Mar 2025, Volume 2 Issue 3-4

    

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• Quantum Computing with Octonions

  Michael Freedman, Modjtaba Shokrian-Zini, Zhenghan Wang

  2019, 2(3-4): 239-273. https://doi.org/10.1007/s42543-019-00020-3

  There are two schools of “measurement-only quantum computation”. The first (Phys. Rev. Lett. 86(22), 5188–5191 (2001)) using prepared entanglement (cluster states) and the second (Phys. Rev. Lett. 101(1), 010501 (2008)) using collections of anyons which, according to how they were produced, also have an entanglement pattern. We abstract the common principle behind both approaches and find the notion of a graph or even continuous family of equiangular projections. This notion is the leading character in the paper. The largest continuous family, in a sense made precise in Corollary 4.2, is associated with the octonions and this example leads to a universal computational scheme. Adiabatic quantum computation also fits into this rubric as a limiting case: nearby projections are nearly equiangular, so as a gapped ground state space is slowly varied, the corrections to unitarity are small.

• Normal Crossings Degenerations of Symplectic Manifolds

  Mohammad Farajzadeh Tehrani, Aleksey Zinger

  2019, 2(3-4): 275-351. https://doi.org/10.1007/s42543-019-00017-y

  We use local Hamiltonian torus actions to degenerate a symplectic manifold to a normal crossings symplectic variety in a smooth one-parameter family. This construction, motivated in part by the Gross–Siebert and B. Parker’s programs, contains a multifold version of the usual (two-fold) symplectic cut construction and in particular splits a symplectic manifold into several symplectic manifolds containing normal crossings symplectic divisors with shared irreducible components in one step.

• Global Steady Prandtl Expansion over a Moving Boundary II

  Sameer Iyer

  2019, 2(3-4): 353-437. https://doi.org/10.1007/s42543-019-00014-1

  This is the second paper in a three-part sequence in which we prove that steady, incompressible Navier–Stokes flows posed over the moving boundary, $y = 0$, can be decomposed into Euler and Prandtl flows in the inviscid limit globally in $[1, \infty ) \times [0,\infty )$, assuming a sufficiently small velocity mismatch. In this paper, we develop a functional framework to capture precise decay rates of the remainders, and prove the corresponding embedding theorems by establishing weighted estimates for their higher order tangential derivatives. These tools are then used in conjunction with a third-order energy analysis, which, in particular, enables us to control the nonlinearity $vu_y$ globally, leading to the main a priori estimate in the analysis.