Proof that the formula $((p\to q)\land (q\to r)\land p)\to r$ is a tautology
Can anyone help me with a solution?
.NET DateTime.Now returns incorrect time when time zone is changed "Mixed declaration and code" warning, is it worth addressing? $[S]$ means linear span of S. Then $[S] = [[S]]$ My definition of span:   Suppose a vector space $(V,+,\cdot)$, and       $$S = \{u_1,\cdots,u_n\}$$      (and $S$ is a subset of $V$, not a subspace)      $$[S]=:\bigcap_{W\subset V, W\supseteq S} W$$      In other words, $[S]$ is, by definition, the intersection of all $W$,   such that $W$ is a subspace of $V$ and $W$ contains $S$.   Well, then I need to prove that   $$[S] = [[S]]$$  My strategy:  $[[S]]$ means the intersection of all subspaces that contain $[S]$. If all the subspaces contains $[S]$, so does their intersection. Therefore, $[S]\subset[[S]]$  However, I'm having trouble proving the converse:  $$[[S]]\subset[S]$$  Any hints? How can I remove duplicates in my .bash_history, preserving order? I really enjoying using control+r to recursively search my command history. I've found a few good options I like to use with it: # ignore duplicate commands, ignore commands starting with a space export HISTCONTROL=erasedups:ignorespace  # keep the last 5000 entries export HISTSIZE=5000  # append to the history instead of overwriting (good for multiple connections) shopt -s histappend  The only problem for me is that erasedups only erases sequential duplicates - so that with this string of commands: ls cd ~ ls  The ls command will actually be recorded twice. I've thought about periodically running w/ cron: cat .bash_history | sort | uniq &gt; temp.txt mv temp.txt .bash_history  This would achieve removing the duplicates, but unfortunately the order would not be preserved. If I don't sort the file first I don't believe uniq can work properly. How can I remove duplicates in my .bash_history, preserving order? Extra Credit: Are there any problems with overwriting the .bash_history file via a script? For example, if you remove an apache log file I think you need to send a nohup / reset signal with kill to have it flush it's connection to the file. If that is the case with the .bash_history file, perhaps I could somehow use ps to check and make sure there are no connected sessions before the filtering script is run? How do I uninstall Ubuntu Wubi? What are the benefits of modeling in one piece, or as separate parts? Is there any benefit of creating model in one piece or I can create separate pieces and just overlap them to create a visually joint part.  For example, the below image were made from two parts 1. Blade 2. Spinner  Is there any benefit to model them (e.g. extrude from the spinner) or it is actually ok to have them separated since visually I can achieve the same effect (See picture 2) $f$ differentiable except in $1$ point If a set is linearly independent, a finite subset of that set is also linearly independent Is the above statement true? If I have already proven that the set is linearly independent, is it automatically given that the subset is also linearly independent? If so, how would I go about showing that as a proof. What is an intuitive explanation for how PCA turns from a geometric problem (with distances) to a linear algebra problem (with eigenvectors)? I've read a lot about PCA, including various tutorials and questions (such as , , , and ).   The geometric problem that PCA is trying to optimize is clear to me: PCA tries to find the first principal component by minimizing the reconstruction (projection) error, which simultaneously maximizes the variance of the projected data.    When I first read that, I immediately thought of something like linear regression; maybe you can solve it using gradient descent if needed.  However, then my mind was blown when I read that the optimization problem is solved by using linear algebra and finding eigenvectors and eigenvalues. I simply do not understand how this use of linear algebra comes into play.  So my question is: How can PCA turn from a geometric optimization problem to a linear algebra problem? Can someone provide an intuitive explanation?  I am not looking for an answer like  that says "When you solve the mathematical problem of PCA, it ends up being equivalent to finding the eigenvalues and eigenvectors of the covariance matrix." Please explain why eigenvectors come out to be the principal components and why the eigenvalues come out to be variance of the data projected onto them  I am a software engineer and not a mathematician, by the way.  Note: the figure above was taken and modified from . I've setup a HTTPS and FTP site in IIS but how can I add users to use the FTP site?  Does they necessarily need an account with password in the computer to be able to use it? Reading an Excel file in PHP I'm trying to read an Excel file (Office 2003). There is an Excel file that needs to be uploaded and its contents parsed.  Via Google, I can only find answers to these related (and insufficient topics): generating Excel files, reading Excel XML files, reading Excel CSV files, or incomplete abandoned projects. I own Office 2003 so if I need any files from there, they are available. It's installed on my box but isn't and can't be installed on my shared host.  Edit: so far all answers point to  and/or  about how to use it. Difference between (in / for / ∅ / during / over) the (past / last) three years Microsoft has steadily improved its operating cash flows in the past three years. Microsoft has steadily improved its operating cash flows in the last three years. Microsoft has steadily improved its operating cash flows for the past three years. Microsoft has steadily improved its operating cash flows for the last three years. Microsoft has steadily improved its operating cash flows for the past three years. (&quot;for&quot; omitted) Microsoft has steadily improved its operating cash flows for the last three years. (&quot;for&quot; omitted) Microsoft has steadily improved its operating cash flows during the past three years. Microsoft has steadily improved its operating cash flows during the last three years. Microsoft has steadily improved its operating cash flows over the past three years. Microsoft has steadily improved its operating cash flows over the last three years.  Which of them are idiomatic and preferred? Are they interchangeable? In my recent study of Artinian modules I have been looking at Prüfer $p$-groups (which I will denote by $\mathbb Z_{p^\infty}$ from now on).   I am attempting to prove that $\mathbb Z_{p^\infty}$ is an Artinian $\mathbb{Z}$-module which is not finitely generated.      To do this I am trying to show that every proper non-trivial submodule of $\mathbb Z_{p^\infty}$ is equal to $\left(\frac{1}{p^n}\mathbb{Z_{}}\right)/\mathbb{Z}$ for some $n$.   I am using the following definition of $\mathbb Z_{p^\infty}$: $$\mathbb Z_{p^\infty}=\left\{x\in \mathbb{Q}/\mathbb{Z}\mid \exists n:p^nx=0 \right\}$$ I have read in several places, Atiyah &amp; MacDonald Example 6.3 and Wikipedia to name a few, that the subgroups of $\mathbb Z_{p^\infty}$ are of this form. This would demonstrate that $\mathbb Z_{p^\infty}$ is Artinian. However, I can't seem to find anywhere a proof or example showing why all the subgroups are of this form. I would like to understand this further, as it is not immediate to me that this would be the case, but my attempts to figure out why have so far been futile. I may be thinking too much and missing a simple explanation. I have looked at starting with an arbitrary subgroup $H$ such that $H\neq 0$ and $H\neq\mathbb Z_{p^\infty}$ and seeing if I can deduce anything from that. I got this far, but it doesn't seem to be going anywhere. I have also looked at Sylow's theorems, but I'm not sure if they will be any use to me as $|\mathbb Z_{p^\infty}|=\infty$ (unless I'm wrong about this!).  I would be extremely grateful if anybody could offer a few hints here as I am slowly losing my sanity trying to figure it out!  Thanks,  Andy. Can't log in using Firefox in Stack Overflow On determinants and common divisors Can an Indian travel to India with 50 days before passport expires? Counting integer partitions of n into exactly k distinct parts size at most M How can I find the number of partitions of $n$ into exactly $k$ distinct parts, where each part is at most $M$?  The number of partitions $p_k(\leq M,n)$ of $n$ into at most $k$ parts, each of size at most $M$, is given by the generating function: $$  \binom{M+k}{k}_{x} = \prod_{j=1}^{k}\frac{1-x^{M+k-j+1}}{1-x^j}= \sum_{n=0}^{kM} p_{k}(\leq M,n) x^n $$  For the number of the partitions $p_k(\mathcal{D},n)$ of $n$ into at most $k$ parts there is the recurrence relationship: $$   p_{k}(\mathcal{D},n) = p_{k}(\mathcal{D},n-k) + p_{k-1}(\mathcal{D},n) $$  But what, if I want to count only the partitions with distinct parts and restricted number of parts and restricted part size?  Update: Now I know the generating function for the number of distinct restricted partitions $p_k(\leq M, \mathcal{D},n)$ of $n$ into exactly $k$ distinct parts, all at most $M$ is $$    \prod_{j=1}^{M} (1+xq^{j}) = \sum_{k,n=0}^{\infty}p_k(\leq M, \mathcal{D},n)x^{k}q^{n} $$ and there is also a recurrence relation $$    p_k(\leq M, \mathcal{D},n) =         p_{k-1}(\leq M-1, \mathcal{D},n-k) +         p_k(\leq M-1, \mathcal{D},n-k) $$ How can I prove this? Could you recommend a book, where I could read about this? I would like to find the correlation between a continuous (dependent variable) and a categorical (nominal: gender, independent variable) variable. Continuous data is not normally distributed. Before, I had computed it using the Spearman's $\rho$. However, I have been told that it is not right.  While searching on the internet, I found that the boxplot can provide an idea about how much they are associated; however, I was looking for a quantified value such as Pearson's product moment coefficient or Spearman's $\rho$. Can you please help me on how to do this? Or, inform on which method would be appropriate?  Would Point Biserial Coefficient be the right option? What is the right way to consistently mangle bash terminal? I'm on OSX 10.11.1 and occasionally my bash terminal gets mangled. It often happens when I accidentally cat a binary file. The result can be seen on the image below. The output becomes weird, and I can't type ascii characters anymore.    Even though this happens occasionally, I couldn't find a way to consistently reproduce the issue. Online search recommends doing cat /bin/*, but that works sporadically, only after a couple dozen tries.  I want to do this so I can find an easy solution how to handle this in tmux.  How do I consistently get bash to a "mangled" state? Is there maybe a magic unicode character that can do this? Stack Exchange doesn't allow me to answer protected questions because I gave away earned rep in bounties