Hybrid nanoparticles based on sulfides, oxides, and carbides

The methods for synthesis of hybrid nanoparticles based on sulfides, oxides, and carbides of heavy and transition metals were considered. The problem of the influence of the method of synthesis of the hybrid nanoparticles on their atomic structure, morphology of the nanomaterials, and functional properties was analyzed. The areas of practical use of the hybrid nanoparticles were proposed. © 2013 Springer Science+Business Media New York

Efficient Homomorphic Comparison Methods with Optimal Complexity

Comparison of two numbers is one of the most frequently used operations, but it has been a challenging task to efficiently compute the comparison function in homomorphic encryption (HE) which basically support addition and multiplication. Recently, Cheon et al. (Asiacrypt 2019) introduced a new approximate representation of the comparison function with a rational function, and showed that this rational function can be evaluated by an iterative algorithm. Due to this iterative feature, their method achieves a logarithmic computational complexity compared to previous polynomial approximation methods; however, the computational complexity is still not optimal, and the algorithm is quite slow for large-bit inputs in HE implementation. In this work, we propose new comparison methods with optimal asymptotic complexity based on composite polynomial approximation. The main idea is to systematically design a constant-degree polynomial $f$ by identifying the \emph{core properties} to make a composite polynomial $f\circ f \circ \cdots \circ f$ get close to the sign function (equivalent to the comparison function) as the number of compositions increases. We additionally introduce an acceleration method applying a mixed polynomial composition $f\circ \cdots \circ f\circ g \circ \cdots \circ g$ for some other polynomial $g$ with different properties instead of $f\circ f \circ \cdots \circ f$. Utilizing the devised polynomials $f$ and $g$, our new comparison algorithms only require $\Theta(\log(1/\epsilon)) + \Theta(\log\alpha)$ computational complexity to obtain an approximate comparison result of $a,b\in[0,1]$ satisfying $|a-b|\ge \epsilon$ within $2^{-\alpha}$ error. The asymptotic optimality results in substantial performance enhancement: our comparison algorithm on encrypted $20$-bit integers for $\alpha = 20$ takes $1.43$ milliseconds in amortized running time, which is $30$ times faster than the previous work

CONSTRUCTIVE APPROXIMATION © 2001 Springer-Verlag New York Inc. On Zeros of Polynomials Orthogonal over a Convex Domain

Abstract. We establish a discrepancy theorem for signed measures, with a given positive part, which are supported on an arbitrary convex curve. As a main application, we obtain a result concerning the distribution of zeros of polynomials orthogonal on a convex domain