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Consultant (environmental stats a specialty) and teacher.


Jan
4
comment How to solve this algorithmic puzzle?
Thinking about this problem recursively looks fruitful: in many cases, making $[a_1, \cdots, a_{n-1}]$ normal will automatically normalize $[a_1, \cdots, a_n]$. Consider, then, how to characterize the lists for which you must operate on $a_n$ in order to normalize them optimally. If your list is one of them, operate on $a_n$. In any event, proceed to recursively solve the problem for $[a_1, \cdots, a_{n-1}]$. The complication to deal with is when $a_n$ is a maximum and the final $T$ elements of the list are not normal.
Jan
3
comment Invariant inner products on infite-dimensional representations
$G$-invariant objects are often constructed by averaging over the group. You can average over a compact group by using its invariant measure (which is finite because the group is compact: the averaging becomes problematic when the measure of the group is infinite), so many statements about representations of finite groups carry over to compact groups, mutatis mutandis, with almost no additional effort.
Jan
3
comment What's a proof that the angles of a triangle add up to 180°?
I see that rschwieb recently sketched the same approach. The figure may help in understanding it and the appeal to properties of the affine group should help make it rigorous, especially insofar as a crucial gap--proving the coincidence of $A_1'$ and $C'$--is concerned.
Jan
3
answered What's a proof that the angles of a triangle add up to 180°?
Dec
31
comment Finding the analytic function
One approach exploits the answers to the closely related question at math.stackexchange.com/questions/267514: consider the real part of the inverse of $f'$.
Dec
27
comment Regressing $Y$ back on the residuals
Hint: When you think of $y$, $x$ and $\hat{e}$ as vectors, the normal equations assert that $\hat{e}$ is orthogonal to $x$. When (after centering $x$ and $y$) you write $y = 1 \cdot \hat{e} + b x$, the residuals--thinking of $\hat{e}$ as the variables this time--are $b x$. If you could somehow show $b x$ is orthogonal to $\hat{e}$, this would demonstrate that you have indeed regressed $y$ against $\hat{e}$.
Dec
27
comment Distribution of the number of children needed so that at least both a girl and a boy are born
Right, but your formula gives a "probability" of $-1$ in the case $k=0$, whereas it should be $0$.
Dec
27
comment Distribution of the number of children needed so that at least both a girl and a boy are born
(1) Take care when $k=0$. (2) What question are you answering? The current one asks for a "distribution" of $N$. (I believe this question needs further clarification in order for the meaning of this request to be properly understood.)
Dec
27
comment Absolute continuity inquality
You may have neglected some additional assumptions: after all, for $u(x)=1$ the rhs is zero while the lhs diverges toward $+\infty$.
Dec
26
comment Why is $\log(\sqrt{x^2+1}+x)$ odd?
It is also crucial that the integral start at $0$ and not someplace else.
Dec
6
comment How to show that the inverse Gaussian density integrates to 1?
One method: look up the CDF (in Wikipedia, say) and differentiate it to prove it is correct. Then evaluate its rightmost limit. Both processes are purely mechanical.
Dec
6
comment How to show that the inverse Gaussian density integrates to 1?
These links seem to address a substantially different question of integrating a Gaussian density rather than the inverse Gaussian.
Nov
30
awarded  Taxonomist
Nov
20
comment is the fixed set of a smooth involution a submanifold?
@Steve It doesn't quite work that way: the z-axis, qua point of the projective plane, also is fixed under this reflection--that's the whole point of the example. To put it another way: an equivalence class (the z-axis) can be fixed without its elements being fixed, so it does not suffice to look only at the fixed points in the covering space. We're not talking about just a single point, either, in general: this example readily generalizes to higher-dimensional projective planes, Grassmannians, flag manifolds, etc., to give a rich set of counterexamples.
Nov
19
comment is it possible to reduce the weight of a best fit line (least squares) given new data points?
Actually, "weight" and "influence" are two completely different things. (See casact.org/pubs/proceed/proceed94/94123.pdf inter alia.) You appear to be looking for the regression version of a time-weighted average. This can be done--as I suggest in an answer to that other question--with a suitable updating algorithm.
Nov
19
comment is it possible to reduce the weight of a best fit line (least squares) given new data points?
A duplicate of this question appeared two years ago at stats.stackexchange.com/questions/6920/… and has some answers there.
Nov
19
comment is it possible to reduce the weight of a best fit line (least squares) given new data points?
@Gerry Actually, if you measure "influence" of a data point as, say, the derivative of the fitted slope with respect to the point's location, then influences vary--a lot. There also is a fairly current and natural definition of "weight" in this context.
Nov
18
comment is the fixed set of a smooth involution a submanifold?
@Steve I don't follow: are you claiming that $(0:0:1)$ lies on the circle of fixed points?
Nov
17
comment is the fixed set of a smooth involution a submanifold?
This argument shows the set is locally a submanifold, but I cannot find a definition of "submanifold" that assures that the entire set will actually be a submanifold: all the definitions I know of are constructed to assure that submanifolds have well-defined dimensions. That's not necessarily the case here.
Nov
17
comment is the fixed set of a smooth involution a submanifold?
The connected components of this subset will each be smooth submanifolds, but their dimensions can vary. As such, their union is not a submanifold. As a simple example, consider the vertical reflection $(x:y:z)\to (x:y:-z)$ of the space of one-dimensional linear subspaces of $\mathbb{R}^3$ (that is, $\mathbb{RP}^2$) to itself, expressed in projective coordinates. It's well-defined and smooth. Its fixed points consist of a codimension-1 submanifold $(\cos(\theta):\sin(\theta):0)$ (a circle of all horizontal lines) together with a codimension-2 submanifold $(0:0:1)$ (the vertical line).