Dominions and Primitive Positive Functions

Let A C such that g and g´ agree on A, we have g = g´. Our main theorem states that if K is closed under ultraproducts, then A dominates b relative to K if and only if there is a partial function F definable by a primitive positive formula in K such that F(a1 ,...,an) = b for some a1,...,an in A. Applying this result we show that a quasivariety of algebras Q with an n-ary near-unanimity term has surjective epimorphisms if and only if SPPu(Q_RSI) has surjective epimorphisms. It follows that if F is a finite set of finite algebras with a common near-unanimity term, then it is decidable whether the (quasi)variety generated by F has surjective epimorphisms.Fil: Campercholi, Miguel Alejandro Carlos. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentin

The Strahler number of a parity game

The Strahler number of a rooted tree is the largest height of a perfect binary tree that is its minor. The Strahler number of a parity game is proposed to be defined as the smallest Strahler number of the tree of any of its attractor decompositions. It is proved that parity games can be solved in quasi-linear space and in time that is polynomial in the number of vertices~n and linear in (d/2k)k, where d is the number of priorities and k is the Strahler number. This complexity is quasi-polynomial because the Strahler number is at most logarithmic in the number of vertices. The proof is based on a new construction of small Strahler-universal trees. It is shown that the Strahler number of a parity game is a robust parameter: it coincides with its alternative version based on trees of progress measures and with the register number defined by Lehtinen~(2018). It follows that parity games can be solved in quasi-linear space and in time that is polynomial in the number of vertices and linear in (d/2k)k, where k is the register number. This significantly improves the running times and space achieved for parity games of bounded register number by Lehtinen (2018) and by Parys (2020). The running time of the algorithm based on small Strahler-universal trees yields a novel trade-off k⋅lg(d/k)=O(logn) between the two natural parameters that measure the structural complexity of a parity game, which allows solving parity games in polynomial time. This includes as special cases the asymptotic settings of those parameters covered by the results of Calude, Jain Khoussainov, Li, and Stephan (2017), of Jurdziński and Lazić (2017), and of Lehtinen (2018), and it significantly extends the range of such settings, for example to d=2O(lgn√) and k=O(lgn−√)

The fossil tragulids of the Siwalik Formations of Southern Asia

Tragulids are common in the Early Miocene through Pliocene Siwalik faunas of the Indian Subcontinent where they are represented by as many as 17 species. Large collections of Siwalik fossils have recently been made by collaborative projects from Harvard University, the University of Arizona, the Geological Survey of Pakistan, and the Pakistan Museum of Natural History. The collections together comprise over 3700 specimens, including dental, cranial, and postcranial elements. Most of this fossil material is from northern and southwestern Pakistan from well-dated stratigraphic sections. The oldest definite tragulids are from the Early Miocene Vihowa Formation and are around 18.7 Ma, while the youngest are in the Pliocene Tatrot Formation and are 3.3 Ma. The fossil tragulids of the Siwaliks differ from the extant species in a number of ways. Importantly, they have a much wider range of body sizes, ranging from 1 to nearly 76 kg. Consequently the small species overlap with the smallest species of extant Tragulus, while the large species approach medium size bovids and cervids. Compared to other ruminants, Siwalik tragulids are also relatively abundant and species rich. Although the status of some described species is uncertain, preliminary analysis indicates there are many as yet undescribed species. Three genera are known and typically at least four species co-exist at any one time during the Miocene. The history of the south Asian tragulids can be correlated to documented environmental changes. The Siwalik deposits formed in a large fluvial system, with mostly forested or wooded low relief floodplains having abundant cover and fruit. Isotopic analyses of tooth enamel and soil carbonates indicate the vegetation was dominated by C3 plants until 9 Ma, after which there was a shift to a more seasonally dry monsoon climate, undoubtedly accounting for a Late Miocene change in the relative abundance of tragulids