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Difference between shadowing and overriding in C#? I've only just begun studying group theory (up to Lagrange) following on from vector spaces and I am still finding them almost frustratingly  arbitrary. I'm not sure what exactly it is about the axioms that motivated them defining groups.  My textbook asks you to list 'common features' of vector spaces and then later defines a set of axioms for vector spaces under addition, scalar multiplication and both, noting that the axioms under addition form an Abelian group. So are groups just a generalisation of vector spaces under any binary operation?  My main problem is that the book notes that 'the axioms may seem rather arbitrary' and links groups with vector spaces but doesn't elaborate.   When introducing groups, you are tasked to complete Cayley tables for the symmetries of an equilateral triangle and square.  Then, similarly to the delivery of vector spaces, notes that the tables have common properties (Closure, Identity, inverse and association) and defines a group as a set of elements under a binary operation that has these features.  So what's so important about these 4 properties? For example, if 1 or 2 the properties were excluded form the  axioms, or we added an extra few properties as axioms how would that cripple the effectiveness of groups?  Are the group axioms ever difficult to work with or do they always work, forgive the crude Littlewood analogy, like a mathematical skeleton key?  What is it about these 4 properties that make groups such a powerful tool in mathematics and physics?  My best guess is that a group is the best way to express our sense of  symmetry and what is symmetric mathematically, but I would prefer some elaboration. Rolle's Theorem and the Mean Value Theorem Let $f\colon \mathbb{R} \to \mathbb{R}$ have derivatives of all orders. Suppose that $a &lt; b$ and that $f(a)=f(b)=f'(a)=f'(b) =0$. Prove that $f'''(c) = 0$ for some $c$ in $(a,b)$. DataTable merge without primary key column What would happen if you used a dead person's hair in Polyjuice Potion? How do I reset my manor so I can rebuild it with different options? How do I find out which dlls an executable will load? If I have a Windows executable, how can I find out which dlls it will load?  I'm just talking about which ones that will be loaded statically, not ones it might load dynamically with something like LoadLibrary. I'm trying to understand the derivation of the equations for the logistic regression. I'm following the cs229 notes:    At some point in the derivation, in the notes we need to maximize the log-likelihood of the parameters:  ℓ(θ) = log L(θ) = y log h(x) + (1 - y) log(1-h(x))  In the notes, after the derivation we get:  $\frac{\partial}{\partial \theta_j}$ = (y $\frac 1 {h(x)}$ - (1 - y) $\frac 1 {1 - h(x)}$) $\frac{\partial}{\partial \theta_j}$h(x)  Shouldn't this be:  $\frac{\partial}{\partial \theta_j}$ = (y $\frac 1 {h(x)}$ + (1 - y) $\frac 1 {1 - h(x)}$) $\frac{\partial}{\partial \theta_j}$h(x)  I've been looking at this for a while and I can't see where this minus sign is coming from...  Thanks for your help! Spambots causing server crashes. How do I block them? Linear independence of $\sin(x)$ and $\cos(x)$ In the vector space of $f:\mathbb R \to \mathbb R$, how do I prove that functions $\sin(x)$ and $\cos(x)$ are linearly independent. By def., two elements of a vector space are linearly independent if $0 = a\cos(x) + b\sin(x)$ implies that $a=b=0$, but how can I formalize that? Giving $x$ different values? Thanks in advance. var functionName = function() {} vs function functionName() {} How to echo a bang! I tried to create a script by echo'ing the contents into a file, instead of opening it with a editor  echo -e "#!/bin/bash \n /usr/bin/command args"  &gt; .scripts/command   The output:     bash: !/bin/bash: event not found   I've isolated this strange behavior to the .  $ echo ! !    $ echo "!" bash: !: event not found  $ echo \#! #!  $ echo \#!/bin/bash bash: !/bin/bash: event not found    Why is bang causing this?   What are these "events" that bash refers to?     How do I get past this problem and print "#!/bin/bash" to the screen or my file? Unable to install "quickly" on Ubuntu GNOME 15.10 When running:  sudo apt-get install quickly   To install quickly I get this output:  Reading package lists... Done Building dependency tree        Reading state information... Done E: Unable to locate package quickly   Why the error and how can I install it then? How do you write a case insensitive query for both MySQL and Postgres? I'm running a MySQL database locally for development, but deploying to Heroku which uses Postgres. Heroku handles almost everything, but my case-insensitive Like statements become case sensitive. I could use iLike statements, but my local MySQL database can't handle that.  What is the best way to write a case insensitive query that is compatible with both MySQL and Postgres? Or do I need to write separate Like and iLike statements depending on the DB my app is talking to? According to specification, to create data that will be signed with server private key for CertificateVerify message, we need to concatenate 64 space characters, the string &quot;TLS 1.3, server CertificateVerify&quot;, null and the hash of all handshake messages to this point. During the handshake process, we choosing the cipher suite and signature scheme... So, if the cipher suite is, for example, TLS_AES_256_GCM_SHA384 and the signature scheme is rsa_pss_rsae_sha256 (different hash algorithms), hash of which algorithm should be used as handshake hash in data to sign, SHA-384 or SHA-256? How big an RSA key is considered secure today? I think 1024 bit RSA keys were considered secure ~5 years ago, but I assume that's not true anymore. Can 2048 or 4096 keys still be relied upon, or have we gained too much computing power in the meanwhile?  Edit: Lets assume an appropriate padding strategy. Also, I'm asking both about security of signatures and security of data encryption. How does the radius of a pipe affect the rate of flow of fluid? Poiseuille's law states that the rate of flow of water is proportionate to $r^4$ where $r$ is the radius of the pipe. I don't see why.  Intuitively I would expect rate of flow of fluid to vary with $r^2$ as the volume of a cylinder varies with $r^2$ for a constant length. The volume that flows past a point is equal to rate of flow fluid, and thus it should vary with $r^2$.  I cannot understand the mathematical proof completely. Is there an intuitive explanation for this? Downloading multiple files with youtube-dl I use youtube-dl to download files from YouTube. I have tried downloading multiple files, but it is not working out for me. I have to open multiple terminals and start afresh every time I want to download a video.   Can you help me to download multiple files with a single terminal window by just mentioning all the URLs at once? I use Ubuntu 12.04 64-bit. In several different contexts we invoke the  to justify whatever statistical method we want to adopt (e.g., approximate the binomial distribution by a normal distribution). I understand the technical details as to why the theorem is true but it just now occurred to me that I do not really understand the intuition behind the central limit theorem.  So, what is the intuition behind the central limit theorem?   Layman explanations would be ideal. If some technical detail is needed please assume that I understand the concepts of a pdf, cdf, random variable etc but have no knowledge of convergence concepts, characteristic functions or anything to do with measure theory. After some gorgeous updates I witness this: