Badminton in google transalation speak like ba~d~mon but I asked philippines friends they always speak like ba~men~ton which one the US use in normal life?
Does That's very sweet of you  in the US sound like That's very sweef you? the of is only speak f and link with swee
Is $7$ the only prime followed by a cube? How to make `sudo` preserve $PATH? I'm struggling a bit with functions(limits, squeeze theorem, etc).  I have done some research and found a list of books on calculus but I'm not sure which one would be better suited for me, so I would appreciate if some experienced people here could advise me which books would be better to master the basics so I can move on with derivatives, etc. Limits is my main problem, so I need a book which explain this in depth so I can master it. Then I'll keep going on with the same book for other chapters.  Below is the list which I found :   Apostol T.M. Calculus Vol. 1 Burn R.P. Numbers and Functions: Steps into Analysis  Courant R. and John F. Introduction to Calculus and Analysis I - One of the better calculus text in print.  Crowell B. Calculus  Dawkins, P Calculus I and Calculus II  Garrett P. Calculus  Ghorpade S.R. and Limaye B.V. A Course in Calculus and Real Analysis  Gill G.S. Calculus Bible  Guichard D. Whitman Calculu.  Hardy G.H. A Course of Pure Mathematics  Hwang A.D. Calculus for Mathematicians, Computer Scientists, and Physicists: An Introduction to Abstract Mathematics  Santos D. Differential Calculus  Shapiro B.E. Calculus and Analysis  Spivak M. Calculus  Strang G. Calculus  Thomas G.B. and Finney R.L. Calculus and Analytic Geometry  Tranter C.J. Advanced Level Pure Mathematics  Tranter C.J. Techniques of Mathematical Analysis  Veeh J.A. Lectures Notes on Calculus  So far I narrowed my list down to Hardy, Spivak and Apostol based on further research, but I haven't got my hands on these books yet to compare them and I'm not sure if these books will be good for me.  Thanks. How to prepare data for ListVectorPlot[]? How many edges does an undirected tree with $n$ nodes have? Difference between spin and polarization of a photon I understand how one associates the spin of a quantum particle, e.g. of a photon, with intrinsic angular momentum. And in electromagnetism I have always understood the polarization of an EM wave as the oscillations of the E and M field, not necessarily being  aligned with the direction of propagation of the wave.     Questions:   But when one talks about the polarization of a photon in Quantum Mechanics, how does it really differ from its spin?  Are they somehow related and what's the physical idea behind photon polarization in contrast to photon-spin? Feel free to use mathematical reasonings as well if you see it fit! $\DeclareMathOperator{\tr}{tr}$Let's consider a $n \times n$ matrix and the sequence of traces $\tr(A)=\tr(A^{2})= \ldots = \tr(A^{n})=0$. How to prove that $A$ is a nilpotent matrix (a matrix so that $A^{k} \times u = 0$ for all $u \in V$ and for some $k$)?  Maybe it would be reasonable to consider $A$'s Jordan form (we assume that the action occurs over an algebraically closed field, such as $\mathbb{C}$). For example, suppose that we have found it and $A'$ consists of $m$ blocks of sizes $r_{1}, \ldots r_{m}$ ($r_{1}+r_{2}+\ldots+r_{m}=m, r_{i} \in \mathbb{Z}_{+}$) with $\alpha_{1}, \ldots \alpha_{m}$ as corresponding eigenvalues to each of the blocks. Then, according to the problem will get the system  $$\begin{align} r_{1}\alpha_{1}+&amp;\ldots+r_{m}\alpha_{m}=0 \\ r_{1}{\alpha_{1}}^{2}+&amp;\ldots+r_{m} {\alpha_{m}} ^{2}=0\\ &amp;\ ~\,\vdots \\ r_{1}{\alpha_{1}}^{n}+&amp;\ldots +r_{m}{\alpha_{m}}^{n}=0  \end{align}$$  But the system's analysis over $\mathbb{C}$ seems to be too much complicated.  Actually, the problem must have more or less simple solution.  Any piece of advice would be much appreciated. By viewing the polynomials as a difference of two squares, factorise the following polynomial: $$x^4+x^2+1.$$ I searched but couldn't find a way to solve this  Edit: By using Hans Lundmark hint, I get:  $$(x^2+1)^2-x^2$$   Is it fully factorized? How to really clear the terminal? I can issue the clear command or press Ctrl + L to clear the current Ubuntu terminal, but this just shifts previous output upwards and if you use mouse scroll or PgUP and PgDown keys it's hard to distinguish where the output of previous command ends and output of current command begins.  Is there a way to really clear the terminal so I won't see previous command results? Is there a way to do proportional editing only on specific vertices? I am getting the following error in the top menu bar:     An error occurred.Please run the Package Manger form the right-click menu or apt-get in a terminal to see what is wrong.The error message was:'Error:BrokenCount>0'.This usually means that your installed packages have unmet dependencies. "Have been having classes" or "have had classes for three months"? Which sentence is correct:  "I have been having classes for three months."  or  "I have had classes for three months." Splitting of primes in the compositum of fields If $L_i/K$ are Galois extensions of number fields, $i=1,\ldots,n$, and $L=L_1\cdots L_n$ is the compositum. Then it's true that a prime $\mathfrak{p}$ of $K$ splits in $L$ if and only if it splits in all of $L_i$. Does this also hold if the $L_i$ are not Galois extensions of $K$?  The proof that I know regarding the compositum uses the fact that the $L_i/K$ are Galois, so is this true in the more general setting and how would one prove it? Can we have a way to access Area 51 on the iOS and android app?  More sites would flourish due to more traffic. Is C strongly typed? To quote :     Two commonly used languages that   support many kinds of implicit   conversion are C and C++, and it is   sometimes claimed that these are   weakly typed languages. However,   others argue that these languages   place enough restrictions on how   operands of different types can be   mixed, that the two should be regarded   as strongly typed languages.   Is there a more definitive answer? Flows/Processes, Version Control and CI I've started using Processes and Flows and am really excited about these features -- but am still wrapping my mind about how to manage them in Version Control (which I use) and Continuous Integration (which I don't yet... but am trying to work in that direction). The underlying challenge is that each time a process or flow is edited, the metadata file is renamed to reflect its update version, e.g. MyFlow-1.flow, MyFlow-2.flow, etc. The Challenge for Version Control If each version of the flow has its own file name, this makes it more complicated to track changes to flows.  It seems like there could be two strategies here:  Rename the files to remove the version number.  I'd need a script to run through my src folder pre-commit, and remove the number so that my file name is always MyFlow.flow.  This is an annoying extra step, but makes version control simpler.  On the other hand, it probably makes deployment much harder.  Track each file in version control with its original name.  We'd need to be diligent about deleting MyFlow-1 in the commit where we add MyFlow-2 so that git would recognize this as a rename/move and show helpful diffs.  Otherwise, the whole file appears to change with every change, which makes version control rather less useful.   The Challenge for Deployment/Continuous Integration From my experiments so far, once a flow has been activated in a given environment, the metadata file can't be deployed again (either it is currently active, or has been active in the past and superseded). That means that I need to prevent my previous flow versions from being included in a deployment package (either by not tracking them at all, or by using a script to delete earlier versions).  I also need some way to determine whether the current version (in my repo) should be included in the deployment payload for a given environment or not. I could imagine a script that (1) downloads flows from the target environment, and removes *.flow files from the payload if they already exist in the target environment, but that feels complicated.  And it's even harder if I've removed the version numbers from my flow files. Are you using flows in VS or CI?  How are you managing these issues? Note:  I'm not sure whether VC and CI should be one question or two - depends on how integrated the strategy for dealing with these two challenges.  If you think these should be separate questions, comment and I'll re-post. I need to factory reset my Macbook but cannot recall the password. So I used Cmd+R during startup and go to terminal and typed 'resetpassword' according to this advice    Now 'Reset Password' application appears but there is nothing where I have to select the volume containing the user account. Why does this happen? Direct proof that nilpotent matrix has zero trace Proving that $d(f,g)=\|f-g\| = \sup \limits_{0\leq x \leq 1} |f(x)-g(x)|$ is a metric on $X=C[0,1]$ Let $X=C[0,1]$ be the set of all continuous functions on $[0,1]$. For any two functions $f,g\in X$, set $$d(f,g)=\|f-g\| = \sup \limits_{0\leq x \leq 1} |f(x)-g(x)|.$$  Prove $(X,d)$ is a metric space.    My attempt:  This question comes down to showing that $d$ is a metric, which means that:   $d(f,g)\geq 0$ $d(f,g)=0\iff f=g$ $d(f,g)=d(g,f)$ Triangular sadness.     1) Nonnegativity is guaranteed by taking the supremum of the absolute value of a sum, since the absolute value of a sum is non-negative, the supremum of non-negative numbers is non-negative.  $\square$    2) For the supremum of $|f(x)-g(x)|$ to be equal to $0$, the greatest number for $|f(x)-g(x)|$ from $0\leq x\leq 1$ must be $0$, hence $|f(x)-g(x)|=0$ for all values on this range, and thus $f(x)=g(x)$ is the constant zero map.  $\square$    3) $|f(x)-g(x)|=|g(x)-f(x)|$ implies that $\sup \limits_{0\leq x \leq 1} |f(x)-g(x)|=\sup \limits_{0\leq x \leq 1} |g(x)-f(x)|$ $\square$    4) Triangular sadness:  We need to prove that $d(f,g)\leq d(f,h) + d(g,h)$  I.e.  $\sup \limits_{0\leq x \leq 1} |f(x)-g(x)|\leq \sup \limits_{0\leq x \leq 1} |f(x)-h(x)|+\sup \limits_{0\leq x \leq 1} |g(x)-h(x)|$  Now I am not sure how I can 'behave' with the sup 'function'. Can I do the following?  $$\sup|f(x)-g(x)| = \sup|f(x)-g(x)+h(x)-h(x)|=\sup|f(x)-h(x)+(h(x)-g(x))|$$ Then use normal triangular inequality:  $$\sup|f(x)-h(x)+(h(x)-g(x))|\leq \sup(|f(x)-h(x)|+|h(x)-g(x)|)=\sup(|f(x)-h(x)|+|g(x)-h(x)|)$$   This is obviously less than $\sup \limits_{0\leq x \leq 1} |f(x)-h(x)|+\sup \limits_{0\leq x \leq 1} |g(x)-h(x)|$, but I don't know how to prove it. How do I finish that one off. In reading I am not sure if the issue below could have something to do with needing a custom wordpress ajax handler? All of these functions are inside my functions.php file. I have a couple of PHP functions that are supposed to pass information from one to another. The an AJAX call to the final function should produce the final result through that call. Its not happening. My count results in 0 and I am not understanding why. If I replace my query with a static array like [44,55,99,33] then it will pass the values ok. Likewise if I keep the query and simply replace the $id in the query and make it a static id such as '20' then it works. I would think that the id was the issue then and it wasnt pulling properly but it seems to work ok just not pass? function idx(){     global $wpdb, $id;       $getId = $_SERVER['REQUEST_URI'];     $ecp = explode('/', $getId);     if($ecp[3] != ''){         $id = $ecp[3];     }         return $id; } add_action( 'after_setup_theme', 'idx' );  =============================================== function Total_Result(){     global $wpdb;     $id = idx();        /*$resultx = [44,88,59,52];*/ ;      $resultx = $wpdb-&gt;get_results( $wpdb-&gt;prepare(&quot;SELECT * FROM {$wpdb-&gt;prefix}custom_form_submission_conversation as cc LEFT JOIN {$wpdb-&gt;prefix}custom_form_submission as cs on cc.fsid ='$id' WHERE fsid='$id'&quot; ));          $resultxs = array($resultx);          $total_result =  count($resultxs[0]);     echo  $total_result; }  when I var_dump $resultx I get: array(3) { [0]=&gt; object(stdClass)#1256 (36) { [“cid”]=&gt; string(3) “167” [“fsid”]=&gt; string(2) “20” [“specified_date”]=&gt; string(3) “yes” [“price_quote”]=&gt; string(4) “1140” [“posted_by”]=&gt; string(9) “dc_vendor” [“limit_count”]=&gt; string(1) “1” [“read_unread”]=&gt; string(1) “1” [“vendor_response”]=&gt; string(4) “zero” [“date_created”]=&gt; string(19) “2021-04-15 11:39:08” }   [1]=&gt; object(stdClass)#1257 (36) { [“cid”]=&gt; string(3) “186” [“fsid”]=&gt; string(2) “20” [“specified_date”]=&gt; string(3) “Yes” [“price_quote”]=&gt; string(3) “602” [“posted_by”]=&gt; string(8) “customer” [“limit_count”]=&gt; string(1) “1” [“read_unread”]=&gt; string(1) “1” [“vendor_response”]=&gt; string(4) “zero” [“date_created”]=&gt; string(19) “2021-04-15 11:39:08” }   [2]=&gt; object(stdClass)#1258 (36) { [“cid”]=&gt; string(3) “188” [“fsid”]=&gt; string(2) “20” [“specified_date”]=&gt; string(3) “yes” [“price_quote”]=&gt; string(4) “1017” [“posted_by”]=&gt; string(9) “dc_vendor” [“limit_count”]=&gt; string(1) “1” [“read_unread”]=&gt; string(1) “0” [“vendor_response”]=&gt; string(4) “zero” [“date_created”]=&gt; string(19) “2021-04-15 11:39:08” } }   so then I try and pass that data like so: function getvendorresponsestatus() {     global $wpdb; $id = idx();          $total_resultx = Total_Result($id);         die; }  Ive tried with and without the die();. The AJAX call is: setInterval(function(){     var ffiidd = jQuery('#ffiidd').val();     var ajaxurl = &quot;&lt;?php echo admin_url('admin-ajax.php'); ?&gt;&quot;;          jQuery.post(ajaxurl, {         action: &quot;vendorAjaxprog&quot;,         ffiidd: ffiidd     }, function(data) {         console.log(data);         jQuery.post(ajaxurl, {             action: &quot;getvendorresponsestatus&quot;, //&lt;&lt;&lt;&lt;&lt;&lt;&lt;&lt; this is the action that calls the function             fsid : ffiidd         },         function(datas) {             console.log('+ datas);             jQuery('.cust_vendor_top').html(datas);         });         if(data == 0){}else{             jQuery('.ajaxreload').html(data);                                     }     });               }, 5000);/*==30 seconds==*/  So the question is why wont it pass along the query array information but it passes a basic array or it will pass if I change the id in the query to be static? I know the query is good because of when I var_dumped it, I have results before I get to the Total_Result() function.