Java: Multiple Loop Function Not returning
How do I compare strings in Java?
Showing that a one-to-one linear transformation maps a linearly independent set onto a linearly independent set Problem 4.3.32 in Linear Algebra, Lay:  Let $V$ and $W$ be vector spaces, let $T:V\to W$ be a linear transformation, and let $\{\mathbf{v}_1, \dots , \mathbf{v}_p \}$ be a subset of $V$. Suppose that $T$ is a one-to-one transformation [...]. Show that if the set of images $\{T(\mathbf{v}_1), \dots ,T(\mathbf{v}_p)\}$ is linearly dependent, then $\{\mathbf{v}_1, \dots , \mathbf{v}_p \}$ is linearly dependent. This fact shows that a one-to-one linear transformation maps a linearly independent set onto a linearly independent set.  All I can think of doing: If $\{T(\mathbf{v}_1), \dots ,T(\mathbf{v}_p)\}$ is linearly dependent, then there exist weights $c_1, \dots, c_p$, not all zero, so that $$c_1 T(\mathbf{v}_1) + \cdots + c_p T(\mathbf{v}_p) = \mathbf{0}$$ Since $T$ is linear: $$c_1 T(\mathbf{v}_1) + \cdots + c_p T(\mathbf{v}_p) = T(c_1\mathbf{v}_1) + \cdots + T(c_p \mathbf{v}_p)=T(c_1 \mathbf{v}_1 +\cdots + c_p \mathbf{v}_p)=\mathbf{0}$$ So the set $\{\mathbf{v}_1, \dots , \mathbf{v}_p \}$ must be linearly dependent. Am I missing something? And why must $T$ be on-to-one for this to be true? An example showing that it is not true if $T$ is not one-to-one would be great! How to quickly and easily create a grid with Adobe Photoshop? Suppose I need to draw a grid in Photoshop, each square being the same size, and a certain amount of pixels tall and wide, and the grid lines having a certain pixel width. Like this:     Currently I do this by drawing one square, copy+pasting+moving+merging it a few times to create one row, copy/pasting that a few times to create more rows above each other.  It works, but feels very clumsy, I'm sure there ought to be a more efficient way to do this?  What are some efficient methods for creating a grid of equally sized squares? Matrix reciprocal positive to prove $λ_{max⁡}=n$ Suppose we have $n\times n$ matrix $A$ having only positive elements and satisfying the property $a_{ij}=1/a_{ji}$ (a matrix satisfying this property is called a reciprocal matrix).  If its largest eigenvalue $λ_{max}$ is equal to $n$, then the matrix $A$ satisfies the property (consistency property) $a_{ij}a_{jk}=a_{ik}$ where $i,j,k=1,2,...,n$.    I already found an example of $5\times 5$ reciprocal matrix  to show this condition:  \begin{bmatrix}1&amp;1/2&amp;1&amp;1&amp;1/4\\2&amp;1&amp;2&amp;2&amp;1/2\\1&amp;1/2&amp;1&amp;1&amp;1/4\\1&amp;1/2&amp;1&amp;1&amp;1/4\\4&amp;2&amp;4&amp;4&amp;1\end{bmatrix}  It has simple characteristic equation $λ^5-5λ^4=0$, so the eigenvalues of matrix $A$ are $λ=0$ and $λ=5$ and it's shown that $λ_{max}=n=5$.  Is there anyone can help me to give another positive reciprocal matrix example  that has simple characteristic equation and integer number as its eigenvalues ($4\times 4, 5\times 5$ or $6\times 6$ matrix)? In the proof of the lemma     For every limit ordinal $\alpha$, $\textrm{cf } \alpha$ is a regular cardinal    Jech says that     It is easy to see that if $\alpha$ is not a cardinal, then using a mapping of $|\alpha|$ onto $\alpha$ one can construct a cofinal sequence in $\alpha$ of length $\leq |\alpha|$, and therefore $\textrm{cf } \alpha &lt; \alpha$.    I've spent a while trying to construct this cofinal sequence and was wondering if I was on the right path:   From a bijection $f$ between $\kappa = |\alpha|$ and $\alpha$ take $S = \{\beta \in \kappa : \ f(\gamma) &lt; f(\beta)$ for all $\gamma &lt; \beta\}$ then if $\xi$ is the order type of $S$ and $g$ is the isomorphism from $\xi$ to $S$ the function $f\circ g$ is a cofinal sequence in $\alpha$ and $\xi &lt; \alpha$.   Is this on the right track, or is there an easier way of doing it? What is the most concise and efficient way to find out if a JavaScript array contains a value? This is the only way I know to do it: function contains(a, obj) {     for (var i = 0; i &lt; a.length; i++) {         if (a[i] === obj) {             return true;         }     }     return false; }  Is there a better and more concise way to accomplish this? Could there be an eighth Horcrux? What does a binomial coefficient with commas mean? I'm reading through  paper (on Dyck Paths). Near the middle of the second page, the author states the following:     Remark For the set $h_n$ there are ${k\choose t_1, t_2, ... , t_m }$ $n + k \choose{k}$ $=$ $ n + k\choose {n, t_1, t_2, ..., t_m}$ different Dyck paths.   What does ${k\choose t_1, t_2, ... , t_m }$ mean? I know what the binomial coefficient is, but I'm not sure how to interpret it when one of the parameters is a set. How can I do a line break (line continuation)? I have a long line of code that I want to break up among multiple lines. What do I use and what is the syntax?  For example, adding a bunch of strings,  e = 'a' + 'b' + 'c' + 'd'   and have it in two lines like this:  e = 'a' + 'b' +     'c' + 'd' Why Smart's attack doesn't work on this ECDLP? 'Possible duplicate' comment not appearing for questions with deleted comments containing the link I posted a comment to a duplicate question on . But I decided to delete it. Afterwards I voted as duplicate and thought a comment would be generated like     possible duplicate of ...   But it didn't. My guess is that the duplicate-comment won't be generated if a comment exists with that link, even if deleted. How many sequences of $n$ tosses of a coin that do not contain two consecutive heads have tails as the first toss? If you toss a coin $n$ times, there are $2^n$ possible sequences of heads and tails. Let $E_n$ be the set of sequences which do NOT contain two consecutive heads and $e_n$, the number of sequences in $E_n$.  Thus $E_3$ $=$ $\{$ $TTT$, $TTH$, $THT$, $HTT$, $HTH$ $\}$ and $e_3$ $=$ $3$.  How many elements of $E_n$ have $T$ as the first toss?  Please can anyone help me out here? I am reading Schwarz QFT and I reached the mass renormalization part. So he introduces, after renormalization, a physical mass, defined as the pole of the renormalized propagator, and a renormalized mass which can be the physical mass, but can also take other values, depending on the subtraction schemes used. Are these masses, other than the physical one, observable in any way experimentally, or are they just ways to do the math easier (using minimal subtraction instead of on-shell scheme, for example)?   Also, in the case of charge renormalization, the explanation was that due to the vacuum polarization, the closer you are to the charge, the more you see of it, so the charge of the particle increases with the momentum you are testing the particle with. However, I am not sure I understand, from a physical point of view, why do you need to renormalize the mass. Is this physical mass (defined as the pole of the renormalized propagator) the same no matter how close you get, or it also changes with energy? And if it changes, what is shielding it at big distances? Is it possible to set the "urgent" flag in Gmail for outgoing mail? I'd like to be able to flag time-sensitive emails as such. I'm giving Deus Ex a quick playthrough as a refresher before jumping into Human Revolution. In the interest of expediency, I've decided to cheat. ^.^  Normally, cheats must be re-enabled for each level when playing the original DX. I've encountered instructions on several sites (word-for-word identical instructions, so clearly copy/pasted) for permanently enabling cheats, but they aren't working for me:     Because you will need to redo this for every level, you can open up 'user.ini' and add 'bCheatsEnabled True' in the [DeusEx.DeusExGameInfo] section of the file to allow cheats to be activated forever.   Not only is that not working for me, it doesn't even look right compared to the rest of the file (space instead of =). With increasing desperation, I've tried adding that line (both verbatim and switching to an equal sign) to every section of both User.ini and DeusEx.ini which looks even remotely plausible — even creating a [DeusEx.JCDentonMale] section, just in case.  Does anyone know how to get this working? I'm running DX:GOTY version 1.112fm from Steam on Windows 7. How can I create a product key for my C# application? How can I create a product key for my C# Application? I need to create a product (or license) key that I update annually.  Additionally I need to create one for trial versions.  Related: In Ubuntu 17.10 ubuntu upgraded Nautilus (File Browser) to version 3.26.0. New nautilus is not using file-roller anymore, it has  to an integrated compression mechanism.     This makes it complicated to create encrypted archives and/or different types compression archives that file-roller does provide.  How to make nautilus use file-roller as before?  Related question: How to list files installed by a snap package? After a snap has been installed with sudo snap install [package] how can I list the files which have been installed by the snap? What is going on with pgfpages and page labels? I’m using a feature of beamer that shows notes beside slides, but the page numbers that are coming out are crazy, making it impossible to extract specific slides into other documents with \includegraphics[page=n]{presentation.pdf}.  For example, in this MWE, there are two frames that are each 1 slide, and then one frame that is four slides. The PDF page labels that show up in Acrobat are, as indicated, 2, 3, 3, 3, 3, and 7. It starts at page 2 instead of 1, and then there are four page 3s, but they don’t even all belong to the same frame!  \documentclass{beamer} \usepackage{pgfpages}  \setbeameroption{show notes on second screen=left} \setbeamertemplate{note page}{frame \insertframenumber \\ \insertnote} \def\pg#1{pgfpages says this is page #1 but it is really page \arabic{page}}  \begin{document}  \setcounter{page}{0}  \frame{hi \note{\pg{2}}}  \frame{there \note{\pg{3}}}  \frame{how \pause are \pause you \pause today \note&lt;1-3&gt;{\pg{3}} \note&lt;4&gt;{\pg{7}}}  \end{document}   Here’s what the output looks like:    I know that the manual says, “pgfpages will produce wrong page numbers in the .aux file” and gives a manual workaround. But the page counter still works, so shouldn’t there be some way to pass that through to the PDF? I’d like to fix this, but don’t know where to start. Specifically, when the manual says that “TeX instantiates the page numbers when writing an .aux file only when the physical page is shipped out,” what exactly is it talking about? Where is the source code for that? Check for missing fonts/characters in XeLaTeX? How can I play encrypted DVD movies?