A famous theorem of D. Orlov describes the derived bounded category of coherent sheaves on projective hypersurfaces in terms of an algebraic construction called graded matrix factorizations. In this article, I implement a proposal of E. Segal to prove Orlov’s theorem in the Calabi–Yau setting using a globalization of the category of graded matrix factorizations (graded D-branes). Let X⊂ℙ be a projective hypersurface. Segal has already established an equivalence between Orlov’s category of graded matrix factorizations and the category of graded D-branes on the canonical bundle Kℙ to ℙ. To complete the picture, I give an equivalence between the homotopy category of graded D-branes on Kℙ and Dbcoh(X). This can be achieved directly, as well as by deforming Kℙ to the normal bundle of X⊂Kℙ and invoking a global version of Knörrer periodicity. We also discuss an equivalence between graded D-branes on a general smooth quasiprojective variety and on the formal neighborhood of the singular locus of the zero fiber of the potential.

We present the first (polynomial-time) algorithm for reducinga given deterministic finite state automaton (DFA) intoa hyper-minimized DFA, which may have fewer states thanthe classically minimized DFA. The price we pay is that thelanguage recognized by the new machine can differ from theoriginal on a finite number of inputs. These hyper-minimizedautomata are optimal, in the sense that every DFA with fewerstates must disagree on infinitely many inputs. With smallmodifications, the construction works also for finite statetransducers producing outputs. Within a class of finitely differing languages, thehyper-minimized automaton is not necessarily unique. There mayexist several non-isomorphic machines using the minimum number ofstates, each accepting a separate language finitely-differentfrom the original one. We will show that there are largestructural similarities among all these smallest automata.