How can I write a double expression below de comand \argmin for explain the range of two variables
Is there a way to force a line break inside a math mode text?  My example:  \[   \underbrace{....}_\text{Some long text that should be multiline} \]   Trying  \text{Some long text that\\ should be multiline}   Didn't do the trick.
What is a word that means to say in a positive sense? For instance, when writers say that a person was sexually assaulted instead of saying raped.  Or people use let go instead of fired.  What is that called? Show that there is a positive integer $n$ such that $G \cong \ker(\phi^{n}) \times \phi^{n}(G)$ Let $G$ be a finite abelian group and let $\phi: G \rightarrow G$ be a group homomorphism. I am trying to show that there is a positive integer $n$ such that $G \cong \ker(\phi^{n}) \times \phi^{n}(G)$.  I know that since $G$ is abelian we have that $\ker(\phi^{k})$ and $\phi^{k}(G)$ are normal subgroups of $G$ for any $k$  I also suspect we have the following towers which stabilize at $m$ and $m'$  $$\phi(G) \unrhd \phi^{2}(G) \unrhd \cdots \unrhd \phi^{m}(G)=\phi^{m+1}(G)=\cdots$$  $$\ker(\phi) \unlhd \ker^{2}(\phi) \unlhd \cdots \unlhd \ker^{m'}(\phi)=\ker^{m'+1}(\phi)=\cdots$$  I know that if given a group of the form $HK$ where $H$ and $K$ are normal subgroups of $HK$ and $H\cap K=1$ then $H \times K \cong HK$. I want to apply this to this situation but I am unable to show how write $G$ as a product $HK$  Resolution One of the comments directed me to the following article:  The last part of which answers the question. Below is what is says:  Choose $n=\max(m,m')$, then we have for $x \in \ker^{n}(\phi) \cap \phi^{n}(x)$, this means that $x=\phi^{n}(y)$ for some $y \in G$. This gives:  $0=\phi^{n}(x)=\phi^{2n}(y)$ which means that $y \in \ker^{2n}\phi=\ker^{n}{\phi}$. and then we have that $0=x=\phi^{n}(y)$.  The answer below kindly points out that every element $x$ is contained in one of the cosets of $G/\ker^{n}(\phi)$, this means that $x=k+g$ for some $k \in \ker^{n}(\phi)$ and $g \in G$. We also have that $g=\phi^{n}(h)$ for some $h \in G$. Writing $x=k+\phi^{n}(h)$ essentially shows that $G=\ker(\phi^{n}) + \phi^{n}(G)$. Using the fact above, the claim follows. "Dangling Participles" My case here is that I was writing something for school, and my teacher explained an error which I really can't see. She referred to articles about &quot;dangling participles&quot;, which was previously unknown to me. I find it hard to relate the examples I've found about it to my situation. So this is a text I've written and the bold part is where the problem arises: (You could read only the bold part if you please)  Wandering, and lost in his thoughts, there is a deep despair occupying the mind of this man. He is barely aware of his surroundings, except for those few moments when the loud blissful voices break through the protective barriers, developed by his immense sadness and grave frustration, into his mind. ”How can they all possibly be so happy?” he thinks to himself. Staring blindly, with his head low, as he walks in a slow pace, whispering ”Mary, Mary, my Mary,” over and over again. Yesterday was her funeral. She had passed away from a severe type of cancer, at a relatively young age. It was his wife. They had been together since high school and were greatly attached to each other. Having spent nearly half his life by her side, we cannot but imagine his grief. Indeed, he could hardly picture a life without her. So, alone he roams in the midst of all the seemingly incessant cheering and noises, drowned by his sorrow.  Her comments on that part was:  we is slightly out of place here – see my comment below These few sentences are about him and his emotions, so ‘we’ seems out of place. If the sentence starts: ‘Having spent nearly half his life by her side’, the reader expects it to continue: ‘he’. After all we have not spent half our lives by her side.  Is she right? If so, could anyone try to explain why? A horizontal line with the same space-filling behaviour as /hfill? Is there a command or package which would allow me to draw a line in the largest available space in the way that \hfill fills up the remaining space in a line?  i.e., a command to draw a line from the end of the text, wherever it happens to fall, to the right edge of the textwidth (leaving space for any additional text indented to be at the end of the line)? How Get Parent Field in Attachment? Please Help me...  Account sObject is Parent of Attachment. Now I want to bring Read-only Field Custom field (isReadOnly__c).  SELECT Id, Name, Account.isReadOnly__c FROM Attachment   How It Gets ?  Thanks in Advance  Manu What is the limiting factor in lens construction when it comes to f-stop values? Why isn't $V= V_1 + V_2$? $V=V_a - V_c = V_a - V_b + V_b - V_c$, $V_a - V_b= V_1$ and $V_b - V_c = V_2$ Doesn't that prove that $V = V_1 + V_2$? Regardless of $V_3$, If i'm wrong , is there a way to obtain $V$ in terms of $V_1$, $V_2$ and $V_3$ Prior to the spell cast on Mjolnir, did Thor control lightning on his own? Use of "to lunch" vs."for lunch" Which is right?      I went to the (mall) to meet my parents for lunch / to lunch   This for a paper I am submitting for a writing assignment. I have heard "to lunch" is right, but is it right in this sentence please? Order of a homomorphism of groups Let $f: G \to H$ be a homomorphism of groups. Assume that $a\in G$ and $\operatorname{ord}(a)=n$. Prove that the order of $f(a)$ is a divisor of $n$. I know that if $H$ is a subgroup of $G$, then $$|G|=|H|\cdot|\text{distinct cosets of $H$}|$$ so $|H|$ must be divisor of $|G|$, but I don't understand how to apply that logic to homomorphisms (or if that will even work). Any help is appreciated! Fibonacci proof question: $f_{n+1}f_{n-1}-f_n^2=(-1)^n$ I mean are we certain that they will be able to provide us a huge improvements (in some tasks) compared to clasical computers? How to splice together several instances of InterpolatingFunction? I know that I have to use some tweaking tool and then a extension for this. But all extensions that are suggested for this(Workspace grid, Workspace matrix), aren't available in the software center. Why doesn't this perpetual motion machine using the buoyant force work? Nice proofs of $\zeta(4) = \frac{\pi^4}{90}$? Clip vector on Geoprocesing QGIS 2.18.13, global name 'features' is not defined I want to clip a vector polygon with a boundary vector polygon but the process is failing. I am getting the error, "global name 'features' is not defined See log for more details". I don't know what is causing this error. The mysql extension is deprecated and will be removed in the future: use mysqli or PDO instead How can I effectively use the Coppersmith? I've been playing Dominion with the base + Intrigue sets for a while now, and one of the cards that I have trouble wrapping my mind around is the Coppersmith.  In theory, it seems like a great card - it allows you to make all of your Coppers into Silvers, and in some cases - even Gold. However, I have yet to build a deck during a game that manages to use it effectively. Usually, in order to pull one or two of those with a nice amount of coppers, I need to create a deck that relies heavily on +cards/+actions chains, and the Coppersmith only begins to affect the game too late for it to be of any use.  So... how can this card be effectively used? How to avoid overflow using \IEEEpubid on the first column? I am trying to write a journal paper using the IEEEtran.cls template.  My problem is that the Publication ID Mark overflows into my thanks section.  This is the bottom of the title page:    From the documentation of IEEEtran.cls we see that:    I included the command \IEEEpubidadjcol into the second column. This prevents the overflow in the second column as you can see in my previous image, but lets the first column run freely.  I tried including \IEEEpubidadjcol into the first column aswell, but that causes compilation errors.  Please, could you give me some hint on how to solve this problem without doing something illegal with respect to the IEEE modus operandi?  EDIT:  As the answer suggests, the problem can be solved using the \IEEEpubidadjcol command twice. When I tried this, I got an error:  Infinite glue shrinkage found in box being split   In spite of this error a correct pdf is produced. The error goes away when I do not include packages fixltx2e and dblfloatfix.