Abstract:
We prove a characterization of a nonhomogeneous A-harmonic equation and describe its generalization. We also point out its connection with 1-Harmonic equation.

Abstract:
We prove a characterization of a nonhomogeneous A-harmonic equation and describe its generalization. We also point out its connection with 1-Harmonic equation.

Abstract:
In this article, we use scaling function interpolation method to solve linear Fredholm integral equations, and we prove a convergence theorem for the solution of Fredholm integral equations. We present two examples which have better results than others.

Wavelet methods are a very useful tool in solving integral equations. Both scaling functions and wavelet functions are the key elements of wavelet methods. In this article, we use scaling function interpolation method to solve Volterra integral equations of the first kind, and Fredholm-Volterra integral equations. Moreover, we prove convergence theorem for the numerical solution of Volterra integral equations and Freholm-Volterra integral equations. We also present three examples of solving Volterra integral equation and one example of solving Fredholm-Volterra integral equation. Comparisons of the results with other methods are included in the examples.

Numerical solutions of singular Fredholm integral
equations of the second kind are solved by using Coiflet interpolation method.
Error analysis of the method is obtained and examples are presented. It turns
out that our method provides better accuracy than other methods.

Abstract:
A freak wave is a wave of very considerable height, ahead of which there is a deep trough. A case study examines some basic properties developed by performing wavelet analysis on a freak wave. We demonstrate several applications of wavelets and discrete and continuous wavelet transforms on the study of a freak wave. A modeling setting for freak waves will also be mentioned.

Abstract:
Regarding R,G, B of color iamge pixel as the three weights of color space vector, two color iamge edge detec- tion algorithms have been proposed using corresponding knowledge of vector field flux and quaternion. Maximizing vec- tor flux based detection algorithm can detect more edge details comparing conventional algorithm, simultaneously, this algorithm has quite self-adaptive ability to different nature iamges. A novel edge detection algorithm also be proposed based on quaternion using the easy quaternion expressing of vector rotating. Experiment result illuminate that the two algorithms have good detection effects to color iamge, especially, quaternion vector rotating based algorithm can detect much more texture details which the conventional algorithm can't realize it.

Abstract:
A compact and simple interdigital capacitor inverted-F antenna (IFA) operated at its quarter-wavelength (λ/4) mode as the fundamental resonant mode for achieving multiband operation in the mobile phone is designed. The proposed antenna consists of a monopole antenna and an IFA. The proposed interdigital capacitor IFA has a simple structure of comprising two meandered radiating strips of length about λ/4 and is fed using an interdigital capacitor coupling feed. The two meandered radiating strips also generate two λ/4 resonance modes at about 900 MHz and 2100 MHz to cover the GSM850/900/1800/1900/DCS/PCS/UMTS bands and the 2.4 GHz WLAN (IEEE 802.11b) band operations. Further, the proposed antenna has a simple planar structure and occupies only a small area of 10×40 mm on the system circuit board of the mobile phone. This proposed antenna with multiband, broadband matched impedance, stable radiation patterns, constant antenna gains, good radiation efficiency and compact size can be suitable for mobile phone applications.

Abstract:
We select a sample of 34 gamma-ray bursts (GRBs) whose $\Gamma_0$ values are derived with the onset peaks observed in the afterglow lightcurves (except for GRB 060218 whose $\Gamma_0$ is estimated with its radio data), and investigate the correlations among $\Gamma_0$, the isotropic peak luminosity ($L_{\rm iso}$), and the peak energy ($E_{\rm p,z}$) of the $\nu f_\nu$ spectrum in the cosmological rest frame. An analysis of pair correlations among these observables well confirms the results reported by the previous papers. More interestingly, a tight correlation among $L_{\rm iso}$, $E_{\rm p,z}$, and $\Gamma_0$ is found from a multiple regression analysis, which takes the form of $L_{\rm iso} \propto E_{\rm p,z}^{1.34\pm 0.14} \Gamma_0^{1.32\pm 0.19}$ or $E_{\rm p,z} \propto L_{\rm iso}^{0.55\pm 0.06}\Gamma_0^{-0.50\pm 0.17}$. Nine other GRBs whose $\Gamma_0$ are derived via the pair production opacity constraint also follow such a correlation. Excluding GRB 060218, the $L_{\rm iso}-E_{\rm p,z}-\Gamma_0$ correlation is valid, and it even holds in the jet co-moving frame. However, GRB 060218 deviates the $L^{'}_{\rm iso}-E^{'}_{\rm p}$ relation of typical GRBs in the jet co-moving frame with $3\sigma$. We argue that the $L_{\rm iso} - E_{\rm p, z} - \Gamma_0$ correlation may be more physical than the $L_{\rm iso} - E_{\rm p,z}$ correlation, since physically the relationship between the observed $L_{\rm iso}$ and $E_{\rm p,z}$ not only depends on radiation physics, but also depends on the bulk motion of the jet. We explore the possible origins of this correlation and discuss its physical implications for understanding GRB jet composition and radiation mechanism.

Abstract:
High power lasers (> +21dBm) have gradually become the common solution for signal transmitting systems including regional cable television, Fiber To The Home (FTTH), and gigabit passive optical networks (G- PONs) due to their ability to generate signals that can be transmitted over long distances. However, if protective design is not implemented in the facility at the client side, users may be exposed to health hazards such as eye damage from these high-power lasers. High-power optical adapters with laser shutter use metal masks to prevent eye exposure to direct laser beams. They have progressively replaced conventional optical adapters and entered the market mainstream. Our study uses the Elastic-Plastic theory together with parametric design to investigate the effect of geometry on the initial spring-back angle of a laser shutter. Once the force stabilizes, the angles of the initial spring back are found to be the same as the simulated results for several attempts. In our study, it is observed that factors including the thickness of the metal masking plate, the initial design angle, the stiffness of the material and the boundary conditions have significant influence on the spring back angle. These can be used as references in design control.