Hall's Conjecture on Extremal Sets for Random Triangles

In this paper we partially resolve Hall's conjecture about the distribution of random triangles. We consider the probability that three points chosen uniformly at random, in a bounded convex region of the plane, form an acute triangle. Hall's conjecture is the "isoprobabilistic inequality" which states that this probability should be maximized by the disk. We first prove that the disk is a weak local maximum for planar regions and that the ball is a weak local maximum in three dimensional regions. We then prove a local $C^{2,1/2}$-type estimate on the probability in the Hausdorff topology. This enables us to prove that the disk is a local maximum in the Gromov-Hausdorff topology (modulo congruences). Finally, we give an explicit upper bound on the isoperimetric ratio of the regions which maximize the probability and show how this reduces verifying the full conjecture to a finite, though currently intractable, calculation. An interesting aspect of our work is the use of tools from outside of geometric probability. We use an autocorrelation integral to provide the appropriate framework for the problem. When we study the problem in $\mathbb{R}^3$, we need non-Abelian harmonic analysis. To set up the proof that the disk is a strong local maximum, we use the Borsak-Ulam theorem.Comment: In this version, we have reorganized the paper to have the technical proofs in the appendix. We have also included links to Mathematica code used to perform some of the calculation

The set of all orthogonal complex structures on the flat 66-tori

In \cite{BSV}, Borisov, Salamon and Viaclovsky constructed non-standard orthogonal complex structures on flat tori $T^{2n}_{\mathbb R}$ for any $n\geq 3$. We will call these examples BSV-tori. In this note, we show that on a flat $6$-torus, all the orthogonal complex structures are either the complex tori or the BSV-tori. This solves the classification problem for compact Hermitian manifolds with flat Riemannian connection in the case of complex dimension three.Comment: 14 page

Effects of Vibrations on Metal Forming Process: Analytical Approach and Finite Element Simulations

Vibration assisted forming is one of the most recent and beneficial technique used to improve forming process. Effects of vibration on metal forming processes can be attributed to two causes. First, the volume effect links lowering of yield stress with the influence of vibration on the dislocation movement. Second, the surface effect explains lowering of the effective coefficient of friction by periodic reduction contact area. This work is related to vibration assisted forming process in viscoplastic domain. Impact of change in vibration waveform has been analyzed. For this purpose, two analytical models have been developed for two different types of vibration waveforms (sinusoidal and triangular). These models were developed on the basis of Slice method that is used to find out the required forming force for the process. Final relationships show that application of triangular waveform in forming process is more beneficial as compare to sinusoidal vibrations in terms of reduced forming force. Finite Element Method (FEM) based simulations were performed using Forge2008®and these confirmed the results of analytical models. The ratio of vibration speed to upper die speed is a critical factor in the reduction of the forming force.ENSA

Impact of social complexity on outcomes in cystic fibrosis after transfer to adult care

Objective This study evaluates the roles of medical and social complexity in health care use outcomes in cystic fibrosis (CF) after transfer from pediatric to adult care. Methods Retrospective cohort design included patients with CF who were transitioned into adult care at Indiana University from 2005 to 2015. Predictor variables included demographic and comorbidity data, age at transition, treatment complexity score (TCS), and an objective scoring measure of their social complexity (Bob's Level of Social Support, BLSS). Outcome variables included outpatient visit rates and hospitalization rates. Pearson's correlations and linear regression were used to analyze the data. Results The median age of the patients (N = 133) at the time of transition was 20 (IQR 19‐23) years. The mean FEV1 % predicted at transition was 69 ± 24%. TCS correlated with outpatient visit rates (r = 0.3, P = 0.003), as well as hospitalization rates (r = 0.4, P < 0.001); while the BLSS only correlated with hospitalization rates (r = 0.7, P < 0.001). After adjusting for covariates, the strongest predictors of post‐transfer hospitalizations are BLSS (P < 0.0001) and pre‐transfer hospitalization rate (P < 0.0001). Conclusion Greater treatment complexity is associated with greater healthcare utilization overall, while greater social complexity is associated with increased hospitalizations (but not outpatient visits). Screening young adults for social complexity may identify high‐risk subpopulations and allow for patient centered interventions to support them and prevent avoidable health care use

Performance Measurement: A Conceptual Framework for Supply Chain Practices

AbstractMe--asurement of Supply Chain (SC) performance with regards to key practices of SC paradigms is the area which is under research. Presently there are no guidance or set rules under which we can measure SC performance. The lack of clarity and comparability concerns in this area creates misunderstanding and makes it more difficult to formulate a clear strategy. The aim of this research is to identify antecedents of existing SC paradigm's practices, as well as antecedents for SC performance measurement to formulate a conceptual framework. Based on this research, new sustainable SC performance measurement conceptual framework is proposed for existing SC paradigms. The detailed analysis presented in this research paper offers a set of characteristics and structure that industry as well as academia could use it as a guidance framework to measure SC performance