Applications of a sequence of points in teaching linear algebra,numerical methods and discrete mathematics

Based on a sequence of points and a particular linear transformation generalized from this sequence, two recent papers (E. Mauch and Y. Shi, Using a sequence of number pairs as an example in teaching mathematics. Math. Comput. Educ., 39 (2005), pp. 198–205; Y. Shi, Case study projects for college mathematics courses based on a particular function of two variables. Int. J. Math. Educ. Sci. Techn., 38 (2007), pp. 555–566) have presented some interesting examples which can be used in teaching high school and college mathematics classes. In this article, we further discuss a few interesting ways to apply this sequence of points in teaching college mathematics courses such as linear algebra, numerical methods in computing, and discrete mathematics. In addition to using them in individual courses, these studies may also be combined together to offer seminars or workshops to college mathematics students. Studies like these are likely to promote student interests and get students more involved in the learning process, and therefore make the learning process more effective.     

The effect of the 4MAT method (learning styles and brain hemispheres) of instruction on achievement in mathematics

The purpose of this study was to determine the efficiency of 4MAT method of instruction in which learning style and cerebral hemispheres are taken into account in teaching the binary operation and its properties in mathematics. The sample of this study comprised 58 ninth grade students in two separate classes in a high school. One of the classes was selected as the experimental group in which 4MAT method of instruction was used; and the other class was selected as the control group in which the traditional teaching was given, and this selection was performed randomly. The data have been obtained primarily from three scales, namely ‘mathematical knowledge test’, ‘mathematical attitude scale’ and ‘knowledge test on binary operation and its properties’. It has been determined that 4MAT method of instruction was more efficient than the traditional method in teaching of the binary operation subject in mathematics.     

Sometimes binary is better

This note uses material involving perfect numbers and Zeno's paradoxes to show that although most students prefer to use base 10 when working with mathematical concepts there are times when the binary system is best.     

Efficient Bayesian Inference for Multivariate Probit Models With Sparse Inverse Correlation Matrices

We propose a Bayesian approach for inference in the multivariate probit model, taking into account the association structure between binary observations. We model the association through the correlation matrix of the latent Gaussian variables. Conditional independence is imposed by setting some off-diagonal elements of the inverse correlation matrix to zero and this sparsity structure is modeled using a decomposable graphical model. We propose an efficient Markov chain Monte Carlo algorithm relying on a parameter expansion scheme to sample from the resulting posterior distribution. This algorithm updates the correlation matrix within a simple Gibbs sampling framework and allows us to infer the correlation structure from the data, generalizing methods used for inference in decomposable Gaussian graphical models to multivariate binary observations. We demonstrate the performance of this model and of the Markov chain Monte Carlo algorithm on simulated and real datasets. This article has online supplementary materials.     

Fatty Acid Ethyl Esters (FAEE): A New,Green and Renewable Solvent for the Extraction of Carotenoids from Tomato Waste Products

Currently there is a drive towards the minimisation and reclamation of valuable materials from the waste products of the food and beverage industry. This can be achieved through the extraction of residual nutraceuticals from such materials. Tomato pomace contains carotenoids and other chemicals which can be extracted directly into edible oils to improve the health-giving properties of such oils. We report here a novel green solvent, fatty acid ethyl esters (FAEE), which is significantly more effective than sunflower oil and hexane for the extraction of lycopene and beta-carotene from tomato skin waste. FAEE are a non-toxic renewable resource that is environmentally friendly and to our knowledge has never been used as a vegetal extraction fluid. The efficiency of FAEE extraction was significantly improved relative to both sunflower oil and hexane under ultrasound-assisted extraction (UAE) conditions. In addition, FAEE have the additional and significant advantage that once enriched with the extracted nutraceuticals can be used directly as a food additive.     

Phase behaviors in a binary mixture of diblock copolymers confined between two parallel walls

The phase behaviors in a binary mixture of diblock copolymers confined between two parallel walls are investigated by using a cell dynamics simulation of the time-dependent Ginzburg-Landau theory.The morphological dependence of the wall-block interaction and the distance between walls(confinement degree) has been systematically studied,and the effect of repulsive interactions between different monomers is also discussed.It is interesting that multiple novel morphological transitions are observed by changing these factors,and various multilayered sandwich structures are formed in the mixture.Furthermore,the parametric dependence and physical reasons for the microdomain growth and orientational order transitions are discussed.From the simulation,we find that much richer morphologies can form in a binary mixture of diblock copolymers than those in a pure diblock copolymer.Our results provide an insight into the phase behaviors under parallel wall confinement and may provide guidance for experimentalists.This model system can also give a simple way to realize orientational order transition in soft materials through confinement.     

Binary-component micelle and vesicle:Free energy and asymmetric distributions of amphiphiles between vesicle monolayers

The real-space two-dimensional self-consistent field theory(SCFT) is employed to study the free energies of micelles and vesicles constituted by binary amphiphilic diblock copolymer AB in homopolymer A.With an increasing volume fraction of copolymer AB,there are morphological transitions from circle micelles to oblate circle-like micelles,to a compound structure with inverted micelles in the inner center and micelles outer layer,and to vesicles.Special attention is paid to the role of the copolymer AB in controlling the free energies of the micelles and vesicles by examining the effect of the length ratio of A/B with the fixed whole chain length of the AB copolymer,the length effect of A or B block with the corresponding fixed length of B or A block,for one component of copolymer,and the effect of different amphiphile compositions for a binary-component copolymer system.The quantity η is provided to describe the asymmetric density distribution of amphiphiles between the inner and outer monolayers of vesicles,and to quantify the relative asymmetric extent of the density distribution between two species of copolymers in binary component vesicles.     

Integrability of an extended （2＋1）-dimensional shallow water wave equation with Bell polynomials

We investigate the extended (2+1)-dimensional shallow water wave equation. The binary Bell polynomials are used to construct bilinear equation, bilinear Bcklund transformation, Lax pair, and Darboux covariant Lax pair for this equation. Moreover, the infinite conservation laws of this equation are found by using its Lax pair. All conserved densities and fluxes are given with explicit recursion formulas. The N-soliton solutions are also presented by means of the Hirota bilinear method.     

Solvatochromism of 2‐(N,N‐dimethylamino)‐7‐nitrofluorene and the natural dye β‐carotene: application for the determination of solvent dipolarity and polarizability

The effects of solvents on chemical phenomena (rate and equilibrium constants, spectroscopic transitions, etc.) are conveniently described by solvation free‐energy relationships that take into account solvent acidity, basicity and dipolarity/polarizability. The latter can be separated into its components by manipulating the UV–vis spectra of two solvatochromic probes, 2‐(N,N‐dimethylamino)‐7‐nitrofluorene (DMANF) and a di‐(tert‐butyl)‐tetramethyl docosanonaen probe (ttbP9) whose synthesis is laborious and expensive. Recently, we have shown that the natural dye β‐carotene can be conveniently employed instead of ttbP9 for the determination of solvent polarizability (SP) of 76 molecular solvents and four ionic liquids. In the present work, we report the polarizabilities of further 24 solvents. Based on the solvatochromism of β‐carotene and DMANF, we have calculated solvent dipolarity (SD) for 103 protic and aprotic molecular solvents, and ionic liquids. The dependence of SD and SP on the number of carbon atoms in the acyl‐ or alkyl group of several homologous series (alcohols; 2‐alkoxyethanols; carboxylic acid‐ anhydrides, and esters, ionic liquids) is calculated and briefly discussed. Copyright © 2013 John Wiley & Sons, Ltd.