I assume implication is the operator that is most of the times used in mathematics, am I right? (i.e. whenever if/then construct is used)  If yes, then I think it is known that it has some   Merely that implication is true if premise is false. Which doesn't make sense in real world because I could say: if pigs can fly, then we can cure any illness. But this is not true in the way we understand if/then in real life, isn't it? Because we can't cure any illness so far.  So my question is more philosophical in nature, if we agree that mathematics is using an operator implication which is problematic from philosophical point of view and doesn't very well align with the real world meaning of if/then, then doesn't this cause problems for mathematics when we try to apply some results of mathematics to real world? More particularly because the operator (implication) that was used in math may not be the one that reflects the one in real life? This means we can't use some results from math which relied on implication, in real life?
I studied propositional logic, and everyday I see applications of what I learned on the internet, in mathematical books and miscelaneous resources.        One particular case is sentences in the form $a \rightarrow b$ , that is, material implications. While I fully understand the mathematical logic approach to the study of those kinds of formulas, sometimes I get confused about what someone ( using implication sentences in the ordinary language ) is trying to say.        In ordinary language, I often see affirmations in the form of an implication.  One example would be someone affirming that "the moon is made of cheese" $\rightarrow 2+2=4$. [ Kleene, Mathematical Logic ].      I perfectly understand that if we choose the interpretation that gives the antecedent as false ( and the consequent as true ) we get that the whole implication is also true, but I would like to understand EXACTLY what he is trying to say when he AFFIRMS a sentence in the form of an implication. Is he saying that under the commonly agreed interpretation of that sentence ( antecedent = false, consequent = true ), that implication becomes true?      Another example, totally different, would be someone affirming that $x \in \mathbb{N} \rightarrow x \in \mathbb{N}$ or $x \in \mathbb{Z}_{&lt;0}$. Is he now affirming the same kind of sentence or saying something totally different?      When he affirms that sentence, is he saying that he finds the sentence to be a tautology?   Why didn't he say that?          Then, with these two examples we have an affirmation of the same kind of sentence (implication) meaning completely different things, in one case he affirms to say that under the commonly agreed interpretation of the sentence, it is true. In another case, he affirms to say that there's no interpretation that makes the sentence false.       Am I missing something? There is really that ambiguity in ordinary language, when people are affirming sentences?  Can't I just take some general instance of the affirmation of a sentence in the form $a \rightarrow b$ in ordinary language and know exactly what he's trying to say, regardless of the propositions he chooses the substitute $a$ and $b$ for?       Thanks a bunch.
What is projection of WGS84 in GIS software? WGS84 is a reference ellipsoid that determines coordinates of every point on Earth using latitude, longitude and height about the its surface.  This ellipsoid is often used as a base to make maps. Though, to make a map on a flat paper or a flat digital screen you need a projection too. An example of a popular projection would be UTM.   Mainstream GIS software give plethora of projection alternatives to display your data layers. Among them, they also let you display your data using WGS84 only.  I think they are using a certain projection but what projection is that? Why do wet objects become darker? What is the ideal data type to use when storing latitude / longitude in a MySQL database? Bearing in mind that I'll be performing calculations on lat / long pairs, what datatype is best suited for use with a MySQL database? Getting error message when trying to start a virtual machine I have been using VMWare on Windows for a long time. But after a long wait, I moved to VirtualBox on Ubuntu 11.10. I installed Ubuntu, 32 Bit, installed all available updates and installed Virtual Box. When I try to create a new Windows installation inside VirtualBox, I got the following error messages.  VirtualBox - Error  Failed to open a session for the virtual machine Windows XP. The virtual machine '**Windows XP**' has terminated unexpectedly during startup with exit code 1. Details  Result Code: NS_ERROR_FAILURE (0x80004005) Component: Machine Interface: IMachine {5eaa9319-62fc-4b0a-843c-0cb1940f8a91}   2nd error dialogue  Virtualbox - Error in suplibOsinit Kernal driver not installed (rc--1908) Please install the virtualbox-dkmbs package and execute 'modprobe vboxdrv' as root.    I have already tried reinstalling VirtualBox. Google result seem to indicate the the problem happens due to Kernel updates.   Is there anyway I can get this working? I need this for malware analysis and if VirtualBox is going to crash on me all the time, then I won't be able to use Ubuntu for work.  Output to dpkg -l | grep virtual   server rc  virtualbox                             4.1.2-dfsg-1ubuntu1                     x86 virtualization solution - base binaries rc  virtualbox-qt                          4.1.2-dfsg-1ubuntu1                     x86 virtualization solution - Qt based user interface  cute 'modprobe vboxdrv' as root.&lt;p&gt; Why won't Blender run on Windows XP? I downloaded Blender 64 bit and the OS indicated that the program was not compatible with my computer chip. I then downloaded the 32 bit version which seemed to allow me to install. However the blender Icon in windows when pressed informs me that the program is not a window application. I have an Intel(R) Core (TM)2 Duo CPU E7300 @2.66Ghz if that is important and I am running XP. I am fairly sure I am doing something wrong here- can anyone help please? Data   I want operator users on this machine to mount their own cifs shares The sudoers file already contains the /bin/mount -t cifs //*/* /media/* -o username=* command for all operators I want the users to mount a cifs share through a script typing the password only once, not twice. The sudo password and the cifs password are identical.   What I already have  This script works:  #!/bin/bash sudo 'mount -t cifs //192.168.1.1/home /media/$USER/home -o username=$USER'   ...but it requires the users to type the same password twice!     Once for sudo Once for the mount itself   This would also work:  #!/bin/bash echo -n Password:  read -s szPassword echo $szPassword | sudo -S sh -c 'echo $szPassword | mount -t cifs //192.168.1.1/home /media/$USER/home -o username=$USER'   ...but this would require me to allow all operator users to be able to sudo sh (major security problem)  Question  How to mount a cifs share in bash¹ without putting sh in the sudoers file nor creating a permanent/temporary file???  Note 1: no python, perl, C, Go, ... please? Note 2: I know I can just remove the password through the sudoers file, but I'm trying to tighten security, not loosen it, without giving up convenience... Simple example of non-arithmetic ring (non-distributive ideal lattice) Can anyone provide a simple concrete example of a non-arithmetic commutative and unitary ring (i.e., a commutative and unitary ring in which the lattice of ideals is non-distributive)? Let $x_1,...,x_n$ be distinct integers. Prove that  $$\prod_{i&lt;j} \frac{x_i-x_j}{i-j}\in \mathbb Z$$   I know there is a solution using determinant of a matrix, but I can't remember it now. Any help will be appreciated. Transpose Ranges repeated by row I would like to convert several range of columns into rows, and keeping the association to the original col/row source  Sample source sheet  --------------------------------------------------- | Data 1 | 1/1/2018 | 1/2/2018 | 1/3/2018 | etc...365 days | --------------------------------------------------- | name 1  |    25    |   30     |    5     | etc...|  --------------------------------------------------- | name 2  |    2     |          |    4     | etc...| --------------------------------------------------- | etc ... 100 rows   Expected Results  --------------------------------------------------- | Data 1 |   Date      |  Value   |  --------------------------------------------------- | name 1  |  1/1/2018   |   25     |    --------------------------------------------------- | name 1  |  1/2/2018   |   30     |    --------------------------------------------------- | name 1  |  1/3/2018   |   5      |    --------------------------------------------------- | name 1  |  etc.. 365 days --------------------------------------------------- | name 2  |  1/1/2018   |   2      |    --------------------------------------------------- | name 2  |  1/2/2018   |          |    --------------------------------------------------- | name 2  |  1/3/2018   |   4      |    --------------------------------------------------- | name 2  |  etc.. 365 days --------------------------------------------------- | etc ... for each change of name pull all the col dates/values   Current results  --------------------------------------------------- | Data 1 |   Date      |  Value   |  --------------------------------------------------- | name 1  |  1/1/2018   |   25     |    --------------------------------------------------- |         |  1/2/2018   |   30     |    --------------------------------------------------- |         |  1/3/2018   |   5      |    --------------------------------------------------- |         |  etc.. 365 days                      I have to copy down to bring in "name 1" for each date  --------------------------------------------------- | name 2  |  1/1/2018   |   2      |             now update the formula to bring in the second item for the sequence of dates --------------------------------------------------- |         |  1/2/2018   |          |    --------------------------------------------------- |         |  1/3/2018   |   4      |    --------------------------------------------------- |         |  etc.. 365 days --------------------------------------------------- | etc ...     =Sheet1!A:A I can bring in the name from the source list, but I have to copy it down for the number of dates.  How can I have it match the number of dates?  and be dynamic to update as dates/values are added Edit: trying to use split and rept to count the number of times to repeat, then transpose.. =transpose(rept(split(arrayformula(concatenate(Sheet1!A:A&amp;"1")),"1",True, True),COUNTA(Sheet1!I1:ZZ1)))  =TRANSPOSE(Sheet1!$C$1:$Z$1) I can use transpose to get the dates/values, but it's fixed range.  How can I make it dynamic to import dates as they are added? My machine has a SSD (fairly small) and a Hard disk (large). I want to have windows and a couple of programs on the SSD but I want to have the documents folders (such as my documents and my downloads) on the HD. I also want the default install directory pointing at the larger HD. Can anyone advise or point me to a relevent guide? Thanks. How can aperture be f/11 on a lens with an aperture range designation of 3.5-5.6? I have a dataset in csv file format that I'm loading into pgfplotstable. I can't change the format of the original csv, and being financial data it has a column of item names and two or three explanatory data points at the top of the table. e.g.  \documentclass{article} \usepackage{array,datatool,pgfplotstable}   \begin{filecontents}{sample.csv}     item,2008,2009,2010,2011      ,Actual,Actual,Forecast,Forecast      ,GBP,GBP,GBP,GBP      ,('000),('000),('000),('000) rounded     Area 1 OP, 1000000,1500000,1750000,250000     Area 2 OP, 400000,500000, 450000,-50000     Area 51 OP, 300000,375000,390000,15000     No P, 1250000, 1000000, 950000, 50000     Residuals, 800000, 80000, 90000, 10000   \end{filecontents}  \pgfplotstableread[col sep=comma]{sample.csv}\mystuff   I'm stumped at the next stage which is to typeset \mystuff so I have comma separators and can add a row that sums the 'Area' columns. At the moment to display the table I have to do something like  \begin{document}  \pgfplotstableread[col sep=comma,header=false]{sample.csv}\mystuff  \pgfplotstabletypeset[   columns/0/.style={string type},   columns/1/.style={string type},   columns/2/.style={string type},   columns/3/.style={string type},   columns/4/.style={string type}]\mystuff  \end{document}   How do I tag the top three rows as string and the rest of the last four columns as integer type so I can format and transform them? Can the default font be changed? I have been playing with the settings for the text object recently, and every time I add a new text object, I need to change the font to my favorite one. (Eurostile LT ExtendedTwo) Is there any way to change the default font so I do not need to change it every time I add a new text object?   Example: Possible to do a MySQL foreign key to one of two possible tables? Well here's my problem I have three tables; regions, countries, states.  Countries can be inside of regions, states can be inside of regions.  Regions are the top of the food chain.  Now I'm adding a popular_areas table with two columns; region_id and popular_place_id.  Is it possible to make popular_place_id be a foreign key to either countries OR states.  I'm probably going to have to add a popular_place_type column to determine whether the id is describing a country or state either way. How to make IntelliJ IDEA recognise code created by macros? My linear algebra professor proved that      Diagonalizable matrices with complex values are dense in set of $n \times n$ complex matrices.   He defined a metric (I believe) that was somehow related to the usual metric on $\mathbb{R}^{n^2}$.  Then somehow proved that diagonalizable matrices were dense because for any matrix $A$ if $\det(A - \lambda I) = 0$ on an open subset, then $\det(A - \lambda I)$ was the zero polynomial.  I Googled around a bit and found some stuff talking about the Zariski topology, and I am not sure this is what I want. Most of what I have found on this topology is much more general than what he was doing.  Does anyone know any resources that would explain how to go about proving this statement?  Edit: I found out how to prove this the way my professor did.  We define the norm of a matrix by $$ |A| = \max\{|Ax| : |x| = 1 \}.$$ Now we have a distance $d(A, B) &lt; \epsilon$. You can prove that if $(A - B)_{ij} &lt; \epsilon/n$, then $|A - B| &lt; \epsilon$.  Let $p(x)$ be the characteristic polynomial of $A$, an $n \times n$ matrix.  $$p(x) = \prod_1^n (x - x_i) = x^n + (-1)\sigma_1 x^{n - 1} + \cdots + (-1)^n\sigma_n$$ where $\sigma_1, \ldots, \sigma_n$ are the elementary symmetric polynomials of $x_1, \ldots, x_n$ which are the eigenvalues.  The discriminant of the characteristic polynomial is a symmetric polynomial, therefore it can be written in terms of the elementary symmetric polynomials, which in turn can be written in terms of the entries of the matrix.  But since the discriminant is a polynomial, it only has finitely many roots. Therefore for $\epsilon &gt; 0$, by changing the entries of the matrix less than $\epsilon / n$ we can find a new matrix $B$ such that $|B - A| &lt; \epsilon$ and the discriminant is not zero.   The discriminant being not zero means $B$ has distinct eigenvalues, thus has a basis of eigenvectors. Therefore, it is diagonalizable.  Thus the set of diagonalizable matrices is dense in the set of matrices with respect to that metric. Proving the Lorentz invariance of the Lorentz invariant phase space element I have been looking around for a satisfactory answer to prove that  $$\frac{d^3\vec{p}}{2E_{\vec{p}}}$$  where $E_{\vec{p}}=+\sqrt{(|\vec{p}|c)^2+(mc^2)^2}$, is Lorentz invariant. The standard answer seems to be that the above measure is equal to  $$d^4p \, \delta (p^2-m^2)\theta(p_0)$$  where $p^2=p_0^2-|\vec{p}|^2$, and $\theta(x)$ is the step function. I understand these two are equivalent, but I don't understand why the second has to be Lorentz invariant, in particular why the Dirac delta has to be Lorentz invariant. I have found a  (section 2.1) that proves that $\delta^{(4)}(p-p')$ is Lorentz invariant, but I can't find a way to extend their method successfully here. In fact, all I can seem to get tells me the above isn't Lorentz invariant, and that in fact it should transform to   $$\frac{d^3\vec{p}'}{2\gamma E_{\vec{p}'}}$$  which makes sense from $d^4p$ being Lorentz invaraint and $dp_0$ transforming proportional to $\gamma$, but it isn't what everyone else says. What's the problem here?    An alternative way to "derive" the factor of $1/\gamma$:  $$\delta (p^2-m^2)\theta(p_0)=\frac{1}{E_{\vec p}}\delta(p_0-E_{\vec{p}})$$  Now, as $dp_0$ transforms to $\gamma dp_0$, the $\delta(p_0-E_{\vec{p}})$ should transform to $\gamma^{-1}\delta(p_0-E_{\vec{p}})$, as can be shown with an analogous argument to the one shown in the . The $E_{\vec{p}}$ also transforms to $\gamma(E_{\vec{p}}-v p_x)$ for an $x$-boost, but this doesn't seem to solve it. Why WordPress choose data serialization over json_encode? In my little age with WordPress, I've seen WordPress itself and its friendly plugins are using PHP serialize() in storing data into db in many cases. But in a recent search I found a serious community support for the json_encode() over the serialize().     And I personally tested an associative array with both of 'em, that shows:  serialize() stores 342 chars json_encode() stores 285 chars  Why I'm asking this? I'm on a project while I'm going to store repeating meta fields to a post. Where:  Data would be basically in English, but sometimes can be Bengali Data would be associative array, 3 level deep (I hope I understood levels correctly):   array(     1 =&gt; array(         'key'=&gt;'value',         'key2'=&gt;'value'     ),     2 =&gt; array(         'key'=&gt;'value',         'key2'=&gt;'value'     ) )  I've checked the postmeta table's meta_value field it's a longtext,  a length of 4,294,967,295 chars (4GB). So I need a robust solution into storing things. The set of homotopy classes Appendix in LaTeX