By making some simple assumptions we can establish an equation governing hair growth. The equations shows that generally the rate of hair growth decreases exponentially and hair length is bounded by a constant which depends on the initial assumptions.
Before beginning I need to clarify what hair length means since not all hairs have the same length. I have treated hair length as some sort of average over all hairs. The nature of the average is a little vague but hopefully won’t affect the result in any significant way.
Here are some simple assumptions which I have used to derive the equations. Let A(t) denote hair length at time t.
Assumption 1. The rate of increase in the length of each individual hair is a constant, say g. g is the same for each hair. This says that input into the ‘hair system’ is constant over time.
Assumption 2. The number of active hair follicles, say N, is fixed. This means that there is no hair loss.
Assumption 3. The number of hairs shed per unit of time , say L, is constant. The shed hairs will be replaced over time in accordance with assumption 2 above.
We derive the following first order differential equation from the above assumptions:
dA/dt = g – L*A/N
If L = 0 (no shedding of hair) then the general solution is
A = g*t + c
for some constant c. Thus the average hair length increases linearly over time without bound. If L is not zero then the general solution is
A = gN/L + cexp(-L*t/N)
for some constant c which is determined from the initial conditions. If c > 0 then average hair length decreases over time. If c < 0 then average hair length increases over time. In both cases A approaches g*N/L as t approaches infinity and the rate at which A approaches this bound decreases exponentially. For a fixed initial hair length and fixed N, c would be > 0 when L is large compared to g and c would be < 0 when L is small compared to g. You would expect this since if L >> g the natural hair growth rate cannot on average replace the hair that is being shed so the average length should decrease over time. Conversely if L << g then growth is outstripping shedding and the average hair length should increase over time.
Wednesday, March 06, 2013
“I first met Hugo Chávez in New York City in September 2006, just after his infamous appearance on the floor of the UN General Assembly, where he called George W. Bush the devil. “Yesterday, the devil came here,” he said, “Right here. Right here. And it smells of sulfur still today, this table that I am now standing in front of.” He then made the sign of the cross, kissed his hand, winked at his audience and looked to the sky. It was vintage Chávez, an outrageous remark leavened with just the right touch of detail (the lingering sulfur!) to make it something more than bombast, cutting through soporific nostrums of diplomatese and drawing fire away from Iran, which was in the cross hairs at that meeting”.
To read the rest of the article go to:
An early analysis of the upcoming Venezuelan election. http://rt.com/news/maduro- capriles-will-chavismo-survive-chavez-883/
Here is the link:
There is a lot of technical detail in the statement. Manning also gives his reasons for releasing the various documents. It is good to finally get some detailed information from Manning himself about what he did. He certainly makes clear that his actions were guided by his sense of morality which may not help him win his court case but I hope will garner him more support from the public. Despite the enormous pressure put on him to implicate Wikileaks and Assange he goes out of his way to clear them from responsibility:
“Although I stopped sending documents to WLO, no one associated with the WLO pressures me into giving more information. The decisions that I made to send documents and information to the WLO and the website were my own decisions, and I take full responsibility for my actions”.
He must be aware of the effect that this statement will have on his chances for leniency.
Decline of the English Murder
by George Orwell
Tribune, 15 February 1946
It is Sunday afternoon, preferably before the war. The wife is already asleep in the armchair, and the children have been sent out for a nice long walk. You put your feet up on the sofa, settle your spectacles on your nose, and open theNews of the World. Roast beef and Yorkshire, or roast pork and apple sauce, followed up by suet pudding and driven home, as it were, by a cup of mahogany-brown tea, have put you in just the right mood. Your pipe is drawing sweetly, the sofa cushions are soft underneath you, the fire is well alight, the air is warm and stagnant. In these blissful circumstances, what is it that you want to read about?