Abstract characterization of a conditional expectation operator on the space of measurable sections

A conditional expectation operator plays an important role in geometry of Banach spaces. However, the main issue is with regards to the existence of a conditional expectation operator that permits other objects to be considered such as martingales and martingale convergence theorems. Thus, the purpose of this study is to provide an abstract characterization of a conditional expectation operator on a space of measurable sections

Abstract characterization of a conditional expectation operator on the space of measurable sections

A conditional expectation operator plays an important role in geometry of Banach spaces. However, the main issue is with regards to the existence of a conditional expectation operator that permits other objects to be considered such as martingales and martingale convergence theorems. Thus, the purpose of this study is to provide an abstract characterization of a conditional expectation operator on a space of measurable sections

Introduction to lattice statistical mechanics

This book is an introduction to statistical mechanics suitable for readers knowing basic measure theory. The mathematical prerequisites are summarized in Chapter 1. It is hoped the reader will be ready to tackle research papers after reading the book. The first part of the text (Chapters 2-4) is concerned with 1-D Ising model. Some examples are described. The second part (Chapter 5) of the text focuses on Ising model on the Cayley tree and 2-D Ising model. Limit Gibbs measure and problem of phase transition for these models are also given. The third part (Appendix) outlines some applications of Ising model and Islamic aspects of the considered subject

Tests of quantum mechanics versus local realism using an atomic deuterium source

Measurements of the polarization correlation of the two-photons emitted simultaneously by metastable atomic deuterium in a true second order radiative process, are described. Two experiments, one involving the use of a half-wave plate and two linear polarizers, and the other three linear polarizers. have been carried out to investigate the validity of quantum mechanics in hitherto unexplored situations and to test local realistic theories of the type recently proposed by Garucclo and Selleri. The result, of the experiment. provide clear evidence against local realistic theories in which the probability of detection is postulated to be dependent on the angle between a polarization vector I. and a detection vector λ- The quantum mechanical predictions for such experiments, in which non-ideal analysers, were employed, have also been established and have been shown to be in agreement with the experimental results. In addition. the results of circular polarization measurements confirm the conservation of angular momentum along the common axis of detection and give further verification of the correctness of the form of the state vector describing the state of polarization of the two-photons. In another experiment, the effect of an electric field on the polarization properties of the two-photons has been measured for the first time. The Stoke's parameters characterizing the two-photon beam have been determined and the results suggest that the two-photon radiation is partially polarized at 22.5o with respect to the electric field vector. It is suggested that the affect is due to the removal of the degeneracy of the virtual Intermediate P states of the deuterium atom

Physico-chemical stability of flaxseed oil with natural antioxidant mixtures during heating

An optimization study has been carried-out to evaluate synergistic effects of natural antioxidants on physico-chemical characteristics of flaxseed oil during heating. Results showed that the use of oleoresin rosemary extract, sage extract and citric acid improved the stability of the oil during 20 times of frying. This study also revealed that the addition of these antioxidants effectively retarded flaxseed oil deterioration after as early as 5 times of frying of potato chips. Analyses from Response Surface Methodology (RSM) indicated that for all physico-chemical characteristics of flaxseed oil evaluated, all mathematical models or equations could be developed with high confidence, where all R2 values greater than 0.87. From this study, the highest R2 (0.98) was shown by yellow colour, followed by absorbance at 268 nm (0.97), saturated /unsaturated fatty acid ratio (0.95), absorbance at 232 nm (0.93), and AnV (0.92). R2 values for four other parameters were 0.88 (PV and IV) and 0.87 (FFA and red colour). This meant that the R2 values obtained from this study were all satisfactory and considered accurate enough, not only for prediction purposes, but also for optimization purposes

Determination of lard in mixture of body fats of mutton and cow by Fourier transform infrared spectroscopy

Fourier transform infrared (FTIR) spectroscopy provides a simple and rapid means of monitoring lard when blended with mutton and cow body fats. The spectral bands associated with mutton, cow body fats and their lard blends were recorded, interpreted and identified by relating them to those spectroscopically representative pure mutton, cow and lard body fats as references. A qualitative approach is proposed by comparing the pure animal fat with blended ones. Partial least squares (PLS) regression was applied for quantitative determination of the percent of lard in its blend with mutton body fat (MBF) using FTIR spectral data at frequency regions 3010 - 3000, 1220 - 1095 and 968 - 965 cm-1. The equation y = 1.151x - 0.1882, coefficient of determination (R2) = 0.9866 with standard error (SE) of 2.01. For the blend with cow body fat (CBF) frequency regions 1419 - 1414 and 968 - 965 cm-1 were used for qualitative and quantitative determination. PLS approach was used to create the equation y = 0.7239 x + 3.1369 with R2 = 0.9749 and SE = 1.86. The PLS calibration models were cross-validated and the standard deviation of difference (SDD) for repeatability and accuracy and R2 were computed

Optimization of the regularization of the solution of the plate heat transfer problems

Optimization of the regularization of the Fourier series in case of steady state heat transfer plate problem and heat transfer insulated plate problems are investigated and the regularization of the series solutions at a fixed point of the plates are studied at initial time and critical index. ****************************************************************************************** Pengoptimuman aturan siri Fourier telah dikaji pada keadaan tetap masalah plat pindah haba dan plat penebat pindah haba. Penyelesaian bersiri secara aturan pada titik tetap plat telah dikaji pada masa mula dan indeks penting

Equiconvergence in summation associated with elliptic polinomial

We compare the Fourier integral with the Fourier series in summation associated with elliptic polinomial

Extraction and characterization of gelatin from different marine fish species in Malaysia

Gelatins from the skin of four local marine fish, namely “kerapu” (Epinephelus sexfasciatus), “jenahak” (Lutjianus argentimaculatus), “kembung” (Rastrelliger kanagurta), and “kerisi” (Pristipomodes typus) have been successfully extracted by acid extraction. Results characterization showed that the fish gelatins were comparable to the fish gelatins from other fish species previously reported. They appeared snowy white in color with crystal-like and light texture. The gelatine extracted from “kerapu” had the strongest fishy odor, followed by the gelatines derived from “jenahak”, “kembung” and “kerisi”. In terms of bloom strength, the gelatin extracted from “kerapu” was found to be the strongest one compared to others, with the bloom value of more than 2000 g. The gelatins developed in this study contained almost all essential amino acids, with glycine being the most predominant one

Synergistic effect of rosemary, sage and citric acid on fatty acid retention of heated flaxseed oil

A study to optimize the use of oleoresin rosemary extract (OR), sage extract (OS) and citric acid (CA) in flaxseed oil during heating was performed using response surface methodology (RSM). Results showed that the natural antioxidants used in this study retarded oil deterioration, as evidenced by retention of fatty acids profiles. Results from this study revealed that the addition of OR and OS effectively retarded flaxseed oil deterioration after 20 times of frying of potato chips. Five type of fatty acids, namely C16:0, C18:0, C18:1, C18:2 and C18:3 were found in the oil, with all fatty acids having very good correlations with addition of natural antioxidants used during the frying. Mathematical models to predict the fatty acid composition of flaxseed oil after the frying could be developed with high confidence for all types of fatty acids. Therefore, for optimization purposes, the use of the ratio of saturated to unsaturated fatty acids was chosen because it covers all individual fatty acids present in the oil. After 20 fryings, the ratio best predicted the efficacy of natural antioxidants in preserving flaxseed oil with R2 of 0.954. Further analysis showed that after 20 fryings, all three antioxidants had a significant effect on the ratio. Based on these results, a combination of 0.064% oleoresin rosemary extract, 0.061 sage extract and 0.041% citric acid can be recommended for use in flaxseed oil before deep-fat frying