On the Number of Sum-Free Triplets of Sets

We count the ordered sum-free triplets of subsets in the group $\mathbb{Z}/p\mathbb{Z}$, i.e., the triplets $(A,B,C)$ of sets $A,B,C \subset \mathbb{Z}/p\mathbb{Z}$ for which the equation $a+b=c$ has no solution with $a\in A$, $b \in B$ and $c \in C$. Our main theorem improves on a recent result by Semchankau, Shabanov, and Shkredov using a different and simpler method. Our proof relates previous results on the number of independent sets of regular graphs by Kahn; Perarnau and Perkins; and Csikvári to produce explicit estimates on smaller order terms. We also obtain estimates for the number of sum-free triplets of subsets in a general abelian group.

Hyo-Mental Angle and Distance: An Important Adjunct in Airway Assessment of Adult Mucopolysaccharidosis

Background: Mucopolysaccharidosis (MPS) is a rare congenital lysosomal storage disorder with complex airways. High anterior larynx is assessed by thyromental distance (TMD) nasendoscopy. A simpler method to assess this hyoid bone is described. The distance between the central-hyoid and symphysis of the mandible (hyo-mental distance; HMD) and inclination of this line to the horizontal axis (hyo-mental angle; HMA) in neutrally positioned patients is investigated. Methods: HMA, HMD in MPS, and non-MPS were compared, and their correlation with height and weight were assessed. Results: 50 adult MPS patients (M = 32, F = 18, age range = 19–66 years; mean BMI = 26.8 kg/m2) of MPS I, II, III, IV, and VI were compared with 50 non-MPS (M = 25, F = 25; age range = 22–84 years; mean BMI = 26.5 kg/m2). Mean HMA in MPS was 25.72° (−10 to +50) versus 2.42° (−35 to +28) in non-MPS. Mean HMD was 46.5 (25.7–66) millimeters in MPS versus 41.8 (27–60.3) in non-MPS. HMA versus height and weight showed a moderate correlation (r = −0.4, p < 0.05) in MPS and no significant correlation (r < 0.4, p > 0.05) in non-MPS. HMD versus height and weight showed no correlation (r < 0.4, p > 0.05) in both groups. Conclusions: HMA seems more acute in MPS despite nearly the same HMD as non-MPS, signifying a high larynx, which may be missed by TMD.

Usefulness of monthly rainfall normals in agroclimatic mapping

Rainfall has the most dominating influence on Indian agriculture. It is, however, commonly argued that monthly rainfall data are not adequate for agricultural planning, for which intervals of 10 days, week or shorter would be appropriate, 'The paper reviews some of the attempts made in this direction, particularly for the state of Maharashtra in Peninsular India. It then explains how the results of such studies can be derived by a simpler method of mapping which involves only monthly normals of rainfall and number of rainy days.

Domesticating Descartes, Renovating Scholasticism

This article studies the academic context in which Cartesianism was absorbed in Germany in the mid-seventeenth century. It focuses on the role of Johann Clauberg (1622-1665), first rector of the new University of Duisburg, in adjusting scholastic tradition to accommodate Descartes’s philosophy, thereby making the latter suitable for teaching in universities. It highlights contextual motivations behind Clauberg’s synthesis of Cartesianism with the existing framework such as a pedagogical interest in Descartes as offering a simpler method, and a systematic concern to disentangle philosophy from theological disputes. These motivations are brought into view by situating Clauberg in the closely-linked contexts of Protestant educational reforms, and debates around the proper relation between philosophy and theology. In this background, it argues that Clauberg nevertheless retains an Aristotelian conception of ontology for purely philosophical reasons, specifically, to give objective foundations to Descartes’s metaphysics of substance. In conclusion, Clauberg should not be assimilated either to Aristotelianism or to Cartesianism or, indeed, to syncretic labels such as ‘Cartesian Scholastic’. Instead, he should be read as transforming both schools by drawing on a variety of elements in order to address issues local to the academic milieu of his time.

A simpler method affords evaluation of π stabilization by phenylalanine of several biochemical carbocations

Simple models based on measurements taken from X-ray structures of relevant active sites are used to evaluate π stabilization by phenylalanine of several biochemical carbocations.

Performance Improvements of Krylov Subspace Methods in Numerical Heat Transfer and Fluid Flow Simulations

In recent years, advancements in computational hardware have enabled massive parallelism that can significantly reduce the duration of many numerical simulations. However, many high-fidelity simulations use serial algorithms to solve large systems of linear equations and are not well suited to exploit the parallelism of modern hardware. The Tri-Diagonal Matrix Algorithm (TDMA) is one such example of a serial algorithm that is ubiquitous in numerical simulations of heat transfer and fluid flow. Krylov subspace methods for solving linear systems, such as the Bi-Conjugate Gradients (BiCG) algorithm, can offer an ideal solution to improve the performance of numerical simulations as these methods can exploit the massive parallelism of modern hardware. In the present work, Krylov-based linear solvers of Bi-Conjugate Gradients (BCG), Generalized Minimum Residual (GMRES), and Bi-Conjugate Gradients Stabilized (BCGSTAB) have been incorporated into the SIMPLER algorithm to solve a three-dimensional Rayleigh-Bénard Convection model. The incompressible Navier-Stoke’s equations, along with the continuity and energy equations, are solved using the SIMPLER method. The computational duration and numerical accuracy for the Krylov-solvers are compared with that of the TDMA. The results show that Krylov methods can improve the speed of convergence for the SIMPLER method by factors up to 7.7 while maintaining equivalent numerical accuracy to the TDMA.