2. Derivatives 10 2.1. Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2. Hasse-Schmidt derivatives ∂ z . . . . . . . . . . . . . . . . . . . . . 11 2.3. Hasse-Schmidt derivations in general . . . . . . . . . . . . . . . . . 18 2.4. Hasse-Schmidt derivations from A to B as algebra maps A→ B [[t]] 22 2.5. Extending Hasse-Schmidt derivations to localizations . ....

Abstract We study model theory of fields with actions a fixed finite group scheme. prove the existence and simplicity companion such actions, which generalizes our previous results about truncated iterative Hasse–Schmidt derivations [13] Galois [14]. As an application methods, we obtain new complete on imperfection degree.

The behavior of the Hasse–Schmidt algebra under étale extension is used to show that the Hasse–Schmidt algebra of a smooth algebra of finite type over a field equals the ring of differential operators. These techniques show that the formation of Hasse–Schmidt derivations does not commute with localization, providing a counterexample to a question of Brown and Kuan; their conjecture is reformula...

Let G be a group and let V an algebraic over algebraically closed field. We introduce subshifts ? ? VG which generalize both the class of sofic finite alphabets. When is polycyclic-by-finite group, we show that satisfies descending chain condition notion subshifts, type are all equivalent. Thus, obtain extensions well-known results Kitchens Schmidt to cover case many non-compact

Schmidt numbers of bipartite mixed states ([1]) characterize the minimum Schmidt ranks of pure states that are needed to construct such mixed states. It is the minimum number of degrees of freedom of a bipartite mixed state entangled between two parties. We give a lower bound of Schmidt numbers of low rank bipartite mixed states and conclude that generic (i.e., all outside a measure zero set) l...