In 1978, Rivest, Adleman and Dertouzos asked for algebraic systems for which useful privacy homomorphisms exist. To date, the only acknownledged result is noise based encryption combined with bootstrapping. Before that, there were several failed attempts. We prove that fully homomorphic schemes are impossible for several algebraic structures. Then we develop a characterisation of all fully homo...

in this paper, based on sinh-cosh method and sinh-gordon expansion method,families of solutions of (2+1)-dimensional breaking soliton equation are obtained.these solutions include jacobi elliptic function solution, soliton solution,trigonometric function solution.

The High-Dimensional Model Representation (HDMR) technique is a family of approaches to efficiently interpolate high-dimensional functions. RS(Random Sampling)-HDMR is a practical form of HDMR based on randomly sampling the overall function, and utilizing orthonormal polynomial expansions to approximate the RS-HDMR component functions. The determination of the expansion coefficients for the com...

In this article, we propose a method for analyzing the spatial variations in the range expansion of the pine processionary moth (PPM), an invasive species in France. Based on binary measurements - the presence or absence of PPM nests - the proposed method allows us to infer the local effect of the environment on PPM population expansion. This effect is estimated at each position x using a param...

background: in vitro model studies are becoming increasingly popular for experimental research designs. they include isolation and expansion of cells of a particular tissue, such as the nervous tissue which contributes to understanding the underlying mechanisms in many pathologies. it enables  the scrutinization of intracellular signaling pathways responsible for cell death. objectives: in the ...

In this paper, a new generalized F-expansion method is proposed to seek exact solutions of nonlinear evolution equations. With the aid of symbolic computation, we choose the (2+1)-dimensional dispersive long wave equations to illustrate the validity and advantages of the proposed method. As a result, many new and more general exact non-traveling wave and coefficient function solutions are obtai...

Consider a power series f ∈ R[[z]], which is obtained by a precise mathematical construction. For instance, f might be the solution to some differential or functional initial value problem or the diagonal of the solution to a partial differential equation. In cases when no suitable method is beforehand for determining the asymptotics of the coefficients fn, but when many such coefficients can b...