to find the radius of the sphere
Sphere inside a conical vessel touches the lid
A book suggests that we should not interchange "so" and "so that". "So" means "therefore", and "so that" means "in order that". However, it seems to me that in many cases they don't actually have a difference. For example:     Alice got up at 5:00 A.M., so that she could drive her son to school.   If I substitute "so that" with "so", this sentence presents the exact same causal relation to me:     Alice got up at 5:00 A.M., so she could drive her son to school.   I must have missed something here, since English is my second language. Could somebody explain the difference? Dual-system Windows and Ubuntu I did dual boot with Windows 7 &amp; Ubuntu 14.04. I get both in boot menu but can't boot into Windows 7. The primary drive where Windows 7 was configured now shows to be logical drive with all files in it. How to resolve it ?  This is what it shows:  irshad@irshad-desktop:~$ lsblk NAME   MAJ:MIN RM   SIZE RO TYPE MOUNTPOINT sda      8:0    0 465.8G  0 disk      ├─sda1   8:1    0  58.6G  0 part /media/irshad/B2E08E69E08E339F     ├─sda2   8:2    0     1K  0 part      ├─sda5   8:5    0 195.3G  0 part /media/irshad/8E90A6A690A6946D     ├─sda6   8:6    0 182.8G  0 part /media/irshad/7650B08250B04B1F    ├─sda7   8:7    0    25G  0 part /    └─sda8   8:8    0     4G  0 part      sr0     11:0    1  1024M  0 rom     I tried to run update-grub but I get the following result.  $ sudo update-grub  Generating grub configuration file ...  /etc/grub.d/06_custom: 31: /etc/grub.d/06_custom: function: not found  /etc/grub.d/06_custom: 34: /etc/grub.d/06_custom: save_env: not found  /etc/grub.d/06_custom: 36: /etc/grub.d/06_custom: Syntax error: "}" unexpected What system monitoring tools are available? I am looking for system monitoring tools which are GUI and CLI or web-based which include basic functions such as:   CPU Usage Ram Usage Swap Usage Disk Usage ( Space / I/O ) Heat Monitoring   I know there are many tools I can use, but I am looking for a single tool that has these basic functions. Relationship between SVD and PCA. How to use SVD to perform PCA? How can I make a Random Number Generator with vanilla redstone? How is the minimum of a set of IID random variables distributed? If $X_1, ..., X_n$ are independent identically-distributed random variables, what can be said about the distribution of $\min(X_1, ..., X_n)$ in general? Finite choice without AC Can anyone explain how we choose one sock from each of finitely many pairs without the axiom of choice?  I mean : To choose one sock from each of infinitely many pairs of socks requires the Axiom of Choice, but for shoes the Axiom is not needed.  The idea is that the two socks in a pair are identical in appearance, and so we must make an arbitrary choice if we wish to choose one of them. For shoes, we can use an explicit algorithm -- e.g., "always choose the left shoe." Why does Russell's statement mention infinitely many pairs? Well, if we only have finitely many pairs of socks, then AC is not needed -- we can choose one member of each pair using the definition of "nonempty," and we can repeat an operation finitely many times using the rules of formal logic (not discussed here). Joel Spolsky asked what rhymes with orange. The official answer is, "Nothing," although a creative poet can get close by using half words, just the -nge part or resorting to place names and foreign words.  Does orange somehow violate the basic phonotactics of English? (And hence we wouldn't expect other English words to rhyme with it?) Prove by induction: $n! \ge 2^{(n-1)}$ for any $n \ge 1$ How much money do I need to make in order to generate 1k/month for the rest of my life? Is that even possible? If yes, in what way? Edit: I am 30 years old. learn how to [verb] vs. learn to [verb] "learn to [verb]" "learn how to [verb]"   Is [1] merely a less formal version of [2]? If not, does [1] communicate something subtly different? Consider the following:  In [2], the object of learning is explicitly "how to [verb]" (for instance, "I will learn how to speak French").  Is the object of learning in [1] "to [verb]" or is it something implicit and abstract that allows the subject "to [verb]"? For instance, in "I will learn to speak French", the object of learning would be the prerequisite elements for speaking French: "French phonemes, grammar, syntax, etc.". How do I remove repeated elements from ArrayList? I have an ArrayList&lt;String&gt;, and I want to remove repeated strings from it. How can I do this? SELECT * EXCEPT Is there any RDBMS that implements something like SELECT * EXCEPT?  What I'm after is getting all of the fields except a specific TEXT/BLOB field, and I'd like to just select everything else.  Almost daily I complain to my coworkers that someone should implement this...  It's terribly annoying that it doesn't exist.  Edit: I understand everyone's concern for SELECT *.  I know the risks associated with SELECT *.  However, this, at least in my situation, would not be used for any Production level code, or even Development level code; strictly for debugging, when I need to see all of the values easily.  As I've stated in some of the comments, where I work is strictly a commandline shop, doing everything over ssh.  This makes it difficult to use any gui tools (external connections to the database aren't allowed), etc etc.  Thanks for the suggestions though. Show that $\sum_{n=1}^{\infty}(n^{3}\sin\left(\frac{1}{n}\right)-n^{2}+\frac{1}{6})$ converges Let $t\in \mathbb R$. Prove that $\displaystyle{\sum_{n=1}^{\infty}n^{3}\sin\left(\frac{1}{n}\right)-n^{2}+t}$ converges iff $t=\frac{1}{6}$.  While the $(\Rightarrow)$ direction was fairly easy (as $a_n$ should converge to $0$, forcing $t=\frac{1}{6}$), i'm facing some hard time trying to prove the second direction, that is $\displaystyle{\sum_{n=1}^{\infty}(n^{3}\sin\left(\frac{1}{n}\right)-n^{2}+\frac{1}{6})}$ converges. I did notice that $a_n=a_{-n}$ and tried to write this series as a Leibnitz series, but had no luck since the sequence doesn't decrease. I also tried to prove absolute convergence but then again, no luck. Thanks. Can the off-regional pokémon hatch from eggs? Some people claim you can get the pokémon that are not in your region from eggs. Is this only a rumour or actually true? I've seen off-regional pokémon in gyms nearby but I can't rule out that the owner had been abroad or maybe even used location spoofing. Let $\alpha,\beta,\gamma$ be the roots of $x^3-3x^2+1 = 0$. Prove that $$\alpha^n+\beta^n+\gamma^n \equiv 4^n+5^n+(-6)^n \pmod{17}$$ where $n$ is an integer.   It sort of makes sense why they are congruent since we can say $x^3-3x^2+1 \equiv (x-4)(x-5)(x+6) \pmod{17}$, but since $\alpha,\beta,\gamma$ aren't integers, how do we prove this? I have this input:  sdkxyosl 1 safkls 2 asdf--asdfasxy_asd 5 dkd8k  jasd 29 sdi44sw 43 asasd afsdfs 10 rklyasd 4   I need this output:  sdi44sw 43 dkd8k  jasd 29 asasd afsdfs 10 asdf--asdfasxy_asd 5 rklyasd 4 safkls 2 sdkxyosl 1   So i need to sort the lines by the last column.  I don't know how many columns are in one line.  I just can't figure it out, how to do it. I don't have "perl powers". I just have ~average scripting powers with sed, awk, cut, etc..   Does somebody know how to do it? What does "rigorous proof" mean? I have heard several times that some mathematician has given another and more rigorous for an established theorem, but I don't know what does it really mean and what differences makes it to be more 'rigorous'. My understanding of a proof is that a proof is some explanation to convincing others that a statement is true. If some proof is not rigorous, does it mean that while the proof is acceptable for many but it is not for everybody? If so, the proof is not correct since it going to be rejected by some people and it is not a proof at all. Or, does rigorous mean it uses more advanced tools? If so, why not any alternative (but simpler) proof is also acceptable as rigorous (which is more 'valuable' as well since it's simpler!)?   Saying those limited insight of mine, my questions are:   $1-$ What does rigorous proof mean, in general definition?  $2-$ In order to understand your answers, as an example I am re-writing three of the proofs for the compactness of the torus. Which of the following proofs are most rigorous, 'normal' and least rigorous; and why so?    Theorem: The torus $T^2$ is compact.  First proof: The torus is homeomorphic to the product space $S^1 \times S^1$. The circle $S^1$ is compact. Therefore the torus is the product of two compact spaces, and thus it is compact.   Second proof: The torus $T$ is the subspace of $\mathbb{R}^3$ obtained by rotating a circle about the $z$-axis. The torus is closed since given any point in its complement, there is an open ball of sufficiently small radius centered at that point and disjoint from the torus. Also, the torus is bounded since it is contained in the ball of radius $4$ centered at the origin. Since the torus is closed and bounded in $\mathbb{R}^3$, it is compact.   Third proof: The torus is homeomorphic to a space obtained as a quotient  space of the square by identifying opposite edges. The square is a closed and  bounded subset of $\mathbb{R}^3$ and therefore is compact. Furthermore, a quotient map on the square is a continuous function, and the image of a compact space under a continuous function is compact. Therefore the torus is compact.   Thanks a lot for your guidance. Einstein's first postulate implies the second? I have been stuck for several days on this old Analysis problem (I am doing some study on my own).  I have tried several things (which I'll indicate below), but I cannot seem to figure it out.  Here is how the problem is presented: Problem: &quot;Let $f$ be a continuous real-valued function on $[a,b]$.  Suppose there exists a constant $M \geq 0 $ such that $$|f(x)| \leq M \int_a^x |f(t)| dt$$ for all $x \in [a,b]$.  Show that $f(x)=0$ for all $x \in [a,b]$.&quot; My Thoughts: I have tried using the mean value theorem iteratively, but that seems to always lead me down a dead end road.  I deduced that $f(a)=0$.  If only $f$ were assumed to be differentiable, then maybe I could play with trying to get the derivative to be $0$, but unfortunately it's only continuous.  My other thought was to (somehow) use the condition to show that $\int_a^b |f(x)| dx = 0$.  I also played around a bit with contradiction, but to no avail.  Even if one of these hair-brained thoughts is correct, I am not really sure what to do next. If you have any ideas, suggestions, or solutions, I would really appreciate it if you are willing to share them.  Thank you for your time.