOR SPACES OVER FINITE FIELDS
BENNETT, MICHAEL,CHAPMAN, JEREMY,COVERT, DAVID,HART, DERRICK,IOSEVICH, ALEX,PAKIANATHAN, JONATHAN Korean Mathematical Society 2016 대한수학회지 Vol.53 No.1
Let $E{\subset}{\mathbb{F}}^d_q$, the d-dimensional vector space over the finite field with q elements. Construct a graph, called the distance graph of E, by letting the vertices be the elements of E and connect a pair of vertices corresponding to vectors x, y 2 E by an edge if ${\parallel}x-y{\parallel}:=(x_1-y_1)^2+{\cdots}+(x_d-y_d)^2=1$. We shall prove that the non-overlapping chains of length k, with k in an appropriate range, are uniformly distributed in the sense that the number of these chains equals the statistically correct number, $1{\cdot}{\mid}E{\mid}^{k+1}q^{-k}$ plus a much smaller remainder.
Long paths in the distance graph over large subsets of vector spaces over finite fields
Michael Bennett,Jeremy Chapman,David Covert,Derrick Hart,Alex Iosevich,Jonathan Pakianathan 대한수학회 2016 대한수학회지 Vol.53 No.1
Let $E \subset {\mathbb F}_q^d$, the $d$-dimensional vector space over the finite field with $q$ elements. Construct a graph, called the distance graph of $E$, by letting the vertices be the elements of $E$ and connect a pair of vertices corresponding to vectors $x,y \in E$ by an edge if $||x-y||:={(x_1-y_1)}^2+\dots+{(x_d-y_d)}^2=1$. We shall prove that the non-overlapping chains of length $k$, with $k$ in an appropriate range, are uniformly distributed in the sense that the number of these chains equals the statistically correct number, $1 \cdot {|E|}^{k+1}q^{-k}$ plus a much smaller remainder.
( Michael C Bennett ),( Amit Patel ),( Nitin Sainani ),( Dan Wang ),( Gregory S Sayuk ),( C Prakash Gyawali ) 대한소화기기능성질환·운동학회(구 대한소화관운동학회) 2018 Journal of Neurogastroenterology and Motility (JNM Vol.24 No.3
Background/Aims Breaks in the peristaltic contour on esophageal high-resolution manometry (HRM) may be associated with bolus retention in the esophagus. We evaluated the relationship between peristaltic breaks and esophageal symptoms, reflux exposure, and symptom outcomes in a prospective patient cohort. Methods Two hundred and eighteen patients (53.2 ± 0.9 years, 68.3% female) undergoing both pH-impedance testing and HRM over a 5-year period were prospectively evaluated. Demographics, symptom presentation, acid exposure time, symptom association probability, and symptom burden scores were collected. Outcomes were assessed on follow-up using changes in symptom scores. Presence of long breaks (≥ 5 cm) on HRM was assessed by a blinded author. Relationships between breaks, reflux parameters, presenting symptoms, and outcomes were assessed. Results Patients with long breaks were more likely to have cough as a presenting symptom than those without (43.4% vs 28.6%, P = 0.024); statistical differences were not demonstrated with other symptoms (P ≥ 0.3). Numbers of swallows with long breaks were higher in patients with cough compared to those without (2.4 ± 0.3 vs 1.6 ± 0.2, P = 0.021); differences were not found with other symptoms (P ≥ 0.4). Long breaks were not associated with age, gender, race, reflux burden, symptom association, or changes in symptom metrics (P ≥ 0.1 for all comparisons). Among patients with cough, the presence of long breaks predicted suboptimal symptom improvement with antireflux therapy (P = 0.018); this difference did not hold true for other symptoms (P ≥ 0.2). Conclusions Long breaks in esophageal peristaltic integrity are associated with cough. The presence of long breaks is associated with suboptimal benefit from antireflux therapy. (J Neurogastroenterol Motil 2018;24:387-394)
김영민,John Bennett Cologne,Harry Michael Cullings 한국데이터정보과학회 2017 한국데이터정보과학회지 Vol.28 No.3
Mediation models are widely used in many fields of research and have recently gained attention in epidemiology. The mediation proportion is a standard measure to evaluate what part of the total exposure effect on an outcome may be explained by a particular mediator and to examine how important that pathway is relative to the overall exposure effect. A common question is how large a sample size is needed to achieve high statistical power or, equivalently, what magnitude of effect can be detected. Current power and sample size calculations for mediation analysis are limited and additional research is needed. We therefore propose a computer-intensive power analysis using the mediation proportion. We conduct simulation studies to calculate statistical powers and sample sizes. And then, we illustrate our power analysis using an example from the Adult Health Study of atomic-bomb survivors and demonstrate that the method is relatively straightforward to understand and compute.
Kim, Young Min,Cologne, John Bennett,Cullings, Harry Michael The Korean Data and Information Science Society 2017 한국데이터정보과학회지 Vol.28 No.3
Mediation models are widely used in many fields of research and have recently gained attention in epidemiology. The mediation proportion is a standard measure to evaluate what part of the total exposure effect on an outcome may be explained by a particular mediator and to examine how important that pathway is relative to the overall exposure effect. A common question is how large a sample size is needed to achieve high statistical power or, equivalently, what magnitude of effect can be detected. Current power and sample size calculations for mediation analysis are limited and additional research is needed. We therefore propose a computer-intensive power analysis using the mediation proportion. We conduct simulation studies to calculate statistical powers and sample sizes. And then, we illustrate our power analysis using an example from the Adult Health Study of atomic-bomb survivors and demonstrate that the method is relatively straightforward to understand and compute.