inf u 6 Ttu(x) 6 u(x) 6 Ťtu(x) 6 supu for each t > 0 and each x ∈ H. A function u : H −→ R is called k-semi-concave, k > 0, if the function x −→ u(x)−‖x‖2/k is concave. The function u is called k-semi-convex if −u is k-semiconcave. A bounded function u is t-semi-concave if and only if it belongs to the image of the operator Tt, this follows from Lemma 1 and Lemma 3 below. A function is called s...

Improvement of light extraction efficiency of InGaN light emitting diodes (LEDs) using polydimethylsiloxane (PDMS) concave microstructures arrays was demonstrated. The size effect of the concave microstructures on the light extraction efficiency of III-Nitride LEDs was studied. Depending on the size of the concave microstructures, ray tracing simulations show that the use of PDMS concave micros...

A class of probabilistic constrained programming problems are considered where the probabilistic constraint is of the form P{gi(x, ξ) ≥ 0, i = 1, . . . , r} ≥ p and the functions gi, i = 1, . . . , r are concave. It is shown that the x-function on the left hand side is logarithmic concave provided ξ has a logarithmic concave density. Special cases are mentioned and algorithmic solution of probl...

We establish global rates of convergence for the Maximum Likelihood Estimators (MLEs) of log-concave and s-concave densities on ℝ. The main finding is that the rate of convergence of the MLE in the Hellinger metric is no worse than n-2/5 when -1 < s < ∞ where s = 0 corresponds to the log-concave case. We also show that the MLE does not exist for the classes of s-concave densities with s < -1.

R d is log-concave if p = e where φ :Rd → [−∞,∞) is concave. We denote the class of all such densities p on R by Pd,0. Log-concave densities are always unimodal and have convex level sets. Furthermore, log-concavity is preserved under marginalization and convolution. Thus, the classes of log-concave densities can be viewed as natural nonparametric extensions of the class of Gaussian densities. ...

We introduce an algorithm design technique for a class of combinatorial optimization problems with concave costs. This technique yields a strongly polynomial primal-dual algorithm for a concave cost problem whenever such an algorithm exists for the fixedcharge counterpart of the problem. For many practical concave cost problems, the fixed-charge counterpart is a well-studied combinatorial optim...