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Dec
22
answered How many k-subsets of the integers {1,…,n} sum to N?
Dec
21
comment Where do Set Theory and Number Theory meet together?
Why would a number theorist be interested in these sequences?
Dec
16
awarded  Necromancer
Dec
3
comment Geometric interpretation of the average of two independent Cauchy distributions
This also shows why the law of large numbers must fail for the Cauchy distribution.
Nov
30
comment Probability question involving simulations of picking balls from a bag
a total of $101$. This version of the problem is more common. You can see the expression (I think due to de Moivre) in a few places though in some there are errors such as in the upper limit of the summation. See quora.com/… or the top of page 30 here stat.ualberta.ca/~schmu/stat371/371notes.pdf.
Nov
30
comment Probability question involving simulations of picking balls from a bag
Choosing $70$ $R$s and $30$ $B$s is equivalent to choosing $31$ nonnegative numbers which add up to $70$. This is the stars and bars argument. en.wikipedia.org/wiki/Stars_and_bars_%28combinatorics%29 By adding $1$ to each, you get $31$ positive integers adding up to $101$. If you use the restriction that you don't have a streak of $10$ or more $R$s in a row, this corresponds to $31$ nonnegative numbers less than $10$ adding up to $70$. Adding $1$, you have $31$ numbers from $1$ to $10$ adding up to $101$. This is like asking how many ways there are to roll $31$ $10$-sided dice to get
Nov
30
comment Probability question involving simulations of picking balls from a bag
There is a standard way to count the number of patterns with no run of $R$s longer than some value. The expression is not closed form, but it can be written as a single sum. Dividing by $ 100 \choose 30 $ gives the probability that no run of $R$s is longer than the given value. Offhand, I don't know how to restrict the runs of $R$ and $B$ simultaneously in a closed form expression, but you can use dynamic programming to compute the count rapidly.
Nov
30
comment Probability question involving simulations of picking balls from a bag
$RBRBRBR$ has $7$ runs under the other definition. $(RB)^{30}R^{40}$ has $61$.
Nov
30
comment Probability question involving simulations of picking balls from a bag
The maximum number of runs becomes $61$ with the other definition.
Nov
30
comment Probability question involving simulations of picking balls from a bag
With the other definition of runs, James Martin's calculation applies, and the average number of runs is $43$.
Nov
30
comment Probability question involving simulations of picking balls from a bag
How important is that definition of runs? I think it is more common to say that runs of red and blue have to alternate, and to call $RRRBRRB$ $R_3B_1R_2B_1$. That makes the analysis easier. Also, the expected number of runs doesn't have to be an integer. Do you really need the conditional distribution of the run lengths, conditioned on the median/mode number of runs? It might be easier to bound the unconditional probability that there is a run of length greater than some $L$.
Nov
30
comment Probability question involving simulations of picking balls from a bag
@James Martin: The basic idea of using linearity of expectation should work, but it is not the case that the $k$th ball begins a new run whenever it is a different color from the $(k−1)$st. The definition of run in this problem may be nonstandard. $RRRBRRB$ would be two runs, $R_3R_2$ even though there are three changes. Question 2 seems to be about the distribution of the lengths of the runs, and this is not uniform. Runs of length $1$ are much more common than runs of length $50$.
Nov
30
comment Hamiltonian paths in subgraphs of rectangular lattice graphs
@user21820: There are some problems in higher dimensions where fitting a combinatorial structure into a grid can take a super-polynomial amount of expansions, even though it seems intuitive that it should be easy. See the animal problem and the word problem for groups. Again, I don't think this is the type of problem where you need a super-polynomial expansion but it has to be worked out, and I gave an argument for how you might simplify a graph using the Koebe-Andreev-Thurston theorem. If you think it is obvious without this type of simplification, are you sure your intuition is solid?
Nov
30
comment Hamiltonian paths in subgraphs of rectangular lattice graphs
@user21820: Do you disagree with my statement that some work needs to be done? What is your proof of the $O(n^4)$ statement, and wouldn't that count as work?
Nov
25
comment The view from inside of a mirrored tetrahedron
It helps to build intuition to ask what it looks like one dimension lower in a triangle with angles that are not $\pi/n$. You see only part of a universal cover of a sphere with $3$ punctures, and while angles of $\pi/n$ (or sometimes $2\pi/n$) makes some of the pieces on either side of a puncture fit together, these pieces don't fit together in general. Imagine a telephone pole in the middle of a photo where the scene on the right behind the pole doesn't fit the scene on the left. Analogues happen with the reflecting tetrahedron because the dihedral angles are irrational multiples of $\pi$.
Nov
22
comment Robotics, Cryptography, and Genetics applications of Grothendieck's work?
@Daniel Litt: The configuration Bob Connelly showed me was made of $32$ regular triangles. Start with a stellated cube, and pull out one square pyramid, placing a square antiprism between the cube and the square pyramid. The apparent motion is an arc. You can do the same on the opposite side to get a two dimensional infinitesimal motion. There are simpler versions if you can fix some points. Connect two unit segments with endpoints fixed at $(0,1)$ and $(0,-1)$. Their connection must be at $(0,0)$ but the condition is $y=0, x^2=0$, giving an infinitesimal motion in the $x$ direction.
Nov
21
revised Robotics, Cryptography, and Genetics applications of Grothendieck's work?
Emphasized question. Deleted sentences about matrices. Added algebraic geometry tag even though some of Grothendieck's work was outside. Made trivial changes.
Nov
19
comment Robotics, Cryptography, and Genetics applications of Grothendieck's work?
I used to carry around Polydrons to show the following example, which I learned from Bob Connelly: You can build a nonconvex polyhedron from triangles that is provably rigid, so the configuration space mod rigid motions of space is locally just a point. However, the model flexes in your hand. This is because the constraints give a condition like $x^4=0$ instead of $x=0$, and so even small tolerances for error produce macroscopic motions. I think this is natural to describe using schemes.
Nov
19
comment Robotics, Cryptography, and Genetics applications of Grothendieck's work?
@Dylan Yott: That is not a mathematical scheme, of course, but it is natural to consider a scheme structure on some configuration spaces.
Nov
19
comment Robotics, Cryptography, and Genetics applications of Grothendieck's work?
Please do not hastily close this question. The use of matrices in robotics is not relevant (as far as I know), but people do study configuration spaces of linkages and other models of robots. I'm not qualified to talk about them, but some people consider a Grothendieck ring of configuration spaces. See Topological Robotics, and this section: books.google.com/…