$$ -1=(-1)^1=(-1)^{2\frac{1}{2}}=((-1)^2)^{\frac12}=1^\frac12=1 $$ I can't find a mistake. I think it might be that $$a^{mn}=(a^m)^n$$ doesn't hold for all $$a,m,n$$ but I don't know exactly. Can anyone help?
I know there must be something unmathematical in the following but I don't know where it is: \begin{align} \sqrt{-1} &amp;= i \\\\\ \frac1{\sqrt{-1}} &amp;= \frac1i \\\\  \frac{\sqrt1}{\sqrt{-1}} &amp;= \frac1i \\\\  \sqrt{\frac1{-1}} &amp;= \frac1i \\\\  \sqrt{\frac{-1}1} &amp;= \frac1i \\\\  \sqrt{-1} &amp;= \frac1i \\\\  i &amp;= \frac1i \\\\  i^2 &amp;= 1 \\\\  -1 &amp;= 1 \quad !!? \end{align}
Is there a possibility to change the font-style in the table of contents? Everything I wrote down in my document was written with Times. However, the ToC does not have the same font family. Is there a way to change the font of the ToC to Times New Roman? Thank you : )    \documentclass[a4paper, fontsize=11pt]{scrbook} \usepackage{geometry} \geometry{a4paper, top=25mm, left=25mm, right=25mm, bottom=20mm, headsep=10mm, footskip=12mm}  \usepackage[ngerman]{babel}  \usepackage[T1]{fontenc}        \usepackage[latin1]{inputenc} \usepackage{acronym}      \usepackage{anyfontsize}  \begin{document}  \tableofcontents  \chapter{Test} Is there a possibility to change the font-style in the table of contents? Everthing I wrote down in my document was written with Times. However, the ToC does not have the same font family. Is there a way to change the font of the ToC to Times New Roman?  \end{document} Solving minimization problem $L_2$ IRLS (Iteration derivation) In the article ''' Chartrand, Rick, and Wotao Yin. "Iteratively reweighted algorithms for compressive sensing." Acoustics, speech and signal processing, 2008. ICASSP 2008. IEEE international conference on. IEEE, 2008. '''  It is given that one iteration of  iterative reweighed least squares problem can be written as $\min \sum_{i=1}^N w_i u_i^2 $ subject to $\Phi u = b  $ which has a closed form solution given as   $u^{(n)} = Q_n \Phi^T (\Phi Q_n \Phi^T)^{-1} b$. where $Q_n$ is the diagonal matrix containing entries $1/(|u_i^{(n-1)}|^2 + \epsilon )^{1/2} $ And the closed form equation can be derived from the Euler Lagrange equation. Can someone please help me to get the derivation of the iteration?\ I understood till $$\min_u u^TWu+λ|| \Phi u−b||_2^2 $$  which can be written as $(\bigtriangledown=0)$ which gives $$ W u + \lambda (\Phi^T \Phi u - \Phi^T b) =0$$ A TV show, from the 90's or 80's, very weird Does anyone remember a show, where:   An alien girl lives with a family She comes from the moon She is looking for a new king because their king (a giant baby) is dying Her spaceship its a VW van They had a doorframe, yes a doorframe, that could open to other alternative universes or dimentions HTML favicon won't show on google chrome Using Virtual Box, how can I install an OS to a secondary, physical disk, and boot it in both (at separate times) Virtual Box, and as a typical secondary OS install? When should I use exposure compensation? When should I use exposure compensation, rather than ISO, shutter speed or aperture? Suppose I have BaseClass with public methods A and B, and I create DerivedClass through inheritance.  e.g.  public DerivedClass : BaseClass {}   Now I want to develop a method C in DerivedClass that uses A and B.  Is there a way I can override methods A and B to be private in DerivedClass so that only method C is exposed to someone who wants to use my DerivedClass? Is division of matrices possible? Should I postpone my trip to Paris in 2nd week of Jan in light of recent protests? I’ve a date and I really waited long to make this happen. My gifrfriend is a bit worried about the security issue in Paris. Which locals think that the media is exagerrating the issue and the issue is small and non-threatening for tourists? The question is $$\text{If }\, p, \ q\in \mathbb{N}, \;1-\frac12+\frac13-\frac14-\dotsb-\frac{1}{1318}+\frac{1}{1319}=\frac{p}{q}.\qquad \text{Prove that } 1979\mid p.$$ So my solution went like this:  $$S=1-\frac12+\frac13-\frac14-\dotsb-\frac{1}{1318}+\frac{1}{1319}\Rightarrow S=\sum_{k=1}^{1319}\frac{1}{k}-2\sum_{k=1}^{659}\frac{1}{2k}=\sum_{k=660}^{1319}\frac{1}{k}$$ Now this is equivalent to finding the sum of the roots of the equation $$\left(x-\frac{1}{660}\right)\left(x-\frac{1}{661}\right)\dots \left(x-\frac{1}{1319}\right)=0$$. Rewriting this equation, we get $(660x-1)(661x-1)\dotsc(1319x-1)=0$ (we can discard the $\displaystyle \frac{1319!}{659!}$ since $1979$ is a prime). Calculating the sum of the roots, we get the required value to be equal to $(1320+659)\times 330$ which is obviously divisible by $1979$ (again discarding the coeffecient of $x^{660}$).  Now my question is: Using the same method of converting the problem into finding the sum of roots of a polynomial, we get from the original equation something like this: $$(x-1)\left(x+\frac{1}{2}\right)\dots \left(x-\frac{1}{1319}\right)=0 \Rightarrow (x-1)(2x+1)\dotsc(1319x-1)=0$$ So, the sum of roots would be $-1+2-3+4-\dotsc+1318-1319=-660$ But this is not divisible by $1979$. Why does the former solution work while the latter doesn't? Is there a flaw in my reasoning? Note: I know that the question can be solved by some other methods. Please only explain my given doubts and don't instead give another solution  I realised my mistake after some hours of pondering over it. (In the 1st equation) To get the required sum I would need to take calculate $\displaystyle -\frac{\text{coeffecient of }x^{659}\,}{\text{coeffecient of }x^{660}}$. However what I've done is calculate this:  $\displaystyle -\frac{\text{ coeffecient of }x}{\text{coeffecient of }x^{660}}$ which is actually equal to $\text{sum of roots taken }659 \text{ at a time} $. It was just a coincidence that the value I calculated was divisible by $1979$. The correct way, indeed, would be through the link supplied Personalize monitor position before login I'm using multiple monitors.  When logged in everything is fine: The left monitor is positioned to the left and the right monitor is positioned to the right, but before I log in, the login-screen positions them the other way around.  This is also the position presented by bios and then the grub menu.  I don't mind where the login is displayed, but when moving the mouse, the boundary is the wrong way around and that's annoying. Why do the eigenvectors for two similar matrixes differ by a large amount Let $A,B,C$ be sets, and $B \cap C=\emptyset$. Show $|A^{B \cup C}|=|A^B \times A^C|$ Let $A,B,C$ be sets, and $B \cap C=\emptyset$. Show $|A^{B \cup C}|=|A^B \times A^C|$ by defining a bijection $f:A^{B \cup C} \rightarrow A^B \times A^C$.  Any hints on this one?  Thank you! We all know that the amount of current flowing b/w the plates is independent of the frequency. If I were to ask why doesn't it depend then you'd probably say that it depends on the number of electrons rather than the frequency or you might show me a current vs frequency graph.  But here is my theoretical argument. Electric current is defined as the amount of charge flowing through a point/space/wire per unit time. Let I be current due to photo electrons ejected by photons of some frequency. Now if I would increase the frequency, the kinetic energy of the ejected electron would increases, therefore it will move faster and therefore contribute more current (charge will through the same point many times per unit time) . Thereby, giving a net current greater than I.  What is wrong with my argument?  After thinking a lot, I came up with a possible reason. However fast the electrons may move, but when they hit the metal plate, they would lose their velocity and start moving in drift velocity. Now there is a problem with this argument. If this were to be true then I should expect to see a big chunk of extra electrons getting stuck at the anode.  If my reasoning was correct, then why can't I see a big charge buildup at the anode. Renaming hundreds of files at once for proper sorting I have a ton of files, all named stuff like 1.jpg, 2.jpg, 3.jpg, and so on up to 1439.jpg, however, I have a problem with one of my projects and alphabetizing. It will usually go in the order 1.jpg, 10.jpg, 11.jpg and so on.  What I need is some way to name the files so they are in the format such as 00001.jpg all the way up to 01439.jpg.  How would I be able to do this quickly and efficiently? Maximum and minimum distance of two points Consider six distinct points in a plane. Let $m$ and $M$ denote respectively the minimum and the maximum distance between any pair of points.  Show that $M/m \geqslant \sqrt{3}$. Physical meaning of this integral in $k$-space I am trying to solve an integral in $k$-space where I want to calculate the ratio of the average energy of an arbitrary electronic band ($E(k)=k^n)$ to the square of average group velocity of electrons($v_g   = k^{n-1})$ approximately in the form of this expression given by $$L=\frac{\frac{\sum_{\mathbf{k}}k^{n}}{\sum_{\mathbf{k}}1}}{\left(\frac{\sum_{\mathbf{k}}k^{n-1}}{\sum_{\mathbf{k}}1}\right)^2}\;.$$  Now, I want to do this for $d$-dimensions, so I convert this sum to integral.  $$L=\frac{\dfrac{\int_{0}^{k_f}k^{n} d^d k}{\int_{0}^{k_f}1d^d k}}{\left(\dfrac{\int_{0}^{k_f}k^{n-1}d^d k}{\int_{0}^{k_f}1d^d k}\right)^2}$$ where $d^dk  = \dfrac{d \pi^{d/2}}{\Gamma(\frac{d}{2}+1)}k^{d-1}dk$ is the volume element in $d$-dimensional space.  I want to get a simple expression for $L$ and how it depends with $k_f,n$ and $d$. any physical insight into this integral could also be helpful. $f$ is surjective iff it has a right inverse: using the axiom of choice and errors in ProofWiki Was Quirrell the Defense against the Dark Arts teacher for more than one year? It's well seen throughout the series that there is a “curse” on the DADA teacher position, that no one can hold that post for more than a year. But it seems like Quirrell has been around for longer than a year. What's up with that? I think there's an unintended motivation for users to post questions to higher-traffic Stack Exchange sites rather than their more specific or appropriate lower-traffic ones.  I think I have a solution.  What motivated this suggestion for me was that I am confident that if users were to post Android design questions to  beta and/or , that they would likely not be noticed by the appropriate audience compared to the vast army of Android experts here on Stack Overflow.    There are plenty of Android questions on SO that really should be on one or both of the aforementioned sites.  Presently there is little motivation for users to start questions like this on the appropriate site.  Is there something we could do to cross-pollinate related sites?  For instance, what if we let certain tags span certain sites?  Obviously the selection would be controlled by high-reputation users.   There could also be a feature to show (or explicitly hide) certain questions from other sites regardless of the tags.  This could be controlled by lower-reputation users than the global tag since it wouldn't have much statistical impact.  For instance, the android-layout tag could by default span sites, but OPs and other users could eliminate from the Graphic Design site questions that are of a purely technical nature.  Users interested in Android tagged questions would see related questions on design and UE sites--whether they are searching, posting, or viewing a question through the related items.  This would help eliminate cross-site duplicates, and encourage people to post on the correct site in the first place.   For related question lists and search results, there could even be a specific cross-site icon  next to an off-site question to show that it's from UX, for example.