Does the existence of continuous injections $f: A\rightarrow B$ and $g: B\rightarrow A$ imply the existence of a bicontinuous bijection between A and B (ie topological equivalence)? If not, what is a good counter example?
Let $A$ , $B$ be topological spaces such that there for some subset $D$ of $B$ there is a homeomorphism form $A$ to $D$ and for some subset $E$ of $A$ there is a homeomorphism form $B$ to $E$ ; then must there exist a homeomorphism from $A$ to $B$ ?
How is Bohr's model related to electron cloud models via the correspondence principle? What's the easiest way to use a linked list in python? In scheme, a linked list is defined simply by '(1 2 3 4 5). Python's lists, [1, 2, 3, 4, 5], and tuples, (1, 2, 3, 4, 5), are not, in fact, linked lists, and linked lists have some nice properties such as constant-time concatenation, and being able to reference separate parts of them. Make them immutable and they are really easy to work with! Let $(a_n)_{n≥0}$ and $(b_n)_{n≥0}$ be two sequences of a nomed algebra such that $\sum{\| a_n\|}$ and $\sum{\| b_n\|}$ converge, and$ $$$\forall n, \  a_n, b_n \neq 0$$  Show that $(\forall  k \in \mathbb N^*,  \ \sum_{n=0}^{\infty}{a_n}^k = \sum_{n=0}^{\infty}{b_n}^k) \implies \exists\ \sigma \in{\mathfrak S}_{\mathbb N} \ ,\ (a_n) = (b_{\sigma(n)})$  The vector-space should be a finite-dimensional $\mathbb R$-vector-space.   I do not see how to deal with the hypothesis. If you have ideas/hints...  It would be relevant to deal with complex sequences even if it is less general. But I don't think one can generalize using real and complex cases. How to obtain optimal hyperparameters after nested cross validation? In Euclidean geometry texts, it is always mentioned that point is undefined, it can only be described. Consider the following definition: "A point is a mathematical object with no shape and size." I do not understand what is the problem with this definition. Please give the detailed reasons. Thanks in advance! Would time freeze if you could travel at the speed of light? I read with interest about Einstein's Theory of Relativity and his proposition about the speed of light being the universal speed limit.    So, if I were to travel in a spacecraft at (practically) the speed of light, would I freeze and stop moving?  Would the universe around me freeze and stop moving?  Who would the time stop for? Integral inequality with a strange condition Let $f$ be a continuously differentiable real valued function on $[0,1]$.  It is given that $\displaystyle \int_{\frac{1}{3}}^{\frac{2}{3}}f(x) dx=0$     Find the minimum value of $\dfrac{\int_{0}^{1} (f'(x))^2 dx}{\left( \int_{0}^{1} f(x) dx \right)^2}$   I tried to use Cauchy-Schwarz to show that  $$\frac{\int_{0}^{1} (f'(x))^2 dx}{\left( \int_0^1 f(x) dx \right)^2} \ge \frac{\left( \int_0^1 \bigl| f(x)f'(x) \bigr| dx \right)^2}{ \left( \int_0^1 f(x) dx \right)^2} \ge \frac{ f(1)^2 - f(0)^2}{2 \left( \int_0^1 f^2 (x) dx \right)^2}$$ But I can't proceed from here.  Also, I don't know how to use the condition  $\int_{1/3}^{2/3}f(x) dx=0$ What would be the best approach to get the recordtype id I know of two approaches to get the recordtype id of a particular record type   Use SOQL   RecordType RecType = [Select Id From RecordType  Where SobjectType = 'Account' and DeveloperName = 'Business'];    Benifit here is if the admin changes the name of the record type, the code still works as we can use developer name   Using Describe  Map&lt;String,Schema.RecordTypeInfo&gt; rtMapByName = d.getRecordTypeInfosByName(); Schema.RecordTypeInfo rtByName =  rtMapByName.get('Business');    Saves on SOQL limits but uses a describe call.  What would be your choice and why   EDIT : Usage is for setting up recordtype while inserting records in apex or trigger When should I use a struct rather than a class in C#? I clean-installed Ubuntu 11.10 today, and then installed VirtualBox.  This required me to add myself to the vboxusers group, and since 11.10 seems to no longer have a graphical app to add users to a group, I ran the following command:  sudo usermod -G vboxusers stephane   This is a problem.  I now see what I should have run instead is:  sudo usermod -aG vboxusers stephane   The end result is I'm no longer in the groups I should be in.  Including whatever group is required to run "sudo".  When I run any command as sudo now, I get the following:  $ sudo ls [sudo] password for stephane:  stephane is not in the sudoers file.  This incident will be reported.   Is there a way to fix this, or do I need to re-install from scratch again? Which edit actions can move a post into Community Wiki mode? Surprisingly, I couldn't find a straight answer to this.  It's not entirely clear in the .  There are (at least?) four different types of edit actions:   Body edits Title edits Retags Rollbacks   And a few actions that show up in the history but aren't really "edits":   Close/Reopen Delete/Undelete   The FAQ seems to indicate that only body edits are important as far as "wikification" is concerned, but I'm not sure if that's really correct or if I'm just interpreting the post too literally.  The FAQ also says that rollbacks don't reverse the calculations - fine, but are they considered to be additional edits or are they ignored?  Can anyone tell me which of these actually count toward the wikification of a post, and whether the rules are any different for owner edits vs. multiple editors? To flag or not to flag question that don't show own effort? I am a little confused about what do with questions that are just asking      "I want this and that (...) and I want it to look exactly like that   (...) please help"   without showning ANY own effort?  According to  those kind of answers are not welcome  also the  says those questions should not be asked:     Don't ask about...   Questions you haven't tried to find an answer for (show your work!)   But does this mean that I should flag those kind of "helpdesk" questions or not? And if yes as what? low quality ? flag for moderator attention ? Or is just adding a kind comment asking for the asker's own attempts enough? How to revert uncommitted changes including files and folders? How to obtain the sum of the following series? $\sum_{n=1}^\infty{\frac{n^2}{2^n}}$ How to mount a NTFS partition in /etc/fstab? Why do you need to know who I know on Google? I just created a SE account using my Google account. It said that it needs to "Know who you are on Google" (I guess my connections). As far as I know, SE is not a social network. I can't even follow other people (seeing when they post questions and answers).  What is the reason you need to know my Google connections?  This was on code review. Actually I have an account there, but for some reason it asked me to reconnect. This is the same on MSO; one sees this explained as "This app is requesting permission to associate you with your public Google profile": Java - when to use 'this' keyword Basically the question is how to get from this:   foo_id   foo_name 1        A 1        B 2        C   to this:   foo_id   foo_name 1        A B 2        C How to switch render source (VSE or Camera)? If you create a scene for the Video Sequence Editor and start working on a project Blender will render the output from the VSE. But if you want to later switch it to render your 3D scene again (the camera output) where is the setting to select the source?  Usually I keep a separate .blend for VSE edits so I don't need to switch, but I feel like I ought to learn how to do this as it's pretty basic. I would expect to find this in the Render context but I see no place to specify the source.  Thanks. Why is the minimum size of a generating set for a finite group at most $\log_2 n$? It seems to be known that the minimum size of a generating set for a finite group of order $n$ is at most $\log_2 n$. Can someone explain why this is true?  Edit: noted that the logarithm is base 2, and $n$ represents the order of the group.