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If a,b,c,d are positive integers with a sum of 63, what is the maximum value of ab + bc + cd ? (By using calculus to solve it)) If $a,b,c,d$ are positive integers with    $$a+b+c+d=63,$$   what is the maximum value of $$ab+bc+cd?$$   I treated this as an optimization problem and tried to solve it but I did not make any progress. C# Settings.Default.Save() not saving? the problem of using "~\\" and blank line Recently,I found the code that cotains the"~\\" and blank line always shows the warning imformation "Underfull \hbox (badness 10000) in paragraph at lines"  \documentclass{ctexart} \usepackage{color} \usepackage{amsmath} \usepackage{amssymb} \usepackage{graphicx} \usepackage{epstopdf} \hfuzz=\maxdimen   $(A \pm  B)^{-1} \neq A^{-1} \pm B^{-1}$,$(A \pm  B)^* \neq A^* \pm B^*$~\\  $|(A^*)^*|=|A|^{n^2-2 n+1}$~\\  \end {document}   I add the code:  \tolerance=10000 \hbadness=10000 \begin{document}   However,fail to solve it! Just as background, I should say I am a mathematics grad student who is trying to learn some physics. I've been reading "The Theoretical Minimum" by Susskind and Hrabovsky and on page 134, they introduce  transformations. Here's the first example they use:  Consider a particle moving in the x,y plane under the influence of a potential, $V$, which depends only on the radius, with Lagrangian:   $L=\frac{m}{2}(\dot{x}^2+\dot{y}^2)-V(x^2+y^2)$  This is clearly invariant under rotations:  $x \rightarrow x\cos \theta + y\sin \theta$  $y \rightarrow -x\sin \theta + y\cos \theta$  All well and good. Now they say "consider what happens ... when the angle $\theta$ is replaced by an infinitesimal angle $\delta$." Already I could say "What the heck is $\delta$ really?", but I'm willing to play along with my intuition. Since $\delta$ is infinitesimal, we work to first order and say   $\cos \delta=1$ and $\sin \delta= \delta$.  Plugging this into our rotation formulas above, we obtain:  $x \rightarrow x+y\delta$  $y \rightarrow y-x\delta$  By differentiating, we see that:  $\dot{x} \rightarrow \dot{x}+\dot{y}\delta$  $\dot{y} \rightarrow \dot{y}-\dot{x}\delta$  Plugging these into the Lagrangian and ignoring terms higher than first order, we see that the Lagrangian is invariant under this transformation.  My main problem with all of this is that I don't understand what the physical nature of an infinitesimal transformation actually is. All I got from the above was that if you do this formal calculation by following rules like "only work to first order in $\delta$," then the Lagrangian is invariant. This is in contrast to the case where we have an actual transformation, like a rotation, where there is no question about what is going on physically.  I would also like to know how all of this relates to rigorous mathematics. In mathematics, I can't recall ever using infinitesimals in an argument or calculation, so it would be useful if there was some way of formulating the above in terms of limits/ derivatives/ differential forms (for example). I sense a connection to Lie Algebras, as the infinitesimal version of the rotation is $(I+A)$ where $I$ is the identity matrix and $A$ is an element of the Lie Algebra of $SO(2)$.   Here are some questions whose answers I believe may be useful to me (feel free to answer some or all):  -What is an infinitesimal quantity like $\delta$ to the physicist?   -Why do physicists argue using infinitesimals rather than "standard" calculus?  -What is the physical meaning of infinitesimal transformation? How does it relate to Lie Algebras?   -Is there a rigorous theoretical apparatus for justifying the computations shown above?  -What is meant by the Lagrangian being invariant under infinitesimal transformations?  If any of the questions seem too vague, please say so. Thanks in advance for your insights! propositional calculus problem, how to prove this right or wrong? $A$$\rightarrow$$(B$ $\vee$ $C$ ) , $B$ $\rightarrow$ $C$ $\vDash$ $A$ $\rightarrow$ $D$  I think it's wrong but I have no idea how to prove. How do I enter a file or directory with special characters in its name? I want to enter the following folder in the terminal:  Milano, Torino (Jan)-Compressed   How should I write the command cd to enter this directory?  Spaces and several other special characters like \, *, ), ( and ? cause problems when I try to use them in the command line or scripts, e.g.:    $ cd space dir bash: cd: space: No such file or directory  $ cat space file cat: space: No such file or directory cat: file: No such file or directory  $ cat ( bash: syntax error near unexpected token `newline'  $ echo content &gt;\ &gt; ^C  $ ls ? (  )  *  ?  \   How do I enter file or directory names that contain special characters in the terminal in general? As we all know, immediate actions are just like swift actions, but can be performed outside of one's turn.     However, unlike a swift action, an immediate action can be performed at any time - even if it's not your turn.   However, it is still unclear if one can use an immediate action during someone else's action.  For example:  A monster swings at a conjurer.    Can conjurer abrupt jaunt outside monster's reach right after the attack was rolled, thus wasting an attack?  What about right after the damage was rolled, but not yet applied?  What about casting celerity and baleful transposition on yourself and monster's ally so the monster would hit his ally, given high enough attack roll?   Let's complicate a bit with a complex, full-round action on a monster's side.    Can one use immediate action in between attacks in full attack? What about after movement part of a charge? Tag synonym suggestion textbox is overlapped by "Suggest" button Is a   select *  from myView   faster than the query itself to create the view (in order to have the same resultSet):  select * from ([query to create same resultSet as myView])   ?  It's not totally clear to me if the view uses some sort of caching making it faster compared to a simple query. "Error: Invalid signature" when configuring dual boot Ubuntu 14.04 and Windows 10 What is the opposite of postpone? I am running k-fold cross validation on my training data, and then choosing the best set of hyper parameters, re-training on the training data and testing on a new (unseen) testing data. I am getting the following testing results (results on the unseen data):  R^2: 1.000 Adj R^2: 1.000 RMSE: 0.081 MSE: 0.007 MAE: 0.045 MAPE: 0.058   Can $R^2$ have a value of 1.00 but RMSE have a  value of 0.081? Doesn't an $R^2$ of value 1 indicate no errors at all? Would a Cup of Tea Be Hotter If you Add the Milk Before or After Boiling Water? This is a bit of dispute between work colleagues. An answer would be greatly appreciated.  My argument is as follows:  If you add X amount of milk at a temperature of M to a mug at room temperature R before adding X amount of water at temp W, the result would be a cooler cup of tea than if you'd added the hot water first. This would be due to the milk reducing the overall temperature of the mug in the time that has been in it resulting in the addition of boiling water having less of an effect and resulting in a slightly cooler cup of tea/coffee.   Conversely, if you add the hot water first, due to the mug being heated, the effect of adding the cool milk will be less and therefore the end result will be a hotter cup of tea.  My Colleagues arguement is simply that it would not make a difference but without any form of justification.   If anyone can give real scientific light to this issue it would be greatly appreciated :) Remote login with graphical display manager (GDM, LightDM) Is it possible to remotely (via ssh) make a login to GDM? I'll explain a situation in which I'd need this: I'm on my laptop and connect to my desktop via ssh. I realize there are some updates available. I install them all, but I need to reboot the computer. The problem is that once the desktop has rebooted, there will be no GDM session available, since it's necessary to log in. Can I somehow perform the GDM login remotely from my laptop?  Edit: I can't find FreeNX for Maverick, and I haven't been able to configure x11vnc + vncviewer to work. Isn't there an easy way? The passive with "let" Is there a way to unfold a mesh? Is there a way to take a mesh, cut it with seams or sharp edges and flatten it, so that the faces stay the same shape and size?  Making paper models is one aspect, where you could use that technique.  No, I'm afraid SZ0Return is not going to do it properly.  And misusing the UV-unwrap is only a last option.    Note, while it's an option to revert to Blender 2.49 which had a script to do this, I'd like a solution that I can use inside the current workspace. How do I add a kernel boot parameter? I need to add the boot parameter foo=bar to my kernel. How can I do this once for testing, and permanently if testing was a success? I want to clear a element from a vector using the erase method. But the problem here is that the element is not guaranteed to occur only once in the vector. It may be present multiple times and I need to clear all of them. My code is something like this:  void erase(std::vector&lt;int&gt;&amp; myNumbers_in, int number_in) {     std::vector&lt;int&gt;::iterator iter = myNumbers_in.begin();     std::vector&lt;int&gt;::iterator endIter = myNumbers_in.end();     for(; iter != endIter; ++iter)     {         if(*iter == number_in)         {             myNumbers_in.erase(iter);         }     } }  int main(int argc, char* argv[]) {     std::vector&lt;int&gt; myNmbers;     for(int i = 0; i &lt; 2; ++i)     {         myNmbers.push_back(i);         myNmbers.push_back(i);     }      erase(myNmbers, 1);      return 0; }   This code obviously crashes because I am changing the end of the vector while iterating through it. What is the best way to achieve this? I.e. is there any way to do this without iterating through the vector multiple times or creating one more copy of the vector? Most statements regarding Riemann integrals (at least the ones that I have encountered) begin with the statement "for $f(x)$ bounded on $[a,b]$." I am wondering if Riemann integrability implies boundedness. I think that this has to be the case, but I am not sure. If Riemann integrability does imply boundedness, are improper integrals considered Riemann integrals? I would think that improper integrals wouldn't be Riemann integrals since improper integrals are allowed to be equal to $+\infty$ or $-\infty$. Or are improper integrals that are not equal to $\infty$ considered Riemann integrals?   I am confused. I know that new arXiv papers usually appear in Google Scholar within a few days or weeks. My question is: if someone submits a replacement of his/her arXiv paper, does the new version of the paper also appear in Google Scholar?