Floquet phases of matter have attracted great attention due to their dynamical and topological nature that are unique to nonequilibrium settings. In this work, we introduce a generic way of taking any integer qqth-root of the evolution operator UU that describes Floquet topological matter. We further apply our qqth-rooting procedure to obtain 2^n2nth- and 3^n3nth-root first- and second-order non-Hermitian Floquet topological insulators~(FTIs). There, we explicitly demonstrate the presence of multiple edge and corner modes at fractional quasienergies \pm(0,1,...2^{n})\pi/2^{n}±(0,1,...2n)π/2n and \pm(0,1,...,3^{n})\pi/3^{n}±(0,1,...,3n)π/3n, whose numbers are highly controllable and capturable by the topological invariants of their parent systems. Notably, we observe non-Hermiticity induced fractional-quasienergy corner modes and the coexistence of non-Hermitian skin effect with fractional-quasienergy edge states. Our findings thus establish a framework of constructing an intriguing class of topological matter in Floquet open systems.

Stereotactic operations have long been used to relieve agitation and muscle rigidity in Parkinson disease. Because of its high-density resolution, CT is quite effective to portray the location, size, shape and density of the thermocoagulative focus. The foci were classified into different types according to their changes in different time periods. Correlation with pathological changes was discussed.

One-dimensional Floquet topological superconductors possess two types of degenerate Majorana edge modes at zero and π quasieneriges, leaving more room for the design of boundary time crystals and quantum computing schemes than their static counterparts. In this work, we discover Floquet superconducting phases with large topological invariants and arbitrarily many Majorana edge modes in periodically driven Kitaev chains. Topological winding numbers defined for the Floquet operator and Floquet entanglement Hamiltonian are found to generate consistent predictions about the phase diagram, bulk-edge correspondence and numbers of zero and π Majorana edge modes of the system under different driving protocols. The bipartite entanglement entropy further show non-analytic behaviors around the topological transition point between different Floquet superconducting phases. These general features are demonstrated by investigating the Kitaev chain with periodically kicked pairing or hopping amplitudes. Our discovery reveals the rich topological phases and many Majorana edge modes that could be brought about by periodic driving fields in one-dimensional superconducting systems. It further introduces a unified description for a class of Floquet topological superconductors from their quasienergy bands and entanglement properties.

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