|We continue the analysis of (r; q)-polycycles, i.e., planar graphs G that admit a realization on the plane such that all internal vertices have degree q, all boundary vertices have degree at most q, and all internal faces are combinatorial r-triangles; moreover, the vertices, edges, and internal faces form a cell complex. Two extremal problems related to chemistry are solved: the description o...

We give a criterion for when idempotents of a ring R which commute modulo the Jacobson radical J(R) can be lifted to commuting idempotents of R. If such lifting is possible, we give extra information about the lifts. A “half-commuting” analogue is also proven, and this is used to give sufficient conditions for a ring to have the internal exchange property. In particular, we show that if R/J(R) ...

We study how external versus internal innovations promote economic growth through a tractable endogenous growth framework with multiple innovation sizes, multi-product firms, and entry/exit. Firms invest in external R&D to acquire new product lines and in internal R&D to improve their existing product lines. A baseline model derives the theoretical implications of weaker scaling for external R&...

Fifty isolates of dry root rot pathogen Rhizoctonia bataticola collected from different agro-climatic zones in India were evaluated for the degree of subdivision in isolates level. The isolates showed variability in pathogenicity test. Genetic characteristics were analyzed based on the sequence of the rDNA-internal transcribed spacer (ITS) region. The phylogenetic tree based on rDNA-ITS analysi...