Why when writing a congruence relation, do we have to add $\pmod{m}$ at the end. I understand it's notation, but $b$ does not have to be equal to $a\bmod b$ if I understand correctly. The definition states that "$a$ and $b$ are said to be congruent modulo $m$, if their difference $a − b$ is an integer multiple of $m$." If that is true, then one can write $a\equiv b \pmod{m}$  So, for example, if we let $a=10$ and $m = 2$, then the set $b_n$, where $b_n$ contains all the integers that satisfy $a\equiv b\pmod{m}$ contains $2,4,6,8\dots$ But $10\bmod 2 = 0$. So why do we write $a\equiv b\pmod{m}$ to indicate a congruence relation? Thanks, any help is greatly appreciated.
I am trying to write the Euclidean algorithm in the following way:  $A = \lfloor A \div B \rfloor \times B + (\text{remainder of}) \: A \div B $   Now is there any symbol I can use  to say "remainder of A $\div$ B"? I know that in the C programming language there is the operator % for modulus; is that a valid symbol in maths? Can I write A % B? Or is there some other way?
What happens if I get the same skill from both my background and my class? Is there a 'biblatex in a nutshell' guide out there? I'd like if someone explained to me the essentials on how to use biblatex (what lines I have to write in my document, which files I have to have, how many times and what I have to compile), so then I can go to  to customise it further. Pop up option are not popping up after snipping the area How to apply a regex on a column fetched from DB I am using IBM DB2.  So, I am retrieving a column named SSN from a table. I have to apply the following regex on that column:  ([1-57-8][0-9]{2} |0([1-9][0-9] |[0-9][1-9]) |6([0-57-9][0-9] |[0-9][0-57-9])) ([1-9][0-9]|[0-9][1-9])([1-9]\d{3}|\d[1-9]\d{2}|\d{2}[1-9]\d|\d{3}[1-9])   Someone suggested to me that I have to write a function/procedure in the query to validate that the SSN picked from the db follows this regex else it should not be stored in the result set of the query.  I don't know how to write a function or a procedure in SQL, can you please help me out here? Will I lose previous purchases when I change country in iTunes I lived in country A for a few years, and made a number of purchases on the iTunes store for that country. I have now moved to country B, and want to start using the iTunes store from country B. If I change the country in my account settings, will I lose all my previous purchases, or will I still be able to re-download my purchases from country A from iCloud? How is MSE divided by $\sigma^2$ always a chi-square distribution? Why didn't anybody Summon the Triwizard cup in the maze? I have an issue which I could not solve, although I tried and I got some help on R forum.  I am trying to calculate Mahalanobis distances on a data.frame, where I have several hundreds of groups and several hundreds of variables. Whatever I do, I get the system is computationally singular: reciprocal condition number error.   It is clear that it is singular, but is there any way to get rid of it and run Mahalanobis? Should I forget solve this using another approach? If yes, then what else to use?  I have uploaded the data file to my :  It is a tab delimited txt file with no headers.  I was working with the R StatMatch Mahalanobis (also tried stats Mahalanobis) function. I have a deadline for this project (not a homework!), and I could always use this function, so I thought I will be able to keep the calculations short, but now I am lost.  Migrated . for loop wget directory creation error As a Ubuntu beginner. I am trying to download links from multiple text files with wget2 in batch. The -P &quot;$f&quot; creates directory name with .txt suffix. Is it possible to ignore .txt for -P &quot;$f&quot;. Any modification suggestions. for f in *.txt;do wget2 -i &quot;$f&quot; -P &quot;$f&quot;;done What is probabilistic programming? For the past year, I've been hearing a lot about Probabilistic Programming (PP) frameworks like  and , and how great PP is. And today, someone shared this link with me:    However, I don't really follow what is special about it since it feels like whatever you can do in PP you can do in any other general purpose language. I'm sure there are technical aspects about PP that make it attractive (e.g. parallel computing), but this aside, is PP really any different from any other language?  Question: I was wondering if there was a consensus of what PP was and how it differs from other statistical-focused software like R, Matlab, Mathematica. It should be noted that PyMC3 and Stan are focused on more Bayesian analysis.  Doing a little research on Google, I came across the following two definitions. The first more abstract, and the second more about PP's technical characteristics.     1.2. Probabilistic Programming Is      Instead, probabilistic programming is a tool for statistical modeling. The idea is to borrow lessons from the world of programming languages and apply them to the problems of designing and using statistical models. Experts construct statistical models already—by hand, in mathematical notation on paper—but it's an expert-only process that's hard to support with mechanical reasoning. The key insight in PP is that statistical modeling can, when you do it enough, start to feel a lot like programming. If we make the leap and actually use a real language for our modeling, many new tools become feasible. We can start to automate the tasks that used to justify writing a paper for each instance.      Here's a second definition: a probabilistic programming language is an ordinary programming language with rand and a great big pile of related tools that help you understand the program's statistical behavior.      Both of these definitions are accurate. They just emphasize different angles on the same core idea. Which one makes sense to you will depend on what you want to use PP for. But don't get distracted by the fact that PPL programs look a lot like ordinary software implementations, where the goal is to run the program and get some kind of output. The goal in PP is analysis, not execution (added emphasis).   --   I'd like to know if the general statistical community agrees with these two definitions of PP, and if there are any other characteristics this definition may be missing. example of compact operator Let $E=l^p=\left\{x=\left(x_n\right): x_n\in\mathbb{R}, \sum_{1}^{\infty}\left|x_n\right|^p&lt;\infty\right\}$ with $1\le p&lt;\infty$, $\left\|x\right\|_E=\left(\sum_{1}^{\infty}\left|x_n\right|^p\right)^{1/p}.$ Let $\left(\lambda_n\right)$ is a bounded sequence in $\mathbb{R}$ and consider the operator $T$ defined by $$T(x)=\left(\lambda_1x_x,\ldots,\lambda_nx_n,\ldots\right)$$ where $x\in l^p$. Prove that $T$ is a compact operator from $E$ to $E$ iff $\lambda_n\rightarrow 0$.  Can anyone give me some hints to solve this problem? Thank you! PrintWriter append method not appending Alternate exits in Bit Trip Runner 2 Is there any point in taking the alternate exits in Bit Trip Runner 2?  I can't see any benefit to doing so in terms of unlocking a different path in the overworld, so is it just for your own benefit? Is the usage of text embedded in an image a bad practice, for very controlled uses? What are your thoughts on writing your text in images?  I'm not saying throughout the whole website... But in certain key locations like image carousels. And I'm not talking about the special cases where you have font types that browsers can't render. I'm talking about text that uses standard fonts and is not too big (3/4 bullets).  Should I embed the text in the images that compose the carousel? Or should I apply the text to the images as "real text" HTML?  What are the advantages/ disadvantages of using text in an image? Does Ubuntu come with a preinstalled or automatic firewall? If not, do I need one?   I've read some articles about the advantages of Linux over other operating systems about security (no need to have antivirus, ...) but would like to be sure. What is going on in front of and behind a fan? Why is it that when you drop paper behind a fan, it drops, and is not blown/sucked into the fan, whereas if you drop paper in front of a fan, it is blown away? Is there any utility to check whether my laptop battery is in healthy condition or not? Is there any utility available on Ubuntu to check whether laptop battery is in healthy condition or not? Determine $4$ specific digits in $34!$ How to fix "W: Duplicate sources.list entry"? I'm using the ieee style for biblatex and when inserting citations I want     [1,2,6-10]   Right now I'm getting     [1],[2],[6-10].   How can I change this?