@article{Lewandowski2022,
  author    = {Lewandowski, Max},
  title     = {Hadamard states for bosonic quantum field theory on globally hyperbolic spacetimes},
  series = {Journal of mathematical physics},
  volume    = {63},
  journal   = {Journal of mathematical physics},
  number    = {1},
  publisher = {American Institute of Physics},
  address   = {Melville},
  issn      = {0022-2488},
  doi       = {10.1063/5.0055753},
  pages     = {34},
  year      = {2022},
  abstract  = {According to Radzikowski's celebrated results, bisolutions of a wave operator on a globally hyperbolic spacetime are of the Hadamard form iff they are given by a linear combination of distinguished parametrices i2(G˜aF-G˜F+G˜A-G˜R) in the sense of Duistermaat and H{\"o}rmander [Acta Math. 128, 183-269 (1972)] and Radzikowski [Commun. Math. Phys. 179, 529 (1996)]. Inspired by the construction of the corresponding advanced and retarded Green operator GA, GR as done by B{\"a}r, Ginoux, and Pf{\"a}ffle {Wave Equations on Lorentzian Manifolds and Quantization [European Mathematical Society (EMS), Z{\"u}rich, 2007]}, we construct the remaining two Green operators GF, GaF locally in terms of Hadamard series. Afterward, we provide the global construction of i2(G˜aF-G˜F), which relies on new techniques such as a well-posed Cauchy problem for bisolutions and a patching argument using Čech cohomology. This leads to global bisolutions of the Hadamard form, each of which can be chosen to be a Hadamard two-point-function, i.e., the smooth part can be adapted such that, additionally, the symmetry and the positivity condition are exactly satisfied.},
  language  = {en}
}
@article{StolleMichaelisRauberg2016,
  author    = {Stolle, Claudia and Michaelis, Ingo and Rauberg, Jan},
  title     = {The role of high-resolution geomagnetic field models for investigating ionospheric currents at low Earth orbit satellites},
  series = {Earth, planets and space},
  volume    = {68},
  journal   = {Earth, planets and space},
  publisher = {Springer},
  address   = {Heidelberg},
  issn      = {1880-5981},
  doi       = {10.1186/s40623-016-0494-1},
  pages     = {10},
  year      = {2016},
  abstract  = {Low Earth orbiting geomagnetic satellite missions, such as the Swarm satellite mission, are the only means to monitor and investigate ionospheric currents on a global scale and to make in situ measurements of F region currents. High-precision geomagnetic satellite missions are also able to detect ionospheric currents during quiet-time geomagnetic conditions that only have few nanotesla amplitudes in the magnetic field. An efficient method to isolate the ionospheric signals from satellite magnetic field measurements has been the use of residuals between the observations and predictions from empirical geomagnetic models for other geomagnetic sources, such as the core and lithospheric field or signals from the quiet-time magnetospheric currents. This study aims at highlighting the importance of high-resolution magnetic field models that are able to predict the lithospheric field and that consider the quiet-time magnetosphere for reliably isolating signatures from ionospheric currents during geomagnetically quiet times. The effects on the detection of ionospheric currents arising from neglecting the lithospheric and magnetospheric sources are discussed on the example of four Swarm orbits during very quiet times. The respective orbits show a broad range of typical scenarios, such as strong and weak ionospheric signal (during day- and nighttime, respectively) superimposed over strong and weak lithospheric signals. If predictions from the lithosphere or magnetosphere are not properly considered, the amplitude of the ionospheric currents, such as the midlatitude Sq currents or the equatorial electrojet (EEJ), is modulated by 10-15 \% in the examples shown. An analysis from several orbits above the African sector, where the lithospheric field is significant, showed that the peak value of the signatures of the EEJ is in error by 5 \% in average when lithospheric contributions are not considered, which is in the range of uncertainties of present empirical models of the EEJ.},
  language  = {en}
}
@article{ChajadaDeneckeHalas1999,
  author    = {Chajada, I. and Denecke, Klaus-Dieter and Halas, R.},
  title     = {Algebras induced by hypersubstitutions},
  year      = {1999},
  language  = {en}
}
@phdthesis{ArwornDenecke1999,
  author    = {Arworn, Srichan and Denecke, Klaus-Dieter},
  title     = {Sets of hypersubstitutions and set-solid varieties},
  year      = {1999},
  language  = {en}
}
@article{DeneckeKoppitz1999,
  author    = {Denecke, Klaus-Dieter and Koppitz, J{\"o}rg},
  title     = {Normal forms of hypersubstitutions},
  year      = {1999},
  language  = {en}
}
@article{DeneckeLeeratanavalee1999,
  author    = {Denecke, Klaus-Dieter and Leeratanavalee, Sorasak},
  title     = {Weak hypersubstitutions and weakly derived algebras},
  year      = {1999},
  language  = {en}
}
@article{ArwornDenecke1999,
  author    = {Arworn, Srichan and Denecke, Klaus-Dieter},
  title     = {Left-edges solid varieties of differential groupoids},
  year      = {1999},
  language  = {en}
}
@article{DeneckeMruczek2000,
  author    = {Denecke, Klaus-Dieter and Mruczek, Krysztyna},
  title     = {P-compatible Hypersubstitutions},
  year      = {2000},
  language  = {en}
}
@book{ArwornDeneckePoeschel1998,
  author    = {Arworn, Srichan and Denecke, Klaus-Dieter and P{\"o}schel, Reinhard},
  title     = {Closure operators on complete lattices},
  series = {Preprint MATH-ALG / Technische Universit{\"a}t Dresden},
  volume    = {1998, 05},
  journal   = {Preprint MATH-ALG / Technische Universit{\"a}t Dresden},
  publisher = {Techn. Univ.},
  address   = {Dresden},
  year      = {1998},
  language  = {en}
}
@article{DeneckeKoppitz1998,
  author    = {Denecke, Klaus-Dieter and Koppitz, J{\"o}rg},
  title     = {Finite monoids of hypersubstitutions of type € = (2)},
  year      = {1998},
  language  = {en}
}