Showing that $R[x_1,...,x_n]$ is the only $R$-algebra up to isomorphism that satisfies the universal property of the polynomial ring.
Proving the universal mapping property for polynomial rings
What gear do I need for a newborn photography session? I am an amateur photographer and I have a newborn session coming up for a family friend.   My current gear is:   Canon T5i (crop sensor) Canon 18-55 IS STM Kit lens 85mm f/1.8 lens   I plan on buying a reflector for the shoot and will be using natural light as my light source.  So my question is: is this enough gear to get some nice photos? Should I get another prime lens besides the 85mm? I don't really use the kit lens.  I also have another 8 month old baby shoot coming up and an informal wedding shoot coming up so it would be nice to use the lenses in all situations. I don't need a telephoto at the moment and will probably not use one any time soon. How should I validate an e-mail address? How to delete a file with a path too long to be deleted Create model from XYZ data points How to stop Tkinter Frame from shrinking to fit its contents? This is the code that's giving me trouble.   f = Frame(root, width=1000, bg="blue") f.pack(fill=X, expand=True)  l = Label(f, text="hi", width=10, bg="red", fg="white") l.pack()   If I comment out the lines with the Label, the Frame displays with the right width.  However, adding the Label seems to shrink the Frame down to the Label's size.  Is there a way to prevent that from happening? Shell Script for logging into a ssh server I tried writing a shell script which can do automatic login into a ssh server using password which is mentioned in the script. I have written the following code:  set timeout 30 /usr/bin/ssh -p 8484 root@172.31.72.103 expect  {    "root@172.31.72.103's password"     {       send "password\r"     } }   This code is not running properly, still it is asking for the password. Can somebody please help me in solving this Should form 'Continue' button be disabled if validation is incomplete? In forms we often see the 'continue' button inactive until all the required fields are complete:         &ndash; Wireframes created with   Is this actually a help to the user, or a hindrance?  I imagine the Pro for this is that it's a visual indicator to the user that they haven't finished filling in the form or dealing with errors (because even with inline validation a user won't be informed if they've never actually interacted with a field so won't show any errors for those at that point).  But then the main Con for this (which I've also seen borne out in usability testing) is that users will be confused why they can't click the button.  Is the pattern of disabling the Continue button until validation is complete actually providing a negative user experience? The equivalent electric field of a magnetic field I know that Lorentz force for a charge $q$, with velocity $\vec{v}$ in magnetic field $\vec{B}$ is given by  $$\vec{F} =q \vec{v} \times \vec{B}$$  but there will exist a frame of reference where observer move at same velocity with that of charge $q$, so according to him $v=0$. hence he will see no magnetic force is exerted on charge $q$. I have work on this problem for a while and found that the special relativity predicts equivalent electric force will acting upon charge instead. I want to know the relationship between this equivalent electric force and magnetic force.  Thanks in advance What is the pythonic way to avoid default parameters that are empty lists? Sometimes it seems natural to have a default parameter which is an empty list. Yet . If for example, I have a function: def my_func(working_list=[]):     working_list.append(&quot;a&quot;)     print(working_list)  The first time it is called the default will work, but calls after that will update the existing list (with one &quot;a&quot; each call) and print the updated version. So, what is the pythonic way to get the behavior I desire (a fresh list on each call)? What is the best way to remove duplicate rows from a fairly large SQL Server table (i.e. 300,000+ rows)?  The rows, of course, will not be perfect duplicates because of the existence of the RowID identity field.  MyTable  RowID int not null identity(1,1) primary key, Col1 varchar(20) not null, Col2 varchar(2048) not null, Col3 tinyint not null C# object references and Action types Prison problem: locking or unlocking every $n$th door for $ n=1,2,3,...$ I have a problem called "The Prison Problem" that I need to explain to my 9-year-old cousin. I would think that he has just started learning about divisors and perfect squares, and as such, I have a proposed solution. Any input from you guys would be welcome, as to what is the best way to go about this.  Problem:  There was a jail with 100 cells in it, all in a long row. The warden was feeling very jolly one night and told his assistant that he wanted to give all the prisoners a wonderful surprise. While they were sleeping, he wanted the assistant to unlock all the cells. This should be done, he told the assistant, by putting the key in each lock, turning it once. The assistant did this, then came back to report that the job was done. Meanwhile, however, the warden has second thoughts. "Maybe I shouldn't let all the prisoners free," he said. "Go back and leave the first cell open, but lock the second one (by putting the key in and turning it once). Then leave the third open, but lock the fourth, and continue in this way for the entire row." The assistant wasn't very surprised at this request as the warden often changed his mind. After finishing this task, the assistant returned, and again the warden had other thoughts. "Here's what I really want you to do," he said. "Go back down the row. Leave the first two cells as they are, and put your key in the third cell and turn it once. Then leave the fourth and fifth cells and turn the key in the sixth. Continue down the row this way." The assistant again did as instructed. Fortunately, the prisoners were still asleep. As a matter of fact, the assistant was getting pretty sleepy, but there was no chance for rest yet. The warden changed his mind again, and the assistant had to go back again and turn the lock in the fourth cell and in every fourth cell down the row. This continued all through the night, next turning the lock in every fifth cell, and then in every sixth, and on and on, until on the last trip, the assistant just had to turn the key in the hundredth cell. When the prisoners finally woke up, which ones could walk out of their cells?  Proposed solution:  Ten cells are left open after this process. Every cell that is a perfect square will remain open (1, 4, 9, 16, 25, 36, 49, 64, 81, and 100). If a number is not a perfect square, then it has an even number of divisors, therefore it will be "toggled" an even number of times and end up where it started (closed). Perfect squares have an odd number of divisors, so they will end up the opposite of where they started (open). Possible Duplicate:          There are several tarballs (e.g. Firefox, Eclipse, Zend Studio and ...) that have executable files in them, that we can extract and run. but this executable will be available only for the user that extracts it.  If we want to make it available for all users we should move it to some location like /usr/share, /usr/local/share/, /opt/, ... and give appropriate permissions, make an executable file in /usr/bin/ and so on...  I want to make a deb installer file that does them all! How to do it? Firefox application associations not working in the 'Downloads' window For every vector space $V$ does there exist a linear functional $f$ ( a linear map from $V$ to $F$ the underlying field ) such that for some $ \vec v \in V$ , $f(\vec v) \ne 0$ ? If it does exist , can we prove the existence without the "axiom of choice " ? Is the existence equivalent to axiom of choice ? 'Should've seen it glow' or 'should've seen it glowing'? Which one of the following is the correct one?   I should have seen it glow. I should have seen it glowing.   Or are both correct? Would you parse them please? Possible Duplicate:          I was wondering if there's any good reason to still use blob fields in a database. A couple of years ago I worked with a DB with a bunch of images in it, the DB was very slow and I couldn't see any good reason to keep the images inside the DB, so I got the images out and stored the filenames instead.  Was this a smart move? What would you do in my place? How do I configure Apache 2 to run Perl CGI scripts? I would like to configure Apache 2 running on Kubuntu to execute Perl CGI scripts. I've tried some steps that I came across by googling, but nothing seems to work.  What is the right way of achieving this? Edelstein Theorem Let $(M,d)$ be a compact metric space and $d(f(x),f(y)) &lt; d(x,y) $ for all $ x\neq y$ Prove that if $f$ is a continuous fuction then there is a unique $x_0 \in M $ such that $f(x_0)=x_0$ I know that since $g(x) = d((f(x),f(y))$ is continuous then a continuous function in a compact space reaches its maximum and minimum. Then by proving that the minimum of $g(x)$ = 0 can I conclude that $x_0$ is a fixed point? I'd like to assign a frequently used string to a shortcut key, so when I press it, it types it in whatever application I'm in. So for example, when I type Shift-F12, it types "This is wonderful!" in whatever Gnome app I'm in at that time.