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facebook ipo valuation could top $100 billion

girl math: bsb harmonic range

yesterday i was taking a trip down memory lane and watching backstreet boys music videos on youtube.  one of my favorites:

in mathematics, a harmonic range is the distance between four harmonic conjugates. 

let's say our backstreet boys are forming collinear points by standing on a line:

brian and nick are harmonic conjugates with respect to a.j. and howie if: 

|brian * a.j.|	=	|brian * howie|
|a.j.* nick|		|howie * nick|

further, a.j. and howie are harmonic conjugates with respect to brian and nick. 

in short, our boys divide the line in the same ratio.  line brian-nick is divided harmonically at a.j. (internal ratio) and at howie (external ratio).  if | brian * howie | = 1, then:

|brian * a.j.|	=	a ( 1 - a )		|brian * nick|	=	2a
		1 + a				a + 1

all four boys form a harmonic range within our harmonic line segment! 

but what if kevin wants to rejoin the group?  no problem, he can stand as the midpoint between brian and nick:

then:

kevin * nick^2 = (kevin * a.j.) (kevin * howie) 

who doesn't love a good 1998 bsb harmony?

made in china or trade with china?

the economist recently featured a story on the economic relationships betwixt africa and china.  the chinese have been interested in the continent since the cold war and have been investing in africa since the mid-90s.  the chart below shows the exponential trade growth between the two over the past 15 years.

i normally take data sets and publications from the heritage foundation with a grain of salt and a stiff drink.  but this was a discussion in one of my grad school courses a couple years ago and i've heard relatively little about it since. 

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but one of the more interesting things in the second chart is that china's greatest outward investment is in the americas (ex-u.s.) at 19.5%.  isn't that supposedly our investment turf?  and keep in mind, these numbers don't include bond (debt) transactions. 

china's portfolio is pretty diverse on the surface, but the majority of their investments (66.9%) are in up-and-coming economies.  if current trends continue, i'm hypothesizing china will have significant underlying influence in shaping the global economy over the next century - more so than "just being" china.

my favorite 3 x 9 comic of the day

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spotlight: tontine

the tontine is the "wolf pack" system for raising capital.  individuals pool their money into a permanent common fund and receive dividend payments based on their investment and the fund's performance. 

until death. 

if a member of the wolf pack dies, his or her shares are divided among the rest of the pack.  whoever outlives the rest of the pack takes the whole pot. 

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lorenzo de tonti created this system in italy and france during the mid-17th century.  it was later used by french, british, and u.s. governments for public initiatives, such as funding buildings and promoting life insurance sales.  the state would recoup the capital pot once all the investors died.

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since tontines create a "live-er takes all" incentive, today, they are prohibited in most of the u.s. and britain; the french still use a limited tontine concept. 

girl math: topology at tiffany's

i recently accompanied a friend to tiffany & co. in search of these somerset knot earrings:

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mathematically, knots are a set of closed, non-intersecting curves which exist in three-dimensional space.  the earrings above are math knots.

this is also a math knot:

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this is not a math knot:

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topology ("rubber sheet" geometry) focuses on geometrical properties that are unchanged by an infinite amount of twisting, bending, pulling, shrinking, or stretching.  angles, length, width, etc. don't matter - everything is elastic.  a sphere is topologically equivalent to a cube (scube?); a donut is topologically equivalent to a coffee mug.  because the heart of "rubber sheet" geometry is the relationship between an object's points and these properties, topology is reliant upon set theory and the theories of limits and continuum.

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knot theory, a branch within topology, centers on the shape of our tiffany's earrings: links and knots.  an ambient isotopy (stretching, shrinking, pulling, bending, twisting, or magic tricking the knot) cannot reduce the curve to an unknot (something that isn't a knot). 

oh, and cutting the knot is forbidden.  but why would you want to?  the earrings are so pretty as is.