Derivatives of (2R)-N-Glyoxylbornane-10,2-sultam and Their Use as auxiliaries in the diastereoselective spirocyclization of 2-substituted tryptamines

A crude hydrate 6 and a crystalline hemiacetal 7 of glyoxylamide 4 were prepared from crotonamide 5 (Scheme 2). Particularly hemiacetal 7 , but also 6 and the ‘dimer’ 8 (obtained from 7 ) may serve as homochiral auxiliaries. The structure of 8 was determined by X-ray analysis. By arenesulfonyl halides, tryptimines 12–14 of 4 were diastereoselectively transformed into spirotricycles 15–17 and 19 .     

Moderate Deviations for Longest Increasing Subsequences: The Lower Tail

We derive a moderate deviation principle for the lower tail probabilities of the length of a longest increasing subsequence in a random permutation. It refers to the regime between the lower tail large deviation regime and the central limit regime. The present article together with the upper tail moderate deviation principle in Ref. 12 yields a complete picture for the whole moderate deviation regime. Other than in Ref. 12, we can directly apply estimates by Baik, Deift, and Johansson, who obtained a (non-standard) Central Limit Theorem for the same quantity.     

On Two Measures of Problem Instance Complexity and their Correlation with the Performance of SeDuMi on Second-Order Cone Problems

We evaluate the practical relevance of two measures of conic convex problem complexity as applied to second-order cone problems solved using the homogeneous self-dual (HSD) embedding model in the software SeDuMi. The first measure we evaluate is Renegar's data-based condition measure C(d), and the second measure is a combined measure of the optimal solution size and the initial infeasibility/optimality residuals denoted by S (where the solution size is measured in a norm that is naturally associated with the HSD model). We constructed a set of 144 second-order cone test problems with widely distributed values of C(d) and S and solved these problems using SeDuMi. For each problem instance in the test set, we also computed estimates of C(d) (using Peña’s method) and computed S directly. Our computational experience indicates that SeDuMi iteration counts and log (C(d)) are fairly highly correlated (sample correlation R = 0.675), whereas SeDuMi iteration counts are not quite as highly correlated with S (R = 0.600). Furthermore, the experimental evidence indicates that the average rate of convergence of SeDuMi iterations is affected by the condition number C(d) of the problem instance, a phenomenon that makes some intuitive sense yet is not directly implied by existing theory.