Improving the Cauchy–Schwarz Inequality

Improving the Vakulenko-Kapitanskii type inequality for Hopfions

The Vakulenko-Kapitanskii type inequality between the energy of the Faddeev-Skyrme O(3) non-linear sigma model and the Hopf index is discussed. It is shown that the configurations obtained by Aratyn, Ferreira and Zimerman yield an inequality better than that conjectured by Ward.

The Brunn-Minkowski-Type Inequality

On the metric triangle inequality

A non-contradictible axiomatic theory is constructed under the local reversibility of the metric triangle inequality. The obtained notion includes the metric spaces as particular cases and the generated metric topology is T$_{1}$-separated and generally, non-Hausdorff.

Improving Upon a Geometric Inequality of Third Order

We show that the best possible positive constant k in a certain geometric inequality of third order lies in the interval [0.14119, 0.14364], which improves upon a previous known result where k = 0. We also consider a comparable question concerning a fourth order version of the inequality.

Discussion of Improving Knowledge of Changes in Poverty and Inequality

Discussion on improving our knowledge of changes in global poverty and inequality focused on three main questions. First, should international comparisons of living standards across countries use PPP estimates, or market exchange rates to compare the purchasing power of citizens in different countries? Second, if PPP estimates are used, is a single national measure of aggregate PPP adequate to ...