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Apr
6
comment What's the probability that 90% of a sample of people are above average at skill X?
It is possible for $90\%$ of a sample to be above average, if the distribution is highly skewed to the left and the average is taken to be the mean. For example, if nine people scored $100$ on a test and one scored $0$, the mean is $90$ and nine people are above the mean. If the average is taken to be the median, however, it is not possible for more than $50\%$ of a sample to be above the median.
Apr
2
comment What is mathematical definition of a fluid?
A fluid is a physical concept rather than a mathematical one. There are mathematical models of fluids, certainly, but any mathematical definition one might provide would be specific to the context of a particular model rather than a general universal definition of "fluid".
Mar
31
comment Proving that if $A=QR$ then $A \sim RQ$
$A = QR = QR(QQ^{-1}) = Q(RQ)Q^{-1}$
Mar
31
revised Proving that if $A=QR$ then $A \sim RQ$
edited title
Mar
29
comment What is this shape ? $|x|+|y|+|z|=1$
Another way to think about it is that in the positive octant $x,y,z\ge0$ the equation is equivalent to $x+y+z=1$, which leads to the equilateral triangle with vertices at $(1,0,0)$, $(0,1,0)$, and $(0,0,1)$. By symmetry we have a shape with eight equilateral triangles and six vertices lying on the coordinate axes.
Mar
29
comment How can I add 2 squares geometrically to get a bigger square?
Cut one square along both diagonals, and attach the four pieces to the edges of the other square.
Mar
27
comment Least sum of power of distances
When $q=2$, this is the centroid $\frac1n(A_1+A_2+\cdots+A_n)$. When $q=1$ it is the geometric median. When $q\to\infty$ it is the center of the smallest enclosing circle. For other $q$ there is no closed form solution in general, so I don't think an "intuitive" method is possible.
Mar
24
comment How would I apply an Exponential Moving Average to Quaternions?
Fair enough; it depends on whether one views EMA as a local filter that happens to have a moving-average interpretation, or as a particular moving average that happens to have a simple implementation as a local filter. I guess that's why we disagree on what the natural generalization is.
Mar
24
comment How would I apply an Exponential Moving Average to Quaternions?
That's a good point. However, the reason the EMA filter works is that the exponentially weighted average of all the data is equivalent to a weighted average of the previous EMA output and the current value. That's no longer true for weighted averages of quaternions.
Mar
24
revised How would I apply an Exponential Moving Average to Quaternions?
added 24 characters in body
Mar
24
comment How would I apply an Exponential Moving Average to Quaternions?
$\mathbf q$ is a placeholder variable denoting the unknown quaternion that you're maximizing over. That is, the average quaternion $\bar{\mathbf q}$ is the value of $\mathbf q$ that maximizes $\mathbf q^T\mathbf M\mathbf q$. In practice, what you have to do is take $\mathbf M$ and find its eigenvector with the largest eigenvalue, and that'll be $\bar{\mathbf q}$.
Mar
24
answered How would I apply an Exponential Moving Average to Quaternions?
Mar
24
comment How would I apply an Exponential Moving Average to Quaternions?
The NASA paper linked in that thread supports weighted averages.
Mar
24
comment How would I apply an Exponential Moving Average to Quaternions?
See "Average" of multiple quaternions?
Mar
23
comment Design an algorithm - Median, computer science
@Oria: If the only thing preventing you from applying the "Select" algorithm is that some of the values in $S'$ may be identical, then instead of replacing every negative element $x$ with $-x$, you can replace a positive $x$ with $2x$ and a negative $x$ with $-2x-1$.
Mar
22
comment Dual curve of the lemniscate of Bernoulli?
The lemniscate of Bernoulli is a plane curve. Why do you have three arguments for $F$?
Mar
21
comment Why do we generally round 5's up instead of down?
$5$s are "rounded up" the same way $6$s and $9$s are: $-1.5002$ and $-1.9$ are both rounded to $-2$.
Mar
20
comment In a linear program, how to add a conditional bound to x?
This is not possible in linear programming. If you consider $k=1$ and add another linear constraint $x\le1$, the problem becomes $0$-$1$ integer linear programming, which is NP-hard.
Mar
20
comment Rotate object around a fixed coordinate axis
It sounds like you're trying to implement a virtual trackball. Perhaps take a look at existing formulations that people have studied?
Mar
17
comment Finding a system of equations whose solution set is the walls of a 1×1×1 cube?
$\max(|x|,|y|,|z|)=1/2$