The reduction of structural Fe in smectite is mediated either abiotically, by reaction with dithionite, or biotically, by Fe-reducing bacteria. The effects of abiotic reduction on clay-surface chemistry are much better known than the effects of biotic reduction. Since bacteria are probably the principal agent for mediating redox processes in natural soils and sediments, further study is needed to ascertain the differences between biotic and abiotic reduction processes. The purpose of the present study was to compare the effects of dithionite (abiotic) and bacteria (biotic) reduction of structural Fe in smectites on the clay structure as observed by infrared spectroscopy. Three reference smectites, namely, Garfield nontronite, ferruginous smectite (SWa-1), and Upton, Wyoming, montmorillonite, were reduced to similar levels by either Shewanella oneidensis or by pH-buffered sodium dithionite. Each sample was then analyzed by Fourier transform infrared spectroscopy (FTIR). Parallel samples were reoxidized by bubbling O2 gas through the reduced suspension at room temperature prior to FTIR analysis. Redox states were quantified by chemical analysis, using 1, 10-phenanthroline. The reduction level achieved by dithionite was controlled to approximate that of the bacterial reduction treatment so that valid comparisons could be made between the two treatments. Bacterial reduction was achieved by incubating the Na-saturated smectites with S. oneidensis strain MR-1 in a minimal medium including 20 mM lactate. After redox treatment, the clay was washed four times with deoxygenated 5 mM NaCl. The sample was then prepared either as a self-supporting film for OH-stretching and deformation bands or as a deposit on ZnSe windows for Si-O stretching bands and placed inside a controlled atmosphere cell also fitted with ZnSe windows. The spectra from bacteria-treated samples were compared with dithionite-treated samples having a similar Fe(II) content. The changes observed in all three spectral regions (OH stretching, M2-O-H deformation, and Si-O stretching) for bacteria-reduced smectite were similar to results obtained at a comparable level of reduction by dithionite. In general, the shift of the structural OH vibration and the Si-O vibration, and the loss of intensity of OH groups, indicate that the bonding and/or symmetry properties in the octahedral and tetrahedral sheets changes as Fe(III) reduces to Fe(II). Upon reoxidation, peak positions and intensities of the reduced smectites were largely restored to the unaltered condition with some minor exceptions. These observations are interpreted to mean that bacterial reduction of Fe modifies the crystal structures of Fe-bearing smectites, but the overall effects are modest and of about the same extent as dithionite at similar levels of reduction. No extensive changes in clay structure were observed under conditions present in our model system.

In 2003, Bohman, Frieze, and Martin initiated the study of randomly perturbed graphs and digraphs. For digraphs, they showed that for every $\alpha \gt 0$, there exists a constant $C$ such that for every $n$-vertex digraph of minimum semi-degree at least $\alpha n$, if one adds $Cn$ random edges then asymptotically almost surely the resulting digraph contains a consistently oriented Hamilton cycle. We generalize their result, showing that the hypothesis of this theorem actually asymptotically almost surely ensures the existence of every orientation of a cycle of every possible length, simultaneously. Moreover, we prove that we can relax the minimum semi-degree condition to a minimum total degree condition when considering orientations of a cycle that do not contain a large number of vertices of indegree $1$. Our proofs make use of a variant of an absorbing method of Montgomery.

Given a family $\mathcal{F}$ of bipartite graphs, the Zarankiewicz number $z(m,n,\mathcal{F})$ is the maximum number of edges in an $m$ by $n$ bipartite graph $G$ that does not contain any member of $\mathcal{F}$ as a subgraph (such $G$ is called $\mathcal{F}$ -free). For $1\leq \beta \lt \alpha \lt 2$ , a family $\mathcal{F}$ of bipartite graphs is $(\alpha,\beta )$ -smooth if for some $\rho \gt 0$ and every $m\leq n$ , $z(m,n,\mathcal{F})=\rho m n^{\alpha -1}+O(n^\beta )$ . Motivated by their work on a conjecture of Erdős and Simonovits on compactness and a classic result of Andrásfai, Erdős and Sós, Allen, Keevash, Sudakov and Verstraëte proved that for any $(\alpha,\beta )$ -smooth family $\mathcal{F}$ , there exists $k_0$ such that for all odd $k\geq k_0$ and sufficiently large $n$ , any $n$ -vertex $\mathcal{F}\cup \{C_k\}$ -free graph with minimum degree at least $\rho (\frac{2n}{5}+o(n))^{\alpha -1}$ is bipartite. In this paper, we strengthen their result by showing that for every real $\delta \gt 0$ , there exists $k_0$ such that for all odd $k\geq k_0$ and sufficiently large $n$ , any $n$ -vertex $\mathcal{F}\cup \{C_k\}$ -free graph with minimum degree at least $\delta n^{\alpha -1}$ is bipartite. Furthermore, our result holds under a more relaxed notion of smoothness, which include the families $\mathcal{F}$ consisting of the single graph $K_{s,t}$ when $t\gg s$ . We also prove an analogous result for $C_{2\ell }$ -free graphs for every $\ell \geq 2$ , which complements a result of Keevash, Sudakov and Verstraëte.

Birnbaum (2020) reanalyses the data from Butler and Pogrebna (2018) using his ‘true and error’ test of choice patterns. His results generally support the evidence we presented in that paper. Here we reiterate the reasons for our agnosticism as to the direction any cycles might take, even though the paradox that motivated our study takes a ‘probable winner’ direction. We conclude by returning to the potential significance of predictably intransitive preferences for decision theory generally.

The transitivity axiom is common to nearly all descriptive and normative utility theories of choice under risk. Contrary to both intuition and common assumption, the little-known ’Steinhaus-Trybula paradox’ shows the relation ’stochastically greater than’ will not always be transitive, in contradiction of Weak Stochastic Transitivity. We bespoke-design pairs of lotteries inspired by the paradox, over which individual preferences might cycle. We run an experiment to look for evidence of cycles, and violations of expansion/contraction consistency between choice sets. Even after considering possible stochastic but transitive explanations, we show that cycles can be the modal preference pattern over these simple lotteries, and we find systematic violations of expansion/contraction consistency.

This article examines large-time behaviour of finite-state mean-field interacting particle systems. Our first main result is a sharp estimate (in the exponential scale) of the time required for convergence of the empirical measure process of the N-particle system to its invariant measure; we show that when time is of the order $\exp\{N\Lambda\}$ for a suitable constant $\Lambda > 0$ , the process has mixed well and it is close to its invariant measure. We then obtain large-N asymptotics of the second-largest eigenvalue of the generator associated with the empirical measure process when it is reversible with respect to its invariant measure. We show that its absolute value scales as $\exp\{{-}N\Lambda\}$ . The main tools used in establishing our results are the large deviation properties of the empirical measure process from its large-N limit. As an application of the study of large-time behaviour, we also show convergence of the empirical measure of the system of particles to a global minimum of a certain ‘entropy’ function when particles are added over time in a controlled fashion. The controlled addition of particles is analogous to the cooling schedule associated with the search for a global minimum of a function using the simulated annealing algorithm.

For a subgraph $G$ of the blow-up of a graph $F$ , we let $\delta ^*(G)$ be the smallest minimum degree over all of the bipartite subgraphs of $G$ induced by pairs of parts that correspond to edges of $F$ . Johansson proved that if $G$ is a spanning subgraph of the blow-up of $C_3$ with parts of size $n$ and $\delta ^*(G) \ge \frac{2}{3}n + \sqrt{n}$ , then $G$ contains $n$ vertex disjoint triangles, and presented the following conjecture of Häggkvist. If $G$ is a spanning subgraph of the blow-up of $C_k$ with parts of size $n$ and $\delta ^*(G) \ge \left(1 + \frac 1k\right)\frac n2 + 1$ , then $G$ contains $n$ vertex disjoint copies of $C_k$ such that each $C_k$ intersects each of the $k$ parts exactly once. A similar conjecture was also made by Fischer and the case $k=3$ was proved for large $n$ by Magyar and Martin.

In this paper, we prove the conjecture of Häggkvist asymptotically. We also pose a conjecture which generalises this result by allowing the minimum degree conditions in each bipartite subgraph induced by pairs of parts of $G$ to vary. We support this new conjecture by proving the triangle case. This result generalises Johannson’s result asymptotically.

We prove that most permutations of degree $n$ have some power which is a cycle of prime length approximately $\log n$. Explicitly, we show that for $n$ sufficiently large, the proportion of such elements is at least $1-5/\log \log n$ with the prime between $\log n$ and $(\log n)^{\log \log n}$. The proportion of even permutations with this property is at least $1-7/\log \log n$.

Over the moduli space of smooth curves, the double ramification cycle can be defined by pulling back the unit section of the universal jacobian along the Abel–Jacobi map. This breaks down over the boundary since the Abel–Jacobi map fails to extend. We construct a ‘universal’ resolution of the Abel–Jacobi map, and thereby extend the double ramification cycle to the whole of the moduli of stable curves. In the non-twisted case, we show that our extension coincides with the cycle constructed by Li, Graber, Vakil via a virtual fundamental class on a space of rubber maps.

The voting paradox occurs when a democratic society seeking to aggregate individual preferences into a social preference reaches an intransitive ordering. However it is not widely known that the paradox may also manifest for an individual aggregating over attributes of risky objects to form a preference over those objects. When this occurs, the relation ‘stochastically greater than’ is not always transitive and so transitivity need not hold between those objects. We discuss the impact of other decision paradoxes to address a series of philosophical and economic arguments against intransitive (cyclical) choice, before concluding that intransitive choices can be justified.

A natural ∼1450-yr global Holocene climate periodicity underlies a portion of the present global warming trend. Calibrated basal radiocarbon dates from 71 paludified peatlands across the western interior of Canada demonstrate that this periodicity regulated western Canadian peatland initiation. Peatlands, the largest terrestrial carbon pool, and their carbon-budgets are sensitive to hydrological fluctuations. The global atmospheric carbon-budget experienced corresponding fluctuations, as recorded in the Holocene atmospheric CO2 record from Taylor Dome, Antarctica. While the climate changes following this ∼1450-yr periodicity were sufficient to affect the global carbon-budget, the resultant atmospheric CO2 fluctuations did not cause a runaway climate–CO2 feedback loop. This demonstrates that global carbon-budgets are sensitive to small climatic fluctuations; thus international agreements on greenhouse gasses need to take into account the natural carbon-budget imbalance of regions with large climatically sensitive carbon pools.

Let $R$ be a commutative ring with 1. In a 1995 paper in J. Algebra, Sharma and Bhatwadekar defined a graph on $R$, $\Gamma \left( R \right)$, with vertices as elements of $R$, where two distinct vertices $a$ and $b$ are adjacent if and only if $Ra\,+\,Rb\,=\,R$. In this paper, we consider a subgraph ${{\Gamma }_{2}}\left( R \right)$ of $\Gamma \left( R \right)$ that consists of non-unit elements. We investigate the behavior of ${{\Gamma }_{2}}\left( R \right)$ and ${{\Gamma }_{2}}\left( R \right)\backslash \text{J}\left( R \right)$, where $\text{J}\left( R \right)$ is the Jacobson radical of $R$. We associate the ring properties of $R$, the graph properties of ${{\Gamma }_{2}}\left( R \right)$, and the topological properties of $\text{Max}\left( R \right)$. Diameter, girth, cycles and dominating sets are investigated, and algebraic and topological characterizations are given for graphical properties of these graphs.