The story is set on a planet where time passes much more slowly at the pole. A soldier is near one of the poles and a major offensive seems to be building up with his side firing into some kind of grey curtain, a horizon caused by the extreme time "gradient".  The other side is firing back.  He is fetched from his post as his discharge has come up and travels to the equator where he starts a new life. 20 years pass when 2 officers approach him, they had left to get him back only a few minutes after he has left and he returns to his post some 10 or so minutes after he had left.  He realizes that the weapon fire they are getting from across the void is just there own stuff coming back.  I have always taken it as an analogy for lots of tit for tat wars, but cannot find the story.
Trying to remember a short story (most likely from the sixties).  It starts with a soldier at the front line fighting an unknown enemy on the other side of the front line.  He then gets withdrawn and you realise that time is passing slower the further he moves from the front. He then spends years in peace away from the front, before being sent back to the front where only a short time (hours or days perhaps) have passed.      I seem to remember their was an inference that they were actually fighting against themselves, with their attacks coming back at themselves, which was perhaps the entire cause of the war, and an interesting critique on war.  I can't remember if this all this was stated explicitly, or just left for the reader to infer.
Let $f : \mathbb{R}^n → \mathbb{R}_∞$ be a function and let $C ⊂ dom f$ be a convex set.  $$**Part I**$$ Prove that $f$ is a convex function if and only if $f$ is convex over every line $L_{v,x_0}$ where $L_{v,x_0} := \{x_0 + tv : t \in \mathbb{R}\}$.  i.e. f is convex iff when restricted to any line, it be convex on that certain line.  So essentially to handle this proof I need to show that convexity for $f$ holds when its convex over any arbitrary line, and on the converse it does not hold when $f$ is not convex over that arbitrary line. If $f$ is convex then,  $$\forall x,y \in C, a \in [0,1]$$ it holds that $$f(ax+(1-a)y) \leq af(x) + (1-a)f(y) \space \space\space\space\space\space\space\space\space\space\space\space (1)$$  EDIT: $$**Part II**$$  Let $f : \mathbb{R}^n \to \mathbb{R}_∞$ be a function.   I want to prove that $f$ is convex over the line $L_{v,x_0}$ iff $\psi : \mathbb{R} \to \mathbb{R}_∞$  $\psi(t) := f (x_0 + tv)$, is convex over $I (x_0, v )$.  where $L_{v,x_0}:= \{x_0 + tv : t ∈ \mathbb{R}\}$, and $I(x_0,v):=\{t \in \mathbb{R} : x_0 + tv \in C\}$  Any help is very much appreciated Near the beginning of the definition of the amsmath environments aligned and gathered (and also mathtools' multlined) there's an explicit thin space \,. This often  has to be cancelled  with  \!, as has been observed  various  times:         I'd like to know why this explicit \, is there &ndash; what would go wrong without it?    From amsmath.dtx:  \newcommand{\start@aligned}[2]{%     \RIfM@\else         \nonmatherr@{\begin{\@currenvir}}%     \fi     \savecolumn@ % Assumption: called inside a group %    \end{macrocode} %    The \cs{null} here is to keep the \cs{,} glue from causing the %    invocation of the clause in \tex/'s built-in tag placement %    algorithm that can cause an equation to be shifted all the way over %    to the margin. %    \begin{macrocode}     \null\,% Stack Overflow is a site of passers-by They says that Stack Overflow is a community-driven web-site. But, as a matter of fact, it is not.  There is no community at all. It's more like an imageboard, as a number of participants makes them pretty much like anonymous to each other despite of registration.   Stack Overflow is a site of passers-by.  Passers-by who want to solve their silly typo problem and passers-by who eager to share their vast knowledge in spotting silly typos. And - oh - sometimes to perform a civic duty and vote to close a question they have no idea what to answer to and thus calling it "not a real question".  So, as a funny consequence, whatever question can get   answered upvoted downvoted closed deleted   depends not on its quality but only on the fluctuation happened to "community" at the given moment, at a pretty random rate.  It is not only quite discouraging. It makes the very idea of collecting and improving knowledge quite impossible. How to get the ASCII value of a character How do I get the  value of a character as an int in ? Let $\alpha, \beta \in \mathbb{C}$, such that $\alpha + \beta$, and $\alpha\beta$ are algebraic. Show that $\alpha$ and $\beta$ are algebraic. Let $\alpha, \beta \in \mathbb{C}$,  such that $\alpha + \beta$, and $\alpha\beta$ are algebraic. Show that $\alpha$ and $\beta$ are algebraic.   attempt: Suppose $\alpha, \beta \in \mathbb{C}$ such that $\alpha + \beta$, and $\alpha\beta$ are algebraic.  Let $\alpha + \beta$, $\alpha\beta$ are algebraic be algebraic over $F$. And let $\alpha, \beta \in K$, where $K$ is an extension of $F$.  then $\alpha + \beta $ and $\alpha\beta$ lie in the extension $K = F(\alpha,\beta)$, which is finite over $F$.  so $[F(\alpha, \beta) : F] = [F(\alpha, \beta): F(\alpha + \beta, \alpha\beta)][F(\alpha + \beta, \alpha\beta) : F] \leq 2[F(\alpha + \beta, \alpha \beta) : F] $, since $\alpha + \beta$ and $\alpha \beta$ are roots .   So $[F(\alpha + \beta) : F]$ is finite and in the extension $F(\alpha,\beta)/F$ the elements must be algebraic over $F$, then $\alpha , \beta$ are algebraic.  Can someone please verify this? Or help me give a better approach. Thank you! Some questions might be asked on three different sites, each site needing three different answers. For example, a question about a text editor.  So perhaps there should be a way to link the questions on the three different sites. Adding a link manually in the questions is certainly possible, but perhaps a feature officially supporting this will be beneficial. $A$ and $B$ are $n\times n$ matrices and $A+B=AB$. I have an interesting proof that this implies $A$ commutes with $B$, but the proof only works when $||B|| \lt 1$. I'm looking for a way to salvage the proof so that it works for all $B$.  $$A=AB-B$$ $$A=(A-I)B$$ by repeatedly substituting $A$ with $(A-I)B$ on the RHS, $$A= (A-I)B^N - \sum_{i=1} ^{N-1} B^i$$  Since the space of $n\times n$ matrices is Banach (hence complete) under the standard operator norm, the limit $\sum_{i=1}^{\infty}B^i$ is well defined given $||B|| \lt 1$. Hence, taking the limit as $N \rightarrow \infty$, $$A =  -\sum_{i=1}^{\infty}B^i$$  Now the LHS commutes with B, hence so does the RHS.    Is there some trick to extend the method to work with any $B$ where $||B|| \geq 1$? I know that the result is true due to another simple algebraic proof, but I'd like to see if this method can be salvaged. Mountain Bike on Paved Roads I'm an occasional biker. I bought a mountain bike (nothing fancy) with the idea that I might use it off-road or on mountain trails (of which there are many here in Calgary). However, since I've bought it I've done nothing but bike in the city on streets and paved bike trails.  Is there anything worth doing to my bike to convert it to something more suited to urban biking? In particular, the tires have me wondering; they have large rubber off-roading lugs that surely aren't doing me any favors on pavement. I'm not looking to spend a tons of money on a retrofit, but will do so if its worthwhile. Navigating lookup relationships with dot notations, to retrieve field values I need to access the value from a roll-up field on my Opportunities, from it's related Opportunity Product.  When I use oli.Opportunity.Software_Line_Item__c the value retrieved is always null.  So I've also tried oli.OpportunityId.Software_Line_Item__c which causes an error in the compiler     Invalid foreign key relationship: OpportunityLineItem.OpportunityId   &amp; just to make sure that it's not the roll-up field calculation that's not working as expected, I've tested oli.Opportunity.Name which is also returning null.  I know this is basic but I don't have Eclipse installed so I can't check the correct reference to use. Could someone please let me know how I can navigate this relationship, to retrieve the value from the Software_Line_Item__c field? As Scott Myers wrote, you can take advantage of a relaxation in C++'s type-system to declare clone() to return a pointer to the actual type being declared:  class Base {     virtual Base* clone() const = 0; };  class Derived : public Base {     virtual Derived* clone() const };   The compiler detects that clone() returns an pointer to the type of the object, and allows Derived to override it to return a pointer to derived.  It would desirable to have clone() return a smart pointer that implies transfer of ownership semantics, like the following:  class Base {    virtual std::auto_ptr&lt;Base&gt; clone() const = 0; };  class Derived : public Base {     virtual std::auto_ptr&lt;Derived&gt; clone() const; };   Unfortunately, the relaxation of the conventions does not apply to templated smart pointers, and the compiler will not allow the override.  So, it seems I am left with two options:   Have clone() return a "dumb" pointer, and document that clients are responsible for disposing of it. Have clone() return a smart base pointer, and have clients use dynamic_cast to save them to a Derived pointer if they need it.   Is one of these approaches preferred?  Or is there a way for me to eat my transfer of ownership semantics and have my strong type safety too? I have defined my own custom vim file type with highlighting etc.  I would like to open it using the terminal based vim when I double click on it.  I am using mac os x.  Any pointers on how to start on this? Sum over all non-evil numbers I'm working on the following contest math problem:     Define an evil number to be any positive integer that contains the   digit $9$. Show that      $$ \sum_{x} \frac{1}{x} &lt; 80 $$      where the sum is over all non-evil positive integers $x$.   I'm very confused on where to begin. Initially, I tried to consider this sum as part of the sum of $1/x$ over all positive integers, namely by noting that  $$ \sum_{n=1}^{\infty} \frac{1}{n} = \sum_{x} \frac{1}{x} + \sum_{y} \frac{1}{y} $$  where the first summation on the right side is the same as in the problem statement and the second is the sum over all evil numbers. However, I can't seem to find a way to use this since the sum of the left diverges.  Could anyone lend a helping hand? Using apt-get to remove packages and dependencies Prove that $\frac{1}{f'(a)} = \frac{1}{2\pi i} \int_C \frac{1}{f(z)-f(a)} dz$ Let $f$ be analytic on an open set $U$, let $a \in U$, and $f'(a)\neq 0$.  Show that  \begin{align*} \frac{1}{f'(a)} = \frac{1}{2\pi i} \int_C \frac{1}{f(z)-f(a)} dz \end{align*} where $C$ is a small circle centered at a.  I thought that this is very similar with Cauchy's integral formula. But, I can't apply the Cauchy's integral formula. How to apply that?   Any help is appreciated..   Thank you! How do I survive in the Underworld? How to create csv files in controller method from list I want to create csv files from lists and send an email attaching them, how to do that in a controller method? How to clone ArrayList and also clone its contents? How can I clone an ArrayList and also clone its items in Java?  For example I have:  ArrayList&lt;Dog&gt; dogs = getDogs(); ArrayList&lt;Dog&gt; clonedList = ....something to do with dogs....   And I would expect that objects in clonedList are not the same as in dogs list. $$\sum_{k=1}^{\infty} \frac{1}{k^{\ln k}}.$$  I have some problems with this sum, I have tried most of the common methods, such as the ratio/root test, integral test, etc. I do believe it looks like a problem easily solvable by a direct comparison with some p-series, however, I do not know how to formulate the inequality.   I did try using the common fact that $\ln(x) \ll x^p$ for large $x$ but that of course only bounds it from below with a convergent series saying nothing about ours. Also I tried rewriting the 'summand' as $e^{\ln(k^{\ln k})}=e^{\ln k \cdot \ln k}$ but this did not really yield anything.   Any answer or hint would be greatly appreciated. Is drinking water during or after a meal better? Round function in Excel, worksheet function vs VBA I had an application for returning closest matches to certain values in a large  cluster of values( as in my earlier )   and I chose a VBA solution. While in the course of using the said application, I observed that the results for the value of 0.5 were not correct. I had been using the VBA Round function which I found to be returning 0 for 0.5 rounded to integer whereas the worksheet round function returns 1. Strangely, the VBA round function returns 2 for 1.5. I had to substitute the worksheet function in place of the VBA one.  Am I missing something?