Building upon the previous research that solved neutron diffusion equations in simplified slab geometry, this study advances the field by addressing the more complex cylindrical geometry, focusing on neutron diffusion equations that are coupled with delayed neutrons in cylindrical reactors of fractional order. The method of solving used integrates the technique of residual power series (RPS) with the Laplace transform (LT) method. Anomalous neutron behavior is explained by examining the non-Gaussian scenario with various fractional parameters α. The LRPSM Laplace transform and residual power series method employed in this approach eliminates the complex difficulties. This simplicity makes the method particularly coherent with different fractional calculus applications. To validate the proposed method, numerical simulations are conducted with two different initial conditions representing distinct scenarios. The obtained results are presented in suitable tables and figures. It should be emphasized that this system is solved for the first time utilizing fractional calculus techniques. The outcomes are consistent with those achieved using the Adomian decomposition method.

In the present paper, we investigate the geometry and topology of warped product Legendrian submanifolds in Sasakian space forms ${\mathbb{D}}^{2n+1}\left(ϵ\right)$ and obtain the first Chen inequality that involves extrinsic invariants like the mean curvature and the length of the warping functions. This inequality also involves intrinsic invariants ($\delta$-invariant and sectional curvature). In addition, an integral bound is provided for the Bochner operator formula of compact warped product submanifolds in terms of the gradient Ricci curvature. Some new results on mean curvature vanishing are presented as a partial solution to the well-known problem given by S.S. Chern.

Let ℜ be a unital algebra over a field $\mathbb{k}$ with $char\left(\mathbb{k}\right)\ne 2$, and let $\mathsf{ϝ},\xi ,\zeta :\Re \to \Re$ be linear mappings. We say that $\mathsf{ϝ}$ is a $\{\xi ,\zeta \}$-derivation if $\mathsf{ϝ}\left(\vartheta \varsigma \right)=\xi \left(\vartheta \right)\varsigma +\vartheta \zeta \left(\varsigma \right)=\zeta \left(\vartheta \right)\varsigma +\vartheta \xi \left(\varsigma \right)\mathrm{for}\mathrm{all}\vartheta ,\varsigma \in \Re .$ The mapping $\mathsf{ϝ}$ is said to be a Lie $\{\xi ,\zeta \}$-derivation if $\mathsf{ϝ}\left(\right[\vartheta ,\varsigma \left]\right)=\left[\xi \right(\vartheta ),\varsigma ]+[\vartheta ,\zeta (\varsigma \left)\right]\mathrm{for}\mathrm{all}\vartheta ,\varsigma \in \Re ,$ where $[\vartheta ,\varsigma ]=\vartheta \varsigma -\varsigma \vartheta$ denotes the Lie product. In this paper, we prove that if every Lie $\{\xi ,\zeta \}$-derivation on ℜ is necessarily a $\{\xi ,\zeta \}$-derivation, then the same property holds for the tensor product algebra $\Re \otimes \Im$, where ℑ is any commutative unital algebra. Moreover, every Lie $\{\xi ,\zeta \}$-derivation of a semiprime algebra is a $\{\xi ,\zeta \}$-derivation. As a consequence, Lie derivations on tensor products of semiprime algebras with commutative algebras reduce to derivations in the classical sense.

In this paper we study superderivations and Jordan superderivations of incidence superalgebras of locally finite ${\mathbb{Z}}_2$-graded posets. We prove that every superderivation can be expressed as a sum of an inner superderivation, an additive superderivation, and a ring superderivation; if the coefficient ring has inner derivations only, then every superderivation is inner. We also show that every Jordan superderivation decomposes uniquely as a sum of a superderivation and a proper Jordan superderivation that vanishes on the even part. These results extend classical results on incidence algebras to the supersetting.

This work presents a graded investigation of $\mathrm{\Theta }$-superderivations within the framework of Lie superalgebras, generalizing the $\mathrm{\Theta }$-derivation concept from ordinary Lie algebras to graded settings. For a given Lie superalgebra $\mathfrak{g}$, a linear mapping $\phi$ qualifies as a $\mathrm{\Theta }$-superderivation where $\mathrm{\Theta }$ is a superderivation such that $\phi \left([\eta ,\xi ]\right)=[\phi \left(\eta \right),\xi ]+{(-1)}^{\left|\phi \right|\left|\eta \right|}[\eta ,\mathrm{\Theta }\left(\xi \right)]$ for all homogeneous elements $\eta ,\xi \in \mathfrak{g}$. This formulation simultaneously encompasses ordinary superderivations and even components of graded centroids. We demonstrate that the collection ${sDer}^{*}\left(\mathfrak{g}\right)$ of all $\mathrm{\Theta }$-superderivations naturally carries the structure of a Lie superalgebra and admits the decomposition ${sDer}^{*}\left(\mathfrak{g}\right)=sDer\left(\mathfrak{g}\right)+C_{\overline{0}}\left(\mathfrak{g}\right)$, where $sDer\left(\mathfrak{g}\right)$ denotes the superderivation algebra and $C_{\overline{0}}\left(\mathfrak{g}\right)$ represents the even part of the graded centroid. For perfect or centerless Lie superalgebras, this sum becomes direct. In the particular case of finite-dimensional simple Lie superalgebras over algebraically closed fields of characteristic zero, we establish ${sDer}^{*}\left(\mathfrak{g}\right)=ad\left(\mathfrak{g}\right)\oplus \mathbb{F}·{id}_{\mathfrak{g}}$. Furthermore, a semidirect product decomposition ${sDer}^{*}\left(\mathfrak{g}\right)\cong sDer\left(\mathfrak{g}\right)⋉C_{\overline{0}}\left(\mathfrak{g}\right)$ holds whenever the center vanishes. Concrete illustrations involving the Heisenberg superalgebra, the super-Virasoro algebra, and low-dimensional examples are provided, complete with explicit matrix representations. Our findings extend classical derivation and centroid theories to the superalgebraic realm, laying groundwork for future implications in deformation theory and supersymmetric quantum mechanics.