Find all functions $f:\mathbb{R^*}\to\mathbb{R^*}$ that satisfy $$f(x)f(\frac{1}{x})=1\hspace{1 em}\forall x\in\mathbb{R^*}$$  I only found that every function $f(x)=x^n, n\in\mathbb{N^*}$ is a solution. Can you find all the solutions to this?
Find the functions $f:\Bbb R_*^+ \to \Bbb R_*^+$ such that:  $$f(x)f \left(\frac{1}{x}\right)=1$$
How to reference the caller object ("this") using attachEvent I have read several introductory texts about LSTM online, but none of them give a rigorously mathematical explanation of what the model actually does. For instance, why does the forget gate forget any more than any of the other gates in the LSTM unit?  Which papers and/or books should I read if I want to get a deeper understanding of LSTM? I've seen them maneuver. They definitely can go backwards. How fast can they go in reverse? Is warp speed possible, or is reverse limited to impulse power? If it is limited to impulse power, can they travel backwards under impulse power as fast as they travel forwards under impulse power? GParted can't resize partition, but unallocated partition is available When does a locational distinction change its suffix and capitialization in a proper name? I'm trying to create a signature and date line in a document.  I need the date line to always be in the same spot on the page no matter what name is under the signature line, such as:  ----------------------   ---------------------- Representative           Date  ----------------------   ---------------------- Program Manager          Date   Here is the code I currently have:  \makebox[2.5in]{\hrulefill} \hspace {1.0in}\makebox[2.5in]{\hrulefill} \\ User Representative \makebox[2.5in][r]{Date} \\  \vspace{.2in} \makebox[2.5in]{\hrulefill} \hspace {1.0in}\makebox[2.5in]{\hrulefill} \\ Program Manager \makebox[2.5in][r]{Date} \   I want to be able to change the names, such as change Program Manager to Joe Foo and have the word 'Date' stay in the exact place.  Any help would be greatly appreciated.  I'm still a newbie at TeX so please forgive my ignorance. Expansion of $(1+\sqrt{2})^n$ Can we know the fundamental nature of space and time? Recently I found a claim saying that  $$  \int_0^\infty \left( \frac{\sin ax}{x}\right)\left( \frac{\sin bx}{x}\right) \mathrm{d}x= \pi \min(a,b)/2  $$ from what I can see this seems to be true. I already know that $\int_{0}^\infty \operatorname{sinc}xy\,\mathrm{d}y = \pi/2$, and so independant of $y$. My suspicion is that this is closely related to the integral above.   Can someone give me some suggestions for evaluating the integral above? Also are there any generalizations for the integral? Eg $$ \int_{0}^{\infty} \left( \prod_{k=1}^N \frac{\sin (a_k \cdot x)}{x} \right) \,\mathrm{d}x $$ Where $a_k, \cdots, a_N$ are arbitrary positive constants. It seems related to the , but there are some subtle differences. Can you grab the blue shirts and socks?   Is the above sentence stating that both the shirts and the socks are blue? Or only the shirts?  At this stage, I am leaning towards the earlier (only the shirts) — though writing "Can you grab the blue shirts and blue socks?" seems redundant. How do I resume a release upgrade? Yesterday I started the upgrade from 12.10 to 13.04, but I interrupted it because it was taking too long. Now if I restart it, it doesn't proceed.  Is there any possibility to clean the mess it left when I interrupted it and make a clean restart of this update? Why XML-Serializable class need a parameterless constructor Do electrons move around a circuit? We could imagine a simple electronic circuit composed by a power source and a resistor.   It is usual find descriptions as "The moving charged particles in an electric current are called charge carriers. In metals, one or more electrons from each atom are loosely bound to the atom, and can move freely about within the metal. These conduction electrons are the charge carriers in metal conductors." (wikipedia ). That made me imagine electrons going from power -, running by the wires, crossing the resistor and returning to power +, where they suffer a new kick to power -. All that at near the speed of light.  In this model, it is easy to define "electrical current intensity" as proportional to the number of electrons per second that cross one specific point of the circuit.  However, looking for the physics of resistivity, I can read the phrase "The actual drift velocity of electrons is very small, in the order of magnitude of a meter per hour." (wikipedia ). This is a complete different point of view, where electrons are more or less fixed around a fixed position (or moving very slowly), an electricity is a wave of "collisions" or "jumps" of electrons, like in a Newton's cradle (or "do the wave" as public in a football match). Obviously, we do not need severals hours from the switch on to the light on.  So, the question is, do electrons move around a circuit ? References to band theory are welcome.  A related question is: if electrons doesn't moves along the circuit, what is "electrical current intensity"?  Note: I known there are similar questions already posted, but answers of the ones I've read are not very detailed (could be I've not found the best ones). Please, be a few patient before "close vote". Which topics are essential for QGIS beginners? What topics do you think are essential in an introduction for QGIS?  The target audience is those with zero or rudimentary knowledge in mapping and geospatial software. Is the electric force a vector or a vector field? Today I got what I presume was the update that should take care of Spectre and Meltdown. That didn't go very well. What do I do now? See screen photo.    I can't boot the 4.4.0-108-generic kernel because of a kernel panic. I'm running 16.04 on an Intel Core i5-4670K CPU @ 3.40GHz × 4. Generalization of Hölder inequality Make Google stop sending 'new sign in from ...' emails I get at least two emails from Google every time I sign in to my account and I want it to stop.  I tried changing the notification settings in Gmail, but they're still coming and they're useless.  How can I stop this? How to automatically create large number of columns in Excel 2013 I would like to start a new excel spreadsheet, create a column named Column1, and quickly fill in increments, (Column2, Column3 etc) up to Column500. Is there an easy way to do this?   Also is there an easy way to afterwards select all 500 columns and add 10 rows of data to all of them? Why is gravity stronger than other forces at the macroscopic level, yet weaker than other forces at the quantum level? Is there an explanation?