The results obtained in this paper are a correction of the main results obtained in [14], for which we also give an alternative proof and improvement. We also study some new monotonic conditions under which various generalizations of the Hermite-Hadamard inequality are valid. Furthermore, we give an improvement of the obtained results. Mathematics subject classification (2010): 26A48, 26A51, 26...

We demonstrate that the Penrose inequality is valid for spherically symmetric geometries even when the horizon is immersed in matter. The matter field need not be at rest. The only restriction is that the source satisfies the weak energy condition outside the horizon. No restrictions are placed on the matter inside the horizon. The proof of the Penrose inequality gives a new necessary condition...

For almost as long as there have been IQ tests, there have been psychologists who believe that it is possible to construct "culture free" tests (Jensen, 1980). The desire for such tests springs directly out of the purposes for which tests of general intellectual ability were constructed in the first place: to provide a valid, objective, and socially unbiased measure of intellectual ability. Our...

It is shown that the laws of thermodynamics are extremely robust under generalizations of the form of entropy. Using the Bregman-type relative entropy, the Clausius inequality is proved to be always valid. This implies that thermodynamics is highly universal and does not rule out consistent generalization of the maximum entropy method.

Intersection cuts were introduced by Balas and the corner polyhedron by Gomory. Balas showed that intersection cuts are valid for the corner polyhedron. In this paper we show that, conversely, every nontrivial facet-defining inequality for the corner polyhedron is an intersection cut.

We consider the problem of generating a lattice-free convex set to find a valid inequality that minimizes the sum of its coefficients for 2-row simplex cuts. Multi-row simplex cuts has been receiving considerable attention recently and we show that a pseudo-polytime generation of a lattice-free convex set is possible. We conclude with a short numerical study.

We present some recent developments involving inequalities for the ADM-mass and capacity of asymptotically flat manifolds with boundary. New, more general proofs of classic Euclidean estimates are also included. The inequalities are rigid and valid in all dimensions, and constitute a step towards proving the Riemannian Penrose inequality in arbitrary dimensions.

A well-known result of Beukers [3] on the generalized Ramanujan-Nagell equation has, at its heart, a lower bound on the quantity |x2 − 2n|. In this paper, we derive an inequality of the shape |x3 − 2n| ≥ x4/3, valid provided x3 6= 2n and (x, n) 6= (5, 7), and then discuss its implications for a variety of Diophantine problems.

Intersection cuts were introduced by Balas and the corner polyhedron by Gomory. It is a classical result that intersection cuts are valid for the corner polyhedron. In this paper we show that, conversely every nontrivial facet-defining inequality for the corner polyhedron is an intersection cut.

and Applied Analysis 3 Remark 1.5. If f x /x is increasing on I and 1.6 holds, then the Petrović-type functional P1 is nonnegative, that is, inequality 1.4 is valid. Conversely, if f x /x is increasing on I and conditions as in 1.7 are fulfilled, then relation 1.5 holds. In order to define another Petrović-type functional, we cite the following Petrović-type inequality involving a convex functi...