By Waclaw Sierpinski, I. N. Sneddon, M. Stark

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**Additional resources for A Selection of Problems in the Theory of Numbers **

**Sample text**

Just as we did for proportionality and strong proportionality, we note that because a point’s location in the IPS determines whether partitions associated with that point are Pareto maximal, Pareto minimal, or neither, it makes sense to refer to “Pareto maximal points” or “Pareto minimal points” in the IPS. 6 Suppose p = ( p1 , p2 ) ∈ IPS. a. p is a Pareto maximal point if and only if there is no q = (q1 , q2 ) ∈ IPS such that q1 ≥ p1 and q2 ≥ p2 , with at least one of these inequalities being strict.

6c shows an IPS corresponding to neither m 1 nor m 2 being absolutely continuous with respect to the other. , for each of these figures, there exists a cake C and corresponding measures m 1 and m 2 such that the given figure is the IPS corresponding to C, m 1 , and m 2 . 6a the point (1, 0) is Pareto maximal but not Pareto minimal, and the point (0, 1) is Pareto minimal but not Pareto maximal. 6b, the situation is exactly the opposite. 6c, neither the point (1, 0) nor the point (0, 1) is Pareto maximal or Pareto minimal.

6a shows an IPS corresponding to m 1 not absolutely continuous with respect to m 2 , but m 2 absolutely continuous with respect to m 1 . 6b shows an IPS corresponding to the reverse situation. 6c shows an IPS corresponding to neither m 1 nor m 2 being absolutely continuous with respect to the other. , for each of these figures, there exists a cake C and corresponding measures m 1 and m 2 such that the given figure is the IPS corresponding to C, m 1 , and m 2 . 6a the point (1, 0) is Pareto maximal but not Pareto minimal, and the point (0, 1) is Pareto minimal but not Pareto maximal.

### A Selection of Problems in the Theory of Numbers by Waclaw Sierpinski, I. N. Sneddon, M. Stark

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