Improving bounds on packing densities of 4-point permutations

We consolidate what is currently known about packing densities of 4-point permutations and in the process improve the lower bounds for the packing densities of 1324 and 1342. We also provide rigorous upper bounds for the packing densities of 1324, 1342, and 2413. All our bounds are within $10^{-4}$ of the true packing densities. Together with the known bounds, this gives us a fairly complete picture of all 4-point packing densities. We also provide new upper bounds for several small permutations of length at least five. Our main tool for the upper bounds is the framework of flag algebras introduced by Razborov in 2007.Comment: journal style, 18 page

Cutting A Pie Is Not A Piece Of Cake

Is there a division among n players of a cake using n-1 parallel vertical cuts, or of a pie using n radial cuts, that is envy-free (each player thinks he or she receives a largest piece and so does not envy another player) and undominated (there is no other allocation as good for all players and better for at least one)? David Gale first asked this question for pies. We provide complete answers for both cakes and pies. The answers depend on the number of players (two versus three or more players) and whether the players' preferences satisfy certain continuity assumptions. We also give some simple algorithms for cutting a pie when there are two or more players, but these algorithms do not guarantee all the properties one might desire in a division, which makes pie-cutting harder than cake-cutting. We suggest possible applications and conclude with two open questions

Envy-free cake divisions cannot be found by finite protocols

No finite protocol (even if unbounded) can guarantee an envy-free division of a cake among three or more players, if each player is to receive a single connected piece

Shock and vibration tests of a SNAP-8 NaK pump

The pump used for reactor cooling in the SNAP 8 space power system was subjected to the expected vehicle launch vibration, and shock loading in accordance with the SNAP 8 environmental specification. Subsequent disassembly revealed damage to the thrust bearing pins, which should be redesigned and strengthened. The unit was operational, however, when run in a test loop after reassembly

Cutting a pie is not a piece of cake

Is there a division among n players of a cake using n-1 parallel vertical cuts, or of a pie using n radial cuts, that is envy-free (each player thinks he or she receives a largest piece and so does not envy another player) and undominated (there is no other allocation as good for all players and better for at least one)? David Gale first asked this question for pies. We provide complete answers for both cakes and pies. The answers depend on the number of players (two versus three or more players) and whether the players' preferences satisfy certain continuity assumptions. We also give some simple algorithms for cutting a pie when there are two or more players, but these algorithms do not guarantee all the properties one might desire in a division, which makes pie-cutting harder than cake-cutting. We suggest possible applications and conclude with two open questions.Fair division; cake-cutting; pie-cutting; divisible good; envy-freeness; allocative efficiency

Cutting the same fraction of several measures

We study some measure partition problems: Cut the same positive fraction of $d+1$ measures in $\mathbb R^d$ with a hyperplane or find a convex subset of $\mathbb R^d$ on which $d+1$ given measures have the same prescribed value. For both problems positive answers are given under some additional assumptions.Comment: 7 pages 2 figure

Challenging empowerment: AIDS-affected southern African children and the need for a multi-level relational approach

Critics of empowerment have highlighted the concept's mutability, focus on individual transformation, one-dimensionality and challenges of operationalisation. Relating these critiques to children's empowerment raises new challenges. Drawing on scholarship on children's subjecthood and exercise of power, alongside empirical research with children affected by AIDS, I argue that empowerment envisaged as individual self-transformation and increased capacity to act independently offers little basis for progressive change. Rather it is essential to adopt a relational approach that recognises the need to transform power relationships at multiple levels. This analysis has implications for our wider understanding of empowerment in the 21st century. © The Author(s) 2013.This research was funded by DFID

The School Experience of Gay, Lesbian and Bisexual Students

A content analysis was conducted on of the public school experience of 3 youth from gay-straight alliance groups, who self identify as gay, lesbian or bisexual. The major themes found are the importance of caring and involved teachers, the positive experience of gay-straight alliance groups, and the affects of peer harassment. Implications for social work practice and policy are discussed