Sub-exact sequence of rough groups

Rough Set Theory (RST) is an essential mathematical tool to deal with imprecise, inconsistent, incomplete information and knowledge Rough Some algebra structures, such as groups, rings, and modules, have been presented on rough set theory. The sub-exact sequence is a generalization of the exact sequence. In this paper, we introduce the notion of a sub-exact sequence of groups. Furthermore, we give some properties of the rough group and rough sub-exact sequence of groups.

THE FIRST U-EXTENSION MODULE AS CLASSES OF SHORT U-EXACT SEQUENCES

Inspired by the notions of the U-exact sequence introduced by Davvaz and Parnian-Garamaleky in 1999, and of the chain U-complex introduced by Davvaz and Shabani-Solt in 2002, Mahatma and Muchtadi-Alamsyah in 2017 developed the concept of the U-projective resolution and the U-extension module, which are the generalizations of the concept of the projective resolution and the concept of extension module, respectively. It is already known that every element of a first extension module can be identified as a short exact sequence. To the simple, there is a relation between the first extension module and the short exact sequence. It is proper to expect the relation to be provided in the U-version. In this paper, we aim to construct a one-one correspondence between the first U-extension module and the set consisting of equivalence classes of short U-exact sequence.Keywords: Chain U-complex, U-projective resolution, U-extension module

Equivariant splitting of the Hodge–de Rham exact sequence

Let X be an algebraic curve with a faithful action of a finite group G over a field k. We show that if the Hodge–de Rham short exact sequence of X splits G-equivariantly then the action of G on X is weakly ramified. In particular, this generalizes the result of Köck and Tait for hyperelliptic curves. We discuss also converse statements and tie this problem to lifting coverings of curves to the ring of Witt vectors of length 2.

Heimdallarchaea encodes profilin with eukaryotic-like actin regulation and polyproline binding

It is now widely accepted that the first eukaryotic cell emerged from a merger of an archaeal host cell and an alphaproteobacterium. However, the exact sequence of events and the nature of the cellular biology of both partner cells is still contentious. Recently the structures of profilins from some members of the newly discovered Asgard superphylum were determined. In addition, it was found that these profilins inhibit eukaryotic rabbit actin polymerization and that this reaction is regulated by phospholipids. However, the interaction with polyproline repeats which are known to be crucial for the regulation of profilin:actin polymerization was found to be absent for these profilins and was thus suggested to have evolved later in the eukaryotic lineage. Here, we show that Heimdallarchaeota LC3, a candidate phylum within the Asgard superphylum, encodes a putative profilin (heimProfilin) that interacts with PIP2 and its binding is regulated by polyproline motifs, suggesting an origin predating the rise of the eukaryotes. More precisely, we determined the 3D-structure of Heimdallarchaeota LC3 profilin and show that this profilin is able to: i) inhibit eukaryotic actin polymerization in vitro; ii) bind to phospholipids; iii) bind to polyproline repeats from enabled/vasodilator‐stimulated phosphoprotein; iv) inhibit actin from Heimdallarchaeota from polymerizing into filaments. Our results therefore provide hints of the existence of a complex cytoskeleton already in last eukaryotic common ancestor.

The New Results in n-injective Modules and n-projective Modules

In this paper, we introduce and clarify a new presentation between the n-exact sequence and the n-injective module and n-projective module. Also, we obtain some new results about them.