An infinite matrix is called totally positive if its minors of all orders are nonnegative. A nonnegative sequence (an)n≥0 is called log-convex (logconcave, resp.) if aiaj+1 ≥ ai+1aj ( aiaj+1 ≤ ai+1aj , resp.) for 0 ≤ i < j . The object of this talk is to study various positivity properties of Riordan arrays, including the total positivity of such a matrix, the log-convexity of the 0th column an...

We present a number of eecient parallel algorithms for constructing 2-dimensional convex hulls on a randomized CRCW PRAM. Speciically, we show how to build the convex hull of n pre-sorted points in the plane in O(1) time using O(n log n) work, with n-exponential probability, or, alternately, in O(log n) time using O(n) work, with n-exponential probability. We also show how to nd the convex hull...

Abstract We show that the sequence of moments order less than 1 averages i.i.d. positive random variables is log-concave. For at least 1, we conjecture log-convex and this holds eventually for integer (after neglecting first $p^2$ terms sequence).

We prove a tight asymptotic bound of Θ(δ log(n/δ)) on the worst case computational complexity of the convex hull of the union of two convex objects of sizes summing to n requiring δ orientation tests to certify the answer. Our algorithm is deterministic, it uses portions of the convex hull of input objects to describe the final convex hull, and it takes advantage of easy instances, such as thos...

We consider the problem of maintaining a maximum matching in a convex bipartite graph G = (V,E) under a set of update operations which includes insertions and deletions of vertices and edges. It is not hard to show that it is impossible to maintain an explicit representation of a maximum matching in sub-linear time per operation, even in the amortized sense. Despite this difficulty, we develop ...

The purpose of this note is to present several aspects of concentration phenomena in high dimensional geometry. At the heart of the study is a geometric analysis point of view coming from the theory of high dimensional convex bodies. The topic has a broad audience going from algorithmic convex geometry to random matrices. We have tried to emphasize different problems relating these areas of res...

We develop and analyze a variant of Nesterov’s accelerated gradient descent (AGD) for minimization of smooth non-convex functions. We prove that one of two cases occurs: either our AGD variant converges quickly, as if the function was convex, or we produce a certificate that the function is “guilty” of being non-convex. This non-convexity certificate allows us to exploit negative curvature and ...

The purpose of this paper is to extend and systematize known results in log-concave and log-convex properties of life distributions. Also, to discuss the closure property of increasing generalized failure rate (IGFR) distributions with respect to mixing operation.