The gallium analogue of the soluble Prussian blue with the formula KGa[Fe(CN)6]·nH2O is synthesized and structurally characterized. A simple aqueous synthetic procedure for preparing nanoparticles of this novel coordination polymer is reported. The stability, in vitro ion exchange with ferrous ions, cytotoxicity, and cellular uptake of such nanoparticles coated with poly(vinylpyrrolidone) are investigated for potential applications of delivering Ga(3+) ions into cells or removing iron from cells.

In this article, we introduce two families of novel fractional θ-methods by constructing some new generating functions to discretize the Riemann-Liouville fractional calculus operator I α with a second order convergence rate. A new fractional BT-θ method connects the fractional BDF2 (when θ = 0) with fractional trapezoidal rule (when θ = 1/2), and another novel fractional BN-θ method joins the fractional BDF2 (when θ = 0) with the second order fractional Newton-Gregory formula (when θ = 1/2). To deal with the initial singularity, correction terms are added to achieve an optimal convergence order. In addition, stability regions of different θ-methods when applied to the Abel equations of the second kind are depicted, which demonstrate the fact that the fractional θ-methods are A(ϑ)-stable. Finally, numerical experiments are implemented to verify our theoretical result on the convergence analysis.

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We prove new criteria for normality for holomorphic mappings into the complex projective space using the generalized Zalcman lemma. This improves previous results in one complex variable. An example is included to complement our theory.

This paper describes a mechanism design methodology that draws plane curves which have finite Fourier series parameterizations, known as trigonometric curves. We present three ways to use the coefficients of this parameterization to construct a mechanical system that draws the curve. One uses Scotch yoke mechanisms for each of the terms in the coordinate trigonometric functions, which are then added using a belt or cable drive. The second approach uses two-coupled serial chains obtained from the coordinate trigonometric functions. The third approach combines the coordinate trigonometric functions to define a single-coupled serial chain that draws the plane curve. This work is a version of Kempe's universality theorem that demonstrates that every plane trigonometric curve has a linkage which draws the curve. Several examples illustrate the method including the use of boundary points and the discrete Fourier transform to define the trigonometric curve.

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We study expansion of reproducing kernels for Hilbert spaces of holomorphic functions on the unit ball in C n with values in the antisymmetric tensor of the tangent and the cotangent spaces. As an application we find the composition series for the analytic continuation of certain families of holomorphic discrete series.

A simple one-step method for preparing biocompatible nanoparticles of gadolinium ferrocyanide coordination polymer KGd(H2O)2[Fe(CN)6]·H2O is reported. The crystal structure of this coordination polymer is determined by X-ray powder diffraction using the bulk materials. The stability, cytotoxicity, cellular uptake, and MR phantom and cellular imaging studies suggest that this coordination-polymer structural platform offers a unique opportunity for developing the next generation of T1-weighted contrast agents with high relaxivity as cellular MR probes for biological receptors or markers. Such high-relaxivity MR probes may hold potential in the study of molecular events and may be used for in vivo MR imaging in biomedical research and clinical applications.