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I teach math at St. John Fisher College in Rochester.


1d
comment The set $E= \{x\in [0,1]: \sum_{j=1}^\infty t^j|x−q_j|^{-r} <\infty\}$ does not contain all irrational numbers in $[0,1]$
$[0,1]$ \ $\mathbb{Q}$ means the irrationals in the interval. So in part (a) you're to show $E$ is not exactly the set of irrationals. That means you can either find a rational number which is in $E$, or find an irrational number which is not in $E$.
2d
revised May directed graph be embedded into manifold?
added 764 characters in body
Jul
27
answered May directed graph be embedded into manifold?
Jul
25
revised Let $a_n$ be the $nth$ term of the sequence $1, 2, 2, 3, 3, 3, 4, 4, 4, 4, \dots$
added 118 characters in body
Jul
25
answered Let $a_n$ be the $nth$ term of the sequence $1, 2, 2, 3, 3, 3, 4, 4, 4, 4, \dots$
Jul
25
comment Solve for $x$, $\tan x +\sec x = 2\cos x$ ; $−∞ < x < ∞$
Next cross multiply, treat where cos=0 separately.
Jul
17
comment Inverted Circle?
Perhaps this would lead to a rational parametrization of the curve, using the Weierstrass substitution. (+1)
Jul
17
comment How to solve this elementary counting problem?
Yes, further look didn't seem clear, and your example makes it impossible anyway.
Jul
17
comment How to solve this elementary counting problem?
I don't know if it works, but the most natural try would be to send a pair $(a,b)$ from the first set to the pair $(k-b,k-a)$ for the second set. When I tried this there were some interesting relations and it looked like using the congruence restrictions it might be a bijection...
Jul
16
comment How to turn arbitrary fractions into arbitrary egyptian fractions?
It may be that $n/m=3/4$ only has one way to do it. But in part 2 of the problem they're asking about $n/m=7/12.$
Jul
16
comment Cyclic system of equations
@VishwaIyer I started by putting $x=m,y=n+1$ so the first equation became $m^2+n^2=a.$ Then I thought to make two solutions, keeping the same $n$ but changing sign on $m$ for the other solution. By trial and error I then found a possible $z$ that works. This involved using the idea that one can change sign inside a square and/or switch the order of two squares in any of the equations. I no longer have the details available, but that's the basic approach I used.
Jul
16
comment Randomness in a sequence
Waffle: I just tried an edit to make it look better at this site. It's still a bit vague in terms of what "most" positive $x$ should mean, hence I put that in quotes. Anything may be changed of course, to be as you want it.
Jul
16
revised Randomness in a sequence
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Jul
16
comment Randomness in a sequence
@This I think there is a sense in which this question is a math question, about setting up a biased random walk where the step size depends on the value of a given term, etc. and it asks for an optimal choice of $f$ for which one can in the long run guarantee the most cases wherein it goes to the right of where it is (at $x$ steps later), for hopefully a "large" set of positive $x$.
Jul
16
comment Inflection question
Looks right, but maybe for completeness come up with examples satisfying hypothesis but where the conclusions of A,B are false. As for D, ruling it out means just knowing definition of inflection.
Jul
15
comment How to express a contour
@user137090 -- No problem. These things can be hard to get straight at first, and even after being familiar with the "tricks". Your idea of using $e^{it}$ in some range was already on the right track. If only one could express $t$ going from $+\pi/2$ and ending at $-\pi/2$ it would work, but for that one would have to be viewing the $t$ interval "mod $2\pi$" and it wouldn't be the usual way.
Jul
15
comment How to express a contour
@user137090 If you keep the t range starting at $-\pi/2$ and increasing until $+\pi/2$ and you want the start to be $i$ and the end of the path to be $-i$ (and also want the path to the left of the imaginary axis), then $f(t)=-e^{it}$ works, and I don't see anything else which does, and also moves at a constant speed. BTW what do you mean in your comments by "the answer"?
Jul
15
revised How to express a contour
added 341 characters in body
Jul
15
answered How to express a contour
Jul
14
comment When differentiability of the product implies differentiability of the individual terms?
@Jeppe The requirement of OP was only differentiable, and this $h$ is so. I'll try for a smooth $h$ only because it could be considered a "better" example. --Actually I see that copper.hat's example has its $h(x)$ analytical, even a polynomial (with the domain restricted to $|x|\le 1/2$).