Abstract. We will determine (up to equivalence) all of the integral positive definite Hermitian lattices in imaginary quadratic fields of class number 1 that represent all positive integers.

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The Eurasian water frog species and their geographic ranges have undergone considerable changes in the last four decades, but the Iranian populations have largely remained unknown. All the Iranian populations of water frogs, despite their vast distribution range have attributed to a single species: Rana ridibunda. In order to understand the phylogenetic relationships and taxonomic status of water frogs of Iran, we collected samples from many populations across the country and used the mitochondrial DNA sequence variation. A data set with a final sequence length of 616 nucleotides was generated for Cyt b from 70 individuals of Pelophylax in which there are 422 invariable sites, 174 variable sites of which 123 were parsimony informative. In total, 43 haplotypes were found (Hd: 0.9752). The result demonstrated that, two major clades with strong support can be identified within the Iranian water frogs. One of these clades that include north western and southwestern populations forms a monophyletic group along with P. bedriagae samples from Turkey. The second clade consists of water frog populations of north and northeastern parts of Iran which in turn is subdivided into two subclades. Inclusion of water frog samples from adjacent areas showed that the second clade of our study is, most likely, a distinct taxonomic entity at species rank with its two subclades indicating two diagnosable subspecies for the clade. In conclusion, we suggest that two distinct species, P. bedriagae and Pelophylax sp., with its two subspecies, should be identified as water frogs of Iran. In Addition, another traditionally reported water frog of Iran, P.ridibundus, most likely should be excluded from the Iranian water frog's checklist.

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