These are notes from introductory survey lectures given at the Institute for Studies in Theoretical Physics and Mathematics (IPM), Teheran, in 2008 and 2010. We present the definition and the fundamental properties of Fomin-Zelevinsky’s cluster algebras. Then we introduce quiver representations and show how they can be used to construct cluster variables, which are the canonical generators of c...

Reticulate, or non-bifurcating, evolution is now recognized as an important phenomenon shaping the histories of many organisms. It appears to be particularly common in plants, especially in ferns, which have relatively few barriers to intra- and interspecific hybridization. Reticulate evolutionary patterns have been recognized in many fern groups, though very few have been studied rigorously us...

This study was aimed to document the floristic attributes of pteridophytes District Tor Ghar; Khyber Pakhtunkhwa, Pakistan. Total 41 species distributed in 20 genera and 10 families were recognized. Dryopteridaceae largest family with 12 (29.26%). Second third large is Pteridaceae Thelypteridaceae (24.39%) 5 (12.19%) respectively. Like-wise, Dryopteris genus 6 (14.63%) species. The main Aspleni...

The current research was carried out to characterize the phytosociology of forests one Pakistan’s most valuable tree species (Deodar) across its native range. In this context, our main hypothesis that, along altitudinal gradient, we would find different plant communities that be driven by environmental variables (climatic, edaphic, and physiographic). Therefore, assess vegetation structure pure...

This paper summarizes a large number of simulations which relax the "symmetry" assumptions in the Goldie-Coldman treatment model, which was symmetrical with respect to two drugs considered for use together. The results are that, under relevant violations of the assumptions, nonalternating treatment schedules frequently outperform alternation and combination substantially. The magnitudes of thes...

There are currently different techniques, such as CGH arrays, to study genetic variations in patients. CGH arrays analyze gains and losses in different regions in the chromosome. Regions with gains or losses in pathologies are important for selecting relevant genes or CNVs (copy-number variations) associated with the variations detected within chromosomes. Information corresponding to mutations...