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The sectional nonassociativity of a metrized (not necessarily associative or unital) algebra is defined analogously to the sectional curvature of a pseudo-Riemannian metric, with the associator in place of the Levi-Civita covariant derivative. For commutative real algebras nonnegative sectional nonassociativity is usually called the Norton inequality, while a sharp upper bound on the sectional nonassociativity of the Jordan algebra of Hermitian matrices over a real Hurwitz algebra is closely related to the B\"ottcher-Wenzel-Chern-do Carmo-Kobayashi inequality. These and other basic examples are explained, and there are described some consequences of bounds on sectional nonassociativity for commutative algebras. A technical point of interest is that the results work over the octonions as well as the associative Hurwitz algebras., Comment: v3: Added missing "nontrivial" in the statement of Lemma 8.6 and eliminated from its proof an incorrect argument using an unnecessary hypothesis that has also been removed from its statement. The change has no further ramifications in the paper

With each Steiner triple system there is associated a one-parameter family of commutative, nonassociative, nonunital algebras that are by construction exact, meaning that the trace of every multiplication operator vanishes, and these algebras are shown to be Killing metrized, meaning the Killing type trace-form is nondegenerate and invariant (Frobenius), and simple, except for certain parameter values. The definition of these algebras resembles that of the Matsuo algebra of the Steiner triple system, but they are different. For a Hall triple system, the associated algebra is a primitive axial algebra for a $\mathbb{Z}/2\mathbb{Z}$-graded fusion law.

Analogues of the classical affine-projective correspondence are developed in the context of statistical manifolds compatible with a radiant vector field. These utilize a formulation of Einstein equations for special statistical structures that generalizes the usual Einstein equations for pseudo-Riemannian metrics and is of independent interest. A conelike radiant structure is a not necessarily flat affine connection equipped with a family of surfaces that behave like the intersections of the planes through the origin with a convex cone in a real vector space. A radiant structure is a torsion-free affine connection and a vector field whose covariant derivative is the identity endomorphism. A radiant structure is conelike if for every point and every two-dimensional subspace containing the radiant vector field there is a totally geodesic surface tangent to the subspace at the point. Such structures exist on the total space of any principal bundle with one-dimensional fiber and on any Lie group with a quadratic structure on its Lie algebra. The affine connection of a conelike radiant structure can be canonically normalized to have antisymmetric Ricci tensor. The compatibility of radiant and conelike structures with metrics is investigated and yields connections for which the symmetrized Ricci curvature is a constant multiple of a compatible metric that generalizes well-known constructions of Riemannian and Lorentzian Einstein-Weyl structures over K\"ahler-Einstein manifolds having nonzero scalar curvature. Einstein equations for special statistical manifolds are given that generalize the Einstein-Weyl equations and encompass these more general examples. There are constructed left-invariant conelike radiant structures on a Lie group endowed with a left-invariant nondegenerate bilinear form, and the case of three-dimensional unimodular Lie groups is described in detail., Comment: v4: Introduction revised to provide more context. Section 9 revised to improve readability and some material added for completeness

There are described equations coupling a completely symmetric conformal Killing or Codazzi tensor to the Einstein equations for a metric, in a manner analogous to that used to obtain the Einstein-Maxwell equations by coupling a two-form to the metric. Examples of solutions are constructed from mean curvature zero immersions, affine spheres, isoparametric polynomials, and regular graphs. There are deduced some constraints on the scalar curvature of the metric occurring in a solution. Along the way, there are reviewed Weitzenb\"ock formulas, vanishing theorems, and related results for conformal Killing and divergence free Codazzi tensors., Comment: Comments/criticisms welcomed

A commutative algebra is exact if its multiplication endomorphisms are trace-free and is Killing metrized if its Killing type trace-form is nondegenerate and invariant. A Killing metrized exact commutative algebra is necessarily neither unital nor associative. Such algebras can be viewed as commutative analogues of semisimple Lie algebras or, alternatively, as nonassociative generalizations of \'etale (associative) algebras. Some basic examples are described and there are introduced quantitative measures of nonassociativity, formally analogous to curvatures of connections, that serve to facilitate the organization and characterization of these algebras., Comment: 58 pages. Comments welcomed. v2: Removed what had been lemma 8.6 because it was trivial

There is considered the problem of describing up to linear conformal equivalence those harmonic cubic homogeneous polynomials for which the squared-norm of the Hessian is a nonzero multiple of the quadratic form defining the Euclidean metric. Solutions are constructed in all dimensions and solutions are classified in dimension at most $4$. Techniques are given for determining when two solutions are linearly conformally inequivalent., Comment: v5: A nowhere used stupidity in the statement of Lemma 6.1 is corrected. This has no effect on other claims. v4: Correction to published version (although Lemma 6.14 was correct as stated, the proof given for it was trivially wrong.)

The space of tensors of metric curvature type on a Euclidean vector space carries a two-parameter family of orthogonally invariant commutative nonassociative multiplications invariant with respect to the symmetric bilinear form determined by the metric. For a particular choice of parameters these algebras recover the polarization of the quadratic map on metric curvature tensors that arises in the work of Hamilton on the Ricci flow. Here these algebras are studied as interesting examples of metrized commutative algebras and in low dimensions they are described concretely in terms of nonstandard commutative multiplications on self-adjoint endomorphisms. The algebra of curvature tensors on a three-dimensional Euclidean vector space is shown isomorphic to an orthogonally invariant deformation of the standard Jordan product on three by three symmetric matrices. This algebra is characterized up to isomorphism in terms of purely algebraic properties of its idempotents and the spectra of their multiplication operators. On a vector space of dimension at least four, the subspace of Weyl (Ricci-flat) curvature tensors is a subalgebra for which the multiplication endomorphisms are trace-free and the Killing type trace-form is a multiple of the nondegenerate invariant metric. This subalgebra is simple when the Euclidean vector space has dimension greater than four. In the presence of a compatible complex structure, the analogous result is obtained for the subalgebra of K\"ahler Weyl curvature tensors. It is shown that the anti-self-dual Weyl tensors on a four-dimensional vector space form a simple five-dimensional ideal isometrically isomorphic to the trace-free part of the Jordan product on trace-free $3 \times 3$ symmetric matrices., Comment: v4: Completely reorganized and rewritten. Greatly shortened; excised material will appear in a companion paper. Certain sign conventions changed. Some results strengthened

Two-dimensional layered semiconductors have recently emerged as attractive building blocks for next-generation low-power non-volatile memories. However, challenges remain in the controllable sub-micron fabrication of bipolar resistively switching circuit components from these novel materials. Here we report on the scalable experimental realisation of lateral on-dielectric memtransistors from monolayer single-crystal molybdenum disulfide (MoS2) utilising a focused helium ion beam. Site-specific irradiation with the probe of a helium ion microscope (HIM) allows for the creation of charged defects in the MoS2 lattice. The reversible drift of these locally seeded defects in the applied electric field modulates the resistance of the semiconducting channel, enabling versatile memristive functionality on the nanoscale. We find the device can reliably retain its resistance ratios and set biases for hundreds of switching cycles at sweep frequencies of up to 2.9 V/s with relatively low drain-source biases. We also demonstrate long-term potentiation and depression with sharp habituation that promises application in future neuromorphic architectures. This work advances the down-scaling progress of memristive devices without sacrificing key performance parameters such as power consumption or its applicability for synaptic emulation., Comment: 37 pages, 5 main figures, 14 supplementary figures