Maps and twists relating U(sl(2))U(sl(2)) and the nonstandard Uh(sl(2))U_{h}(sl(2)): unified construction

A general construction is given for a class of invertible maps between the classical $U(sl(2))$ and the Jordanian $U_{h}(sl(2))$ algebras. Different maps are directly useful in different contexts. Similarity trasformations connecting them, in so far as they can be explicitly constructed, enable us to translate results obtained in terms of one to the other cases. Here the role of the maps is studied in the context of construction of twist operators between the cocommutative and noncocommutative coproducts of the $U(sl(2))$ and $U_{h}(sl(2))$ algebras respectively. It is shown that a particular map called the `minimal twist map' implements the simplest twist given directly by the factorized form of the ${\cal R}_{h}$-matrix of Ballesteros-Herranz. For other maps the twist has an additional factor obtainable in terms of the similarity transformation relating the map in question to the minimal one. The series in powers of $h$ for the operator performing this transformation may be obtained up to some desired order, relatively easily. An explicit example is given for one particularly interesting case. Similarly the classical and the Jordanian antipode maps may be interrelated by a similarity transformation. For the `minimal twist map' the transforming operator is determined in a closed form.Comment: LaTeX, 13 page

Assessing extinction risk across the geographic ranges of plant species in Europe

Societal Impact Statement Plants play fundamental roles in ecosystems, yet merely 10% of species have an assessment of their global extinction risk. Through the integration of national Red Lists and comprehensive global plant distribution data, we identify previously unassessed plant species in Europe that are threatened throughout their geographic range and thus at risk of global extinction. Our workflow can be replicated to facilitate the integration of disparate national monitoring efforts around the world and help accelerate global plant risk assessments. Summary • A comprehensive extinction risk assessment for plant species is a global biodiversity target. However, currently, only 10% of plant diversity is assessed in the global Red List of Threatened Species. To guide conservation and restoration actions in times of accelerated species extinction, plant risk assessments must be expedited. • Here, we examine the extinction risk of vascular plant species in Europe through the integration of two data streams: (1) national Red Lists and (2) global plant distribution data from Kew's Plants of the World Online database. For each species listed on a national Red List, we create a list of countries that form part of its range and indicate the threat status in these countries, allowing us to calculate the percentage of the range in which a given species is listed as threatened. • We find that 7% to 9% of European vascular plant diversity is threatened in its entire range, the majority of which are single-country endemics. Of these globally threatened species, 84% currently have no assessment in the global Red List. • With increasing national biodiversity monitoring commitments shaping the post- 2020 policy environment, we anticipate that integrating national Red Lists with global plant distribution data is a scalable workflow that can help accelerate global risk assessments of plants

New connection formulae for the q-orthogonal polynomials via a series expansion of the q-exponential

Using a realization of the q-exponential function as an infinite multiplicative sereis of the ordinary exponential functions we obtain new nonlinear connection formulae of the q-orthogonal polynomials such as q-Hermite, q-Laguerre and q-Gegenbauer polynomials in terms of their respective classical analogs.Comment: 14 page