To determine the aqueous humor (AH) exposure to travoprost free acid (TFA) and the in vivo elution rate of travoprost over a 24-month period in subjects with open-angle glaucoma administered a travoprost intracameral implant, 75 µg. In this prospective, single-center, open-label study, 210 subjects (7 cohorts of 30 subjects each) were administered a travoprost intracameral implant and followed for 3–24 months. At pre-determined timepoints (3, 6, 12, 15, 18, 21, and 24 months), AH was collected, a new implant was administered, and the prior implant removed. AH samples were assayed for TFA concentrations using a validated liquid chromatography-tandem mass spectrometry method. Explants were analyzed for remaining travoprost using a validated high-performance liquid chromatography method. Mean AH concentrations of TFA were 5.0, 3.7, 5.6, 2.0, 2.2, 3.8, and 3.3 ng/mL at 3, 6, 12, 15, 18, 21, and 24 months, respectively, post-administration. Mean percent travoprost remaining in explants was approximately 79

We study the problem of detecting infeasibility of large-scale linear programming problems using the primal-dual hybrid gradient (PDHG) method of Chambolle and Pock [J. Math. Imaging Vision, 40 (2011), pp. 120--145]. The literature on PDHG has focused chiefly on problems with at least one optimal solution. We show that when the problem is infeasible or unbounded, the iterates diverge at a controlled rate toward a well-defined ray. In turn, the direction of such a ray recovers infeasibility certificates. Based on this fact, we propose a simple way to extract approximate infeasibility certificates from the iterates of PDHG. We study three sequences that converge to certificates: the difference of iterates, the normalized iterates, and the normalized average. All of them are easy to compute and suitable for large-scale problems. We show that the normalized iterates and normalized averages achieve a convergence rate of O \bigl( k-1\bigr) . This rate is general and applies to any fixed-point iteration of a nonexpansive operator. Thus, it is a result of independent interest that goes well beyond our setting. Finally, we show that, under nondegeneracy assumptions, the iterates of PDHG identify the active set of an auxiliary feasible problem in finite time, which ensures that the difference of iterates exhibits eventual linear convergence. These results provide a theoretical justification for infeasibility detection in the newly developed linear programming solver PDLP.

We present PDLP, a practical first-order method for linear programming (LP) that can solve to the high levels of accuracy that are expected in traditional LP applications. In addition, it can scale to very large problems because its core operation is matrix-vector multiplications. PDLP is derived by applying the primal-dual hybrid gradient (PDHG) method, popularized by Chambolle and Pock (2011), to a saddle-point formulation of LP. PDLP enhances PDHG for LP by combining several new techniques with older tricks from the literature; the enhancements include diagonal preconditioning, presolving, adaptive step sizes, and adaptive restarting. PDLP improves the state of the art for first-order methods applied to LP. We compare PDLP with SCS, an ADMM-based solver, on a set of 383 LP instances derived from MIPLIB 2017. With a target of 10^-8 relative accuracy and 1 hour time limit, PDLP achieves a 6.3x reduction in the geometric mean of solve times and a 4.6x reduction in the number of instances unsolved (from 227 to 49). Furthermore, we highlight standard benchmark instances and a large-scale application (PageRank) where our open-source prototype of PDLP, written in Julia, outperforms a commercial LP solver.

An asymmetric covering D(n,R) is a collection of special subsets S of an n-set such that every subset T of the n-set is contained in at least one special S with |S| − |T | ≤ R. In this paper we compute the smallest size of any D(n, 1) for n ≤ 8. We also investigate “continuous” and “banded” versions of the problem. The latter involves the classical covering numbers C(n, k, k−1), and we determine the following new values: C(10, 5, 4) = 51, C(11, 7, 6, ) = 84, C(12, 8, 7) = 126, C(13, 9, 8) = 185 and C(14, 10, 9) = 259. We also find the number of nonisomorphic minimal covering designs in several cases.

We describe our solution for the KDD Cup 2011 track 2 challenge. Our solution relies heavily on ensembling together diverse individual models for the prediction task, and achieved a final leaderboard/Test 1 misclassification rate of 3.8863%. This paper provides details on both the modeling and ensemble creation steps.

As IP becomes the predominant choice for video delivery, storing the ever increasing number of videos for delivery will become a challenge. In this paper we focus on how to take advantage of user viewing patterns to place content in provider networks to reduce their storage and network utilization. We first characterize user viewing behavior using data collected from a nationally deployed Video-on-Demand service. We provide proof that users watch only a small portion of videos (not just for short clips, but even with full-length movies). We use this information and a highly flexible Mixed Integer Programming (MIP) formulation to solve the placement problem, in contrast to traditional popularity-based placement and caching strategy. We perform detailed simulations using real traces of user viewing sessions (including stream control operations such as Pause, Skip, etc.). Our results show that the use of a segment-based placement yields substantial savings both in storage as well as network bandwidth. For example, compared to a simple caching scheme using full videos, our MIP-based placement using segments can achieve up to 71% reduction in peak link bandwidth usage.

Multidimensional distributions are often used in data mining to describe and summarize different features of large datasets. It is natural to look for distinct classes in such datasets by clustering the data. A common approach entails the use of methods like k -means clustering. However, the k -means method inherently relies on the Euclidean metric in the embedded space and does not account for additional topology underlying the distribution. In this paper, we propose using Earth Mover Distance (EMD) to compare multidimensional distributions. For a n -bin histogram, the EMD is based on a solution to the transportation problem with time complexity O(n 3 log n ). To mitigate the high computational cost of EMD, we propose an approximation that reduces the cost to linear time. Given the large size of our dataset a fast approximation is crucial for this application. Other notions of distances such as the information theoretic Kullback-Leibler divergence and statistical χ 2 distance, account only for the correspondence between bins with the same index, and do not use information across bins, and are sensitive to bin size. A cross-bin distance measure like EMD is not affected by binning differences and meaningfully matches the perceptual notion of "nearness". Our technique is simple, efficient and practical for clustering distributions. We demonstrate the use of EMD on a real-world application of analyzing 411,550 anonymous mobility usage patterns which are defined as distributions over a manifold. EMD allows us to represent inherent relationships in this space, and enables us to successfully cluster even sparse signatures.

Summary What might arithmetic look like on an island that eschews carry digits? How would primes, squares and other number theoretical concepts play out on such an island?

The 3x + 1 function T(x) takes the values (3x+l)/2 if x is odd and x/2 if x is even. Let a be any integer with a £ 0 (mod 3). If na(x) counts the number of n with |«| < x which eventually reach a under iteration by T, then for all sufficiently large x , na(x) > xsx . The proof is based on solving nonlinear programming problems constructed using difference inequalities of Krasikov.

We investigate what arithmetic would look like if carry digits into other digit position were ignored, so that 9 + 4 = 3, 5 + 5 = 0, 9 X 4 = 6, 5 X 4 = 0, and so on. For example, the primes are now 21, 23, 25, 27, 29, 41, 43, 45, 47, ... .

We investigate what arithmetic would look like if carry digits into other digit position were ignored, so that 9 + 4 = 3, 5 + 5 = 0, 9 X 4 = 6, 5 X 4 = 0, and so on. For example, the primes are now 21, 23, 25, 27, 29, 41, 43, 45, 47, ... .

We investigate what arithmetic would look like if carry digits into other digit position were ignored, so that 9 + 4 = 3, 5 + 5 = 0, 9 X 4 = 6, 5 X 4 = 0, and so on. For example, the primes are now 21, 23, 25, 27, 29, 41, 43, 45, 47, ... .

A two-dimensional arrangement of toothpicks is constructed by the following iterative procedure. At stage 1, place a single toothpick of length 1 on a square grid, aligned with the y-axis. At each subsequent stage, for every exposed toothpick end, place an orthogonal toothpick centered at that end. The resulting structure has a fractal-like appearance. We will analyze the toothpick sequence, which gives the total number of toothpicks after n steps. We also study several related sequences that arise from enumerating active cells in cellular automata. Some unusual recurrences appear: a typical example is that instead of the Fibonacci recurrence, which we may write as

We describe a computer code and data that together certify the optimality of a solution to the 85,900-city traveling salesman problem pla85900, the largest instance in the TSPLIB collection of challenge problems.