Hadamard's optimized inequality

Mubs and Hadamards of Order Six

We report on a search for mutually unbiased bases (MUBs) in 6 dimensions. We find only triplets of MUBs, and thus do not come close to the theoretical upper bound 7. However, we point out that the natural habitat for sets of MUBs is the set of all complex Hadamard matrices of the given order, and we introduce a natural notion of distance between bases in Hilbert space. This allows us to draw a ...

Education , Inequality and Income Inequality

In this paper we propose to measure inequality of educational achievements by constructing a Gini index on educational attainments. We then use the proposed measure to analyse the relationship between inequality in incomes and educational achievements (in terms of both the average attainments and its concentration). Even if theoretical considerations suggest a non-linear relationship between th...

Corruption And Inequality

Human behavior under economic inequality shapes inequality.

In his 1926 short story, “The Rich Boy,” F. Scott Fitzgerald observes: “Let me tell you about the very rich. They are different from you and me. They possess and enjoy early, and it does something to them, makes them soft where we are hard, and cynical where we are trustful, in a way that, unless you were born rich, it is very difficult to understand. They think, deep in their hearts, that they...

An Operator Inequality Related to Jensen’s Inequality

For bounded non-negative operators A and B, Furuta showed 0 ≤ A ≤ B implies A r 2BA r 2 ≤ (A r 2BA r 2 ) s+r t+r (0 ≤ r, 0 ≤ s ≤ t). We will extend this as follows: 0 ≤ A ≤ B ! λ C (0 < λ < 1) implies A r 2 (λB + (1− λ)C)A r 2 ≤ {A r 2 (λB + (1 − λ)C)A r 2 } s+r t+r , where B ! λ C is a harmonic mean of B and C. The idea of the proof comes from Jensen’s inequality for an operator convex functio...