Suppose that $A \geq B \geq 0$ where $A$ and $B$ are two symmetric n by n matrices.  $A \geq B$ stands for $A-B$ is positive semi-definite. Then, is it able to show that $det(A)\geq det(B)$.  I don't find any such statement. So I conjecture this could be wrong, but how to construct a simple counterexample?
Exercise 12 from SEC. 82 of Finite-Dimensional Vector Spaces - 2nd Edition by Paul R. Halmos.  If $A$ and $B$ are linear transformations on a finite-dimensional inner product space, and if $\textbf{0} \le A \leq B$, then $\det A \leq \det B$.  Notation: $0 \le A$ means that $A$ is positive.  If $\det B = 0$, choose $x \in \ker B$, then $0 \le ((B-A)x,x) = (Bx, x) - (Ax, x) = -(Ax, x) \le 0$. Therefore $\ker B \subseteq \ker A$ and $0 = det A \le det B = 0$. How to prove for the case when $B$ is invertible? Intuitively, I feel that since both $A$ and $B$ are positive their proper values are also positive and $B$'s proper values should be greater the $A$'s and same for determinants.
How do you convert Html to plain text? I have snippets of Html stored in a table. Not entire pages, no  tags or the like, just basic formatting.  I would like to be able to display that Html as text only, no formatting, on a given page (actually just the first 30 - 50 characters but that's the easy bit).  How do I place the "text" within that Html into a string as straight text?  So this piece of code.  &lt;b&gt;Hello World.&lt;/b&gt;&lt;br/&gt;&lt;p&gt;&lt;i&gt;Is there anyone out there?&lt;/i&gt;&lt;p&gt;   Becomes:  Hello World. Is there anyone out there? Entry to the US with an expired Nexus Is it possible to enter the US through the Nexus lane while my Nexus card is being renewed? And how? If I am shooting the 8-ball for a win, and my opponent has a ball near the pocket I'm aiming for, can I call something like "8-ball off the 2 ball into the corner pocket"?  I thought this was allowed, but a friend says this is an automatic loss and that the 8-ball must never touch another ball before going into the pocket.  Various pool rules aren't clear for me on this scenario. How do I change my username? Some time ago, when I installed Ubuntu, I chose a rather stupid username for my account that I do not want to use anymore. How do I change this (including the name of my home directory, and the name in the terminal) without losing settings for applications? How do I keep permissions and my keys for various authentification (e.g. email, SSH, GPG and more)? What settings could possibly get lost if I changed my username? How do I put /home in a different partition? Yeah, so I have 2 hard drives, on with 2 partitions(backups and windows), and I have another that also has 2 partitions with only one used. I have Ubuntu on hard drive 2:partition one, but I have come to a point when the /home folder can't hold any more files, but the partition is only about 15% filled up. So I did some researching and figured out that you can put it on a separate partition, but most of the guides were vague on what I needed to do. Could someone give me step-by-step instructions on what I need to do, or even tell me a, easier or different way to fix this /home folder capacity problem? Calculation of $\int_0^{\pi} \frac{\sin^2 x}{a^2+b^2-2ab \cos x}\mathrm dx$ Calculate the definite integral      $$ I=\int_0^{\pi} \frac{\sin^2 x}{a^2+b^2-2ab \cos x}\;\mathrm dx $$      given that $a&gt;b&gt;0$.   My Attempt:  If we replace $x$ by $C$, then  $$ I = \int_{0}^{\pi}\frac{\sin^2 C}{a^2+b^2-2ab\cos C}\;\mathrm dC $$  Now we can use the Cosine Formula ($A+B+C=\pi$). Applying the formula gives  $$ \begin{align} \cos C &amp;= \frac{a^2+b^2-c^2}{2ab}\\ a^2+b^2-2ab\cos C &amp;= c^2 \end{align} $$  From here we can use the formula $\dfrac{\sin A}{a} = \dfrac{\sin B}{b} = \dfrac{\sin C}{c}$ to transform the integral to  $$ \begin{align} I &amp;= \int_{0}^{\pi}\frac{\sin^2 C}{c^2}\;\mathrm dC\\ &amp;= \int_{0}^{\pi}\frac{\sin^2A}{a^2}\;\mathrm dC\\ &amp;= \int_{0}^{\pi}\frac{\sin^2 B}{b^2}\;\mathrm dC \end{align} $$  Is my process right? If not, how can I calculate the above integral? Comparing Arrays of Objects in JavaScript When running sudo apt-get update I get these following errors:  Err archive.ubuntu.com quantal InRelease Err archive.ubuntu.com quantal-updates InRelease Err archive.ubuntu.com quantal-backports InRelease Err archive.ubuntu.com quantal-security InRelease Err archive.ubuntu.com quantal Release.gpg   Unable to connect to archive.ubuntu.com:http: [IP: 91.189.92.177 80] Err archive.ubuntu.com quantal-updates Release.gpg   Unable to connect to archive.ubuntu.com:http: [IP: 91.189.92.177 80] Err archive.ubuntu.com quantal-backports Release.gpg   Unable to connect to archive.ubuntu.com:http: [IP: 91.189.92.177 80] Err archive.ubuntu.com quantal-security Release.gpg   Unable to connect to archive.ubuntu.com:http: [IP: 91.189.92.177 80] Reading package lists... Done W: Failed to fetch archive.ubuntu.com/ubuntu/dists/quantal/InRelease   W: Failed to fetch archive.ubuntu.com/ubuntu/dists/quantal-updates/InRelease   W: Failed to fetch archive.ubuntu.com/ubuntu/dists/quantal-backports/InRelease   W: Failed to fetch archive.ubuntu.com/ubuntu/dists/quantal-security/InRelease   W: Failed to fetch archive.ubuntu.com/ubuntu/dists/quantal/Release.gpg  Unable to connect to archive.ubuntu.com:http: [IP: 91.189.92.177 80] W: Failed to fetch archive.ubuntu.com/ubuntu/dists/quantal-updates/Release.gpg  Unable to connect to archive.ubuntu.com:http: [IP: 91.189.92.177 80] W: Failed to fetch archive.ubuntu.com/ubuntu/dists/quantal-backports/Release.gpg  Unable to connect to archive.ubuntu.com:http: [IP: 91.189.92.177 80] W: Failed to fetch archive.ubuntu.com/ubuntu/dists/quantal-security/Release.gpg  Unable to connect to archive.ubuntu.com:http: [IP: 91.189.92.177 80] W: Some index files failed to download. They have been ignored, or old ones used instead.   I am using Ubuntu 12.04. This is the command I used after installing Ubuntu from window installer. I am sure my laptop is connected to internet. What can I do now? What it's the best way to generate a string of \t's in C#  I am learning C# and experimenting with different ways of saying the same thing.    Tabs(uint t) is a function that returns a string with t amount of \t's  For example Tabs(3) returns "\t\t\t"  Which of these three ways of implementing Tabs(uint numTabs) is best?  Of course that depends on what "best" means.     The LINQ version is only two lines, which is nice.  But are the calls to Repeat and Aggregate unnecessarily time/resource consuming? The StringBuilder version is very clear but is the StringBuilder class somehow slower? The string version is basic, which means it is easy to understand.   Does it not matter at all?  Are they all equal?   These are all questions to help me get a better feel for C#.     private string Tabs(uint numTabs) {     IEnumerable&lt;string&gt; tabs = Enumerable.Repeat("\t", (int) numTabs);     return (numTabs &gt; 0) ? tabs.Aggregate((sum, next) =&gt; sum + next) : "";  }    private string Tabs(uint numTabs) {     StringBuilder sb = new StringBuilder();     for (uint i = 0; i &lt; numTabs; i++)         sb.Append("\t");      return sb.ToString(); }    private string Tabs(uint numTabs) {     string output = "";     for (uint i = 0; i &lt; numTabs; i++)     {         output += '\t';     }     return output;  } I have two checkboxes in a lightning component, when we click on the first checkbox it marks true to the other checkbox, it works well only with the UI tags but doesn't work when I convert it to lightning tag  &lt;ui:inputCheckbox aura:id="box3" change="{!c.selectAll}"/&gt;  &lt;ui:inputCheckbox aura:id="boxPack" value=""                                                          change="{!c.checkboxSelect}"/&gt;   below are the lightning tags  &lt;lightning:input type="checkbox" aura:id="box3" onclick="{!c.selectAll}"/&gt;  &lt;lightning:input type="checkbox" aura:id="boxPack" value=""                                                          onclick="{!c.checkboxSelect}"/&gt;   controller  checkboxSelect: function(component, event, helper) {         // get the selected checkbox value           var selectedRec = event.getSource().get("v.value");         // get the selectedCount attrbute value(default is 0) for add/less numbers.          var getSelectedNumber = component.get("v.selectedCount");         // check, if selected checkbox value is true then increment getSelectedNumber with 1          // else Decrement the getSelectedNumber with 1              if (selectedRec == true) {             getSelectedNumber++;         } else {             getSelectedNumber--;         }         // set the actual value on selectedCount attribute to show on header part.          component.set("v.selectedCount", getSelectedNumber);     },      selectAll: function(component, event, helper) {         //get the header checkbox value           var selectedHeaderCheck = event.getSource().get("v.value");         // get all checkbox on table with "boxPack" aura id (all iterate value have same Id)         // return the List of all checkboxs element          var getAllId = component.find("boxPack");         // If the local ID is unique[in single record case], find() returns the component. not array            if(! Array.isArray(getAllId)){             if(selectedHeaderCheck == true){                  console.log('true');                 component.find("boxPack").set("v.value", true);                 component.set("v.selectedCount", 1);             }else{                 component.find("boxPack").set("v.value", false);                 component.set("v.selectedCount", 0);             }         }else{             // check if select all (header checkbox) is true then true all checkboxes on table in a for loop               // and set the all selected checkbox length in selectedCount attribute.             // if value is false then make all checkboxes false in else part with play for loop              // and select count as 0              if (selectedHeaderCheck == true) {                 for (var i = 0; i &lt; getAllId.length; i++) {                     component.find("boxPack")[i].set("v.value", true);                     component.set("v.selectedCount", getAllId.length);                 }             } else {                 for (var i = 0; i &lt; getAllId.length; i++) {                     component.find("boxPack")[i].set("v.value", false);                     component.set("v.selectedCount", 0);                 }             }          }        }, Show that is $\int_A f$ exists, then so does $\int_A |f|$ Let $f$ be a real-valued function on $A$ in $E^n$. Show that is $\int_A f$ exists, then so does $\int_A |f|$ and $\left | \int_A f\right | \le \int_A |f|$.   I am trying to finish Rosenlicht's Introduction to Analysis and the last chapter is awful. But I do want to finish it because I have come this far. Could someone show me how to do this, so I can at least have some example problems to understand the material. This chapter (Multiple Integrals) really just left more confused than before I started reading the chapter.  Thank you in advance. Why are .NET timers limited to 15 ms resolution? I have an interesting mathematical/electrical dilemma and I'm not sure where to start.  I am considering buying  (specs on page).  I will be hopefully buying a battery pack with it to support a long-road trip (a few months) wherein I'll keep medication in the fridge (powered by the battery pack when parked and by the car when driving). The medication needs to be keep between 2 and 8 degrees C at all times.  Here's the  I'm thinking of getting:  Can someone please help me understand what I need to do in order to calculate whether or not this fridge + battery combo will work, and if so, how much time will I be able to power the fridge for.  Ideally, I need a combination of fridge and battery which will allow me to have the fridge constantly powered, assuming I sleep in the car for approx. 8 hours a night and drive for the rest of the day (although, there will be times which I'm away from the car during the day for several hours + times where I hope to have access to a wall outlet).   Specs:   Fridge:   Input voltage (AC 220-240 V Input frequency 50/60 Hz Rated input power (AC) 64 W Rated input power (DC) 46W DC12V/50W DC24V W Input voltage (DC) 12/24 V   Battery:   Capacity: 453Wh Battery Type: Lithium-ion Input Voltage: AC 110-220 V, 50/60 Hz; Solar Power: 18 V; Car Input: 12 V Output Voltage: AC 110V 60 Hz; DC Output: 4 * 12V 4 A (Max.10 A); USB Output: 4 ports: 2* 5V 2.1 A and 2* 5V 1 A Power Output: Rate 500 W Peak/Surge 1000 W How can I close a terminal without killing its children (without running `screen` first)? During iPhone update to iOS 10.3.1, I was asked for my password using the correct machine name for my Mac mini. To be clear, the request was not for my iCloud password, my actual local user account password for macOS. This can also happen in the other direction - setting up a Mac may ask for verification for you to enter your iPhone / iPad passcode (and not your iCloud password)  I could not find any information about this kind of access. I was thinking it was security for iCloud and possibly related to two step authorization or two factor authorization. This is the first time I can think of that I entered a desktop password outside of that actual desktop.  I don't have a screen shot but I recall an implication that some files in my Mac are encrypted and need my password. '... need to enter the password for .'   Are there settings I can I control this assuming it's a legitimate part of iCloud setup process or iOS setup process? Suppose there are two users that have commented on an answer. Both have the same user name, and I want to reply to one of them. How can I do this?  It appears that, under the current system, there is no way for me to do this. Specifically, from , I see that:     4. Matching is performed in reverse chronological order, so if five people named John are participating, @john will match the most recent John   So, consider this a request to add this feature using an -n suffix or something similar; e.g. @john-2 could be used to reply to the second John from the top.  Update Thanks to  for nice answer of @name(userid) based auto-complete suggestion which has no loophole. It involves no counting from top &amp; has no consequences when a comment is deleted. But, I think @name(userid) should be used directly in comment as all form inputs at SE are text (nothing embedded).  Now, I'm trying to finalize it based on it: The system should allow optional @name(userid) format for reply. Provided userid with this format, system should ignore name. And, in the situation of duplicate name based confusion, system should demand this format as mandatory to submit comment.  How to get userid? In case of duplicate name based confusion, system should append (userid) in hypertext to their profile. It'd work for mobile version too. On full version, for easy fill-up, a hover reply icon would be fine (like hover vote up icon). Upon clicking it, system should fill @name(userid) in comment text box.  From reply reader side: Multiple Johns could be confused even if they didn't come from notification (because nobody remembers his/her own userid). So, highlight the background of comments which are replied to them. How do I deploy an OpenStack cloud with Juju? I'd like to get a test cloud up and running in preparation for a possible deployment of 12.04. I messed with the UEC a bit, but have had a much harder time finding anything helpful as a primer for the new stack. There is generic documentation on Diablo, but there are likely to be a lot of pieces already handy in Ubuntu that a specific guide could save time with.  Also, as much as I would love to, my budget doesn't include the $4-6k to get a small cluster running with the Canonical support. I recently tried to change a bulb using an adaptor (converting from a E27 to B22 fitting). After switching on the light fitting the fuse had blown. After inspection it turns out the adapter is faulty and must have caused a short-circuit.  After removing the bulb and resetting the switch in the main fuse box, I am now unable to operate the switch on the wall to turn the fitting off (the switch physically won't click).   Is there something I can do to fix the problem?  Thanks, "Ten times fewer the number" versus "one-tenth the number" Lately I've been hearing and reading statistics that are communicated in wording that, frankly, confuses me.  Forgive me for not citing specific instances, but I can give a hypothetical statistic that exemplifies the kind I'm referring to:     Comparing the number of tax returns the IRS audited in 2002 and 2010, the number of unlucky folks who had their returns audited in 2002 was ten times fewer than the number in 2012.    There is something about that wording that bothers me, and I'm not sure why.    After a fruitless search on the internet using numerous combinations/permutations of words and phrases such as "X times fewer," "X times less," "mathematical expressions of 'times fewer' as opposed to 'a fraction fewer,' or 'a fraction less,'" and more, I came up empty. Perhaps this question is more appropriate on a math website, but in the off-chance members of EL&amp;U might give their imprimatur to this question, here goes:  Here's a second hypothetical example.  Is it grammatical—not to mention mathematical—to say the following?:     There are ten times fewer pollinating honeybees worldwide today than there were in 1912.  [Though this is a made-up statistic, there has been a dramatic reduction in the number honey bees in America of late!]   Should not the expression be:     There are one-tenth fewer honeybees worldwide today than there were in 1912.     On the surface, the "ten times fewer" locution seems to me to be a contradiction in terms.  How can something less be 10 times fewer?    I can understand readily how, for example, 2012's bee population of, say, 90 million can be one-tenth (point one) less than 1912's population of 100 million, but ten times fewer?   I'm confused.  Which expression is "more" correct? In a  of Amstat News, the authors (Mark van der Laan and Sherri Rose) stated that "We know that for large enough sample sizes, every study—including ones in which the null hypothesis of no effect is true — will declare a statistically significant effect.".  Well, I for one didn't know that. Is this true? Does it mean that hypothesis testing is worthless for large data sets?