Representation of Cyclotomic Fields and Their Subfields

Let \K be a finite extension of a characteristic zero field \F. We say that the pair of $n\times n$ matrices $(A,B)$ over \F represents \K if \K \cong \F[A]/ where \F[A] denotes the smallest subalgebra of M_n(\F) containing $A$ and $$ is an ideal in \F[A] generated by $B$. In particular, $A$ is said to represent the field \K if there exists an irreducible polynomial q(x)\in \F[x] which divides the minimal polynomial of $A$ and \K \cong \F[A]/. In this paper, we identify the smallest circulant-matrix representation for any subfield of a cyclotomic field. Furthermore, if $p$ is any prime and \K is a subfield of the $p$-th cyclotomic field, then we obtain a zero-one circulant matrix $A$ of size $p\times p$ such that (A,\J) represents \K, where \J is the matrix with all entries 1. In case, the integer $n$ has at most two distinct prime factors, we find the smallest 0-1 companion-matrix that represents the $n$-th cyclotomic field. We also find bounds on the size of such companion matrices when $n$ has more than two prime factors.Comment: 17 page

Comparative study of antibacterial activity of two different earthworm species, Perionyx excavatus and Pheretima posthuma against pathogenic bacteria

Disease outbreaks are being increasingly recognized as a significant constraint on aquaculture production and trade affecting the economic development of the sector in many countries. Extracting and using biologically active compounds from earthworms has traditionally been practiced by indigenous people throughout the world. The aim of the present study was to shown antimicrobial activity through earthworm extract against fish bacterial pathogens. In total, 8 bacterial strains i.e. 6 gram negative viz. Aeromonas hydrophila, Pseudomonas aeruginosa, P. fluorescens, E.coli, Enterobacter aerogens and Shigella sp. and 2 gram positive viz. Staphylococcus aureus and Micrococcus luteus were identified. The extract of earthworm Perionyx excavatus, Pheretima posthuma were prepared and antimicrobial activity of the extract was determined by antimicrobial well diffusion assay. After 24 hrs of incubation period, it was observed that earthworm extract showed antibacterial activity against isolated bacterial strains. Among earthworm extract of two different species, the maximum zone of inhibition was shown against A. hydrophila by Perionyx excavatus (18.33± 0.66 mm) and P. posthuma (16.66±0.33). P. excavatus showed antibacterial activity against all pathogenic bacteria except Shigella spp. However on the other hand, P.posthuma showed antibacterial activity against A. hydrophila, P. fluorescens, E.coli, and S. aureus. The study has proved that earthworm extract can be effectively used for suppression of bacterial infection in fishes and that it can used as potential antimicrobial drug against commercial antibiotic resistance bacteria

Understanding the Modus Operandi of Class II KNOX Transcription Factors in Secondary Cell Wall Biosynthesis

Lignocellulosic biomass from the secondary cell walls of plants has a veritable potential to provide some of the most appropriate raw materials for producing second-generation biofuels. Therefore, we must first understand how plants synthesize these complex secondary cell walls that consist of cellulose, hemicellulose, and lignin in order to deconstruct them later on into simple sugars to produce bioethanol via fermentation. Knotted-like homeobox (KNOX) genes encode homeodomain-containing transcription factors (TFs) that modulate various important developmental processes in plants. While Class I KNOX TF genes are mainly expressed in the shoot apical meristems of both monocot and eudicot plants and are involved in meristem maintenance and/or formation, Class II KNOX TF genes exhibit diverse expression patterns and their precise functions have mostly remained unknown, until recently. The expression patterns of Class II KNOX TF genes in Arabidopsis, namely KNAT3, KNAT4, KNAT5, and KNAT7, suggest that TFs encoded by at least some of these genes, such as KNAT7 and KNAT3, may play a significant role in secondary cell wall formation. Specifically, the expression of the KNAT7 gene is regulated by upstream TFs, such as SND1 and MYB46, while KNAT7 interacts with other cell wall proteins, such as KNAT3, MYB75, OFPs, and BLHs, to regulate secondary cell wall formation. Moreover, KNAT7 directly regulates the expression of some xylan synthesis genes. In this review, we summarize the current mechanistic understanding of the roles of Class II KNOX TFs in secondary cell wall formation. Recent success with the genetic manipulation of Class II KNOX TFs suggests that this may be one of the biotechnological strategies to improve plant feedstocks for bioethanol production