Proving $\arg(𝑧𝑤)=\arg(𝑧)+\arg(𝑤)$?
Proving $\arg(zw)=\arg(z)+\arg(w)$
using the \lstlistoflistings always named as Contents in table of contents.  I did look into   and  but the solutions there did not work.  Example: (my complete preamble)  \documentclass[11pt, oneside]{Thesis}  \usepackage{tikz} \usepackage{pgfplots} \usepackage{pgfplots,filecontents} \usepackage{caption} \usepackage{subcaption} \usepackage{booktabs,colortbl} \usepackage{graphicx} \usepackage{rotating} \usepackage{listings} \usepackage{color} \usepackage{comment} \usepackage[official]{eurosym} \usepackage{colortbl} \usepackage{algorithm}  \usepackage{algpseudocode} \usepackage{indentfirst} \usepackage{latexsym} \usepackage{titlesec} \usepackage[square, numbers, comma, sort&amp;compress]{natbib}   \hypersetup{urlcolor=blue, colorlinks=true}  \renewcommand{\lstlistlistingname}{List of Code}   \begin{document}  \tableofcontents \lstlistoflistings \lstinputlisting[language=python,firstline=1,lastline=68,caption=some caption here., label=code:getbb]{Code/getBoundingBox.py} \end{document}   The renew command changes the header in the page but not in the table of contents. Any ideas?  EDIT x2:  Managed to get the naming correct by editing this in the Thesis.cls   }  \renewcommand\lstlistoflistings{  \btypeout{List of Listings}  \addtotoc{List of Listings}  \begin{spacing}{1}{ \setlength{\parskip}{1pt} \if@twocolumn   \@restonecoltrue\onecolumn \else   \@restonecolfalse \fi \chapter*{\lstlistlistingname   \@mkboth{       \MakeUppercase\lstlistlistingname}{\MakeUppercase\lstlistlistingname}} \@starttoc{lot} \if@restonecol\twocolumn\fi \cleardoublepage  }\end{spacing}   Now the name appears correctly in the TOC but the Listings do no show in the Listings page, its just empty..  A MWE can be found here: Breadcrumb hyperlink header Do animagi get to pick their animal? Say that I want to become an Animagus, and am really hoping to be a certain animal (say, a stag). I study a whole bunch and manage to perfect the technique. Do I get to pick the animal I wind up as, so that I wind up being a stag like I want? Or is it something where I just find out what I turn into (and if I wind up as a dung beetle, tough luck)?    This is not a duplicate of  because that question is about which animals an Animagus can potentially transform into. Mine is about if I can decide which potential animal I wind up being. Regex to validate date format dd/mm/yyyy How to cite a book chapter written by an author who is not the editor or author of the book with BibTeX? I don't know how to cite a chapter written by an author who is not the editor or author of the book.   How can I get the following citation?     Grandstrand, O. (2004), 'Innovation and Intellectual Property Rights', in J. Fagerberg, D.C. Mowery and R.R. Nelson (Eds.), Oxford Handbook of Innovation. Oxford University Press: Oxford. How can I set conditional formatting on a particular cell that depends on another cell's value? What is the correct way to use infinitive after the verb "help": with or without "to"? What is the correct way to use infinitive after the verb "help": with or without "to"?  For example:     Please, help me to understand this.   or:     Please, help me understand this. How to make a Rigid Body fall off a conveyor belt? I've created conveyor belt and and two objects with rigid body physics. But the first object won't fall and second object falls backwards.  How can I fix it? Radial Schrodinger equation with inverse power law potential Recently I read a  about solving radial Schrodinger equation with inverse power law potential.  Consider the radial Schrodinger equation(simply set $\mu=\hbar=1$):  $$\left(-\frac{1}{2}\Delta+V(\mathbf{r})\right)\psi(\mathbf{r})=E\psi(\mathbf{r}).$$  A  gives a one-dimension equation:  $$-\frac{1}{2}D^2\phi(r)+\left(V(r)+\frac{1}{2}\frac{l(l+1)}{r^2}\right)\phi(r)=E\phi(r),$$  where $D=\dfrac{d}{dr}$, and $l$ is the azimuthal quantum number.  If we only consider the ground state, then $l=0$, so  $$-\frac{1}{2}D^2 \phi(r)+V(r)\phi(r)=E\phi(r).$$  We want to find eigenvalue $E$ such that $\phi(0)=\phi(+\infty)=0$.  The central potential discussed in the paper is of this form:  $$V(r)=\alpha r^{-\beta}.$$  It states (see page 4) that if $\beta&gt;2$ then the potential is repulsive (i.e. $\alpha&gt;0$).  My questions are:    Is this conclusion (i.e. If $\beta&gt;2$ then we must have $\alpha&gt;0$) generally valid in physics? What would happen if $\alpha&lt;0$ and $\beta&gt;2$? Is there 'ground state' in this condition? What about the condition $\alpha&gt;0$ and $\beta=1,2$? I have tried to solve the equation numerically with $\alpha=1,\beta=1,2$, and the ground state energy in this two conditions seem to be $0$, is my result correct?   P.S. When I try to find ground state when  $\alpha=-1,\beta=2$, the energy seem to be $-\infty$, which is qualitatively different from $\alpha=-1,\beta=1$. proof of functional completeness of logical operators Dividing an equilateral triangle into N equal (possibly non-connected) parts It’s easy to divide an equilateral triangle into $n^2$, $2n^2$, $3n^2$ or $6n^2$ equal triangles. But can you divide an equilateral triangle into 5 congruent parts? Recently M. Patrakeev  an awesome way to do it — see the picture below (note that the parts are non-connected — but indeed are congruent, not merely having the same area). So an equilateral triangle can also be divided into $5n^2$ and $10n^2$ congruent parts.  Question. Are there any other ways to divide an equilateral triangle into congruent parts? (For example, can it be divided into 7 congruent parts?) Or in the opposite direction: can you prove that an equilateral triangle can’t be divided into $N$ congruent parts for some $N$?                                               (Naturally, I’ve tried to find something in the spirit of the example above for some time — but to no avail. Maybe someone can find an example using computer search?..) I’d prefer to use finite unions of polygons as ‘parts’ and different parts are allowed to have common boundary points. But if you have an example with more general ‘parts’ — that also would be interesting. Why should a CTE start with a semi-colon? I was just looking at a post on  where Aaron Bertrand proposes using a CTE instead of a numbers table, which is an elegant way of performing the task at hand.  My question is, why does the first line of the CTE begin with a semi-colon?    ;WITH n AS (SELECT TOP (10000) n FROM    (SELECT n = ROW_NUMBER() OVER     (ORDER BY s1.[object_id])     FROM sys.all_objects AS s1     CROSS JOIN sys.all_objects AS s2   ) AS x ORDER BY n ) SELECT n FROM n ORDER BY n; -- look ma, no gaps!   Is this to ensure the WITH statement does not get parsed into a previous SELECT or something?  I see nothing in SQL Server 2005 BOL about using a semi-colon prior to the WITH. Note: If you're looking for a simple explanation as to why comments aren't required on downvotes, see .   I used to get "upset" (though that is too strong a term) when I got downvoted without comment. If my answer isn't good enough then I'd like to know why. Not only does it improve the answer for the OP but it improves my knowledge too.  Where the down-vote has been explained I've found it useful &amp; it has improved my answer, or forced me to delete the answer if it was totally wrong.  So is there any way we can encourage people to leave a comment? Perhaps they don't lose rep if they explain their down-vote?  I must admit though that I haven't always explained my down-vote either so you could call me a hypocrite. I've also grown a thicker skin over the months of using SO (it seems to have come with the higher rep score ;-)), so I'm less bothered about this now. $\mathbb{Z}[X]/(X^2-1)$ is not isomorphic with $\mathbb{Z}\times \mathbb{Z}$ I have to show that the ring $\mathbb{Z}[X]/(X^2-1)$ is not isomorphic with $\mathbb{Z}\times \mathbb{Z}$.    I know that $(\mathbb{Z}\times\mathbb{Z})^*=\{(\pm1,\pm1)\}$, so I thought I should be looking for elements which have inverses in $\mathbb{Z}[X]/(X^2-1)$ and hopefully find more or less than 4. But I didnt succeed, so I need hints. Thanks. How can I get a pure white background in jewelry photography, without losing sparkle and shine? I've read through several threads similar to this, but I'm looking for a comparison of two options here.   I currently take photos on a raised glass platform against a white foam board background and run about 300w of lights (3 100x LEDs @ 5000k, Home Depot Specials in softboxes and umbrellas). As you can imagine, it takes me about 15 minutes of post processing to achieve a perfect white background, and sometimes I have a particular photo that I am unable to get there.  I want to upgrade to either a photography table with a translucent background that I can place a speedlight under, or a cyclorama infinity background such as the MyStudio MS20. The latter option may require an upgrade to my lighting, which I am planning to do anyways.  The idea of course is to get a blown out 255 background with no post processing, other than color correction on the jewelry or product as needed. This would cut my time after the shoot to a ridiculously small amount of time.   Which option would make my life easier?   For reference, I currently use a D5300 with a Tamron 90mm f2.8 macro lens.  This is the best shot I got after quite a lot of effort, before and after processing:      Note that the silver is not shiny like it should be, and the sparkle on the gems is lost. I am planning to visit Moscow this winter as a solo tourist. I am a Hungarian citizen. I booked airplane tickets myself. I was informed that I will need an invitation letter, and I think I will go this way: (), but now the tourist visa application online form asks:    the options are: None, Travel company. I think I will need to select travel company because the description says:     In the case of a tourist visa select "Travel company" from the list and give the name and address of the travel company. In the case of   business visa select "Organization" from the list and enter in the   "Name of organization" name of the host organization, enter the   address of the organization and TIN in the appropriate fields. In   the case of private visa select “Individual” from the list and enter   the name, birth date and home address of the host. In the case of   transit, select "None" from the list.  Reference number: printed on   the bottom line of approval of a foreign tourist "Референс №». It is   necessary to introduce whole number with leading zeros.       Confirmation number: printed on the top line of confirmation of a   foreign tourist. The number can contain only numbers or numbers with   letters. You should only enter the numbers.   However I do not have a travel company.   What do I do?   Select "None" from the drop down so that I don't have to fill in travel company(this seems to be against the description that says select travel company for tourist visa) Select Travel company and the Invitation letter company will be my travel company and they will provide appropriate details? as in () Apply at some additional travel company, because there is no other way?  Some other option?   I didn't expect the visa application to be this complicated. I hope I get there. Greatest and Least Resistance and Current Characteristics of Parallel vs Series circuits Which type of circuit would have the greater resistance?  Which type would have the greater current? and Why?  My book says that parallel has the greater resistance than current as parallel wiring if thought as water pump tubes has greater resistance during water flow.  Contrary to my book, if I make up numbers to solve for which has the greater resistance, i find that  series has greater resistance.  Equivalent resistance of 2 Ohm resistor in series is greater than that of 2 resistors with 2 ohms in parallel. I Can inner classes access private variables? class Outer {      class Inner {     public:         Inner() {}         void func() ;     };  private:     static const char* const MYCONST;     int var; };  void Outer::Inner::func() {     var = 1; }  const char* const Outer::MYCONST = "myconst";   This errors out when I compile with class Outer::Inner' has no member named `var' Why does space expansion not expand matter? For some reason my iMac keeps turning off around 5 to 10 minutes into boot.  Also just recently it turns on but has shaded lines going down the screen. I have not installed any weird programs and i have not used it in around 3 months.   Any ideas on what the issue is, i don't have any other information about the issue besides whats happening when i use it.  The image i'm showing you was found on google but this looks exactly like mine does.  The model of my imac is the late 2011 model.