Find all functions of defined on the set of all real numbers with real values, such that $$ f\left(x^2+f(y)\right) = y + f(x)^2 $$  My attempt: Putting $x = 0$, $f\big(f(y)\big) = y + f^2(0)$ and putting $y = 0$, $f\left(x^2 + f(0)\right) = f(x)^2$. But I can't go any further.
Let $\mathbb R$ denote the set of all real numbers. Find all function $f: R\to \ R$ such that   $$f\left(x^2+f(y)\right)=y+(f(x))^2$$   It is the problem. I tried to it by putting many at the place of $x$ and $y$ but I can't proceed. Please somebody help me.
I have a 3TB Seagate Backup Plus Desktop USB 3.0 drive, which works fine when in its enclosure, but when I get it off its enclosure and directly plug it as an internal SATA drive, it's just not properly recognized (it works again when used in the enclosure). My systems’s motherboard is an , which has two SATA 6.0 Gb/s and four SATA 3.0 Gb/s ports.  When used as an internal SATA drive Windows 7 asks if I want to format the drive, as if it didn’t have a proper filesystem, and if I use the DISKMGMT.MSC tool (typing that in the start menu) I get completely wrong information about the drive.  It says the drive has 3 partitions (349,31GB of RAW data and two unassigned partitions of 1698,68GB and 746,52GB). This information is plain wrong, since the drive, when used in the USB enclosure, works as a single NTFS partition (2794,52GB). How come it’s shown as 3 partitions without filesystem when connected through SATA?!  Is Seagate using some proprietary way of storing the data when using the drive in its provided USB 3.0 enclosure? I didn’t use any kind of encryption. I just plugged the drive with USB 3.0 and started using it straight away, since it apparently came pre-formatted with an NTFS filesystem. Say a prospective PhD has a high reply rate (90-95%) to first contact emails with potential superviors. He/She has a well thought out e-mail, but a particular professor (here: USA) that is in the top preferences does not reply after two to three e-mails that were sent within a time frame of three months. I also followed the instructions on the professor personal website in the section "prospective PhD students". The student has already contacted some of the advisor students and has received positive feedback about the professor.  Should the student just give up and move on? Should the student keep trying?  Aside from the obvious "I have too many incoming e-mails to answer" or "I am out in the field", what could create a no response behavior from the professor?  I have seen the simplistic answer "move on, bad advisor, doesn't have time, etc". But this seems a contradiction in this case. The PhD students like the professor and say he/she is personable and no e-mail response after a few well written emails. Is there a term for solving a sub-sub problem of a problem so far removed from the main problem the link is not clear? In some situations, trying to solve a problem uncovers another problem. Attempts to fix that uncover yet another. A few more rounds and one can be doing something seemingly completely unrelated to the main problem. Is there an expression for that? Determine $4$ specific digits in $34!$ Find the values of $a,b,c,d\in\mathbb{N}$ such that      $$ 34!=295232799cd9604140847618609643ab0000000 $$   My Attempt:  The factorial of $34$ contains a $3$, so the RHS must be divisible by  $3$. Similarly, it must be divisible by $7$, $11$, $13$, $19$ etc.  But I do not understand how can I calculate $a,b,c,d$ in this equation. How to Mark NTFS filesystem dirty from *NIX/Linux? Deriving Sum of a Geometric Progression Showing that a map is bijective and finding its inverse Let $S$ := {$(x, y) ∈ \mathbb R^2: x^2 + y^2 = 1$} be the unit circle. Let $p = (0, -1)$ ∈ $S$. Define a map $F$ from $\mathbb R$ × {$0$} to $S$ as follows: Given $(t, 0)$, let $F((t, 0)) = (x, y)$ ∈ $S$ be the intersection of $S$ with the line through $p$ and $(t, 0)$. Derive a formula for $F$. Show that $t$ → $F((t, 0))$ is a bijective map from $\mathbb R$ to $S$ \ {$p$}, and also derive a formula for its inverse.  If we express $S$ in polar coordinates: $$S:=\{(r,\varphi):r=1,\varphi∈(-\frac\pi{2},\frac{3\pi}{2})\}$$ then, from simple trigonometric relations, the inverse function $\Phi$ is: $$\Phi(\varphi)=\tan\frac{\frac\pi2-\varphi}2$$ and its inverse $F$ is: $$F(t)=\frac\pi{2}-2\arctan t$$ After changing coordinates back to cartesian, $F$ will look like: $$F(t)=(\cos(\frac\pi{2}-2\arctan t),\sin(\frac\pi{2}-2\arctan t))$$ which can be further simplified to: $$F(t)=(\frac{2 t}{t^2 + 1},\frac{1 - t^2}{t^2 + 1})$$ Is this the formula of $F$? Is it correct? And I know that to show that $\phi$ : $\mathbb R$ →  $S$ \ {$p$} is a bijective map, I have to show that $\phi$ is injective and surjective. But how do I do that? Can someone please help me show that $\phi$ is bijective? and how can I derive a formula for its inverse? 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? How do I install Skype? At page 17 of Munkres' Elements of Algebraic topology it says, referring to fig. 3.6, that "the diagram does not determine [the torus]. It does more than paste opposite edges together":   Is it so? I tried to glue them together, but end up with a normal torus. What is this strange function definition syntax in C? Creating an index of quality from multiple variables to enable rank ordering I have four numeric variables. All of them are measures of soil quality. Higher the variable, higher the quality. The range for all of them is different:  Var1 from 1 to 10  Var2 from 1000 to 2000  Var3 from 150 to 300  Var4 from 0 to 5  I need to combine four variables into single soil quality score which will successfully rank order.  My idea is very simple. Standardize all four variables, sum them up and whatever you get is the score which should rank-order. Do you see any problem with applying this approach. Is there any other (better) approach that you would recommend?  Thanks  Edit:  Thanks guys. A lot of discussion went into "domain expertise"... Agriculture stuff... Whereas I expected more stats-talk. In terms of technique that I will be using... It will probably be simple z-score summation + logistic regression as an experiment. Because vast majority of samples has poor quality 90% I'm going to combine 3 quality categories into one and basically have binary problem (somequality vs no-quality). I kill two birds with one stone. I increase my sample in terms of event rate and I make a use of experts by getting them to clasify my samples. Expert classified samples will then be used to fit log-reg model to maximize level of concordance / discordance with the experts.... How does that sound to you? I have a script that reads a text stream and generates a file of sed commands that is later run with sed -f. The generated sed commands are like:  s/cid:image002\.gif@01CC3D46\.926E77E0/https:\/\/mysite.com\/files\/1922/g s/cid:image003\.gif@01CC3D46\.926E77E0/https:\/\/mysite.com\/files\/1923/g s/cid:image004\.jpg@01CC3D46\.926E77E0/https:\/\/mysite.com\/files\/1924/g   Assume the script which generates the sed commands is something like:  while read cid fileid do     cidpat="$(echo $cid | sed -e s/\\./\\\\./g)"     echo 's/'"$cidpat"'/https:\/\/mysite.com\/files\/'"$fileid"'/g' &gt;&gt; sedscr done   How can I improve the script to ensure all regex metacharacters in the cid string are escaped and interpolated properly? In general relativity (GR), spacetime is viewed as a differentiable manifold of dimension $D$ with a metric of Lorentzian signature $(-,+,+,...,+)$.  My question is why differentiable? Why didn't the Emperor realize that torturing Luke to death in front of Vader might be a bad idea? 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Then there is an $\alpha \in$ ON such that for all $\beta \in $ LIM: $\beta &lt; \alpha$, i.e. $\beta \in \alpha$ and hence LIM $\subset \alpha$.  This means that LIM is a set so we can consider $\bigcup$ LIM. Then we have $\beta &lt; \bigcup$ LIM for all $\beta \in$ LIM by definition of $\bigcup$ LIM. So LIM $\subseteq \bigcup$ LIM.   $\bigcup$ LIM is either a limit or a successor ordinal.  If it is a limit ordinal then $\bigcup$ LIM $\in$ LIM and by transitivity of ON, $\bigcup$ LIM $\subseteq$ LIM, so $LIM = \bigcup$ LIM in this case which would mean that $\bigcup$ LIM $\in$ LIM $= \bigcup$ LIM which would be a contradiction because sets cannot contain themselves. Hence $\bigcup$ LIM cannot be a limit ordinal.  If $\bigcup$ LIM is a successor ordinal then $\bigcup$ LIM = $\beta \cup \{&#160;\beta \}$ for some $\beta \in $ ON.  Now I'm not sure how to proceed from here. I'm also not sure about whether I'm on the right track with this proof because I think there should be a shorter way  to prove this.  Thanks for your help. Modal vs Non-Modal vs Auxiliary modal vs Conjugated Verb Modal vs Non-Modal vs Auxiliary modal vs Conjugated Verb   How are these kinds of verb related to each other? Specially Non-Modals and Conjugated Verb. Is the "Auxiliary modal" just another name for "Auxiliary verb"?