Given is the forgetful functor: $U: Grp \rightarrow Set$. I've already seen that we can construct a left adjoint $F: Set \rightarrow Grp$ to this functor by introducing the free functor, which associates to any set $X \in Set$ the free group $F(X)$.  Now I want to check if there is a right adjoint to our forgetful functor, let's say it is the functor $R: Set \rightarrow Grp$. If we had a right adjoint, we'd have a bijection between hom-sets, namely $Hom_{Grp}(R(X), G) \simeq Hom_{Set}(X, U(G))$ for any set $X \in Set$ and $G \in Grp$.  I guess there is no such a right adjoint to the forgetful functor $U$ but I don't know how to show it. Can anybody help me, please? Thank you very much!
The forgetful functor $\mathsf{Groups} \to \mathsf{Sets}$ admits a left adjoint, namely, "forming the free group" functor.  I was wondering if this has a left adjoint.  This seems unlikely, but I don't have a proof.  Is there a functor $\mathsf{Sets} \to \mathsf{Groups}$ that admits a left adjoint at all?
I'm working on the Fourier transform, but I don't know how to evaluate the integral:  $$I = \int_{ - \infty }^{ + \infty } {\frac{{\cos (ax)}}{{{x^2} + 1}}{e^{ - ibx}}\,\mathrm dx} $$ how to prove $G_1$ and $G_2$ aren't isomorphic? What do you call the symbols list on a map? Every map has symbols that describe things. In the corner of a map, there is a list that explains what each symbol means.  What is the name of that list? (What title should I put above this list)? What's is the origin of Orbital Angular Momentum of electrons in atoms? Consider the Hydrogen 1s electron. We know that, in the quantum picture, the electron isn't orbiting or rotating at all, rather we simply state that the electron is spread over the entire space with the probability of finding it being maximum a radial distance $R$ (=Bohr radius) away from the nucleus. This helps explain why the electron does not radiate EM radiation while in the atom.  But, with this understanding, I do not understand the source of the orbital angular momentum. Is it intrinsic like the spin? Is it safe to fully cook a stuffed turkey the night before thanksgiving? Why two inverters cascaded in multiplexer control signal terminals? Why cascaded inverters are used in the multiplexer when it is possible to realize the circuit using a single inverter. If in a UFD every maximal ideal is principal then it is a PID How many associative binary operations for a set of two elements? Suppose we have a set  $S=\{a,b\}.$ Obviously, the total number of binary operations on $S$ is the number of all the pairs such that $$*(a,b)=a \, \text{or}  \, b,$$ which gives  $ 2 \cdot 2 \cdot 2 \cdot 2=2^4$ possible ones. Is there a way to find which of these $16$ binary operations are associative without writing all the operation matrices and checking it by hand? What is the number of strings at size $n$ that is constructed from ${a,b,c,d}$ and there is an even num of $a$ What is the number of strings at size $n$ that is constructed from ${a,b,c,d}$ and there is an even num of $a$?  I have tried to answer it as recursion formula in the following logic:  In order to build $ A(n) $, I will split into several cases:   if the string starts with $a$, then I will need another $a$ to make an even number of $a's$. so we get $aa[A(n-2)]$ meaning 2 $a's$ at the start + a valid string at the size of $n-2$. if it starts with $b,c,d$ we get $b[A(n-1)]$ + $c[A(n-1)]$ +$d[A(n-1)]$, meaning a one of the letters + a valid string at the size of $n-1$.   So we will get a recursion formula that looks like that:  $A(n) = A(n-2) + 3A(n-1)$  I'm not 100% on my answer, am I right to approach it in that direction? I have text file. Task - get first and last line from file after   $ cat file | grep -E "1|2|3|4" | commandtoprint  $ cat file 1 2 3 4 5   Need this without cat output (only 1 and 5).  ~$ cat file | tee &gt;(head -n 1) &gt;(wc -l) 1 2 3 4 5 5 1   Maybe awk and more shorter solution exist... In order to better understand the SQL Server query processor, I've been thinking about how the ORDER BY clause works, and how SQL Server provides results.  It appears SQL Server will provide results in exactly the same order for any given set of data that remains the same and never, ever changes.  Once you introduce any kind of uncertainty such as developers changing something, the order of results can no longer be expected to be the same.  Simply seeing the results in the same order, each time you press F5 does not guarantee the results you expect.  Try this:  USE TempDB; DROP TABLE TestOrderBy; DROP TABLE TestOrderBy2; CREATE TABLE TestOrderBy (     ID INT NOT NULL CONSTRAINT PK_TestOrderBy PRIMARY KEY CLUSTERED IDENTITY(1,1)     , SomeData varchar(255) ); INSERT INTO TestOrderBy (SomeData) VALUES ('E'); INSERT INTO TestOrderBy (SomeData) VALUES ('D'); INSERT INTO TestOrderBy (SomeData) VALUES ('C'); INSERT INTO TestOrderBy (SomeData) VALUES ('B'); INSERT INTO TestOrderBy (SomeData) VALUES ('A');  CREATE TABLE TestOrderBy2 (     ID INT NOT NULL CONSTRAINT PK_TestOrderBy2 PRIMARY KEY CLUSTERED IDENTITY(1,1)     , SomeData varchar(255) ); INSERT INTO TestOrderBy2 (SomeData) VALUES ('E'); INSERT INTO TestOrderBy2 (SomeData) VALUES ('D'); INSERT INTO TestOrderBy2 (SomeData) VALUES ('C'); INSERT INTO TestOrderBy2 (SomeData) VALUES ('B'); INSERT INTO TestOrderBy2 (SomeData) VALUES ('A');  SELECT SomeData  FROM TestOrderBy UNION ALL SELECT SomeData FROM TestOrderBy2;  CREATE INDEX IX_TestOrderBy_SomeData ON TestOrderBy (SomeData); CREATE INDEX IX_TestOrderBy2_SomeData ON TestOrderBy2 (SomeData);  SELECT SomeData  FROM TestOrderBy UNION ALL SELECT SomeData FROM TestOrderBy2;   The results:    As you see, adding a simple index on the fields selected in the query alters the order of the results.  From here on, I'm adding ORDER BY unless I truly don't care. Thousands of .log files in Exchange 2007 Server folder I'm a bit of an Exchange admin noob so I don't know if this is normal, but on our Exchange 2007 server under C:\Program Files\Microsoft\Exchange Server\Mailbox\First Storage Group\ there are 30,000 1MB .log files.  I don't know if those should be there or not, but they keep multiplying like little jackrabbits.  I've so far resisted deleting them as I'm sure something would break, but it's come to the point now where our 80GB disk is completely full up and Exchange has stopped receiving messages.  My best guess is that these are transaction logs and would be useful in rebuilding the database should something go awry, but 30GB of transaction logging seems a bit excessive.  Is there a way to limit it or is that even a good idea?  And of course did I completely miss the point? Cannot upgrade unsupported version of Skype for Linux (v5) Should Emergency Funds be Used for Infrequent, but Likely, Expenses? Say we break down non-monthly costs into the following three categories.   High level of predictability: This would include oil changes that may be needed every 2 to 4 months depending on usage, car insurance (if paid every 6 month), car registration, etc. Medium level of predictability: There are some expense which one can reasonable assume will occur on an infrequent basis.  For example, car repairs, house repairs, or even doctor visits during flu season. Low level of predictability: For example, losing a job, significant medical expense, etc.   Which of these categories are emergency funds meant to cover?  I think it is pretty clear that a high level of predictability costs should not use emergency funds while a low level of predictability should.  But medium level of predictability costs seem different, and it left me wondering what is generally defined as an "emergency".  While you do not know when these costs will occur or how much they will be, you could devise some pretty good rules of thumb on frequency and cost. Say you know your 2007 Volvo should have a repair cost averaging out to $100 per month.  Conceptually you could save $100 each month and when your car breaks, not have to spend any emergency funds on the repair.  Is doing this "double-saving" and not suggested if you already have the 6 month emergency fund saved up?  Note: this is a question on how to save money for expenses.  It is not a question about if you should spend emergency fund money to fix a car when you have no other money set aside. New post formatting Update 3 It took me longer than I wanted, but I’ve got some updates for you on our s-prose component. You can see the  with my thinking over at our design system’s repo. It does the following:  Spreads s-prose headers out from paragraphs for better grouping. Tightens up s-prose paragraphs a bit, and made sure spacing under headings are more consistent. Replaces all s-prose margins internal to the component with a CSS variable var(--s-prose-spacing) so our users can more easily tinker with it in their browsers and user scripts. Kills trailing margins in s-prose  You can see  of content at our Stacks documentation.  Update 2 I’ve split the difference between the original 1.3 line-height and the proposed 1.6 line-height. We’re now at 1.5, with additional refinements to spacing between elements. I’ve also reduced line height within code blocks back to near the original value.  Update 1 This is now live.  We’re doing a bit of refactoring on our post formatting. Currently, we apply a single class with both layout and styling called .post-text. Our first goal is to separate layout from our text styles. Simple enough! However, our current post styling has a few missing spots and areas for improvements. Using our  as a sandbox, we’ve beefed up our styling and created a new component called .s-prose. It’s designed to offer styling for everything the  spec allows. This should allow us to add Stack Overflow-flavored styling anywhere we need to consume Markdown—questions and answers, of course, but also things like user profiles. Our .s-prose component adds:  Proper formatting for   Further header support and better spacing there  Additional blockquote refinements.  Refinements for revealing spoilers. We now reveal this on click / tap for a more consistent experience across devices—fewer accidental reveals as well, since it’s no longer a :hover interaction. Clicking again does not hide the spoiler. We also now add a badge to the top right corner. You can see how these work on our . It looks something like this:   This is a spoiler  It has two lines   General refinements to spacing including nested lists, images within lists  Sizing variations for different contexts. We offer xs, sm, and md variations of the .s-prose component.   As with everything, our prose component is a work in progress. We think we’ve got a heck of a start to a reusable component. We plan on going live with the new component network wide on Thursday, August 27, 2020. We’ll edit this post if that changes. If you see something funky, let us know. Some things are intended but might feel like regressions—images have a slightly different baseline—but others may have slipped through the cracks. How do I go about backing up/saving installed device drivers in Windows? I want to reinstall Windows on a machine which has some very obscure drivers. I don't much fancy having to hunt round for them (that's if I'm able to locate them at all). Is it possible to save or backup already installed drivers? 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