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We give an explicit description of the mutation classes of quivers of type . Furthermore, we provide a complete classification of cluster tilted algebras of type up to derived equivalence. We show that the bounded derived category of such an algebra depends on four combinatorial parameters of the corresponding quiver.
The coefficients of the Kazhdan–Lusztig polynomials are nonnegative integers that are upper semicontinuous relative to Bruhat order. Conjecturally, the same properties hold for -polynomials of local rings of Schubert varieties. This suggests a parallel between the two families of polynomials. We prove our conjectures for Grassmannians, and more generally, covexillary Schubert varieties in complete flag varieties, by deriving a combinatorial formula for . We introduce drift configurations to formulate a new and compatible combinatorial rule for . From our rules we deduce, for these cases, the coefficient-wise inequality .
The aim of this paper is to describe how to use regularization and renormalization to construct a perturbative quantum field theory from a Lagrangian. We first define renormalizations and Feynman measures, and show that although there need not exist a canonical Feynman measure, there is a canonical orbit of Feynman measures under renormalization. We then construct a perturbative quantum field theory from a Lagrangian and a Feynman measure, and show that it satisfies perturbative analogues of the Wightman axioms, extended to allow time-ordered composite operators over curved spacetimes.
For a large class of isotrivial rational elliptic surfaces (with section), we show that the set of rational points is dense for the Zariski topology, by carefully studying variations of root numbers among the fibers of these surfaces. We also prove that these surfaces satisfy a variant of weak-weak approximation. Our results are conditional on the finiteness of Tate–Shafarevich groups for elliptic curves over the field of rational numbers.