|a bit too strong!|
Isn’t language wonderful!
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|a lot to recycle…|
|Three transformations (shown in black, red, green), shown by their action on the unit square (grey), produce the Sierpinkski triangle|
x := rnd repeat choose transform w_i with probability p_i x := w_i(x) plot xThe resulting plot converges to the relevant fractal.
|Here, the black and green transformations are as before, but the red transformation includes a rotation|
|A pentagonal fractal produced by five transformations|
|A “snowflake” produced from four transformations, one with a smaller "shrinkage" than the others|
|A “coral tree”|
|From pentagon to coral, and back.|
And Magic, Science and Religion and other essays by Bronislaw Malinowski, that could have some rich morsel ripe for the spinning in some fringe show or other, and if it is no use, I can always pass it on to Alan Moore. The Consolation of Philosophy by Boethius? I am sure I will get around to reading that, and if not, it matches all the other Penguin Classics I haven’t read that look nice in a row.And fortunately (or maybe not?), I don’t have to worry what my other half will say about my habit – he’s as bad as me.
p79. Boredom has its place in driving us out from cognitive malaise. Though curiosity inspires our cognitive apparatus into detailed exertion surrounding particular as-of-yet-unexplained regularities, we would scarcely commence toil at all without the dull pain of boredom to keep us from the simple irresponsibility of just doing nothing. If there is no pressing topic to think about, we still think, and incessantly so, because it hurts not to.
p82. Choosing how to behave under uncertainty requires a heuristic choice process. Good heuristics give excellent approximations much of the time. But, in the (restricted-by-design) areas where they fail, they give predictably—even pathologically—poor results. The emotions are rational, but the system is a heuristic driver of behavior that operates on incomplete information; so we must accept that the emotions will fail us in some ways, such as overreactions and addictions, that are irresolvable.
p120. The need, then, is for a timely and reliable system to protect us from the risks entailed by our own cleverness. Discerning and locating these mistakes would have the immediate payoff of allowing current reasoning to progress without an error (before we act on such errors), but would also provide a legacy for the future, keeping a fallacious conclusion from becoming registered as verity in long-term memory. A mechanism for consistency checking is indispensable for a system that depends crucially on data-intensive knowledge structures that are built by processes that have been designed to take chances under time pressure. Undersupervised and of variable reliability, their contributions need to be subjected to frequent “reality checks” if the organism that relies on them is to maintain its sanity.
Norovirus originates in the gastrointestinal system and often causes aching limbs, especially the arms and legs.Interesting: how many other limbs are there?
|this person is only acting ill|
|Data is gathered at 10 minute intervals. The horizontal time axis runs from 3:00am to 9:00pm GMT. The vertical axis runs from zero to 8kW. The orange regions indicate the lower quartile, median, upper quartile, and maximum generation at that time, over the month. (Click to embiggen.)|
|Daily power generation, in kWh (grey bars), along with a running average (lower quartile, median, upper quartile, maximum; orange areas), over the previous 7 day sliding window.|
from numpy import * from matplotlib.pyplot import *We define the number of steps, +++N+++, that we are going to walk. We also define the total number of different positions the walker can be in after +++N+++ steps.
N = 100 # number of random steps P = 2*N+1 # number of positions
coin0 = array([1, 0]) # |0> coin1 = array([0, 1]) # |1>
C00 = outer(coin0, coin0) # |0><0| C01 = outer(coin0, coin1) # |0><1| C10 = outer(coin1, coin0) # |1><0| C11 = outer(coin1, coin1) # |1><1|Quantum operators are unitary matrices. The coin operator, that can be used to flip a quantum coin into a superposition, is:
C_hat = (C00 + C01 + C10 - C11)/sqrt(2.)
ShiftPlus = roll(eye(P), 1, axis=0) ShiftMinus = roll(eye(P), -1, axis=0) S_hat = kron(ShiftPlus, C00) + kron(ShiftMinus, C11)
U = S_hat.dot(kron(eye(P), C_hat))
posn0 = zeros(P) posn0[N] = 1 # array indexing starts from 0, so index N is the central posn psi0 = kron(posn0,(coin0+coin1*1j)/sqrt(2.))
psiN = linalg.matrix_power(U, N).dot(psi0)And we’re done! +++ | \psi \rangle_N+++ is the state of the system after +++N+++ random quantum steps.
prob = empty(P) for k in range(P): posn = zeros(P) posn[k] = 1 M_hat_k = kron( outer(posn,posn), eye(2)) proj = M_hat_k.dot(psiN) prob[k] = proj.dot(proj.conjugate()).real
fig = figure() ax = fig.add_subplot(111) plot(arange(P), prob) plot(arange(P), prob, 'o') loc = range (0, P, P / 10) #Location of ticks xticks(loc) xlim(0, P) ax.set_xticklabels(range (-N, N+1, P / 10)) show()
|probability distribution for a quantum random walk with +++N=100+++, symmetric initial coin|
|probability distribution for a quantum random walk with +++N=100+++, initial coin +++|0\rangle+++|