Identity transformations, used as skip-connections in residual networks, directly connect convolutional layers close to the input and those close to the output in deep neural networks, improving information flow and thus easing the training. In this paper, we introduce two alternative linear transforms, orthogonal transformation and idempotent transformation. According to the definition and pro...

We investigate the extension of modal logics by bisimulation quantifiers and present a class of modal logics which is decidable when augmented with bisimulation quantifiers. These logics are refered to as the idempotent transduction logics and are defined using the programs of propositional dynamic logic including converse and tests. This is a nontrivial extension of the decidability of the pos...

The notion of intuitinitic (T, S)-fuzzy submodule in Module is introduced, and related properties are investigated. Mathematics Subject Classification: 08A72, 13C99

let $r$ be a commutative ring with identity and let $m$ be an $r$-module. a proper submodule $p$ of $m$ is called strongly prime submodule if $(p + rx : m)ysubseteq p$ for $x, yin m$, implies that $xin p$ or $yin p$. in this paper, we study more properties of strongly prime submodules. it is shown that a finitely generated $r$-module $m$ is artinian if and only if $m$ is noetherian and every st...

In this article, we formalize a free Z-module and its property. In particular, we formalize the vector space of rational field corresponding to a free Z-module and prove formally that submodules of a free Z-module are free. Z-module is necassary for lattice problems LLL (Lenstra, Lenstra and Lovász) base reduction algorithm and cryptographic systems with lattice [20]. Some theorems in this arti...

In this paper we characterize the radical of an arbitrary‎ ‎submodule $N$ of a finitely generated free module $F$ over a‎ ‎commutatitve ring $R$ with identity‎. ‎Also we study submodules of‎ ‎$F$ which satisfy the radical formula‎. ‎Finally we derive‎ ‎necessary and sufficient conditions for $R$ to be a‎ ‎Pr$ddot{mbox{u}}$fer domain‎, ‎in terms of the radical of a‎ ‎cyclic submodule in $Rbigopl...