The symmetry of intersection numbers in group theory

For suitable subgroups of a finitely generated group, we define the intersection number of one subgroup with another subgroup and show that this number is symmetric. We also give an interpretation of this number.Comment: 19 pages. Published copy, also available at http://www.maths.warwick.ac.uk/gt/GTVol2/paper2.abs.html . Includes erratum added to the original, published 19 Mar 199

Don\u27t Fall for the Lies Behind Trump\u27s Calls for Unity

Each summer, more than half a million tourists come to Gettysburg, a battlefield in the most divisive conflict in American history. After marching around in the heat, thousands stop in at Mr. G’s for some of the best ice cream in town. As they battle with their melting orange pineapple, they can look across the street to the new Unity Park and the newest of Gettysburg’s 1,300-plus monuments. Most commemorate battle; this one commemorates unity. It encourages a “focus on unity and peace.” (excerpt

Book review: breakout nations: in pursuit of the nextEconomic miracles

The argument of Breakout Nations is that the astonishingly rapid growth over the last decade of the world’s celebrated emerging markets is coming to an end. The era of easy money and easy growth is over. China, in particular, will soon slow, but its place will not necessarily be taken by Brazil, Russia or India, all of which Ruchir Sharma shows have weaknesses and difficulties often overlooked in the inflated expectations and emerging markets mania of the past decade. Scott Levin hopes that Sharma is putting his money where his mouth is, because many readers certainly will be