Simple question that I can't solve
The security guard problem
Why can't polynomials have negative exponents or division by a variable Why can't:  $$2x^{-3} - 3x$$  or  $$\frac{1}{2x}$$  be polynomials too?  Why have a definition that excludes these algebraic forms? Circular References Cause Memory Leak? I'm trying to run down a memory leak in a windows forms application.  I'm looking now at a form which contains several embedded forms.  What worries me is that the child forms, in their constructor, take a reference to the parent form, and keep it in a private member field.  So it seems to me that come garbage-collection time:  Parent has a reference to the child form, via the controls collection (child form is embedded there).  Child form is not GC'd.  Child form has a reference to the parent form, via the private member field.  Parent form is not GC'd.   Is this an accurate understanding of how the garbage collector will evaluate the situation?  Any way to 'prove' it for testing purposes? Evaluate the integral $\int_0^{2a}\int_0^\sqrt{2ax-x^2}\frac{\phi'(y)(x^2+y^2)x dxdy}{\sqrt{4a^2x^2-(x^2+y^2)^2}}$ Change the order of integration in:  $$\int_0^{2a}\int_0^\sqrt{2ax-x^2}\frac{\phi'(y)(x^2+y^2)x }{\sqrt{4a^2x^2-(x^2+y^2)^2}}dxdy$$ and hence evaluate it.    I changed the order of integration and got the limits $y=0$ to $a$ and $x=a-\sqrt{a^2-y^2}$ to $a+\sqrt{a^2-y^2}$. I then tried changing to polar coordinates, but it is only making the integrand more complex. Syntax to add multiple commands to a single command block I am looking to try and set multiple commands in a single command block, but so far everything I have tried just returns "Unknown Command".  What is the format for writing multiple commands in a single command block in Minecraft 1.8?  Can I use either of the following:  Command:(command here) Command:(command here)     /command here; /command here;   Or is there another format? Why NuGet adds app.config with assemblyBinding to LIBRARY projects during a NuGet package update? I will be arriving in Mumbai Terminal T2 on a Turkish Airlines international flight. I have a separately-purchased IndiGo ticket 5 hours later from Terminal 1B (Domestic). In the past I heard of a free bus to transfer.    Is that still the case and if not how do I get to Terminal 1B from Terminal 2? Show that the series $\sum(\exp(\frac{(-1)^n}{n})-1)$converges, but not absolutely. Show that the series converges, but not absolutely. $\sum_{n=1}^{\infty}( $exp$(\frac{(-1)^n}{n})-1)$.   My Try:  Let $a_n=$exp$(\frac{(-1)^n}{n})-1$. I was going to use alternating series test because the sequence $\{a_n\}$ is alternating. But $\{|a_n|\}$ is not decreasing to $0$. So, I am stuck now. Can anybody please give me a hint? Stochastic processes book suggestions. I would like to find a book that introduces me gently to the subject of stochastic processes without sacrificing mathematical rigor. It would be great if the book has lots of examples and that the book is designed for undergraduates. Just like how there are rigorous undergraduate abstract algebra books (I am thinking of GAllian's contemporary modern algebra here).   I already studied measure theory, and prob. theory.  Thank you. Let $x&gt;0$, $y&gt;0$ and $z&gt;0$. Show that   $$x^3y+y^3z+z^3x\ge xyz(x+y+z).$$   I known we can't WLOG: $x\ge y\ge z$, if this, I can use rearrangement inequality, But other I can't it. Thanks? getting exception while putting list data into the map Show that the union $\bigcup A_n$ has empty interior in $X$ What are some grandiloquent, or simply better, ways of expressing "an idea/thought suddenly came to me", or "an idea/thought struck me", or "I was struck by an idea/thought"? Why do RAW images look worse than JPEGs in editing programs? I've found that when you load a RAW image into an editing program such as Lightroom/Aperture, the image is usually worse than if you just took the image as JPEG. Now I understand that the camera does some magic during the JPEG conversion. But I'm trying to understand what that "magic" is.   If I'm looking to implement that "magic" myself on the desktop, what kind of settings should I be trying? I find that the RAW files have more extreme contrast. The dark areas for example are way darker than JPEG. Why is this? And what is the best way to fix this? Which interpolation technique is suitable for a bathymetry of a small lake? Is it correct to write $\Bbb E[X]$ or $\Bbb E_{\theta}[X]$? Suppose that we observe the discrete random variable $ X = (X_1, \dotsc , X_n)$ with state space S, whose distribution we do not know but we are assuming that its joint p.m.f. belongs to a known family $\{f_θ : θ ∈ Θ\}$. We derive a sufficient statistic $T(X)$.  Is it correct to write $\Bbb E[X]$? If not, why? Would it be more correct to write $\Bbb E_{\theta}[X]$? Is GNU's Java Compiler (GCJ) dead? How can I get file extensions with JavaScript? A friend of mine has some very old photos where the deterioration process seems to have increased in the past few years, and the owner is worried about the preservation of these old memories.  They have been stored inside an album within plastic sleeves. The owner is not aware of the display method prior to that.   I have been informed by the owner that since being in their possession, they seem to be looking worse than when first viewed and seem to be falling apart.   He is looking for the best way to preserve these photos inside an album, but not really sure what would be the best method. I have suggested for the albums as a whole to be stored in vacuum sealed bags for now, but I'm not even sure if that is good!  I am looking for any suggestions or personal experiences from anyone who has had to deal with such photos and what they did to preserve their memories. How many solutions for an equation with simple restrictions I'm working on an assignment in which I have to count the number of solutions for this particular equation: $$x_1+x_2+x_3+x_4=20$$for non negative integers with $x_1&lt;8 $ and $x_2&lt;6$   I'm aware that this kind of a task isn't that complicated, but I don't get combinatorics in general that well.  So I've tried two following approaches to get this done.  Firstly I tried to substitute the variable x:  $x_1+x_2+x_3+x_4=20 \Leftrightarrow y_1+y_2+y_3+y_4=34$  in which $y_1=x_1+8$ and $y_2=x_2+6$ (casue $x_1=y_1-8$ and $x_2=y_1-6$) Following this approach the total number of possible solutions would be  $${34+3 \choose 3} $$  But I'm not sure if its the right solution.  The second approach is to sum all of the possible values that $x_1$ and $x_2$ could possibly take, also $x_1=0,1,2,3,4,5,6,7$ and $x_2=0,1,2,3,4,5,6$ And then  count all the possibilities for each of the variables $${20 -x_1-x_2+1\choose 1}$$ and sum them together like this: $${21\choose 1}+{20\choose 1}+{19\choose 1}+{18\choose 1}+... $$ and so on...  I'm sure I'll get the correct number with this one, but I'm not feeling like summing all of this possibilities. There's got to be a better, more elegant way to deal with this.  My professor gave me a hint that I should do it using the complement. How to create automatic dual battery changeover/switching circuit for uninterruptible power? I've search for a solution to my problem both on this site and on Google but I did not find a complete and adequate response.  I need a circuit that switches two 12v sources (one that comes from a lead-acid battery powered by a solar panel and the other source comes from the output of a 220-12v switching power supply) without the final power output being interrupted during switching (this is very important!).  Notes:  - The final power output will be used to power 12v devices (30 Ampere maximum).  - The sources voltage range is 11-14.7v  How can I switch between two 12V sources?