The purpose of this study is to introduce a novel design method of a shim coil array specifically optimized for whole brain shimming and to compare the performance of the resulting coils to conventional spherical harmonic shimming. Methods: The proposed design approach is based on the stream function method and singular value decomposition. Eighty-four field maps from 12 volunteers measured in seven different head positions were used during the design process. The cross validation technique was applied to find an optimal number of coil elements in the array. Additional 42 field maps from 6 further volunteers were used for an independent validation. A bootstrapping technique was used to estimate the required population size to achieve a stable coil design. Results: Shimming using 12 and 24 coil elements outperforms fourth-and fifthorder spherical harmonic shimming for all measured field maps, respectively. Coil elements show novel coil layouts compared to the conventional spherical harmonic coils and existing multi-coils. Both leave-one-out and independent validation demonstrate the generalization ability of the designed arrays. The bootstrapping analysis predicts that field maps from approximately 140 subjects need to be acquired to arrive at a stable design. Conclusions: The results demonstrate the validity of the proposed method to design a shim coil array matched to the human brain anatomy, which naturally satisfies the laws of electrodynamics. The design method may also be applied to develop new shim coil arrays matched to other human organs.

We investigate algebraic and compositional properties of abstract multiway rewriting systems, which are archetypical structures underlying the formalism of the Wolfram model. We demonstrate the existence of higher homotopies in this class of rewriting systems, where homotopical maps are induced by the inclusion of appropriate rewriting rules taken from an abstract rulial space of all possible such rules. Furthermore, we show that a multiway rewriting system with homotopies up to order n may naturally be formalized as an n-fold category, such that (upon inclusion of appropriate inverse morphisms via invertible rewriting relations) the infinite limit of this structure yields an ∞-groupoid. Via Grothendieck’s homotopy hypothesis, this ∞-groupoid thus inherits the structure of a formal homotopy space. We conclude with some comments on how this computational framework of homotopical multiway systems may potentially be used for making formal connections to homotopy spaces upon which models relevant to physics may be instantiated.

We investigate the algebraic and compositional properties of multiway (non-deterministic) abstract rewriting systems, which are the archetypical structures underlying the formalism of the so-called Wolfram model. We demonstrate the existence of higher homotopies in this class of rewriting systems, where these homotopic maps are induced by the inclusion of appropriate rewriting rules taken from an abstract rulial space of all possible such rules. Furthermore, we show that a multiway rewriting system with homotopies up to order n may naturally be formalized as an n-fold category, such that (upon inclusion of appropriate inverse morphisms via invertible rewriting relations) the infinite limit of this structure yields an ∞-groupoid. Via Grothendieck's homotopy hypothesis, this ∞-groupoid thus inherits the structure of a formal homotopy space. We conclude with some comments on how this computational framework of multiway rewriting systems may potentially be used for making formal connections to homotopy spaces upon which models of physics can be instantiated.