We present an explicit solution of the Ar T -system for arbitrary boundary conditions. For each boundary, this is done by constructing a network, i.e. a graph with positively weighted edges, and the solution is expressed as the partition function for a family of non-intersecting paths on the network. This proves in particular the positive Laurent property, namely that the solutions are all Laur...

The solutions to a change problem form restricted partitions. For one particular change problem, we look at the sequence representing the parity of these restricted partition values. It appears that the period of this sequence has not been studied. Through recurrences involving binomial coefficients, we find that the sequence has a period of 200.

For a given weight of a complex simple Lie algebra, the q-analog of Kostant’s partition function is a polynomial valued function in the variable q, where the coefficient of q is the number of ways the weight can be written as a nonnegative integral sum of exactly k positive roots. In this paper we determine generating functions for the q-analog of Kostant’s partition function when the weight in...

We consider the Itzykson-Zuber-Eynard-Mehta two-matrix model and prove that the partition function is an isomonodromic tau function in a sense that generalizes Jimbo-Miwa-Ueno’s [20]. In order to achieve the generalization we need to define a notion of tau-function for isomonodromic systems where the ad–regularity of the leading coefficient is not a necessary requirement.

Growing applications of phenylalanine dehydrogenase (PheDH) enzyme in the medical and pharmaceutical industries encourages researchers to seek simple, fast and economical alternative purification methods. With goal of finding a new technique, the extraction and purification of recombinant Bacillus badius PheDH in polyethylene glycol 6000 (PEG-6000) and ammonium sulfate aqueous two-phase systems...