Cantor-type cylindrical-coordinate method for differential equations with local fractional derivatives

In this Letter, we propose to use the Cantor-type cylindrical-coordinate method in order to investigate a family of local fractional differential operators on Cantor sets. Some testing examples are given to illustrate the capability of the proposed method for the heat-conduction equation on a Cantor set and the damped wave equation in fractal strings. It is seen to be a powerful tool to convert differential equations on Cantor sets from Cantorian-coordinate systems to Cantor-type cylindrical-coordinate systems.     

Notes on the optimal variational iteration method

Here, various variational iteration algorithms are compared. An auxiliary parameter can be introduced in the iteration procedure, and can be identified optimally, which results in Turkyilmazoglu’s optimal variational iteration method. Some unknown auxiliary parameters can be also included in the initial solution, and can be optimally determined, that is Heri?anu and Marinca’s optimal variational iteration method.     

A parallel-multipoint fiber-optic temperature sensor based on Fresnel reflection

A parallel-multipoint fiber-optic temperature sensor for monitoring ambient temperature with a measured temperature resolution of approximately 0.03 °C is presented. The sensor relies on the temperature-dependent Fresnel reflection from each interface between a sensing fiber tip and a thermo-optic medium. An arrayed waveguide grating is used to achieve wavelength-division multiplexed-multipoint sensing. Applying the relative-intensity technique, the errors resulting from fluctuation of the light source and other influences of the environment are eliminated, and the stability for long time measurement can be improved.     

Preliminary report on the energy balance for nonlinear oscillations

In this paper, a reliable technique for calculating angular frequencies of nonlinear oscillators is developed. The new algorithm offers a promising approach by constructing a Hamiltonian for the nonlinear oscillator. Some illustrative examples are given.     

Ratio of a strange quark mass ms to up or down quark mass mu,d predicted by a quark propagator in the framework of the chiral perturbation theory

Based on the fully dressed quark propagator and chiral perturbation theory, we study the ratio of the strange quark mass ms to up or down quark mass mu,d . The ratio is related to the determination of quark masses which are fundamental input parameters of QCD Lagrangian in the Standard Model of particle physics and can not be directly measured since the quark is confined within a hadron. An accurate determination of these QCD free parameters is extremely important for both phenomenological and theoretical applications. We begin with a brief introduction to the non-perturbation QCD theory, and then study the mass ratio in the framework of the chiral perturbation theory (χPT) with a parameterized fully dressed quark propagator which describes confining fully dressed quark propagation and is analytic everywhere in the finite complex p2-plane and has no Lehmann representation so there are no quark production thresholds in any theoretical calculations of observable data. Our prediction for the ratio ms/mu,d is consistent with other model predictions such as Lattice QCD, instanton model, QCD sum rules and the empirical values used widely in the literature. As a by-product of this study, our theoretical results, together with other predictions of physical quantities that used this quark propagator in our previous publications, clearly show that the parameterized form of the fully dressed quark propagator is an applicable and reliable approximation to the solution of the Dyson-Schwinger Equation of quark propagator in the QCD.     

Synthesis,characterization, and photoluminescence properties of difluoroboron complexes with bis-β-diketone ligands

Some novel difluoroboron bis-β-diketonates containing a pyridyl moiety were synthesized from diethyl 2,6-pyridinedicarboxylate via Claisen condensation with the corresponding aryl methyl ketones and followed by complexation with boron trifluoride etherate. Their spectroscopic behaviors were studied by FTIR, ¹H NMR, UV–Vis, and fluorescence spectroscopic techniques. The results indicated that difluoroboron bis-β-diketonates exhibited violet or blue fluorescence emission at 428–454 nm under UV illumination in DMSO and possessed high extinction coefficients. It was found that the nature of the substituents at benzene ring in bis-β-diketone ligands had a significant impact on the photoluminescence behaviors of difluoroboron complexes. The complex 5b exhibited the strongest photoluminescence intensity and highest quantum yield (Φ _u = 0.93), due to two strong electron-donating methoxyl moieties in molecule and the compound 4b displayed the lowest photoluminescence intensity and quantum yield, assigned to the heavy atom effect of the chlorine atom in its molecule. The photoluminescence intensity and quantum yield of these difluoroboron complexes decreased in the sequence, 5b > 2b > 1b > 3b > 4b.     

A Tutorial Review on Fractal Spacetime and Fractional Calculus

This tutorial review of fractal-Cantorian spacetime and fractional calculus begins with Leibniz’s notation for derivative without limits which can be generalized to discontinuous media like fractal derivative and q-derivative of quantum calculus. Fractal spacetime is used to elucidate some basic properties of fractal which is the foundation of fractional calculus, and El Naschie’s mass-energy equation for the dark energy. The variational iteration method is used to introduce the definition of fractional derivatives. Fractal derivative is explained geometrically and q-derivative is motivated by quantum mechanics. Some effective analytical approaches to fractional differential equations, e.g., the variational iteration method, the homotopy perturbation method, the exp-function method, the fractional complex transform, and Yang-Laplace transform, are outlined and the main solution processes are given.     

Geometrical explanation of the fractional complex transform and derivative chain rule for fractional calculus

The fractional complex transform is suggested to convert a fractional differential equation with Jumarie?s modification of Riemann-Liouville derivative into its classical differential partner. Understanding the fractional complex transform and the chain rule for fractional calculus are elucidated geometrically.     

Non-perturbative Aspects of QCD and Parameterized Quark Propagator

Based on the Global Color Symmetry Model, the non-perturbative Q, CD vacuum is investigated in the parameterized fully dressed quark propagator. Our theoretical predictions for various quantities characterized the QCD vacuum are in agreement with those predicted by many other phenomenologieal QCD inspired models. The successful predictions clearly indicate the extensive validity of our parameterized quark propagator used here. A detailed discussion on the arbitrariness in determining the integration cut-off parameter ofμ in calculating QCD vacuum condensates and a good method, which avoided the dependence of calculating results on the cut-off parameter is also strongly recommended to readers.