Gopakumar–Vafa Invariants Do Not Determine Flops

The Donaldson-thomas Invariants under Blowups and Flops

Using the degeneration formula for Doanldson-Thomas invariants, we proved formulae for blowing up a point and simple flops.

Useful Mutations Violate Invariants — Equivalent Mutants do not

Mutation testing measures the adequacy of a test suite by seeding artificial defects (mutations) into a program. If a mutation is not detected by the test suite, this usually means that the test suite is not adequate. However, it may also be that the mutant keeps the program’s semantics unchanged— and thus cannot be detected by any test. Such equivalent mutants have to be eliminated manually, w...

A General Method to Determine Invariants

The determination of invariant characteristics is an important problem in pattern recognition. Many invariants are known: Hu's moment invariants are invariant to shifts, to changes of scale, and to rotations [Hu62]; Burkhardt's Fourier descriptors to shifts, to changes of scale and to rotations [Bu79]; Arbter's Fourier descriptors to a ne transformations [Ar90]; Wang's invariant moments to a ne...

Do Not Copy Do Not Copy Do Not Copy Do Not Copy Do Not Copy Do Not Copy Do Not Copy Do Not Copy Do Not Copy

Design a clocked synchronous state machine with four inputs, G1–G4, that are connected to pushbuttons. The machine has four outputs, L1–L4, connected to lamps or LEDs located near the like-numbered pushbuttons. There is also an ERR output connected to a red lamp. In normal operation, the L1–L4 outputs display a 1-out-of-4 pattern. At each clock tick, the pattern is rotated by one position; the ...

Flops, Type Iii Contractions and Gromov–witten Invariants on Calabi–yau Threefolds