We examine the topological entanglements of polygons confined to a lattice tube and under the influence of an external tensile force f. The existence of the limiting free energy for these so-called stretched polygons is proved and then, using transfer matrix arguments, a pattern theorem for stretched polygons is proved. Note that the tube constraint allows us to prove a pattern theorem for any arbitrary value of f, while without the tube constraint it has so far only been proved for large values of f. The stretched polygon pattern theorem is used first to show that the average span per edge of a randomly chosen n-edge stretched polygon approaches a positive value, non-decreasing in f, as n → ∞. We then show that the knotting probability of an n-edge stretched polygon confined to a tube goes to one exponentially as n → ∞. Thus as n → ∞ when polygons are influenced by a force f, no matter its strength or direction, topological entanglements, as defined by knotting, occur with high probability.

For a fixed permutation τ, let S N (τ) be the set of permutations on N elements that avoid the pattern τ. Madras and Liu (2010) it must equal the Stanley-Wilf limit. We prove the conjecture for every permutation τ of length 5 or less, as well as for some longer cases (including 704 of the 720 permutations of length 6). We also consider permutations drawn at random from S N (τ), and we investigate properties of their graphs (viewing permutations as functions on {1, . . . , N}) scaled down to the unit square [0, 1] 2 . We prove exact large deviation results for these graphs when τ has length 3; it follows, for example, that it is exponentially unlikely for a random 312-avoiding permutation to have points above the diagonal strip |y − x| < , but not unlikely to have points below the strip. For general τ, we show that some neighbourhood of the upper left corner of [0, 1] 2 is exponentially unlikely to contain a point of the graph if and only if τ starts with its largest element. For patterns such as τ = 4231 we establish that this neighbourhood can be extended along the sides of [0, 1] 2 to come arbitrarily close to the corner points (0, 0) and (1, 1), as simulations had suggested.

We consider two self-avoiding polygons (2SAPs) each of which spans a tubular sublattice of ℤ3. A pattern theorem is proved for 2SAPs, that is any proper pattern (a local configuration in the middle of a 2SAP) occurs in all but exponentially few sufficiently large 2SAPs. This pattern theorem is then used to prove that all but exponentially few sufficiently large 2SAPs are topologically linked. Moreover, we also use it to prove that the linking number Lk of an n edge 2SAP Gnsatisfies limn→∞ℙ(|Lk(Gn)| ≥ f(n))=1 for any function [Formula: see text]. Hence the probability of a non zero linking number for a 2SAP approaches one as the size of the 2SAP goes to infinity. It is also established that, due to the tube constraint, the linking number of an n edge 2SAP grows at most linearly in n.

The Level of Service/Case Management Inventory (LS/CMI) is one of the most frequently used tools to assess criminogenic risk–need in justice-involved individuals. Meta-analytic research demonstrates strong predictive accuracy for various recidivism outcomes. In this exploratory study, we applied machine learning (ML) algorithms (decision trees, random forests, and support vector machines) to a data set with nearly 100,000 LS/CMI administrations to provincial corrections clientele in Ontario, Canada, and approximately 3 years follow-up. The overall accuracies and areas under the receiver operating characteristic curve (AUCs) were comparable, although ML outperformed LS/CMI in terms of predictive accuracy for the middle scores where it is hardest to predict the recidivism outcome. Moreover, ML improved the AUCs for individual scores to near 0.60, from 0.50 for the LS/CMI, indicating that ML also improves the ability to rank individuals according to their probability of recidivating. Potential considerations, applications, and future directions are discussed.

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