arXiv subjects
Publications and source records attributed to Virender Singh.
In this paper we prove a normality criterion for the families of meromorphic functions involving sharing of functions. Our result generalizes some of the earlier results on Gu's normality criterion.
When crowdsourcing systems are used in combination with machine inference systems in the real world, they benefit the most when the machine system is deeply integrated with the crowd workers. However, if researchers wish to integrate the crowd with "off-the-shelf" machine classifiers, this deep integration is not always possible. This work explores two strategies to increase accuracy and decrease cost under this setting. First, we show that reordering tasks presented to the human can create a significant accuracy improvement. Further, we show that greedily choosing parameters to maximize machine accuracy is sub-optimal, and joint optimization of the combined system improves performance.
In this paper we prove the result: Let $\mathcal{F}$ be a family of meromorphic functions on a domain $Ω$ such that every pair of members of $\mathcal{F}$ shares a set $S:=\left\{ψ_1(z), ψ_2(z), ψ_3(z) \right\}$ in $Ω$, where $ψ_j(z), \ j=1,2,3$ is meromorphic in $Ω.$ If for every $f\in \mathcal{F}$, $f(z_0)\neq ψ_i (z_0)$ whenever $ψ_i(z_0)=ψ_j(z_0)$ for $i,j\in \left\{1,2,3 \right\}(i\neq j)$ and $z_0\in Ω,$ then $\mathcal{F}$ is normal in $Ω$. This result generalizes a result of M.Fang and W.Hong [Some results on normal family of meromorphic functions, Bull. Malays. Math. Sci. Soc. (2)23 (2000),143-151,] and in particular, it generalizes the most celebrated theorem of Montel-the Montel's theorem.
In this paper, we prove normality criteria for families of meromorphic functions involving sharing of a holomorphic function by a certain class of differential polynomials. Results in this paper extends the works of different authors carried out in recent years.
In this paper, we prove two normality criteria for a family of meromorphic functions. The first criterion extends a result of Fang and Zalcman[Normal families and shared values of meromorphic functions II, Comput. Methods Funct. Theory, 1(2001), 289 - 299] to a bigger class of differential polynomials whereas the second one leads to some counterexamples to the converse of the Bloch's principle.