A real polynomial is called log-concave if its coefficients form a log-concave sequence. We give a new elementary proof of the fact that a product of log-concave polynomials with nonnegative coefficients and no internal zero coefficients is again log-concave. In addition, we show that if the coefficients of the polynomial ∏ m∈M(x + m) form a monotone sequence where M is a finite multiset of pos...

In a companion paper, we showed that weak and strong notions of substitutes in economies with discrete goods have different implications for auction theory and equilibrium theory. In contrast, for the divisible goods case with concave valuations, natural extensions of these concepts coincide. Concave substitute valuations are characterized by submodularity of the dual profit function over nonli...

A two step process has been developed for the fabrication of diffraction limited concave microlens arrays. The process is based on the photoresist filling of melted holes obtained by a preliminary photolithography step. The quality of these microlenses has been tested in a Mach-Zehnder interferometer. The method allows the fabrication of concave microlens arrays with diffraction limited optical...

Rubin and Narsimhan [1984] proposed a method for formulating fuzzy priorities in goal programming. They used the concept that the membership function based on nested priorities has to be concave to solve the fuzzy goal programming problem by linear programming method. In this paper, we relax the condition of concave membership function by quasi concave membership function. We showed that the fu...

We investigate models for unsupervised learning with concave log-likelihood functions. We begin with the most well-known example, IBM Model 1 for word alignment (Brown et al., 1993) and analyze its properties, discussing why other models for unsupervised learning are so seldom concave. We then present concave models for dependency grammar induction and validate them experimentally. We find our ...

Moreover, if p , q, then the inequality is strict. A proof: The map u 7→ −u log u is strictly concave on [0, 1]; this follows from the fact that its derivative −(1 + log u) is strictly decreasing on [0, 1]. Now, a sum of concave functions is concave, so we conclude that H is concave. Moreover, if p , q, then they differ in some coordinate; strict concavity of the map u 7→ −u log u applied to th...

We propose a deterministic global optimization approach, whose novel contributions are rooted in the edge-concave and piecewise-linear underestimators, to address nonconvex mixed-integer quadratically-constrained quadratic programs (MIQCQP) to ε-global optimality. The facets of low-dimensional (n≤ 3) edge-concave aggregations dominating the termwise relaxation of MIQCQP are introduced at every ...

Maximum likelihood estimation of a log-concave probability density is formulated as a convex optimization problem and shown to have an equivalent dual formulation as a constrained maximum Shannon entropy problem. Closely related maximum Renyi entropy estimators that impose weaker concavity restrictions on the fitted density are also considered, notably a minimum Hellinger discrepancy estimator ...

Rankboost has been shown to be an effective algorithm for combining ranks. However, its ability to generalize well and not overfit is directly related to the choice of weak learner, in the sense that regularization of the rank function is due to the regularization properties of its weak learners. We present a regularization property called consistency in preference and confidence that mathemati...

let $c_0(alpha)$ denote the class of concave univalent functions defined in the open unit disk $mathbb{d}$. each function $f in c_{0}(alpha)$ maps the unit disk $mathbb{d}$ onto the complement of an unbounded convex set. in this paper, we study the mapping properties of this class under integral operators.