let $r=oplus_{nin bbb n_0}r_n$ be a noetherian homogeneous ring with local base ring $(r_0,frak{m}_0)$, $m$ and $n$ two finitely generated graded $r$-modules. let $t$ be the least integer such that $h^t_{r_+}(m,n)$ is not minimax. we prove that $h^j_{frak{m}_0r}(h^t_{r_+}(m,n))$ is artinian for $j=0,1$. also, we show that if ${rm cd}(r_{+},m,n)=2$ and $tin bbb n_0$, then $h^t_{frak{m}_0r}(h^2_{...

Bacteria and Archaea encode clustered, regularly interspaced, short palindromic repeat (CRISPR) systems to confer adaptive immunity to invasive viruses and plasmids. Recent studies of CRISPR systems revealed that diverse CRISPR-associated (Cas) interference modules often coexist in different organisms but functions of cas genes have not been dissected in any of these systems. The crenarchaeon S...

Exercise 1 Let A be a domain, α an injective endomorphism of A, D = A[∂;α, β] a skew polynomial ring, E ∈ An×n and F ∈ An×n, R = ∂ In − E ∈ Dn×n and R′ = ∂ In − F two matrices with entries in D, M = D1×n/(D1×nR) and M ′ = D1×n/(D1×nR′) two left D-modules respectively finitely presented by R and R′. 1. Describe the left D-modules M and M ′ in terms of generators and relations. 2. Prove that any ...

Laminins are a major constituent of all basement membranes. The polymerisation of laminins at the cell surface is mediated by the three short arms of the cross-shaped laminin heterotrimer. The short arms contain repeats of laminin-type epidermal growth factor-like (LE) domains, interspersed with globular domains of unknown function. A single LF domain is inserted between LE5 and LE6 of the lami...

Protein-based therapeutics feature large interacting surfaces. Protein folding endows structural stability to localised surface epitopes, imparting high affinity and target specificity upon interactions with binding partners. However, short synthetic peptides with sequences corresponding to such protein epitopes are unstructured in water and promiscuously bind to proteins with low affinity and ...

3-methyl-1-sulfonic acid imidazolium tetrachloroferrate {[Msim]FeCl4} as an efficient sulfur catalyst was prepared and applied, as an efficient catalyst, for the cross-aldol condensation reaction between cycloalkanones and arylaldehydes to give α,αʹ- bis(arylidene)cycloalkanones in high yields and short reaction times at 75 °C under solvent-free conditions. 3-methyl-1-sulfonic acid imidazolium...

We investigate a generalization of extending modules relative to a class of modules and a proper class of short exact sequences of modules.