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The space $C^0([0;1],\mathbb{R})$ of all continuous, real-valued functions on $[0,1]$ is not reflexive. Construction of $\mathbb{CP}^2$ from $S^2\times S^2$ Show that $\mathbb{CP}^2$ is homeomorphic to the orbit space of $(S^2\times S^2)/\mathbb{Z}_2$. The action is given by $(x,y) \mapsto (y,x)$. My attempt: I can see that $S^2\times S^2$ is a subset of $S^5$. I am unsuccessfully trying to restrict the action of $S^1$ on $S^5$ to $S^2\times S^2$ to get the required map. Show that there is $\xi$ s.t. $f(\xi)=f\left(\xi+\frac{1}{n}\right)$ Let $f:[0,1]\to\mathbb{R}$ be continuous and $f(0)=f(1)$. Show that for any integer $n\geqslant 2$, there is $\xi\in(0,1)$ s.t.$f(\xi)=f\left(\xi+\frac{1}{n}\right).$  I think this requires the intermediate value theorem somewhere. Let $A$ be an $n\times n$ complex nilpotent matrix. Then we know that because all eigenvalues of $A$ must be $0$, it follows that $\text{tr}(A^n)=0$ for all positive integers $n$.  What I would like to show is the converse, that is,      if $\text{tr}(A^n)=0$ for all positive integers $n$, then $A$ is nilpotent.   I tried to show that $0$ must be an eigenvalue of $A$, then try to show that all other eigenvalues must be equal to 0. However, I am stuck at the point where I need to show that $\det(A)=0$.  May I know of the approach to show that $A$ is nilpotent? I use the scroll function with my mouse and it works fine. But either in script, or in terminal sections of Spyder, I can't use the touchpad for scrolling. I do not experience any problems with my touchpad in other programs like Word, Chrome etc. Is there any option for this in Spyder, specific for touchpad?  I am using Lenovo Legion Y520, Windows 10. Reference for the subgroup structure of $\rm{PSL}_2(q)$ TSA lock reset button stuck and now lock button is also stuck Seems like a stupid question, but I'm after a nice solid answer if possible.  I joined SO a while ago to ask the odd question, but recently I'm starting to use it more often.  I've been reviewing the 40 max items (20 first posts/20 late answers) per day, and editing anything I see wrong with them (lots of +2's)... flagging them... etc.  I've also been  per day.  But other than commenting where I can and answering the odd question, I'm only  (1k at the moment).  I might just be overeager, but I was hoping some high-rep users might be able to write a little something on how they earned their stripes?  I've , and to be honest I was hoping I'd get a hit, because it sounds like a stupid question... anyways, here you go!  Also, I'm not referring to how to gain rep on Meta, since it's a bit of a fluke I've earned the rep I have on here. I'm asking about main SO. Simple regular expression for a decimal with a precision of 2 VirtualBox - how to increase video memory? I have a conjecture that any two positive coprime integers $a$ and $b$ can be added to form any integer greater than or equal to $(a-1)(b-1)$.   This is true for $3$ and $5$, which can form any integer greater than or equal to $8$, where $8=3+5, 9=3+3+3, 10=5+5; k=3n+8$ or $3n+9$ or $3n+10$.  It is true for $4$ and $7$, which can form any integer greater than or equal to $18$, where $18=4+7+7, 19=4+4+4+7, 20=4+4+4+4+4, 21=7+7+7$, etc. How can it be proven that this will work for any two coprime numbers? Where can I find web design projects? I am a recently graduated graphic designer. I would like to know if there are any worthwhile websites that I could present my portfolio and receive or look for job offers.  I have came across the following freelancewebsites by searching the web that seem to meet my criteria:       (the most famous website for people seeking for employers and employees)   I've also found two articles providing reviews on these types of sites:       I am new to the online freelance market. As a result, I don't have any clue which websites are mainstream for graphic designers and illustrators (which are my specialties). It's not immediately discernible to me which sites are the ones that are not a scam and are also good place for establishing myself.  I'm mainly looking for the resources that experienced designers are familiar with. I'm looking to narrow it down to the ones that have been used by those established in the industry. As a result I decided to consult the matter here and ask for input. I told my follower (Serana) to wait. But this is different. I told her to go somewhere and wait so it will activate the enemies. Well it did and I killed them. But Serana seems to be not following me anymore. I can't tell her to follow me. And she is constantly in a squatting position. The only thing I can do is tell her to wait somewhere else to move her. And I can't part ways because I'm doing a main quest and she has to be there. Can anybody give me a tip on what to do? Batch file to create Folder based on Date and Copy Files from One Location to Another Location? how many linearly independed vectors combinations may I extract from a set of vectors Consider the set of $n$ vectors $\mathcal{S}=\{v_1,v_2,\dots,v_n\}$, with $v_i \in \mathbb{R}^n$. Let that set generate a subspace $\mathbb{V}$ of $\mathbb{R}^n$ of dimension $r\le n$. How many different unordered basis of  $\mathbb{V}$ may be constructed by picking up $r$ vectors from $\mathcal{S}$. There is a method to find them? Pandas Merging 101 Short story with aliens who hear their own thoughts as god-like voices I would have read this about 20-25 years ago.  I believe the subject was an alien individual, but it could have been a non-standard human instead.  The hook of the story was that the subject lacked an organ like the  that allows the halves of the brain to communicate efficiently, allowing the illusion of a single consciousness.  The subject is conscious, and has thoughts, but is subject to hearing commands like "Wash your hands!" "Fetch water!" within its head.  I recall the subject (and others like the subject) believing that the voices are their gods talking to them.  To the reader it becomes clear that this is a separate part of the subject's own mind that is responsible for long-term planning and simply can't communicate with the rest of the mind in the way a "normal" human is accustomed to. This is the views section of the site analytics visible to 25k users on one of my sites:    That's, uh, not correct.  Our site has had its usual activity level during this time.  I checked a couple other sites, and the pattern is consistent.  This looks like a problem in the analytics data that's coming in; did Google change their API?  Update: Same site a week later; note that in addition to the brief spike of accuracy, page views have dropped to (near-)zero for the whole period, unlike the earlier report.  (I know this question has been closed as a dupe, but I thought having the two screen shots together might help somebody.) I read an essay about prime numbers. In it the author suggests that a natural first question to ask is,      What is the probability of a random natural number being prime?   but proceeds to dismiss it as "not making sense". I wonder what's wrong with the question. Is its meaninglessness related to the fact that there is an infinite number of naturals? Best File Management Practices when you have thousands of Files We publish over 100+ textbooks and growing (1-color, 2-color, 4-color) maninly designed in CorelDraw &amp; Indesign with associated Image and Font Files. We also have merged multipage PDF &amp; EPS files that we send to our offset printers.Also, some Image files are exclusive to a project while some are shared across projects.   What are the best practices to follow when it comes to management of these files? We are always working on multiple versions of the same files and often with multiple DTP operators.  Right now it is all on a Shared Drive with Individual Folders but that always creates confusion in identifying the latest version of the file and some users save the file to some other folder. Security of the Data is another issue as we work with a lot of freelancers also.  How do big design studios and publishing houses handle versioning, collaboration and the associated files with a project neatly? Do they use a CMS / Document Management System?