Can I get individual man pages for the bash builtin commands?
Reading and searching long man pages
Why is a vector space equal to its tangent space for any point? I'm self-studying Guillemin and Pollack, but I'm stuck on Problem 3 of section 2. It says that if $V$ is a vector subspace of $\mathbb{R}^N$, then $T_x(V)=V$ if $x\in V$.  If $x\in V$, then since $V$ is a manifold, there is a local parametrization $\phi: U\to V$ where $U$ is open in $\mathbb{R}^k$. Without loss of generality, we can require $\phi(0)=x$. Then $T_x(V)$ is defined to be the image of $d\phi_0$ on $\mathbb{R}^k$.   An arbitrary element of $T_x(V)$ looks like $$ d\phi_0(v)=\lim_{t\to 0}\frac{\phi(0+tv)-\phi(0)}{t}=\lim_{t\to 0}\frac{\phi(tv)-x}{t} $$  but this doesn't seem very useful to show $T_x(V)=V$. What is the right approach? Let $\mathbb S = \varnothing$.      Then from the definition: $ \bigcap \mathbb S = \left\{{x: \forall X \in \mathbb S: x \in X}\right\}$      Consider any $x \in \mathbb U$.      Then as $\mathbb S = \varnothing$, it follows that: $\forall X \in \mathbb S: x \in X$ from the definition of vacuous truth.      It follows directly that: $\bigcap \mathbb S = \left\{{x: x \in \mathbb U}\right\}$      That is: $\bigcap \mathbb S = \mathbb U$. ()   Proofwiki uses the above "proof" to "prove" that intersection of the empty set is the whole universe.   My question is, is the use of vacuous truth really allowed in axiomatic set theory, like ZFC? I don't see how the use of vacuous truth is justified.   The next problem I can think of is that we cannot really "define" the elements of empty set (to my knowledge, there is no element in empty set) so how can we then prove as the above proof did? This seems to contradict the use of vacuous truth.  And of course, there is issue of using the whole universe as a set, and I don't think this is allowed.... (Maybe proof above is using a different axiomatic set theory, as I am using ZF-minded thoughts...) Open a Oracle 7.3.x database in Oracle 11 Is there any way to open or convert an old Oracle 7.x database into Oracle 11g? Thanks in advance! Find the exact value of the infinite sum $\sum_{n=1}^\infty \big\{\mathrm{e}-\big(1+\frac1n\big)^{n}\big\}$ How can we find the exact value of the infinite sum  $$ \displaystyle\sum_{n=1}^\infty \left\{\mathrm{e}-\Big(1+\frac1n\Big)^n\right\}? $$  This problem appears in:  , p.114. How do I ensure a piece of code runs only once? Can anyone tell me how to judge if a supervised machine learning model is overfitting or not? If I don't have an external validation dataset, I want to know if I can use ROC of 10 fold cross validation to explain overfitting. If I have an external validation dataset, what should I do next? Can every even integer be expressed as the difference of two primes? If so, is there any elementary proof? The use of "real" in the following cases Possible Duplicate:             If you listen real close...      Can you swing by real quick...   Sentences like the above two are what I often hear in daily life. If I didn't hear them in the real world, I would probably be more ready to say "If you listen really close..." and "Can you swing by really quick...".  Is real being used as an adjective in this scenario? If so, is the usage of adj.+ adj. a common practice in English language? I know that matrix multiplication in general is not commutative. So, in general:  $A, B \in \mathbb{R}^{n \times n}: A \cdot B \neq B \cdot A$  But for some matrices, this equations holds, e.g. A = Identity or A = Null-matrix $\forall B \in \mathbb{R}^{n \times n}$.  I think I remember that a group of special matrices (was it $O(n)$, the ?) exist, for which matrix multiplication is commutative.  For which matrices $A, B \in \mathbb{R}^{n \times n}$ is $A\cdot B = B \cdot A$? If you have an array of Java objects which have a primitive type (for example Byte, Integer, Char, etc).  Is there a neat way I can convert it into an array of the primitive type?  In particular can this be done without having to create a new array and loop through the contents.  So for example, given  Integer[] array   what is the neatest way to convert this into  int[] intArray   Unfortunately, this is something we have to do quite frequently when interfacing between Hibernate and some third party libraries over which we have no control.  It seems this would be a quite common operation so I would be surprised if there's no shortcut.  Thanks for your help! How can you strip non-ASCII characters from a string? (in C#) How can you strip non-ASCII characters from a string? (in C#) "One of them was/were you" If I am talking to somebody about a certain group of people in the third person, and then want to refer to the person I am talking with as one of those people, which do I say?     One of them were you      One of them was you. Show timestamp of a vote in the tooltip when mousing over it Sometimes I come across an old question or answer that I voted on in the past, but I don't clearly remember when (or sometimes even why) I voted it up or down. Being able to see the date/time that I cast a vote would help me remember my mindset at the time.   When mousing over the toggled vote, the tooltip reads the same as if I hadn't voted, with no further information. For example:     This answer is useful (click again to undo)   The only way to get the timestamp is if I try to click the vote to undo it and get the locked vote message:     You last voted on this answer [datetime] Your vote is now locked in unless this answer is edited   However this requires me to guess that the vote is locked - if there was an edit since then I'll actually undo the vote and erase the info I was looking for.  Feature request: Please add the vote timestamp to the tooltip when mousing over a cast vote. This is already done with the green acceptance checkmark for example. I am a Ph.D. dropout with a one year of career gap. I want to return to academia, but now  I have realized that it was perhaps the biggest mistake to take a gap after quitting my Ph.D. The purpose of taking a gap was to work on my communication skills and get a  new perspective on my work and my career in science. This could have been done within a period of 2-3 months but it became one year long because of my own recklessness. I allowed myself to distract from my goal and I started working on my hobby i.e. Painting. I lost myself there and completely lost track of my time. I was really enjoying everything about this new venture. I took it as a challenge to become better at my skills in painting. Eventually, I became the best in my group and got lots of appreciation. But then, I lost my interest while realizing that there was no further challenge and all there was left is to work and work while crafting your art which was never going to be perfect. It made me realize that what I actually love is to take the new challenges and work on myself. To be an artist was never a career option for me, all I wanted was to learn the skills. That's it!   While counting my achievement in science, I  realized that it was an irresponsible thing to take a gap in doing something that has nothing to do with science. Earlier I had a scholarship and a very competitive mind that has led me to get a Ph.D. position in one of the best labs in the country. Now I don't have the scholarship. I want to go back to my world of research where I was accelerating exponentially. I want to tackle the challenges of science with a new perspective and better skills. I want to be productive more than earlier and ready to take all the tough challenges.  But I feel embarrassed about my past mistake. Because of this, I come across as someone who is not sincere about her career. But I can't help it because the truth is that I did behave insincerely by losing track of my time.  I should have been conscious enough to realize the consequences. Now, what do I do? The past has gone and the vision of the present is crying for a better future. How do I make things better like before? When voting to migrate an off-topic question, why is the list of sites limited? Earlier, I voted to close  as off-topic.  The question has since been closed.  There is an , but it didn't show up in the list when I selected "off-topic".  The options were:   Meta Stack Overflow Server Fault Super User Webmasters Programmers   Questions:   Should Wordpress appear in this list? Are there any other sites that should appear in the list?   Should all non-beta sites appear in that list by default? How to re-index the values of an array in PHP? Possible Duplicate:          Is there some rule governing the following, or similar, subject-auxiliary inversions (*"Rarely they do see the light of day", *"Never I have been so insulted")?     In fancy sit-down restaurants, you can order a large meal and halfway through the main course, take a little dead cockroach or a piece of glass out of your pocket and place it deftly on the plate. Jump up astonished and summon the headwaiter. "Never have I been so insulted. I could have been poisoned" you scream slapping down the napkin. You can refuse to pay and leave, or let the waiter talk you into having a brand new meal on the house for this terrible inconvenience. ()      Rarely do they see the light of day; it's generally judged to be in no one's commercial interest that they should. Just occasionally, however, like a seemingly dormant volcano, they explode into the open. () What's this =! operator? What is the purpose of the first test in an inductive proof? Learning about proof by induction, I take it that the first step is always something like "test if the proposition holds for $n = \textrm{[the minimum value]}$"  Like this:       Prove that $1+2+3+...+n = \frac{n(n + 1)}{2}$ for all $n \ge 1$.   Test it for $n = 1$:  $$n = \frac{n(n + 1)}{2} \implies 1 = \frac{2}{2}\textrm{, which holds.}$$  * The rest of the proof goes here *    So, I do it all the time (like a standard). But I never really thought about why. Why do I do such test? I can see that if the test does not hold, it cannot be proven by induction, I guess. But is there another reason we do this? How do I delete Windows.old in Windows?