The Wells exact sequence for the automorphism group of a group extension

AbstractWe obtain an explicit description of the Wells map for the automorphism group of a group extension in the full generality and investigate the dependency of this map on group extensions. Some applications are given

A class of finite pp-groups and the normalized unit groups of group algebras

Let $p$ be a prime and $\mathbb{F}_p$ be a finite field of $p$ elements. Let $\mathbb{F}_pG$ denote the group algebra of the finite $p$-group $G$ over the field $\mathbb{F}_p$ and $V(\mathbb{F}_pG)$ denote the group of normalized units in $\mathbb{F}_pG$. Suppose that $G$ is a finite $p$-group given by a central extension of the form $$1\longrightarrow \mathbb{Z}_{p^n}\times \mathbb{Z}_{p^m} \longrightarrow G \longrightarrow \mathbb{Z}_p\times \cdots\times \mathbb{Z}_p \longrightarrow 1$$ and $G'\cong \mathbb{Z}_p$, $n, m\geq 1$ and $p$ is odd. In this paper, the structure of $G$ is determined. And the relations of $V(\mathbb{F}_pG)^{p^l}$ and $G^{p^l}$, $\Omega_l(V(\mathbb{F}_pG))$ and $\Omega_l(G)$ are given. Furthermore, there is a direct proof for $V(\mathbb{F}_pG)^p\bigcap G=G^p$

The kernels of powers of linear operator via Weyr characteristic

The adjoint of a matrix in the Lie algebra associated with a matrix algebra is a fundamental operator, which can be generalized to a more general operator $\varphi_{AB}: X\rightarrow AX-XB$ by two matrices $A$ and $B$. The well-known dimensional formula of the kernel of the adjoint of a matrix is due to Frobenius. The dimensional formulas for the kernels of each power of the operator $\varphi_{AB}$ were given in terms of the Segre characteristics of these two matrices by the second and third authors in this paper and their collaborators. The referee encourage the authors to try to express the dimensional formulas in terms of the characteristics of Weyr. This paper provides an alternative approach to this problem via Weyr characteristic. We obtain the dimensional formulas for kernels of each power of the operator in terms of the characteristics of Weyr. Furthermore, the basis for kernels of powers of the operator is described explicitly. As a consequence, for arbitrary square matrices $A$ and $B$ over an algebraically closed field, the dimension of kernels of each power of the operator $\varphi_{A-\lambda I,B}$ for eigenvalues $\lambda$ of $\varphi_{AB}$ can be viewed as a similarity invariant of the operator $\varphi_{AB}$, so we characterise the operator within similarity, which should be of interest to a number of people (including Physicists)

Mendelian randomization based on immune cells in diabetic nephropathy

BackgroundDKD, a leading cause of chronic kidney and end-stage renal disease, lacks robust immunological research. Recent GWAS utilizing SNPs and CNVs has shed light on immune mechanisms of kidney diseases. However, DKD’s immunological basis remains elusive. Our goal is to unravel cause-effect relationships between immune cells and DKD using Mendelian randomization.MethodologyWe analyzed FinnGen data (1032 DKD cases, 451,248 controls) with 731 immunocyte GWAS summaries (MP=32, MFI=389, AC=118, RC=192). We employed forward and reverse Mendelian randomization to explore causal links between immune cell traits and DKD. Sensitivity analysis ensured robustness, heterogeneity checks, and FDR correction minimized false positives.ResultsOur study explored the causal link between diabetic nephropathy (DKD) and immunophenotypes using two-sample Mendelian Randomization (MR) with IVW. Nine immunophenotypes were significantly associated with DKD at p<0.05 after FDR correction. Elevated CD24, CD3 in Treg subsets, CD39+ CD4+, and CD33− HLA DR− AC correlated positively with DKD risk, while CD27 in B cells and SSC−A in CD4+ inversely correlated. Notably, while none showed significant protection, further research on immune cells’ role in DKD may provide valuable insights.ConclusionThe results of this study show that the immune cells are closely related to DKD, which may be helpful in the future clinical study

On groups of automorphisms of nilpotent pp-groups of finite rank

summary:Let $\alpha$ and $\beta$ be automorphisms of a nilpotent $p$-group $G$ of finite rank. Suppose that $\langle (\alpha \beta (g))(\beta \alpha (g))^{-1}\colon g\in G\rangle$ is a finite cyclic subgroup of $G$, then, exclusively, one of the following statements holds for $G$ and $\Gamma$, where $\Gamma$ is the group generated by $\alpha$ and $\beta$. \item {(i)} $G$ is finite, then $\Gamma$ is an extension of a $p$-group by an abelian group. \item {(ii)} $G$ is infinite, then $\Gamma$ is soluble and abelian-by-finite

Achieving strength-ductility balance in a casting non-equiatomic FeCoNi based medium-entropy alloy via Al and Ti combination addition

The conflict of strength and ductility has always been a huge challenge when developing advanced structural materials. The introduction of nano-scaled precipitates into metallic materials through composition design is a well-established method for achieving the strength-ductility balance. Herein, a series of non-equiatomic FeCoNi based medium-entropy alloys (MEAs) with Al and Ti combination addition were prepared by vacuum induction melting, and its phase constitution, microstructural changes and mechanical properties were systemically studied. The microstructure of Al/Ti co-doped alloys are composed of BCC phase + FCC phase + L21 nanoparticles structure, in which the L21 nanoparticles are homogeneously distributed and coherent with matrix. By tailoring Al/Ti ratio, the as-cast Ni0.6CoFe1.6 MEA presents a good strength-ductility combination. Among them, the Al0.2Ti0.1 MEA possesses higher yield strength of 837.7 ± 18.9 MPa and ultimate tensile strength of 1305.4 ± 22.0 MPa, as well as maintains an acceptable failure strain of 12.3 ± 0.7%. The improvement of yield strength is originated from the contributions of the precipitation hardening, grain boundary strengthening, and solid solution strengthening. The present study provides a new avenue to design strong yet ductile MEAs for industrial applications

Študija ranljivosti okolja za občino Bohinj

summary:In this paper, we study the structure of polycyclic groups admitting an automorphism of order four on the basis of Neumann's result, and prove that if $\alpha$ is an automorphism of order four of a polycyclic group $G$ and the map $\varphi \colon G\rightarrow G$ defined by $g^{\varphi }=[g,\alpha ]$ is surjective, then $G$ contains a characteristic subgroup $H$ of finite index such that the second derived subgroup $H''$ is included in the centre of $H$ and $C_{H}(\alpha ^{2})$ is abelian, both $C_{G}(\alpha ^{2})$ and $G/[G,\alpha ^{2}]$ are abelian-by-finite. These results extend recent and classical results in the literature