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   Borggaard, Jeff ; Zietsman, Lizette ( July 2020 , Proceedings of the American Control Conference)

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2. G-quadratic, LG-quadratic, and Koszul quotients of exterior algebras

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   McCullough, Jason ; Mere, Zachary ( January 2022 , Communications in Algebra)

   This paper introduces the study of LG-quadratic quotients of exterior algebras, showing that they are Koszul, as in the commutative case. We construct an example of an LG-quadratic algebra that is not G-quadratic and another example that is Koszul but not LG-quadratic. This is only the second known Koszul algebra that is not LG-quadratic and the first that is noncommutative. 

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   Vitral, E. ; Hanna, J. A. ( March 2022 , Mathematics and Mechanics of Solids)

   A nonlinear small-strain elastic theory is constructed from a systematic expansion in Biot strains, truncated at quadratic order. The primary motivation is the desire for a clean separation between stretching and bending energies for shells, which appears to arise only from reduction of a bulk energy of this type. An approximation of isotropic invariants, bypassing the solution of a quartic equation or computation of tensor square roots, allows stretches, rotations, stresses, and balance laws to be written in terms of derivatives of position. Two-field formulations are also presented. Extensions to anisotropic theories are briefly discussed. 

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   Donoghue, John F. ; Menezes, Gabriel ( March 2022 , Il Nuovo cimento C)

   We provide a brief overview of what is known about quadratic grav- ity, which includes terms quadratic in the curvatures in the fundamental action. This is proposed as a renormalizeable UV completion for quantum gravity which contin- ues to use the metric as the fundamental dynamical variable. However, there are unusual field-theoretic consequences because the propagators contain quartic mo- mentum dependence. At the present stage of our understanding, quadratic gravity continues to be a viable candidate for a theory of quantum gravity. 

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