Bulletin of the Iranian Mathematical Society

Let $$G_{2}$$ be a group which acts trivially on an abelian group $$G_{1}$$ . According to the Schreier’s Theorem, each 2-cocycle $$\varepsilon \in Z^{2}(G_{2},G_{1})$$ determines a group $$G_{\varepsilon }$$ which is a central extension of $$G_{1}$$ by $$G_{2}$$ , and we will denote this group by $$G_{1}\underset{\varepsilon }{\times }G_{2} $$ and call it the perturbed direct product of $$G_{1}$$ by $$G_{2}$$ under $$\varepsilon $$ . The aim of this paper is to study properties of the perturbed direct products. For two distinct 2-cocycles $$\varepsilon _{1}$$ and $$\varepsilon _{2}$$ , we find necessary and sufficient conditions for $$G_{1}\underset{\varepsilon _{1}}{\times }G_{2} $$ to be isomorphic to $$G_{1}\underset{\varepsilon _{2}}{\times }G_{2}$$ . Furthermore, we obtain some results about decompositions for a given perturbed direct product $$G_{1}\underset{ \varepsilon }{\times }G_{2}$$ when $$G_{1}$$ or $$G_{2}$$ is a nontrivial direct product.

The concept of topological gyrogroups is a generalization of a topological group. In this work, ones prove that a topological gyrogroup G is metrizable iff G has an $$\omega ^{\omega }$$ -base and G is Fréchet–Urysohn. Moreover, in topological gyrogroups, every (countably, sequentially) compact subset being strictly (strongly) Fréchet–Urysohn and having an $$\omega ^{\omega }$$ -base are all weakly three-space properties with H a closed L-subgyrogroup.

In this article, we use mathematical induction to derive the representation of the solution of conformable impulsive linear differential equations with constant coefficients. We present the existence of solutions to impulsive nonlinear differential equations with constant coefficients under mild conditions on the nonlinear term. In addition, we consider the concepts of Ulam stability for this type of equation and give Ulam–Hyers and Ulam–Hyers–Rassias stability results. Finally, we give examples to verify our theoretical results.

Trong bài báo này, chúng tôi thu được các ước lượng độ dốc Lorentz có trọng số cục bộ như sau: $$\begin{aligned} g^{-1}\left( {\mathcal {M}}_1(\mu ) \right) \in L_{w,{\text {loc}}}^{q,r}(\Omega ) \Longrightarrow |Du| \in L_{w,{\text {loc}}}^{q,r}(\Omega ) \end{aligned}$$ cho các nghiệm yếu của một lớp các phương trình elliptic quasilinear không đồng nhất với dữ liệu đo đạc $$\begin{aligned} -\text {div} ~\! \left( a\left( \left| \nabla u \right| \right) \nabla u \right) = \mu , \end{aligned}$$ trong đó $$g(t)= t a(t)$$ với $$t\ge 0$$ và $$\begin{aligned} {\mathcal {M}}_1(\mu )(x):=\sup _{r>0}\frac{r|\mu |(B_r(x))}{|B_r(x)|}, \quad x\in {\mathbb {R}}^{n}. \end{aligned}$$ Hơn nữa, chúng tôi nhận thấy rằng hai ví dụ tự nhiên và đơn giản của các hàm g(t) trong công trình này là $$\begin{aligned} g(t)=t^{p-1} ~~(p\text{-Laplace } \text{ phương trình) }~~~~~~ \text{ và } ~~~~~~ g(t)=t^{p-1 }\log ^\alpha \big ( 1+t\big ) \quad \text{ cho }~\alpha > 0. \end{aligned}$$ Thực tế, ví dụ tổng quát và thú vị hơn có liên quan đến điều kiện tăng trưởng (p, q) bằng cách ghép nối thích hợp các đa thức. Chúng tôi nhận thấy rằng các kết quả của chúng tôi cải thiện các kết quả đã biết cho các phương trình như vậy.

Barycentric rational finite differences (RFD), as an application of linear barycentric interpolants, have been used to develop numerical integrators for solving ordinary differential equations. The class of SDBDF-type methods based on RFD has been already introduced. In this paper, we are interested in introducing an extended version of this class of the methods using future-step point technique. This extension improves the stability and accuracy properties of the methods as verified theoretically and numerically.

Let (V, 0) be a hypersurface with an isolated singularity at the origin defined by the holomorphic function $$f: (\mathbb {C}^n, 0)\rightarrow (\mathbb {C}, 0)$$ . We introduce a new derivation Lie algebra associated to (V, 0). The new Lie algebra is defined by the ideal of antiderivatives with respect to the Tjurina ideal of (V, 0). More precisely, let $$I = (f, \frac{\partial f}{\partial x_1},\ldots , \frac{\partial f}{\partial x_n})$$ and $$\Delta (I):= \{g\mid g,\frac{\partial g}{\partial x_1},\ldots , \frac{\partial g}{\partial x_n}\in I\}$$ , then $$A^\Delta (V):= \mathcal O_n/\Delta (I)$$ and $$L^\Delta (V):= \textrm{Der}(A^\Delta (V),A^\Delta (V))$$ . Their dimensions as a complex vector space are denoted as $$\beta (V)$$ and $$\delta (V)$$ , respectively. $$\delta (V)$$ is a new invariant of singularities. In this paper we study the new local algebra $$A^\Delta (V)$$ and the derivation Lie algebra $$L^\Delta (V)$$ , and also compute them for fewnomial isolated singularities. Moreover, we formulate sharp lower estimation conjectures for $$\beta (V)$$ and $$\delta (V)$$ when (V, 0) are weighted homogeneous isolated hypersurface singularities. We verify these conjectures for a large class of singularities. Lastly, we provide an application of $$\beta (V)$$ and $$\delta (V)$$ to distinguishing contact classes of singularities.