Why Noether Symmetry of F(R) Theory Yields Three-Half Power Law?

Noether symmetry of F(R) theory of gravity in vacuum or in matter dominated era yields $F(R) \propto R^{\frac{3}{2}}$ . We show that this particular curvature invariant term is very special in the context of isotropic and homogeneous cosmological model as it makes the first fundamental form h _ij cyclic. As a result, it allows a unique power law solution, typical for this particular fourth order theory of gravity, both in the vacuum and in the matter dominated era. This power law solution has been found to be quite good to explain the early stage but not so special and useful to explain the late stage of cosmological evolution. The usefulness of Palatini variational technique in this regard has also been discussed.     

Viability of Noether Symmetry of F(R) Theory of Gravity

Recently, we have explored vices and virtues of $R^{\frac{3}{2}}$ term in the action which has in-built Noether symmetry and anticipated that a linear term might improve the situation (Sarkar et al., arXiv:1201.2987 [astro-ph.CO], 2012). In the absence of a conserved current it is extremely difficult to obtain an analytical solution of the said fourth order theory of gravity in the presence of a linear term. Here, we therefore enlarge the configuration space by including a scalar field in addition and also taking some of the anisotropic models (in the absence of a scalar field) into account. We observe that Noether symmetry remains obscure and it does not even reproduce the one that already exists in the literature (Sanyal, Gen. Relativ. Gravit., 37:407, 2005). However, there exists in general, a conserved current for F(R) theory of gravity in the presence of a non-minimally coupled scalar field (Sanyal, Phys. Lett. B, 624:81, 2005; Mod. Phys. Lett. A, 25:2667, 2010), which simplifies the field equations considerably. Here, we briefly expatiate the non-Noether conserved current and show that indeed the situation is modified.     

The modern origin of matrices and their applications

This paper deals with the modern development of matrices, linear transformations, quadratic forms and their applications to geometry and mechanics, eigenvalues, eigenvectors and characteristic equations with applications. Included are the representations of real and complex numbers, and quaternions by matrices, and isomorphism in order to show that matrices form a ring in abstract algebra. Some special matrices, including Hilbert’s matrix, Toeplitz’s matrix, Pauli’s and Dirac’s matrices in quantum mechanics, and Einstein’s Pythagorean formula are discussed to illustrate diverse applications of matrix algebra. Included also is a modern piece of information that puts mathematics, science and mathematics education professionals at the forefront of advanced study and research on linear algebra and its applications.     

On a new simple method for evaluation of certain multiple definite integrals

This paper deals with a simple method of evaluation of certain multiple difinite integrals. This is followed by two main theorems concerning multiple difinite integrals. Some examples of applications are given.     

Validity of the Generalized Second Law of Thermodynamics in the Logamediate and Intermediate Scenarios of the Universe

In this work, we have investigated the validity of the generalized second law of thermodynamics in logamediate and intermediate scenarios of the universe bounded by the Hubble, apparent, particle and event horizons using and without using first law of thermodynamics. We have observed that the GSL is valid for Hubble, apparent, particle and event horizons of the universe in the logamediate scenario of the universe using first law and without using first law. Similarly the GSL is valid for all horizons in the intermediate scenario of the universe using first law. Also in the intermediate scenario of the universe, the GSL is valid for Hubble, apparent and particle horizons but it breaks down whenever we consider the universe enveloped by the event horizon.     

The Effects of Tachyonic and Phantom Fields in the Intermediate and Logamediate Scenarios of the Anisotropic Universe

In this work, we have analyzed two scenarios namely, “intermediate” and “logamadiate” scenarios for closed, open and flat anisotropic universe in presence of phantom field, normal tachyonic field and phantom tachyonic field. We have assumed that there is no interaction between the above mentioned dark energy and dark matter. In these two types of the scenarios of the Universe, the nature of the scalar fields and corresponding potentials have been investigated. In intermediate scenario, (i) the potential for normal tachyonic field decreases, (ii) the potentials for phantom tachyonic field and phantom field increase with the corresponding fields. Also in logamediate scenario, (i) the potential for normal tachyonic field increases, (ii) the potentials for phantom tachyonic field and phantom field decrease with the corresponding fields.     

Evaluation of spectral phase in spectrally resolved white-light interferometry: Comparative study of single-frame techniques

Spectral phase in a white-light interferogram contains information about the absolute optical path difference (OPD) in the interferometer. Evaluation of spectral phase is therefore important in applications such as profilometry with white light. In spectrally resolved white-light interferometry (SRWLI) the white-light interferogram is spectrally decomposed by a spectrometer in order to determine this phase. Several single-frame techniques in SRWLI have been proposed for the evaluation of the phase including Fourier transform, Hilbert transform, spatial phase shifting, windowed Fourier transform and wavelet transform. In this paper we present a comparative study of these techniques in this application in relation to the temporal phase-shifting technique which is a multi- frame method. Further, we also propose a modified method to remove the influence of source spectrum modulation in Hilbert transform procedure.     

Generalizations of a parameterized Jordan-type inequality,Janous’s inequality and Tsintsifas’s inequality

In this work, we first prove a generalized version of a parameterized Jordan-type inequality. We then use it to prove the generalized versions of Janous’s inequality and Tsintsifas’s inequality which reduce to two inequalities conjectured by Janous and Tsintsifas as special cases.     

Validation of analytical method for furan determination in eight food matrices and its levels in various foods

In this study, an automated solid‐phase microextraction coupled with gas chromatography and mass spectrometry method was developed and validated for the determination of furan in eight matrices including ham, milk, apple juice, rice porridge, peanut butter, flatfish, tuna (canned) and seaweed. The calibration curves were highly linear (r² > 0.990) and the limit of detection and limit of quantification ranged from 0.01?0.02 and 0.04?0.06 ng/g, respectively. The recovery ranged from 77.81?111.47%. The validated method was used to analyse the furan levels in 120 foods. The highest levels of furan were detected in black tea (172.05 ng/g) and red ginseng extract (89.27 ng/g). Whelk (canned) contained a high furan content (21.34 ng/g) among the seafood samples.