Powerful sets: a generalisation of binary matroids

A set $S\subseteq\{0,1\}^E$ of binary vectors, with positions indexed by $E$, is said to be a \textit{powerful code} if, for all $X\subseteq E$, the number of vectors in $S$ that are zero in the positions indexed by $X$ is a power of 2. By treating binary vectors as characteristic vectors of subsets of $E$, we say that a set $S\subseteq2^E$ of subsets of $E$ is a \textit{powerful set} if the set of characteristic vectors of sets in $S$ is a powerful code. Powerful sets (codes) include cocircuit spaces of binary matroids (equivalently, linear codes over $\mathbb{F}_2$), but much more besides. Our motivation is that, to each powerful set, there is an associated nonnegative-integer-valued rank function (by a construction of Farr), although it does not in general satisfy all the matroid rank axioms. In this paper we investigate the combinatorial properties of powerful sets. We prove fundamental results on special elements (loops, coloops, frames, near-frames, and stars), their associated types of single-element extensions, various ways of combining powerful sets to get new ones, and constructions of nonlinear powerful sets. We show that every powerful set is determined by its clutter of minimal nonzero members. Finally, we show that the number of powerful sets is doubly exponential, and hence that almost all powerful sets are nonlinear.Comment: 19 pages. This work was presented at the 40th Australasian Conference on Combinatorial Mathematics and Combinatorial Computing (40ACCMCC), University of Newcastle, Australia, Dec. 201

Mixed Statistics on 01-Fillings of Moon Polyominoes

We establish a stronger symmetry between the numbers of northeast and southeast chains in the context of 01-fillings of moon polyominoes. Let \M be a moon polyomino with $n$ rows and $m$ columns. Consider all the 01-fillings of \M in which every row has at most one 1. We introduce four mixed statistics with respect to a bipartition of rows or columns of \M. More precisely, let $S \subseteq \{1,2,..., n\}$ and $\mathcal{R}(S)$ be the union of rows whose indices are in $S$. For any filling $M$, the top-mixed (resp. bottom-mixed) statistic $\alpha(S; M)$ (resp. $\beta(S; M)$) is the sum of the number of northeast chains whose top (resp. bottom) cell is in $\mathcal{R}(S)$, together with the number of southeast chains whose top (resp. bottom) cell is in the complement of $\mathcal{R}(S)$. Similarly, we define the left-mixed and right-mixed statistics $\gamma(T; M)$ and $\delta(T; M)$, where $T$ is a subset of the column index set $\{1,2,..., m\}$. Let $\lambda(A; M)$ be any of these four statistics $\alpha(S; M)$, $\beta(S; M)$, $\gamma(T; M)$ and $\delta(T; M)$, we show that the joint distribution of the pair $(\lambda(A; M), \lambda(\bar A; M))$ is symmetric and independent of the subsets $S, T$. In particular, the pair of statistics $(\lambda(A;M), \lambda(\bar A; M))$ is equidistributed with (\se(M),\ne(M)), where \se(M) and $\ne(M)$ are the numbers of southeast chains and northeast chains of $M$, respectively.Comment: 20 pages, 6 figure

Two-photon Lithography for 3D Magnetic Nanostructure Fabrication

Ferromagnetic materials have been utilised as recording media within data storage devices for many decades. Confinement of the material to a two dimensional plane is a significant bottleneck in achieving ultra-high recording densities and this has led to the proposition of three dimensional (3D) racetrack memories that utilise domain wall propagation along nanowires. However, the fabrication of 3D magnetic nanostructures of complex geometry is highly challenging and not easily achievable with standard lithography techniques. Here, by using a combination of two-photon lithography and electrochemical deposition, we show a new approach to construct 3D magnetic nanostructures of complex geometry. The magnetic properties are found to be intimately related to the 3D geometry of the structure and magnetic imaging experiments provide evidence of domain wall pinning at a 3D nanostructured junction

Gradient microfluidics enables rapid bacterial growth inhibition testing

Bacterial growth inhibition tests have become a standard measure of the adverse effects of inhibitors for a wide range of applications, such as toxicity testing in the medical and environmental sciences. However, conventional well-plate formats for these tests are laborious and provide limited information (often being restricted to an end-point assay). In this study, we have developed a microfluidic system that enables fast quantification of the effect of an inhibitor on bacteria growth and survival, within a single experiment. This format offers a unique combination of advantages, including long-term continuous flow culture, generation of concentration gradients, and single cell morphology tracking. Using Escherichia coli and the inhibitor amoxicillin as one model system, we show excellent agreement between an on-chip single cell-based assay and conventional methods to obtain quantitative measures of antibiotic inhibition (for example, minimum inhibition concentration). Furthermore, we show that our methods can provide additional information, over and above that of the standard well-plate assay, including kinetic information on growth inhibition and measurements of bacterial morphological dynamics over a wide range of inhibitor concentrations. Finally, using a second model system, we show that this chip-based systems does not require the bacteria to be labeled and is well suited for the study of naturally occurring species. We illustrate this using Nitrosomonas europaea, an environmentally important bacteria, and show that the chip system can lead to a significant reduction in the period required for growth and inhibition measurements (<4 days, compared to weeks in a culture flask)

Spatial symmetry constraint of charge-ordered kagome superconductor CsV3_3Sb5_5

Elucidating the symmetry of intertwined orders in exotic superconductors is at the quantum frontier. Recent surface sensitive studies of the topological kagome superconductor CsV$_3$Sb$_5$ discovered a cascade 4a$_0$ superlattice below the charge density wave (CDW) ordering temperature, which can be related to the pair density modulations in the superconducting state. If the 4a$_0$ phase is a bulk and intrinsic property of the kagome lattice, this would form a striking analogy to the stripe order and pair density wave discovered in the cuprate high-temperature superconductors, and the cascade ordering found in twisted bilayer graphene. High-resolution X-ray diffraction has recently been established as an ultra-sensitive probe for bulk translational symmetry-breaking orders, even for short-range orders at the diffusive limit. Here, combining high-resolution X-ray diffraction, scanning tunneling microscopy and scanning transmission electron microscopy, we demonstrate that the 4a$_0$ superstructure emerges uniquely on the surface and hence exclude the 4a$_0$ phase as the origin of any bulk transport or spectroscopic anomaly. Crucially, we show that our detected 2$\times$2$\times$2 CDW order breaks the bulk rotational symmetry to C2, which can be the driver for the bulk nematic orders and nematic surface superlattices including the 4a$_0$ phase. Our high-resolution data impose decisive spatial symmetry constraints on emergent electronic orders in the kagome superconductor CsV$_3$Sb$_5$

Spectral Line-by-Line Pulse Shaping of an On-Chip Microresonator Frequency Comb

We report, for the first time to the best of our knowledge, spectral phase characterization and line-by-line pulse shaping of an optical frequency comb generated by nonlinear wave mixing in a microring resonator. Through programmable pulse shaping the comb is compressed into a train of near-transform-limited pulses of \approx 300 fs duration (intensity full width half maximum) at 595 GHz repetition rate. An additional, simple example of optical arbitrary waveform generation is presented. The ability to characterize and then stably compress the frequency comb provides new data on the stability of the spectral phase and suggests that random relative frequency shifts due to uncorrelated variations of frequency dependent phase are at or below the 100 microHertz level.Comment: 18 pages, 4 figure

Direct observation of active material concentration gradients and crystallinity breakdown in LiFePO4 electrodes during charge/discharge cycling of lithium batteries

The phase changes that occur during discharge of an electrode comprised of LiFePO4, carbon, and PTFE binder have been studied in lithium half cells by using X-ray diffraction measurements in reflection geometry. Differences in the state of charge between the front and the back of LiFePO4 electrodes have been visualized. By modifying the X-ray incident angle the depth of penetration of the X-ray beam into the electrode was altered, allowing for the examination of any concentration gradients that were present within the electrode. At high rates of discharge the electrode side facing the current collector underwent limited lithium insertion while the electrode as a whole underwent greater than 50% of discharge. This behavior is consistent with depletion at high rate of the lithium content of the electrolyte contained in the electrode pores. Increases in the diffraction peak widths indicated a breakdown of crystallinity within the active material during cycling even during the relatively short duration of these experiments, which can also be linked to cycling at high rate