How can I upload a pygame that I made on a website?
Any way to embed Pygame in webpages?
Imagine that a pebble is placed on a uniformly rotating, frictionless disk.  What will happen to this pebble? Will the disk slide under it and the pebble stay as is? Or will there be a centrifugal force on the pebble and it'll be thrown off the disk? Alternating sum of squares of binomial coefficients Uniform convergence for sequences of functions Windows 7 low and critical battery levels cannot be set lower than 10% What is the formal definition of polynomial ring of several variables? Let's consider a polynomial ring of single variable. One can define them informally by saying $P(X)=\sum_{i=1}^n a_n X^n$ while $X$ is an indeterminate variable. However, since mathematics is based on first-order logic, one can only talk about something that actually exists (as a set). So as its name says, indeterminate variable is not a sentence in $ZFC$. Nevertheless, we can formally define a polynomial ring $R[X]$ as a subset of $R^\omega$ whose support is finite. ($R$ is a commutative ring with unity) Just like a single variable, I want to know what would be the formal definition of a polynomial ring of several variables. What would be a formal definition? === I have figured out a candidate. That is,  Candidate for a &quot;Definition of a polynomial ring $R[X_1,...,X_n]$&quot;. ($R$ is a commutative ring with unity) Let $R[X_1,...,X_n]$ be the set of all functions $f:\omega^n \rightarrow R$ whose support is finite. Define $(f+g)(a,b,c)=f(a,b,c)+g(a,b,c)$ and $(f•g)(a'',b'',c'')=\sum_{a+a'=a''} f(a,b,c)•g(a',b',c')$ (Note that in this way, $+,•$ are well defined.) Then, it can be checked that $(R[X_1,...,X_n],+,•)$ is a ring.  Is this a formal definition of a polynomial ring? Why did Dumbledore drink the potion in the attempt to get a Horcrux? What is the physical explanation for energy transport in simple electrical circuits? Keeping it simple, what is the physics approach to a current in a wire connected to a battery? How does the electric field behave?  I've read somewhere that, contrary to the common conception, electrons don't really flow as water in a hose so, what is really going on there? How is energy transported at the speed of light if electrons don't flow like very fast water? This is my first time through playing Elder Scrolls and initially I grabbed every quest that has come available...  At first I thought I wanted to be part of the Imperial Legion but I feel like I picked the wrong side and am now regretting my choice.  So...is there any way to un-join the Legion and join up with the Stormcloaks, or will I have to wait and join them next time? Biblatex with Biber: Configuring my editor to avoid undefined citations Why is \[ ... \] preferable to $$ ... $$? I've heard that you should use \[ ... \] for displayed equations instead of  $$ ... $$, but why is that?  I'd assumed that it's so that you can more easily tell which are starting and which are ending delimiters, but if I always use a syntax-highlighting text editor, I can see that easily based on the color of the symbols.  Is there any reason for this suggestion? to infinitive as direct object?! (as in : I promised to get there in time.) I'm a passionate Korean English-learner, and I would like to ask you a question about grammar.   It was 3 years ago when I first started to study the to-infinitive as studying grammar with Grammar In Use series and English Grammar series by Longman, having been exposed to English for 3 years. There wasn't any mention of the author saying that the usage of the to-infinitive is as an (direct) object. And for me also, the times I have encountered sentences in which the usage of the to-infinitive, so to speak, as direct objects were used, I've read them as some kind of obligatory adverbial that shows what the verbs are about, not as objects, which means they are noun phases! 100% of Korean textbooks say the usage is as a direct object, but I doubt it. I think it's some kind of obligatory adverbial that shows what the verbs are about.    I thought it was better for me to ask Americans themselves. Do you read this usage of to-infinitive as a noun phrase or an adverbial? Now the cubs are growing, fed on mom's milk, but mom herself hasn't eaten in five months, and is getting thinner by the day.   What does in mean here, and how would the meaning of that piece change if for were used? I like some of the features provided by the Vim-LaTeX suite, but not all of them.  In particular, I dislike all of the mappings like [[ = \left[ \right] and the automatic placeholder stuff with &lt;++&gt;. I am a very fast typist but with that comes more typing mistakes, which I correct almost instantly by pressing backspace. Unfortunately, if something has automatically added a bunch of place-holders or expanded some expression, I then have to move around the document and remove more characters than necessary. For example, if I accidentally press [ twice, instead of only having to delete one character, I have to delete 10.  Having already looked through the documentation, and some of the source, I have managed to disable some of these "features", but not all of them:  In particular:  let g:Tex_SmartKeyBS = 0 let g:Tex_SmartKeyQuote = 0 let g:Tex_SmartKeyDot = 0   (Turn off maps for backspace, double quote and triple dot.)  let g:Imap_UsePlaceHolders = 0   (Turn off &lt;++&gt; placeholder insertion.)  But I can't seem to find a way of turning off the other stuff. Anyone have any suggestions?  Edit: I was able to "disable" some more things by setting:  let g:Tex_Leader = '`tex' let g:Tex_Leader2 = ',tex'   (Which stops some expansions but still not all. It's not really disabling it but just making it less likely that you're going to type in `texa as opposed to the potentially more common `a, for example.)  PS: Personally, I think it is very poor practice to make an optional package that, when installed, greatly changes the default handling of your system. I don't mind some intelligent source highlighting etc, or some new commands (that don't clobber any old ones), but it seems unhelpful to enable this drastic kind of key remapping by default. Can cited works hold grammatical positions in sentences? Though I read this style quite often, I was recently told unambiguously by a reviewer that I was NOT supposed to use citations "as if they are objects in a sentence."  The following sentence is an example of what the reviewer considered unacceptable:       We analyzed the data using the Wilmerding method, guided by [12].   The references section might include the following:      [12] Smith, D., Marshawn, J., &amp; Devenshaw, A. 2011. Techniques and   Procedures for Applying the Wilmerding Method.  Prince Publications,   Inc, New York, NY.   The Wilmerding method1 is not a step-by-step algorithm that can be precisely followed as if by a machine, and thus [12] does not provide a step-by-step algorithmic description but rather guidance for using  the method. [12] is a relatively slim but authoritative textbook about how to use the Wilmerding method.  Within this question, for the purpose of discussion, I have intentionally put [12] in grammatical positions where it's an important element of the sentence and the sentence would make less sense without the reference.  Sometimes that seems to be the most efficient way to communicate the intended message.  Is using a reference as a grammatical sentence element like this OK?   Why might this not be considered acceptable?   Should I be rewording those sentences?  Is it enough of a violation to be worth passing the note on to authors of papers I review?   I considered posting this on  but it seems more specific to academia than general English usage, and the comment came from a content reviewer rather than a copy editor.   is related but it seems to be more about when to put authors names' in vs. outside of the parentheses in an APA-like style.   The numbered citation style (as opposed to following APA, for example) is required by the venue.  1: Fictionalized for the purpose of this discussion How can I convert audio from MP4 or FLV video files to mp3? Is there an application that allows me to extract the audio from a MP4 or FLV video file and store it as an MP3 file for use in a portable media player? Suppose that, for a real application, I have ended up with a sorted list A = {$a_1, a_2, ..., a_{|A|}$} of elements of a certain kind (say, Type-A), and another sorted list B = {$b_1, b_2, ..., b_{|B|}$} of elements of a different kind (Type-B), such that Type-A elements are only comparable with Type-A elements, and likewise for Type-B.  At this point I seek to count the following: in how many ways can I merge both lists together, in such a way that the relative ordering of Type-A and Type-B elements, respectively, is preserved? (i.e. that if $P_M(x)$ represents the position of an element of A or B in the merged list, then $P_M(a_i)&lt;P_M(a_j)$ and $P_M(b_i)&lt;P_M(b_j)$ for all $i&lt;j$)  I've tried to figure this out constructively by starting with an empty merged list and inserting elements of A or B one at a time, counting in how many ways each insertion can be done, but since this depends on the placement of previous elements of the same type, I've had little luck so far. I also tried explicitly counting all possibilities for different (small) lengths of A and B, but I've been unable to extract any potential general principle in this way. Let us consider the following code and the algorithm corresponding to it    \documentclass{article}   \usepackage[utf8]{inputenc}    \usepackage[]{algorithm2e}     \begin{document}     \begin{center}    \begin{algorithm}[H]     \KwData{this text}     \KwResult{how to write algorithm with \LaTeX2e }     initialization\;     \While{not at end of this document}{      read current\;      \eIf{understand}{              go to next section\;       current section becomes this one\;       }{       go back to the beginning of current section\;      }     }     \caption{How to write algorithms}    \end{algorithm}    \end{center}    \end{document}   Which gives the following     I want to add running times for each step to my algorithm as shown below (for example)    So, I need to make my algorithm as a column in the table and one more column for running times of each step, how can I make my algorithm as a column inside a table? How to remove "onclick" with JQuery? Order discrete x scale by frequency/value Help evaluating a gamma function