history – Is there something I’m missing about the Bitcoin price and “extreme volatility”?

I keep hearing about how Bitcoin goes “up and down” in crazy ways all the time and “has done so since it was born”.

Then I look at the historical chart: http://www.livebitcoincharts.com/ (be sure to click “All”)

I see like two spikes in total for the entire time that Bitcoin has been traded.

From what I hear, it sounds like it goes up to the moon and then back down to $0.1/Bitcoin every other day. But that’s obviously not the case. Just look at the chart. Any possible “real gains” that could potentially have been made were in two very specific periods of time, in the end of 2017 and 2020. All of the rest is just very mild ups and downs that don’t really matter unless you are extremely rich and are dealing with thousands of Bitcoins. (Which is not a lot of people.)

So what am I missing? Is that chart lying? Are they talking about trading within the hours and gambling with huge amounts of Bitcoin and being very lucky, consistently, guessing whether it’s going up or down in the very short term?

simulation – Simulate transmission spectrum of extreme ultraviolet laser pulse through laser-dressed finite sample

I am currently participating in research on transient absorption spectroscopy and four wave mixing. In the experimental design, an extreme ultraviolet (XUV) laser pulse and infrared (IR) laser pulse are sent through a finite gas sample with a certain time delay between them. My current task is to develop a function that, given data on the system and input pulses, provides the transmission spectrum of the XUV pulse electric field after exiting the sample.

The Procedure

The gist of the procedure is to take two matrices describing the sample susceptibility, combine them into a single matrix, diagonalize said matrix, normalize the left and right eigenvector matrices according to each other, apply an exponential function to the eigenvalue matrix, and finally compute the exiting spectrum given an input spectrum.

The Code

function E_out = FiniteSampleTransmissionXUV(w, X_0, X_nd, E_in, a, N, tau)
% Function takes as input:
% > w - A vector that describes the frequency spectrum range
% > X_0 - A vector that describes the susceptibility 
%   without the IR pulse
% > X_nd - A matrix that describes the susceptibility with the
%   IR pulse
% > E_in - A vector that describes the frequency spectrum of the 
%   incoming XUV pulse electric field
% > a - A constant that describes the intensity of the IR pulse
% > N - A constant that describes the optical depth of the
%   sample
% > tau - A constant that describes the time delay between the 
%   IR pulse and the XUV pulse
%
% Function provides output:
% > A vector that describes the frequency spectrum of the
%   XUV pulse electric field at the end of the sample


    % determines number of frequency steps from input frequency range
    n = size(w);    % constant
    
    % determines frequency step size
    delta_w = 1 / (n(2) - 1);   % constant
    
    % create matrix sqrt(w_i*w_j)
    sqrt_w = w.^(1/2).' * w.^(1/2);     % nxn matrix

    % combine X_0 and X_nd into total suscptibility matrix
    X_ij = (a * delta_w * sqrt_w .* X_nd) + ...
            diag(diag(sqrt_w).' .* (X_0));      % nxn matrix
    
    
    % diagonalize X_ij where sum(U_R_i^2) = 1
    (U_R, LAMBDA, U_L) = eig(X_ij);     % nxn matrices
    
    % attain the function that scales U_L'*U_R
    F = sum(U_L' * U_R, 1);     % row vector
    
    sqrt_F = F.^2;
    
    % scale U_R and U_L so that U_L'*U_R = 1
    U_R_bar = U_R ./ sqrt_F;    % nxn matrix
    U_L_bar = U_L ./ sqrt_F;    % nxn matrix

    
    % apply exponential transform to eigenvalues      % diagonal nxn matrix
    exp_LAMBDA = diag(exp(1i * (2*pi*N / (3*10^8)) * diag(LAMBDA)));
    
    % create phase shift due to the time delay
    tau_phase = exp(1i * w * tau);  % row vector
    
    % recombine transformed susceptibility matrix
    ULAMBDAU = U_R_bar * exp_LAMBDA * U_L_bar';     % nxn matrix
    
    % apply effects of finite propagation and pulse
    % time delay to input electric field spectrum
    E_out = ULAMBDAU * (tau_phase .* E_in).';   % vector
    
    
end

Testing

The following is a script to test and demonstrate the function.

n = 100;    % number of frequency steps
w = linspace(0,1,n);    % linearly spaced 1D array

X_0 = (1 + 1i) * (sin(5*pi*w)).^2;

X_nd = (1 + 1i) * ((sin(1*pi*w')) * sin(1*pi*w)).^2;

E_in = GaussianSpectrum(w, 1, 0.5, 0.1);


a = 1;
N = 10^8;
tau = 0;
E_out = FiniteSampleTransmissionXUV(w, X_0, X_nd, E_in, a, N, tau);


figure(1)
plot(w, E_in,'b')
hold on
plot(w, abs(E_out),'r')
hold off


function E_w_x0 = GaussianSpectrum(w, E_0, w_0, sigma)
% creates a quasi-gaussian spectrum defined by the input
% frequency range w, amplitude E_0, center frequency w_0,
% and spectral width sigma

    E_w_x0 = E_0 * sin(pi*w) .* ...
             (exp(-((w - w_0) / (2^(1/2) * sigma)).^2));

end

In running the test, we see the frequency spectrum of the XUV pulse before entering the sample in blue with the exiting spectrum in red. One sees that frequencies of the XUV pulse have been absorbed, while new frequencies have been generated.

enter image description here

Please let me know what violates best practices and how the function could be refactored to be more robust and clean. Thank you!

pr.probability – Extreme value distribution for both minimum and maximum at the same time

I am wondering if there is an extreme value distribution that is closed under both the minimum and the maximum operation.
For example, for there is a Gumbel maximum distribution closed under the maximum (provided $beta$ is the same for both distributions). Also there is the Gumbel minimum distribution closed under minimum. However, I am interested in a distribution that is closed under both such that, given two distributions $X_1,X_2$, I can find $Y_1=min(X_1,X_2)$ and $Y_2=max(X_1,X_2)$, which are of the same kind of distribution.
If it only holds in special cases (except the trivial case if iid) that would also be interesting.

I imagine that it is possible that no such kind of distribution exists, in which case my question is why this does not exist / if there is a proof that it cannot exist.

Thank you.

statistics – How do Hard or Extreme rolls vs increasing penalty dice affect the odds of success or failure?

Let’s try modelling this mechanic with AnyDice.

To start off, let me note that the CoC 7e bonus / penalty dice mechanic is a bit awkward to model directly in AnyDice, because the choice of “best” / “worst” tens die depends on the value of the ones die: if you roll a 0 on the ones die, the best possible roll on the tens die is 10, whereas otherwise it’s 00.

However, it turns out that there is a mathematically equivalent variant mechanic that is much easier to work with in AnyDice: instead of numbering the ones die from 0 to 9 and treating 0 + 00 = 100 as a special case, just number the ones die from 1 to 10 and add it directly to the lowest / highest tens die.

(Formal proof of equivalence left as an exercise. As a quick proof sketch, note that there are two cases depending on what you roll on the ones die: if you roll anything but 0/10, the result will be the same using either mechanic anyway; if you do roll a 0/10 on the ones die, you can relabel the sides of the tens dice before rolling them — basically mapping 00 to 90 and subtracting 10 from all other sides — so that the result using the variant mechanic with the relabeled dice will be the same as using the official mechanic with the unrelabeled dice. Since all sides of the dice are supposed to be equally likely and interchangeable, this relabeling will not change the probabilities of the outcomes.)


Using this simplified but equivalent version of the dice rolling mechanic, we can model a CoC 7e roll with N penalty dice in AnyDice simply with:

TENS: 10 * d{0..9}
output d10 + (highest 1 of (N+1)dTENS)

(Conversely, replacing highest with lowest gives a roll with N bonus dice instead.)

To find out the probability of succeeding on such a roll with a given skill or stat, just plot the results of the code above (for the appropriate value of N) in AnyDice using “At Most” mode and look up the probability corresponding to your skill/stat in the plot.


What about hard and extreme rolls, then? You could simply use the same code as for normal rolls, and just look up the probability corresponding to your skill or stat divided by 2 or 5 instead. (You did write those down on your character sheet, right?) But it would be nice to have a direct graphical comparison.

One way to do that is to multiply the result of the roll by 2 or 5 instead of dividing the target value. (Again, showing that this is mathematically equivalent is left as an exercise.) For plotting convenience, it’s also a good idea to clamp the result to at most 100 so that AnyDice’s graph mode won’t needlessly widen the range of the graph.

(Also, since the range of possible rolls starts from 1, AnyDice’s graph plotter gets a bit silly and decides to place tick marks at 1, 11, 21, 31, etc. instead of at 0, 10, 20, 30, etc. One way to fix that is to add a dummy output 0 statement at the end of the code.)

Putting all that together, here’s the final version of the code (for some more or less arbitrary value of “final”, anyway):

TENS: 10 * d{0..9}

output d10 + dTENS named "normal roll"

loop N over {1..5} {
  output d10 + (highest 1 of (N+1)dTENS) named "normal, (N) penalty"
   output d10 + (lowest 1 of (N+1)dTENS) named "normal, (N) bonus" 
}

output (lowest of 100 and 2 * (d10 + dTENS)) named "hard roll"
output (lowest of 100 and 5 * (d10 + dTENS)) named "extreme roll"

output 0 named ""  dummy output to fix graph tick positions 

This code calculates the probability distributions of results for normal rolls with 0 to 5 penalty dice and for hard and extreme rolls with no penalties. (There’s also a line for rolls with 1 to 5 bonus dice as well, but I’ve commented it out with backslashes to reduce clutter. You can uncomment it if you want.)

The output of this program is probably best viewed in Graph + At Most mode (which the link above should take you directly to), and should look something like this:

AnyDice graph screenshot

Looking at where the graphs intersect, you can see that penalty dice are really punishing for low skill / stat values. If your skill / stat is below 50, a normal roll with one penalty die is harder than a hard roll, and if it’s below 20 it’s even harder than an extreme roll. With two penalty dice the crossover points are at about 70 and 44 respectively, and with three penalty dice they’re around 79 and 59 or so.

(If you wanted to know the exact thresholds, you could switch from Graph mode to Export mode and compare the exact numerical probability values. But for most practical purposes just eyeballing it should be good enough, as small differences in probability of a fraction of a percent won’t really be noticeable in play anyway.)

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linear algebra – A question related to integer extreme points of a polyhedron

Let $S$ be a finite set and let $S_1,ldots,S_n$ be subsets of $S$. Let
$q_1,ldots,q_n$ be non-negative rational numbers and suppose that the
function on $S$ defined by $f(s)=sum_{i:sin S_i} q_i$ takes only
integer values.

Prove that there exist non-negative integers $x_1,ldots,x_n$ such
$f(s) = sum_{i:sin S_i} x_i$.

Or provide a counter-example …

extreme tourism – How can I find a guide who will take me safely through the Amazon jungle?

I am an American expat. operating a small adventure tourism company out of Puerto Maldonado, Peru called WildPERU. Our operation is geared more towards adventure-style travel in the S.W. Amazon Basin in Peru including boat travel up-river, camping and trail walking in the forest and Pampas. We call it more of a “dirt-under-your-fingernails” kind of experience, and an opportunity to get off the tourist track in safety and comfort.

We purposely keep our groups small to provide the best experience. For instance, we are planning now for a boat trip up a remote river near to the Brazilian border in Peru next June, 2013. The intent of my clients for this expedition is to survey the rivers fish fauna and research the area for any fish species that may be new to science. We will spend five nights traveling to the rivers headwaters and search the small side streams in order to document any new species that might be found there. This will involve return travel to the area by boat, and camping on the river bank beaches directly in deep forest. Our chances for seeing the regions wildlife are in this way optimized.

Our trips are customizable to the extent that if you have a particular interest or vocation, we can accommodate this by focusing on your particular passion; whether this is Orchids, fish fauna, or Photography. We also have opportunities for volunteering on conservation projects in the region.

I am happy to answer your questions regarding our programs, at your convenience.