The Cauchy problem for the Liouville equation and Bryant surfaces

We give a construction that connects the Cauchy problem for the 2-dimensional elliptic Liouville equation with a certain initial value problem for mean curvature one surfaces in hyperbolic 3-space H³, and solve both of them. We construct the unique mean curvature one surface in H³ that passes through a given curve with a given unit normal along it, and provide diverse applications. In particular, topics such as period problems, symmetries, finite total curvature, planar geodesics, rigidity, etc. are treated for these surfaces.     

Application of Multicriteria Methodology in the Development of Improved RP-LC-DAD for Determination of Rizatriptan and Its Degradation Products

A simple, rapid, and stability-indicating reversed-phase high-performance liquid chromatographic (LC) method for analysis for dutasteride has been successfully developed. Chromatography was performed on a 150 mm × 4.6 mm C₁₈ column with acetonitrile–water 60:40 (v/v) as isocratic mobile phase at 1.0 mL min⁻¹. Ultraviolet detection of dutasteride was at 210 nm. Its retention time was approximately 10 min and its peak was symmetrical. Response was a linear function of concentration over the range 0.2–1 μg mL⁻¹ (R ² = 0.997) and the limits of detection and quantitation were was 0.05 and 0.10 μg mL⁻¹, respectively. The method was validated for linearity, precision, repeatability, sensitivity, and selectivity. Selectivity was validated by subjecting dutasteride stock solution to photolytic, acidic, basic, oxidative, and thermal degradation. The peaks from the degradation products did not interfere with that from dutasteride. The method was used to quantify dutasteride in pharmaceutical preparations.

     

Adaptive Interpolation by Scaled Multiquadrics

We present an adaptive method to extract shape-preserving information from a univariate data sample. The behavior of the signal is obtained by interpolating at adaptively selected few data points by a linear combination of multiquadrics with variable scaling parameters. On the theoretical side, we give a sufficient condition for existence of the scaled multiquadric interpolant. On the practical side, we give various examples to show the applicability of the method.     

Construction of generators of quasi-interpolation operators of high approximation orders in spaces of polyharmonic splines

The paper presents a simple procedure for the construction of quasi-interpolation operators in spaces of m-harmonic splines in R^d, which reproduce polynomials of high degree. The procedure starts from a generator ?₀, which is easy to derive but with corresponding quasi-interpolation operator reproducing only linear polynomials, and recursively defines generators ?₁,?₂,…,?_m−1 with corresponding quasi-interpolation operators reproducing polynomials of degree up to 3,5,…,2m−1 respectively. The construction of ?_j from ?_j−1 is explicit, simple and independent of m. The special case d=1 and the special cases d=2,m=2,3,4 are discussed in details.     

Analysis of optodynamic sources generated during laser-induced breakdown in water

We propose a method for evaluating the size of the laser-induced breakdown region in water based on the detection and analysis of optodynamic waves. The breakdown region is an optodynamic source of pressure waves that propagate into the surrounding liquid as an ultrasonic pulse. In the experiment the optical breakdown was generated by a standard ophthalmic Nd:YAG laser with a pulse duration of 10 ns and a maximum energy per pulse of 10 mJ. The pulses were detected inside the liquid with a laser-beam deflection probe. The waveforms were captured in the far-field and analyzed. The analysis provides information about the apparent size of the optodynamic source, which is directly related to the size of the breakdown region. The proposed method can be adapted for online monitoring.