Lattice uniformities inducing unbounded convergence

A net $(x_\gamma)_{\gamma\in\Gamma}$ in a locally solid Riesz space $(X,\tau)$ is said to be unbounded $\tau$-convergent to $x$ if $|x_\gamma-x|\wedge u\mathop{\overset{\tau}{\longrightarrow}} 0$ for all $u\in X_+$. We recall that there is a locally solid linear topology $\mathfrak{u}\tau$ on $X$ such that unbounded $\tau$-convergence coincides with $\mathfrak{u}\tau$-convergence, and moreover, $\mathfrak{u}\tau$ is characterised as the weakest locally solid linear topology which coincides with $\tau$ on all order bounded subsets. It is with this motivation that we introduce, for a uniform lattice $(L,u)$, the weakest lattice uniformity $u^\ast$ on $L$ that coincides with $u$ on all the order bounded subsets of $L$. It is shown that if $u$ is the uniformity induced by the topology of a locally solid Riesz space $(X,\tau)$, then the $u^*$-topology coincides with $\mathfrak{u}\tau$. This allows comparing the results of this paper with earlier results on unbounded $\tau$-convergence. It will be seen that despite the fact that in the setup of uniform lattices most of the machinery used in the techniques of [M. A. Taylor 2019: Unbounded topologies and uo-convergence in locally solid vector spaces, J. Math. Anal. Appl. \bf{472} no.1, 981--1000] is lacking, the concept of `unbounded convergence' well fittingly generalizes to uniform lattices. We shall also answer Questions 2.13, 3.3, 5.10 of [M. A. Taylor 2019: Unbounded topologies and uo-convergence in locally solid vector spaces, J. Math. Anal. Appl. \bf{472} no.1, 981--1000] and Question 18.51 of [M. A. Taylor 2018: Unbounded convergence in vector lattices, Thesis University of Alberta].Comment: 19 page

Realization of Rectangular Artificial Spin Ice and Direct Observation of High Energy Topology

In this letter, we have constructed and experimentally investigated frustrated arrays of dipoles forming two-dimensional artificial spin ices with different lattice parameters (rectangular arrays with horizontal and vertical lattice spacings denoted by $a$ and $b$ respectively). Arrays with three different ratios $\gamma =a/b = \sqrt{2}$, $\sqrt{3}$ and $\sqrt{4}$ are studied. Theoretical calculations of low-energy demagnetized configurations for these same parameters are also presented. Experimental data for demagnetized samples confirm most of the theoretical results. However, the highest energy topology (doubly-charged monopoles) does not emerge in our theoretical model, while they are seen in experiments for large enough $\gamma$. Our results also insinuate that magnetic monopoles may be almost free in rectangular lattices with a critical ratio $\gamma = \gamma_{c} = \sqrt{3}$, supporting previous theoretical predictions

The mechanical response of cellular materials with spinodal topologies

The mechanical response of cellular materials with spinodal topologies is numerically and experimentally investigated. Spinodal microstructures are generated by the numerical solution of the Cahn-Hilliard equation. Two different topologies are investigated: "solid models," where one of the two phases is modeled as a solid material and the remaining volume is void space; and "shell models," where the interface between the two phases is assumed to be a solid shell, with the rest of the volume modeled as void space. In both cases, a wide range of relative densities and spinodal characteristic feature sizes are investigated. The topology and morphology of all the numerically generated models are carefully characterized to extract key geometrical features and ensure that the distribution of curvatures and the aging law are consistent with the physics of spinodal decomposition. Finite element meshes are generated for each model, and the uniaxial compressive stiffness and strength are extracted. We show that while solid spinodal models in the density range of 30-70% are relatively inefficient (i.e., their strength and stiffness exhibit a high-power scaling with relative density), shell spinodal models in the density range of 0.01-1% are exceptionally stiff and strong. Spinodal shell materials are also shown to be remarkably imperfection insensitive. These findings are verified experimentally by in-situ uniaxial compression of polymeric samples printed at the microscale by Direct Laser Writing (DLW). At low relative densities, the strength and stiffness of shell spinodal models outperform those of most lattice materials and approach theoretical bounds for isotropic cellular materials. Most importantly, these materials can be produced by self-assembly techniques over a range of length scales, providing unique scalability

Vector lattices with a Hausdorff uo-Lebesgue topology

We investigate the construction of a Hausdorff uo-Lebesgue topology on a vector lattice from a Hausdorff (o)-Lebesgue topology on an order dense ideal, and what the properties of the topologies thus obtained are. When the vector lattice has an order dense ideal with a separating order continuous dual, it is always possible to supply it with such a topology in this fashion, and the restriction of this topology to a regular sublattice is then also a Hausdorff uo-Lebesgue topology. A regular vector sublattice of $\mathrm{L}_0(X,\Sigma,\mu)$ for a semi-finite measure $\mu$ falls into this category, and the convergence of nets in its Hausdorff uo-Lebesgue topology is then the convergence in measure on subsets of finite measure. When a vector lattice not only has an order dense ideal with a separating order continuous dual, but also has the countable sup property, we show that every net in a regular vector sublattice that converges in its Hausdorff uo-Lebesgue topology always contains a sequence that is uo-convergent to the same limit. This enables us to give satisfactory answers to various topological questions about uo-convergence in this context.Comment: 37 pages. Minor changes; a few references added. Final version, to appear in J. Math. Anal. App

A view of canonical extension

This is a short survey illustrating some of the essential aspects of the theory of canonical extensions. In addition some topological results about canonical extensions of lattices with additional operations in finitely generated varieties are given. In particular, they are doubly algebraic lattices and their interval topologies agree with their double Scott topologies and make them Priestley topological algebras.Comment: 24 pages, 2 figures. Presented at the Eighth International Tbilisi Symposium on Language, Logic and Computation Bakuriani, Georgia, September 21-25 200