I asked a math professor once about infinity and his answer puzzled me. I asked if i had two sets of numbers:   A =  all the whole numbers in infinity B = all the whole and half numbers (1, 1.5, 2, 2.5 etc...)  Is B twice as large as A?  He chuckled and said no, and then explained something about ordinal and cardinal numbers, that left me confused and still not understanding why my logic that it would be twice as large (and not the same size as he argued) did not hold up. Any good way to explain it to a non-mathematician?
A friend of mine was trying to explain to me how all infinities are equal. For example, they were saying that there are the same amount of numbers between $0$–$1$ as there are between $0$–$2$.  The way they explained it, you could prove that there are the same amount of numbers by getting a match to each number.  For example, for any number in the range from $0$–$2$, you can find a matching number in the range $0$–$1$ by dividing the number by 2. In every case, the ending number will end up between $0$–$1$. In addition, if you multiplied any number in the range of $0$–$1$ by 2, you will always end up with a number between $0$–$2$ Therefore, there should be the same infinite amount of numbers between  $0$–$1$ as there are between $0$–$2$.  Is this principle accurate? The problem that I had with it was, if in-fact there are the same amount of numbers between 0 and 1 as there are between 0 and 2, why is 2 greater than 1?
What is the difference between ubuntu-18.04-server and ubuntu-18.04-live-server? On the  there is a download link for ubuntu-18.04-live-server-amd64.iso, and on the  there is a download link for ubuntu-18.04-server-amd64.iso. Before 18.04, there is only a server version. What is the difference between these 2 images? Detailed Bathymetric/Bathymetry contour data (0-200M) Ice skater increase of energy This may be a very basic question but I am not seeing how it works. Consider the standard example of an ice skate rotating about his/her center of mass and pulling in his/her arms. The torque is zero so we have conservation of angular momentum. This implies that $\omega$ increases to keep $I\omega$ constant, but then $K_{rot}=\frac{1}{2}I\omega^2$ doesn't stay constant, it increases. This implies that there is work done, but what force is doing this work? Proof of linear independence of $e^{at}$ Given $\left\{ a_{i}\right\} _{i=0}^{n}\subset\mathbb{R}$ which are distinct, show that $\left\{ e^{a_{i}t}\right\} \subset C^{0}\left(\mathbb{R},\mathbb{R}\right)$,  form a linearly independent set of functions.   Any tips on how to go about this proof. I tried a working from the definition of an exponential and combining sums but that didn't seem to get me anywhere. I saw a tip on the internet that said write it in the form  $\mu_{1}e^{a_{1}t}+\dots+\mu_{n}e^{a_{n}t}=0$ to try to show $\mu_{1}=\dots=\mu_{n}=0$ considering each term of the left hand side must be positive, but I can't get my head around that because while I understand $e^{x}&gt;0\forall x\in\mathbb{R}$ I cannot see why $\mu_{i}$ must be positive in any case. I have thought about differentiating but that doesn't seem to help. The question did originally ask for a "rigourous" proof but I'll take any hints right now and the provided the solution of 'is obvious' is most unhelpful to me.  Any input would be fantastic. Thank you. Ignore .DS_Store and Icon files in a folder with Cocoa NSFIleManager Why use gradient descent for linear regression, when a closed-form math solution is available? I am taking the Machine Learning courses online and learnt about Gradient Descent for calculating the optimal values in the hypothesis.  h(x) = B0 + B1X   why we need to use Gradient Descent if we can easily find the values with the below formula? This looks straight forward and easy too. but GD needs multiple iterations to get the value.  B1 = Correlation * (Std. Dev. of y/ Std. Dev. of x)  B0 = Mean(Y) – B1 * Mean(X)   NOTE: Taken as in   I did checked on the below questions and for me it was not clear to understand.      The above answers compares GD vs. using derivatives. Shadows along edges of mesh in 3D View Basic mathematical induction regarding inequalities These are just the examples from my textbook, but I don't think it did not explain well.  One of the problem was to prove the inequality   $$n&lt;2^n$$  for all integers $n$.  I understand we assume if $P(n)$ is true, $P(k+1)$ is true.  Thus $k+1&lt; 2^{k+1}$  But in the text book, they added one more line that I don't understand:  $$k+1&lt;2^k+1 \le 2^k+2^k=2\cdot2^k=2^{k+1}$$  I understand  $$k+1&lt;2^{k+1}$$  But how do I know $2^k+1$ is larger than $k+1$, and smaller than $2^{k+1}$?  I have one another example just in case,  proving $2^n&lt;n!$ using induction.  $$2^{k+1}=2\cdot 2^k&lt;2\cdot k!&lt;(k+1)k!=(k+1)!$$  I understand we should start with $2^{k+1}&lt;(k+1)!$, but  Where did $2\cdot k!$ come from? Textbook says it came from inductive hypothesis, but I still don't get it.  Please advise. Can we have an option to disable notifications on community wiki posts? Why do I keep getting told I posted less than 15 characters when I posted more than 15? I tried to post the following comment and it claims it was less than 15 when it was more.  Based on Tim's comment and testing I believe it would not let me post because with @Shadow removed it is less than 15. However, the characters left message indicates I have passed the 15 character minimum thus it is wrong.  This is the bug I'm reporting:  The characters remaining message and the characters left message do not take into account striping out @name prefixes. Biblatex citation order BASH script to set environment variables not working I've written the following script to set some environment variables when needed.  #!/bin/sh export BASE=/home/develop/trees echo $BASE export PATH=$PATH:$BASE echo $PATH   Below the command and the results I can see on my terminal: the script runs, but the variables are not set at the end.  ~$: ./script.sh /home/develop/trees /bin:......:/home/develop/trees ~$: echo $BASE  ~$:    What's wrong? Thanks in advance. Mirko Is it possible to configure Windows 7 to use YYYY-MM-DD instead of DD/MM/YYYY as the date format? How? Where? Reprojecting LiDAR data with libLAS, error: "latitude or longitude exceeded limits"? A thin-walled container of mass $m$ floats vertically at the separation surface of the two liquids of density $ρ_1$ and $ρ_2$  . The whole mass of the container is concentrated in the part of height $h$.  The question is to determine the immersion depth $h'$of the container in the lower liquid if the bottom of the container has a thickness $h$ and an area $S$ and if the container itself is filled with the liquid of density $ρ_1$.    I applied the fundamental principle of the statics on the container  $Balance$ $Sheet$:  The weight of the container ($-mg$)  The buoyancy force applied by the fluid with density $ρ_2$ ($ρ_2gSh$)  The weight of the fluid with density  $ρ_1$  (-$ρ_2gSh'$)  I summed up the forces and set them to zero and find my $h'$  but then  realised that the fluid with density  $ρ_1$  applied also a buoyoncy force on the top of the container , and the fluid with density  $ρ_2$  applied a buoyoncy force on the other fluid.  Where's the problem ? How to find the smallest $n$ such that $n^a\equiv 1 \pmod p$ How can I prove this sequence is decreasing? My problem is how to prove  $$\frac{1+2}{1}  &gt; \frac{1+3+\frac{3^2}{2!}}{1+2} &gt; \frac{1+4+\frac{4^2}{2!}+\frac{4^3}{3!}}{1+3+\frac{3^2}{2!}}&gt;\cdots $$  i am sure that the sequence is decreasing as limit exp(1).  However, the reason why the sequence is decreasing is not evident.  Thanks if you consider this issue and make me relief. I've seen a lot of times the pronoun them used like an article. For example, in the title of the Delta Rhythm Boys Them bones, or in the first sentence of "Money for nothing":     Now look at them yo-yo's, that's the way you do it.   I know that it's not "proper English" (i.e., not something you'll use in a serious writing), but I'd like to know when it's commonly used, and why. Is there any difference between using it and using the? I'm code reviewing a change one of my co-workers just did, and he added a bunch of calls to Date.toMonth(), Date.toYear() and other deprecated Date methods.  All these methods were deprecated in JDK 1.1, but he insists that it's ok to use them because they haven't gone away yet (we're using JDK 1.5) and I'm saying they might go away any day now and he should use Calendar methods.  Has Sun/Oracle actually said when these things are going away, or does @deprecated just mean you lose style points? Let $A, B \in \mathcal M_n(\mathbb C)$, we want to know if there's any relation between their spectral radius and the spectral radius of $A + B$.  The first part of this exercise is giving examples of the three posibilities: $\rho(A) + \rho(B) &lt; \rho(A+B)$ ; $\rho(A) + \rho(B) &gt; \rho(A+B)$ and $\rho(A)+ \rho(B) =\rho(A+B)$. This part was easy because there are no conditions about A or B.  The second part is proving that if A and B are normal matrixes, (i.e. $ AA^* =A^*A$, where $A^*$ is its conjugate transpose), then $ \rho(A+B) \le \rho(A) + \rho(B) $.   So my first attempt was using the Schur's decomposition, so $A=U^{-1}D_AU$ where $D_A$ is a diagonal matrix with the eigenvalues of $A$ in its diagonal and U is an unitary matrix. Similarly, B is similar to a diagonal matrix $D_B$ with the eigenvalues of B in its diagonal. That way I have $\rho(A) = \rho(D_A)$ and $\rho(B) = \rho(D_B)$ so now since $D_A + D_B$ is a diagonal matrix with $\lambda_i + \mu_i$ for $ 1\le i \le n $, where $\lambda_i$ and $\mu_i$ are the eigenvalues of $A$ and $B$ respectively. From here it's easy to see that $ \rho(D_A+D_B) \le \rho(D_A) + \rho(D_B) $, since $ |\lambda_i+\mu_i|\le|\lambda_i|+|\mu_i| \ \forall i$. But I don't see how to include $\rho(A+B)$ in this.  And the last part of the exercise is proving that if $ \{X_i\}$ is a basis of eigenvectors, which is common to A and B, then $ \rho(A+B) \le \rho(A) + \rho(B) $.  Here's my attempt, but I would like to know if it's correct:  Let $\lambda_i$ and $\mu_i$ be eigenvalues of A and B associated to the vector $X_i$, then $AX_i=\lambda_i X_i$ and $BX_i = \mu_iX_i$, and if we sum both equalities, we have $AX_i + BX_i = \lambda_iX_i + \mu_iXi$ so $(A+B)X_i=(\lambda_i+\mu_i)X_i$ so $(\lambda_i+\mu_i)$ is an eigenvalor of $(A+B)$ associated to the eigenvector $X_i$. Then is again clear that $|\lambda_i+\mu_i|\le|\lambda_i|+|\mu_i| \le \rho(A) + \rho(B)\ \forall i$, and particularly, for $\max_i |\lambda_i+\mu_i| = \rho(A+B)$ so $ \rho(A+B)\le \rho(A) + \rho(B)$  Any kind of help is appreciated and thank you very much in advance!