Freshman rules in calculus

In this note we present a class of functions (f, g) that satisfy ‘freshman rules’ in calculus.     

Evaluation of fiber tracking from subsampled q-space data in diffusion spectrum imaging

Diffusion spectrum imaging (DSI) is capable of resolving crossing and touching fiber bundles in a given voxel. Acquisition of DSI data involves sampling large number of points in the q-space which significantly increases scan times. The scan times can be reduced by exploiting the symmetry of the q-space. In this study the fiber pathways for five (fornix, cingulum, superior longitudinal fasciculus, corticospinal tract, and crossing fibers in the centrum semiovale region) fiber bundles derived using three subsampled data sets of different sizes derived from the 257 samples in the q-space are compared. The coefficient of variation of the ratio of the number of fiber pathways for each subsample data set to the original data points, averaged over all the 10 subjects, was used for quantitatively investigating the effect of subsampling on the tractography. The effect of threshold angles on tractography is also investigated. The effect of subsampling on the orientation distribution function (ODF) was quantitatively evaluated using both scalar and vector measures derived from the ODF. A streamline tractography method that improves the curvature problem and reduces the local truncation error to further improve the mapping of fiber pathways is adapted. Analysis of the fiber pathways in ten normal subjects, based on qualitative and quantitative methods, shows that the 129 and 198 q-space points provide very similar result with angle of threshold between 41° and 45°. Based on the scan time advantage, 129 subsampled points appear to be adequate for tractography.     

Solving the Maximum Clique Problem with Symmetric Rank-One Non-negative Matrix Approximation

Finding complete subgraphs in a graph, that is, cliques, is a key problem and has many real-world applications, e.g., finding communities in social networks, clustering gene expression data, modeling ecological niches in food webs, and describing chemicals in a substance. The problem of finding the largest clique in a graph is a well-known difficult combinatorial optimization problem and is called the maximum clique problem. In this paper, we formulate a very convenient continuous characterization of the maximum clique problem based on the symmetric rank-one non-negative approximation of a given matrix and build a one-to-one correspondence between stationary points of our formulation and cliques of a given graph. In particular, we show that the local (resp. global) minima of the continuous problem corresponds to the maximal (resp. maximum) cliques of the given graph. We also propose a new and efficient clique finding algorithm based on our continuous formulation and test it on the DIMACS data sets to show that the new algorithm outperforms other existing algorithms based on the Motzkin–Straus formulation and can compete with a sophisticated combinatorial heuristic.     

Kinetic analysis of dehydroxylation of Ethiopian kaolinite during calcination

Journal of Thermal Analysis and Calorimetry - The calcination process is the dehydroxylation reaction of the kaolinite mineral into the formation of amorphous metakaolinite phase. The...     

Understanding the groundwater dynamics in the Southern Rift Valley Lakes Basin (Ethiopia): multivariate statistical analysis method, oxygen (δ 18O) and deuterium (δ 2H)

Multivariate statistical analysis is very important to classify waters of different hydrochemical groups. Statistical techniques, such as cluster analysis, can provide a powerful tool for analyzing water chemistry data. This method is used to test water quality data and determine if samples can be grouped into distinct populations that may be significant in the geologic context, as well as from a statistical point of view. Multivariate statistical analysis method is applied to the geochemical data in combination with δ ¹⁸O and δ ²H isotopes with the objective to understand the dynamics of groundwater using hierarchical clustering and isotope analyses. The geochemical and isotope data of the central and southern rift valley lakes have been collected and analyzed from different works. Isotope analysis shows that most springs and boreholes are recharged by July and August rainfalls. The different hydrochemical groups that resulted from the multivariate analysis are described and correlated with the geology of the area and whether it has any interaction with a system or not.     

Determination of furan levels in coffee using automated solid-phase microextraction and gas chromatography/mass spectrometry

Furan is a 5-member ring chemical with high volatility. Because it is highly volatile, furan levels in foods are not easily determined with accuracy. In this study, an analytical method for furan analysis using an automated solid-phase microextraction system in combination with gas chromatography/mass spectroscopy is described. The performance of the method was demonstrated by the results obtained from a variety of coffees. Furan was detected at part-per-billion levels in coffee. The limit of detection was 0.3 ng/g and limit of quantitation, 0.8 ng/g. The percent recoveries were between 92 and 102.