Nguyen Thai An

Surface enhanced Raman scattering of melamine on silver substrate: An experimental and DFT study.

Surface enhanced Raman spectroscopy (SERS) is a powerful vibrational spectroscopy technique that allows amplifying weak Raman signals by the excitation of localized surface plasmons. In this paper, we used nanoscale roughened silver surface to enhance Raman signals of melamine analyte. Silver nanoparticles were synthesized by wet-chemical method and employed for SERS measurement. Theoretically, optimized geometries and vibrational frequencies of melamine and melamine absorbed on silver-cluster surface were calculated by using the B3LYP/6-31G(d) method. Then, the predicted spectrograms are compared with the experimental Raman spectra. As a result, Raman signals of melamine were dramatically enhanced by using obtained silver nanoparticles as the substrate. Typical peaks of melamine at 676 and 983cm(-1) were enhanced and could be obviously observed in experiments. The calculated vibrational frequencies seem to relatively coincide with the experimental values. SERS effect of melamine on nanoscale silver was also explained by analyzing molecular frontier orbitals (HOMO and LUMO) melamine-silver complexes.

Minimal Time Functions and the Smallest Intersecting Ball Problem with Unbounded Dynamics

The smallest enclosing circle problem introduced in the nineteenth century by Sylvester asks for the circle of smallest radius enclosing a given set of finite points in the plane. An extension of this problem, called the smallest intersecting ball problem, was also considered recently: given a finite number of nonempty closed subsets of a normed space, find a ball with the smallest radius that intersects all of the sets. In this paper, we initiate the study of minimal time functions generated by unbounded dynamics and discuss their applications to further extensions of the smallest enclosing circle problem. This approach continues our effort in applying convex and nonsmooth analysis to the well-established field of facility location.

Convergence analysis of a proximal point algorithm for minimizing differences of functions

Abstract Several optimization schemes have been known for convex optimization problems. However, numerical algorithms for solving nonconvex optimization problems are still underdeveloped. A significant progress to go beyond convexity was made by considering the class of functions representable as differences of convex functions. In this paper, we introduce a generalized proximal point algorithm to minimize the difference of a nonconvex function and a convex function. We also study convergence results of this algorithm under the main assumption that the objective function satisfies the Kurdyka–ᴌojasiewicz property.

Constructions of Solutions to Generalized Sylvester and Fermat–Torricelli Problems for Euclidean Balls

The classical problem of Apollonius is to construct circles that are tangent to three given circles in the plane. This problem was posed by Apollonius of Perga in his work “Tangencies.” The Sylvester problem, which was introduced by the English mathematician J.J. Sylvester, asks for the smallest circle that encloses a finite collection of points in the plane. In this paper, we study the following generalized version of the Sylvester problem and its connection to the problem of Apollonius: given two finite collections of Euclidean balls, find the smallest Euclidean ball that encloses all the balls in the first collection and intersects all the balls in the second collection. We also study a generalized version of the Fermat–Torricelli problem stated as follows: given two finite collections of Euclidean balls, find a point that minimizes the sum of the farthest distances to the balls in the first collection and shortest distances to the balls in the second collection.

The Log-Exponential Smoothing Technique and Nesterov's Accelerated Gradient Method for Generalized Sylvester Problems

The Sylvester or smallest enclosing circle problem involves finding the smallest circle enclosing a finite number of points in the plane. We consider generalized versions of the Sylvester problem in which the points are replaced by sets. Based on the log-exponential smoothing technique and Nesterov’s accelerated gradient method, we present an effective numerical algorithm for solving these problems.