Wikipedia:Reference desk/Archives/Mathematics/2016 October 17

= October 17 =

Plane Projection of a Hyperbola
Two planes intersect at an angle strictly between 0° and 90°. One of them contains a hyperbola, which we project onto the other intersecting plane. Is the result still a hyperbola ? And if the answer is affirmative, does this mean that the answer to this question is also a `yes`, inasmuch as any triangle can be seen as the projection of an equilateral one onto a plane at a given angle ? — 79.113.235.46 (talk) 06:13, 17 October 2016 (UTC)
 * If one of the axes of the hyperbola is parallel (or normal) to the line of intersection of the two planes: yes (by a change of variables in the conic equation). I suspect it's true in only that case (an example of an obviously non-hyperbolic projection of a hyperbola), and so is not useful for the triangle question.  --Tardis (talk) 00:54, 18 October 2016 (UTC)

Unitary Transformation
Let $$I=\{0,1,\dots,2^n-1\}$$, $$A=\{|y_i\rangle=\sum_{x_i\in I}{(-1)^{a_i}|x_i\rangle}\mid a_i \in I\}$$, $$f(|y_i\rangle)=\oplus a_i$$. (the XOR of the phases' exponents). We say that $$|x\rangle=|y\rangle$$ iff $$\forall i : |x_i|^2=|y_i|^2$$ (two vectors are equivalent iff all of their amplitudes are equal).

Is there any unitary transformation $$U$$ that satisfies: $$\forall |a\rangle,|b\rangle\in A : f(|a\rangle)\ne f(|b\rangle)\implies U|a\rangle\not\equiv U|b\rangle$$?

Thanks in advance! עברית (talk) —Preceding undated comment added 06:27, 17 October 2016 (UTC)
 * The definition of A sounds fishy, but in any case, a prominent notice at the top of the page states: We don't do your homework for you, though we’ll help you past the stuck point. Tigraan Click here to contact me 11:09, 17 October 2016 (UTC)

Measure of angle formed by absolute value graph
Let a be a positive real number. What is the measure of the angle formed by the graph of $$f(x) = a|x|$$? GeoffreyT2000 ( talk,  contribs ) 16:03, 17 October 2016 (UTC)
 * Just compute the angle between $$f(x) = ax$$ and the vertical line, and double that. Are you familiar with tangent (trigonometry)? Tigraan Click here to contact me 17:04, 17 October 2016 (UTC)
 * The angle formed between $$f(x) = ax$$ and the vertical line has the same value as the angle formed between $$f(x) = ax$$ and the horizontal line. Logic dictates that it must be half the angle formed between the horizontal line and the vertical line. 175.45.116.99 (talk) 04:48, 19 October 2016 (UTC)
 * The IP's answer is only true for a = 1. However, for a > 0, the angle between y = ax and the x-axis is tan&minus;1a, the requested angle is 2(&pi; / 2 – tan&minus;1a) = &pi; – 2tan&minus;1a.  As expected, as a &rarr; 0+, the angle &rarr; &pi;.  As a &rarr; +&infin;, angle &rarr; 0, and at a = 1, angle = &pi; / 2.  If you wanted an angle of &pi; / 3, solve for a to get a = &radic;3. EdChem (talk) 05:25, 19 October 2016 (UTC)
 * ... and, of course, all of the above assumes identical scales on the axes.   D b f i r s   15:54, 19 October 2016 (UTC)