Suppose $f$ is a map from a non-empty finite set $X$ to a finite group $G$. Define the map $zeta^f_G: Glongrightarrow mathbb{N}cup {0}$ by $gmapsto |f^{-1}(g)|$. In this article, we show that for a suitable choice of $f$, the map $zeta^f_G$ is a character. We use our results to show that the solution function for the word equation $w(t_1,t_2,dots,t_n)=g$ ($gin G$) is a character, where $w(t_1,...

It is shown that if d ≥ 2, then every map φ : Ω ⊂ R → Rd of class L∞ can be approximated in the Lp-norm by a sequence of orientationpreserving diffeomorphims φn : Ω̄ → φn(Ω̄). These conclusions hold provided that Ω ⊂ Rd is open, bounded, and that 1 ≤ p < +∞. In addition, φn(Ω̄) is contained in the 1/n-neighborhood of the convex hull of φ(Ω). All these conclusions fail for Ω ⊂ R. The main ingredien...

Scalar product protocol aims at securely computing the dot product of two private vectors. As a basic tool, the protocol has been widely used in privacy preserving distributed collaborative computations. In this paper, at the expense of disclosing partial sum of some private data, we propose a linearly efficient even-dimension scalar product protocol (EDSPP) without employing expensive homomorp...

Let S, T be semilattices. Let us assume that if S is upper-bounded, then T is upper-bounded. A map from S into T is said to be a semilattice morphism from S into T if: (Def. 1) For every finite subset X of S holds it preserves inf of X . Let S, T be semilattices. Observe that every map from S into T which is meet-preserving is also monotone. Let S be a semilattice and let T be an upper-bounded ...