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Apr
13
comment if $a_n=\frac{a_{[\frac{n}{2}]}}{2}+\frac{a_{[\frac{n}{3}]}}{3}+\ldots+\frac{a_{[\frac{n}{n}]}}{n}$,then $a_{2n}<2a_{n}$
@china math: I was not eager to get the bounty, but why put a bounty on a question and when you receive the answer forget about it?
Apr
13
comment A basic question on measurability of lim sup and lim inf of a function
Yes, I get what you mean. I don't have the time now to write the proof that if $f$ is measurable then $g(x)=\liminf_{t \to x}f(t)$ is also measurable. You can prove it using the definition of $\liminf$. If someone else doesn't do it, I will write it tonight.
Apr
13
comment what is the dimension of this subspace for given problem
Yes, you are right.
Apr
13
comment A basic question on measurability of lim sup and lim inf of a function
$g$ is the liminf and $h$ is the limsup which you know are measurable. You say you need to prove that the set where $g=h$ is measurable. This is what I did above.
Apr
13
comment Suppose $x_{\theta} = (\cos(\theta),\sin(\theta )) \in \mathbb R^2$. Prove $||x_{\theta + \theta^{'}} - x_{\theta}|| = ||x_{\theta^{'}} - x_{0}||$.
I do not use the fact that $x(\theta)=(\cos \theta,\sin \theta)$, because then my reasoning is circular. I use just the fact that $x(\theta)=(x_1(\theta),x_2(\theta))$ and the usual formula for the scalar product.
Apr
12
comment Suppose $x_{\theta} = (\cos(\theta),\sin(\theta )) \in \mathbb R^2$. Prove $||x_{\theta + \theta^{'}} - x_{\theta}|| = ||x_{\theta^{'}} - x_{0}||$.
For the geometric argument, look here: gradestack.com/CBSE-Class-9/Theorem-1-Equal-chords-of-a-Circl/… The idea is that equal chords oppose equal angles at the center and conversely. Your norm equality says exactly that. There is also a rotation argument mentioned in the comments to the question. As for the derivative, if $x,y$ are vector functions of $\theta$ then the derivative of $\langle x(\theta),y(\theta)\rangle$ is taken just like the derivative of a product. (it is mentioned as property 7 in this page: en.wikipedia.org/wiki/Dot_product)
Apr
12
comment Suppose $x_{\theta} = (\cos(\theta),\sin(\theta )) \in \mathbb R^2$. Prove $||x_{\theta + \theta^{'}} - x_{\theta}|| = ||x_{\theta^{'}} - x_{0}||$.
If you start at $x(0)$ and go to $x(\theta)$ on the circle, you go on a path of length $\theta$ (as a consequence of the arclength parametrization). The definition of the angles in radians tells you that an arc of length $\theta$ on the unit radius circle corresponds to an arc of angle $\theta$ (at the center of the circle). Therefore $x(\theta)$ makes an angle of $\theta$ with $x(0)$ and using right triangles you can prove that $x(\theta)=(\cos\theta,\sin \theta)$.
Apr
12
comment Suppose $x_{\theta} = (\cos(\theta),\sin(\theta )) \in \mathbb R^2$. Prove $||x_{\theta + \theta^{'}} - x_{\theta}|| = ||x_{\theta^{'}} - x_{0}||$.
@NicolasLykkeIversen: I don't think we need anything related to sin and cos (apart of their definitions) for the above proof to work. I still like the geometric argument better.
Apr
12
comment if $a_n=\frac{a_{[\frac{n}{2}]}}{2}+\frac{a_{[\frac{n}{3}]}}{3}+\ldots+\frac{a_{[\frac{n}{n}]}}{n}$,then $a_{2n}<2a_{n}$
Square brackets are floor functions.
Apr
11
comment Suppose $x_{\theta} = (\cos(\theta),\sin(\theta )) \in \mathbb R^2$. Prove $||x_{\theta + \theta^{'}} - x_{\theta}|| = ||x_{\theta^{'}} - x_{0}||$.
@NicolasLykkeIversen: Any proof that does not use geometry must use in some way the properties $\cos$ and $\sin$ for sums/differences, and that is not permitted. If you want to use calculus, you use derivatives, and the proof that $\sin'=\cos$ and $\cos'=-\sin$ uses the identies you want to prove. To prove that the two norms are equal is the same thing as to prove two segments have equal lengths, which is a good and simple geometry problem.
Apr
7
comment Exact Line Search in Projected Gradient Descent
Maybe you could get more help at: scicomp.stackexchange.com
Apr
5
comment Computing $\langle\sin(\gamma_i)\rangle= \int_{(S^2)^N} \sin(\gamma_i)p(\Theta)dS$
Please explain clearly what you mean by $\langle\cdot\rangle$. in the beginning of the question you write it as an integral, then you use other formulas...
Apr
5
comment Computing $\langle\sin(\gamma_i)\rangle= \int_{(S^2)^N} \sin(\gamma_i)p(\Theta)dS$
What about the following remark on the second page: "Please do not disseminate these notes. They are only intended for internal use within the context of the course. This is the first version of these notes"?? Do you think your teacher will appreciate posting the file on a public site?
Apr
4
comment Vector spaces and intersections
Nice counterexample.
Apr
3
comment Can all polygons outside of the largest inscribed rectangle in a convex polygon be concave
Use the applet in the end of this page: cgm.cs.mcgill.ca/~athens/cs507/Projects/2003/DanielSud to see that this rectangle is maximal.
Apr
3
comment All polynomials with no natural roots and integer coefficients such that $\phi(n)|\phi(P(n))$
Is $\phi(n)$ the Euler function?
Mar
31
comment Showing $2^{n_2} + 3^{n_3}+\cdots+9^{n_9}$ is dense in $\mathbb{R}^+$
I used $a^{-\infty}=0$ to keep notations simple. Of course it is intended to be understood as a limit.
Mar
31
comment Showing a piece-wise function is differentiable everywhere/Clarification
See this answer: math.stackexchange.com/a/47978/7327 You have a criteria to verify when a function defined on branches is differentiable.
Mar
31
comment In $\triangle ABC$, I is the incenter. Area of $\triangle IBC = 28$, area of $\triangle ICA= 30$ and area of $\triangle IAB = 26$. Find $AC^2 − AB^2$
@AGoogler: It's not really a method. You analyze what you're given and see what you can obtain out of that.
Feb
23
comment How do I solve $y'=\frac{x-y}{x+y}$?
Look at the relation between the numerator and the denominator in the integral. What happens when you differentiate the denominator?