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visits member for 2 years, 9 months
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I teach math at St. John Fisher College in Rochester.


2h
comment Find equation of ellipse given two tangent lines at given points and a point on ellipse
Maybe in some cases of initial data of points, tangent lines, and third point, the curve turns out to be a different conic like a hyperbola, parabola, etc. [Maybe this is not necessary to say since OP explicitly was looking for ellipses, but for such data there might not be an ellipse] +1 on answer!
2d
comment Do not exist IID random variables $X, Y$ such that $X-Y \sim U[-1,1]$
What is $\phi_Z(t)$ here? (doesn't seem a likely density function for a uniform distribution, which is what the notation $U[-1,1]$ usually means.)
2d
comment Searching for the measure of an angle (circle)
Fix the formatting to get rid of the "\widehat"s and maybe insert missing equal signs, if they were meant to be there. Otherwise it's hard to read your question.
Jan
27
comment Number of real roots of $2 \cos\left(\frac{x^2+x}{6}\right)=2^x+2^{-x}$
@NeerajBhauryal Yes, just realized that, but the title (at least here) was different and did not have the last division by $2$ when I put the comment. (sorry)
Jan
27
comment Number of real roots of $2 \cos\left(\frac{x^2+x}{6}\right)=2^x+2^{-x}$
The equation of the title is different from the one at the start of the posted question.
Jan
26
comment Integer Factorization: Possible progress
@Ilya_Gazman Thanks for adding the example. It still seems like it would be difficult to prove "mathematically" the behaviors you state. But there may be other sites than this one better suited to checking the algorithm, maybe some site dedicated to programming applications to factoring.
Jan
26
comment Parallel postulate from Playfair's axiom
Let us continue this discussion in chat.
Jan
26
comment If $\frac{a+1}{b}+\frac{b}{a}$ is an integer then it is $3$.
Correct is $(a^2+b^2+a)/(ab)$
Jan
26
comment Parallel postulate from Playfair's axiom
@user21820 By the way, are you the person who posted the question, maybe having changed your user name? In what I see that OP (original poster) was "metacompactness", and it is only that user who can "accept" the answer or not.
Jan
26
comment Integer Factorization: Possible progress
I still don't see what is computed from $a$ and $x$ to get the numbers $k_1,k_2,...$ Now you have also a number $b$ included in your formula for floor(a/x), and that number $b$ is not defined except to say it's some 2-power rounded up, but that's not a clear way to define what it is. Again, a specific example where you start with some particular $a,x$ and go through how to get the number $b$ and the $k_i$ might make it clear.
Jan
26
comment Parallel postulate from Playfair's axiom
I cannot see what your complaint is, @user2180 -- what the topic of neutral geometry (also called absolute geometry) is about is to see what can be deduced from Euclid's axioms without the parallel postulate. Saccheri-Legendre Theorem is really the thing to look at, and see if any of the steps in the proof are objectionable to you. Saccheri at least was one of the mathematicians who was trying to prove the parallel postulate from the other Euclid axioma, so was pretty careful not to use them in his work.
Jan
26
awarded  Enlightened
Jan
26
awarded  Nice Answer
Jan
25
comment Why do I generally see real solutions to recurrence relations?
In some cases one can get a pair of nonreal complex conjugate roots, and use a term of the type $A \sin (kn)+B\cos(kn)$ to get the thing to give real results.
Jan
25
comment Is this identify valid?
If both $t$ and $f(t)$ are real, the real part of the left side is $\sin t \cos f(t)$ which doesn't match the right side.
Jan
25
comment Parallel postulate from Playfair's axiom
See en.wikipedia.org/wiki/Absolute_geometry about this issue. Absolute geometry is another term for "neutral geometry" and refers to what can be deduced from using all of Euclid's axioms except for any axiom equivalent to the parallel postulate. One of the theorems in absolute geometry is the Saccheri-Legendre Theorem, that the sum of the three angles of a triangle is at most 180 degrees. In particular if one adds two of the angles of a nondegenerate triangle in neutral geometry, the sum is in fact less than 180 degrees.
Jan
25
comment Volume of a sphere with two cylindrical holes.
IMO the answer should not be deleted, only the restriction added about the cylinder radius not being too large in comparison with the sphere radius. The problem would still have interest, since one might reasonably expect the drilled holes weren't so wide as to make the intersection of cylinders go outside the sphere. (+1 for now, in case you don't end up deleting)
Jan
25
comment Volume of a sphere with two cylindrical holes.
If the sphere radius is $a=1$ and the drilled hole radius is $b$ then in case $b>1/\sqrt{2}$ the intersection of the cylinders (each viewed as extended), which you call $B$, is not entirely inside the sphere. One can take the equations of those cylinders as $x^2+y^2=a^2,\ x^2+z^2=a^2,$ and the intersection then contains the point $(0,a,a)$ lying outside the unit sphere when $2a^2>1.$
Jan
25
comment Integer Factorization: Possible progress
@Ilya_Gazman If it isn't too much trouble, could you include an example of maybe two relatively small numbers for $a$ and $x$, showing how the series of $k_1,k_2$ etc are calculated, and how it works out to equal the floor of a/x?
Jan
25
comment Integer Factorization: Possible progress
What do $k_1,k_2$ etc. represent in your formula for floor(a/x)? Are they computed somehow from $a$ and $x$? Also right after that you refer to the area where $k<3,$ is this $k$ related to the other subscripted $k_j$'s? Anyway these terms need to be defined in your question to get a good response.