On the Order of the Schur Multiplier of a Pair of Finite p-Groups

In 1998, G. Ellis defined the Schur multiplier of a pair $(G,N)$ of groups and mentioned that this notion is a useful tool for studying pairs of groups. In this paper, we characterize the structure of a pair of finite $p$-groups $(G,N)$ in terms of the order of the Schur multiplier of $(G,N)$ under some conditions.Comment: 11 pages, to appear in Journal of Advanced Research in Pure Mathematic

Groups whose non-normal subgroups are either nilpotent or minimal non-nilpotent

Let $\mathfrak{Nil}$ be the class of nilpotent groups and $G$ be a group. We call $G$ a meta-$\mathfrak{Nil}$-Hamiltonian group if any of its non-$\mathfrak{Nil}$ subgroups is normal. Also, we call $G$ a para-$\mathfrak{Nil}$-Hamiltonian group if $G$ is a non-$\mathfrak{Nil}$ group and every non-normal subgroup of $G$ is either a $\mathfrak{Nil}$-group or a minimal non-$\mathfrak{Nil}$ group. In this paper we investigate the class of finitely generated meta-$\mathfrak{Nil}$-Hamiltonian and para-$\mathfrak{Nil}$-Hamiltonian groups.Comment: 16 Page

Circumcision with the Plastibell Device in Hooded Prepuce or Glanular Hypospadias

Purpose. To retrospectively review our experience in infants with glanular hypospadias or hooded prepuce without meatal anomaly, who underwent circumcision with the plastibell device. Although circumcision with the plastibell device is well described, there are no reported experiences pertaining to hooded prepuce or glanular hypospadias that have been operated on by this technique. Materials and Methods. Between September 2002 and September 2008, 21 children with hooded prepuce (age 1 to 11 months, mean 4.6 months) were referred for hypospadias repair. Four of them did not have meatal anomaly. Their parents accepted this small anomaly and requested circumcision without glanuloplasty. In all cases, the circumcision was corrected by a plastibell device. Results. No complications occurred in the circumcised patients, except delayed falling of bell in one case that was removed by a surgeon, after the tenth day. Conclusion. Circumcision with the plastibell device is a suitable method for excision of hooded prepuce. It can also be used successfully in infants, who have miniglanular hypospadias, and whose parents accepted this small anomaly

The impact of the solubilizer of an element on the structure of a finite group

Let $G$ be a finite group, and let $x$ be an element of $G$. Denote by \Sol_G(x) the set of all $y \in G$ such that the group generated by $x$ and $y$ is soluble. We investigate the influence of \Sol_G(x) on the structure of $G$.Comment: 9 pages, research pape

Finite groups with Quaternion Sylow subgroup

In this paper we show that a finite group $G$ with Quaternion Sylow $2$-subgroup is $2$-nilpotent if, either $3\nmid |G|$ or $G$ is solvable and the order of its Sylow $2$-subgroup is strictly greater than $16$

Finite groups with Quaternion Sylow subgroup

In this paper we show that a finite group $G$ with Quaternion Sylow $2$-subgroup is $2$-nilpotent if, either $3\nmid |G|$ or $G$ is solvable and the order of its Sylow $2$-subgroup is strictly greater than $16$

PR-DARTS: Pruning-Based Differentiable Architecture Search

The deployment of Convolutional Neural Networks (CNNs) on edge devices is hindered by the substantial gap between performance requirements and available processing power. While recent research has made large strides in developing network pruning methods for reducing the computing overhead of CNNs, there remains considerable accuracy loss, especially at high pruning ratios. Questioning that the architectures designed for non-pruned networks might not be effective for pruned networks, we propose to search architectures for pruning methods by defining a new search space and a novel search objective. To improve the generalization of the pruned networks, we propose two novel PrunedConv and PrunedLinear operations. Specifically, these operations mitigate the problem of unstable gradients by regularizing the objective function of the pruned networks. The proposed search objective enables us to train architecture parameters regarding the pruned weight elements. Quantitative analyses demonstrate that our searched architectures outperform those used in the state-of-the-art pruning networks on CIFAR-10 and ImageNet. In terms of hardware effectiveness, PR-DARTS increases MobileNet-v2's accuracy from 73.44% to 81.35% (+7.91% improvement) and runs 3.87$\times$ faster.Comment: 18 pages with 11 figure