What do you call a family you create yourself through marriage or having children?   I want to distinguish between the family I was born into (my birth family) versus the one I made.
Is there any lingual way to differentiate between the family that I’m a child of and the family that I am a parent of? (I.e., the first family consists of my parents, my siblings, and me; the second one consists of my spouse, my children and me.)
Let $A_n$ denote an $n \times n$ tridiagonal matrix.  $$A_n=\begin{pmatrix}2 &amp; -1 &amp; &amp; &amp; 0 \\ -1 &amp; 2 &amp; -1 &amp; &amp; \\ &amp; \ddots &amp; \ddots &amp; \ddots &amp; \\ &amp; &amp; -1 &amp; 2 &amp; -1 \\ 0 &amp; &amp; &amp; -1 &amp; 2 \end{pmatrix} \quad\text{for }n \ge 2$$  Set $D_n = \det(A_n)$  Prove that $D_n = 2D_{n-1} - D_{n-2}$ for $n \ge 4$. Does a magic item used as an improvised weapon count as a magic weapon? Was Philip K. Dick able to watch a complete version of Blade Runner before his death or just unedited material?  The Blade Runner release date according to  and IMDb was 25 June, 1982. Philip K. Dick died a few months before, on March 2, 1982. I often find myself in a situation where I am facing multiple compilation/linker errors in a C++ project due to some bad design decisions (made by someone else :) ) which lead to circular dependencies between C++ classes in different header files (can happen also in the same file). But fortunately(?) this doesn't happen often enough for me to remember the solution to this problem for the next time it happens again.  So for the purposes of easy recall in the future I am going to post a representative problem and a solution along with it. Better solutions are of-course welcome.     A.h  class B; class A {     int _val;     B *_b; public:      A(int val)         :_val(val)     {     }      void SetB(B *b)     {         _b = b;         _b-&gt;Print(); // COMPILER ERROR: C2027: use of undefined type 'B'     }      void Print()     {         cout&lt;&lt;"Type:A val="&lt;&lt;_val&lt;&lt;endl;     } };       B.h  #include "A.h" class B {     double _val;     A* _a; public:      B(double val)         :_val(val)     {     }      void SetA(A *a)     {         _a = a;         _a-&gt;Print();     }      void Print()     {         cout&lt;&lt;"Type:B val="&lt;&lt;_val&lt;&lt;endl;     } };       main.cpp  #include "B.h" #include &lt;iostream&gt;  int main(int argc, char* argv[]) {     A a(10);     B b(3.14);     a.Print();     a.SetB(&amp;b);     b.Print();     b.SetA(&amp;a);     return 0; } Don't exclude results that have punctuation just because the query doesn't The new search function allows quoted queries that include punctuation, and this is fantastic, most of the time. However, the search considers the punctuation as part of the word itself, so searches for "a test" will not show results containing a test,, a test., or a test?. The reverse is also true.  Is this the intended behavior ? &lt;--(space added so that behavior will show in results.)  It seems that the logical way to do fix this (while maintaining character search) is for queries with punctuation to match results with punctuation, while queries without punctuation to match either with and without.    Example:  vs     A more practical example is described in Arjan's answer below. Algorithm for offsetting a latitude/longitude by some amount of meters After hitting 80% LTV, my mortgage lender refuses to remove PMI. Any recourse? I recently refinanced my home. At the time based on the appraisal vs. amortization I found I would be stuck with PMI (private mortgage insurance) for about 9 months.  I've paid extra on the mortgage each month and after 7 months I got my LTV (loan-to-value ratio) to be 79%. Yay! So I called Pacific Union and asked for PMI to be removed.  The woman said I'd need to fax in a written request. Then she added that my written request would likely be denied because my account had not been in good standing the last 12 months. I pointed out that I'd only had the loan for 7 months and had never been late, but that didn't seem to matter to her.  So I faxed in my written request. Then I read through all my documentation on my loan. Nowhere did it say that I needed to have the loan for 12 months.  After a week I called them back and they said it would be another week for a decision.  So, I waited another week and called. They now just had a recording saying they were too busy to take a call and I could leave a callback number. I did this and never heard back. I've now tried twice more to call them. Each time I leave a number and each time I don't hear back. So I've fired off a written request again.  At this point we've gone around the horn to the next month and in two short days I'll be paying PMI on March's payment.  Do they have any grounds to not remove PMI? Isn't it federally mandated that PMI must be removed at 80%? Any recourse anyone can think of?  [EDIT] - I heard back from Pacific Union. They state that I did not sign anything agreeing to these terms which is what I was certain of. They state these terms are simply automatic with Fannie Mai backed loans and it doesn't matter if I was informed or not about these terms. They also chose not to hide behind the "no late payments" clause and instead said they were requiring an appraisal to remove PMI. Of course an appraisal is $400 and I'll hit 78% LTV before I would pay that much extra in PMI. I have now contacted a lawyer. I'll update again with whatever opinion the lawyer has and whether I proceed or not.  [EDIT] - I filed a BBB complaint. They waited 30+ days to respond. In their response they indicate that the first woman (and some SE users below) are wrong. The loan does NOT have to be in good standing for 12 months. The loan simply needs no late payments within 12 months. They then hide behind the request for another appraisal as the sole reason to not remove it. I checked with Fannie Mae who backs the loan and was told this is optional per the lender, however the lender can force it and I have no recourse. Additionally, they could have asked for a Broker's Price Option (BPO) or a certificate of value which would cost me hundreds less.   At this point I cannot continue to fight them because the amount of PMI I'll pay until 78% LTV is equal to or less than the appraisal cost of $400.  [EDIT] As of August 2015 I've hit 78% LTV. I called Pacific Union to ensure that the automatic cancellation of PMI had happened. They said no, I'd need to request that in writing. This is a violation of federal law as it's supposed to be automatic. They also said they would still deny it because now they want 24 months of good history. I've sent a threatening letter off and am waiting for a reply. If any lawyers come along and read this I'd sure like to hire you.  [EDIT] After my letter they said they would remove PMI. They did not and I was charged again in September. I called them yet again and complained. They said PMI was going to be removed. They then charged my again in October! I called again. I called again and went berserk on the phone. PMI was finally removed for November 2015. First month without it. How do I install VMware player on a Mac os X? Finite group with isomorphic normal subgroups and non-isomorphic quotients? I know it is possible for a group $G$ to have normal subgroups $H, K$, such that $H\cong K$ but $G/H\not\cong G/K$, but I couldn't think of any examples with $G$ finite. What is an illustrative example? Cannot add custom launcher to Ubuntu Dock ('Add to Favorites' option does not show up) I have a custom .desktop file: [Desktop Entry] Type=Application Terminal=false Icon[en_US]=/home/alex/bin/idea-IC-173.4127.27/bin/idea.png Name[en_US]=IntelliJ Exec=/home/alex/bin/idea-IC-173.4127.27/bin/idea.sh Name=IntelliJ Icon=/home/alex/bin/idea-IC-173.4127.27/bin/idea.png  Output of ls -lha .local/share/applications/intellij.desktop: -rwx--x--x 1 alex alex 248 Dez 30 10:54 .local/share/applications/intellij.desktop  Answers posted to a previous question () don't work for me. I can find this launcher in Activities and I can double click on that file and IntelliJ will start. Nevertheless, I cannot add this launcher as a favorite to the dock. The option does not show up. I'd like to delete a file from my Time Machine backups, but there seems to be no option to do so in the Big Sur version of Time Machine. I go to the Time Machine menu and select &quot;Enter Time Machine&quot;. Then I see the view with the stack of windows. I browse to a file and select it. Now I want to delete it so it doesn't exist and take up space in backups. There is no option to delete it - either the action menu in the toolbar, or the control-click context menu. How do I delete a file from Time Machine backups? There used to be a way to do this. (I'd take a screenshot, but the key shortcuts for screenshots don't work in Time Machine. Not sure if this is just a Big Sur thing or if that's always the case.) Update: I should mention I'm using APFS for my Time Machine volume. That seems to be relevant. Also, I contacted Apple support about deleting entire snapshots (slightly different goal than the one here in this question). They said you can't do it anymore. It seems like there's some bugs here, because I have a &quot;Move to Trash&quot; option if I control-click one of my backups. It seems to work initially but then the backup comes back.  I don't like losing this much control over my files. I may look for another backup solution. Let $R$ be a ring and let $J$ be an ideal of $R$. Assume $J$ contains   a unit of $R$. Prove $J=R$, rigorously.   Again I feel here as if the statement is intuitive so I am not able to make it rigorous. If you could show all steps in the proof that would be nice. Is it possible to prevent an NSURLRequest from caching data or remove cached data following a request? On iPhone, I perform a HTTP request using NSURLRequest for a chunk of data. Object allocation spikes and I assign the data accordingly. When I finish with the data, I free it up accordingly - however instruments doesn't show any data to have been freed!  My theory is that by default HTTP requests are cached, however - I don't want my iPhone app to cache this data.   Is there a way to clear this cache after a request or prevent any data from being cached in the first place?  I've tried using all the cache policies documented a little like below:  NSMutableURLRequest *theRequest = [NSMutableURLRequest requestWithURL:[NSURL URLWithString:url]]; theRequest.cachePolicy = NSURLRequestReloadIgnoringLocalCacheData;   but nothing seems to free up the memory! Highlight tags corresponding to our favorites Each of us have our list of favorite tags.  When a question has tag(s) that match one or more of ours, the background gets highlighted, drawing our attention to the question.  It would also be nice to highlight (perhaps using a bold font) the tags(s) that matched our own, so one could quickly do a further filtering of the question to see if it is of interest at the present time, like the following:    e.g. arduino is in my list of favorites, but the other two in each question are not. How do I change login screen theme or background in Ubuntu 20.04? How can I change the Ubuntu 20.04 login screen theme or background?  In the previous version 19.10, I can change it from the CSS file under Yaru theme but now there is no CSS file. limsup of intersection of events as a subset of intersection of limsups Let $A_1, A_2, \ldots$ and $B_1, B_2, \ldots$ be two sequences of events in some probability triple $(\Omega, \mathcal{F}, \mathbf{P})$. Now, it is true that $\left(\limsup_n A_n\right) \cap \left(\limsup_n B_n\right) \supseteq \limsup_n (A_n \cap B_n)$, and I can think of many different cases where this relation holds true for equality (e.g., let $A_n = B_n$, or $A_n = B_n^c$, or $B_n = \emptyset$, or $B_n = \Omega$). However, I cannot think of any case where $\left(\limsup_n A_n\right) \cap \left(\limsup_n B_n\right) \supset \limsup_n (A_n \cap B_n)$ holds for strict inclusion.  My proof for the relation is simple. Since $A_n \cap B_n \subseteq A_n$ and $A_n \cap B_n \subseteq B_n$, then we can perform some set algebra as follows:  $$\bigcup_{k=n}^\infty (A_k \cap B_k) \subseteq \bigcup_{k=n}^\infty A_k$$ $$\bigcap_{n=1}^\infty \bigcup_{k=n}^\infty (A_k \cap B_k) \subseteq \bigcap_{n=1}^\infty \bigcup_{k=n}^\infty A_k$$  Then we can use the same logic for $B_n$, and come to the conclusion as stated in the beginning. Scripting `rm` and `rm -i` I want to create a script that whenever there are more than three files being removed, it invokes rm -i by default. I know it will go in initialization files but can't seem to get it working.  So far:  if [$file -ge 3]; then   rm -i    exit 0  else   rm exit 1 For whatever reason, the "Open With" context menu in Finder is always listing every app four times (exactly).     I've read about (and tried) rebuilding the Launch Services database to restore the context menu to its initial state, but nothing seems to work. I've also used Onyx.app to clear user and system caches, but I'm still left with four repetitions of each app under the "Open With" service. I have restarted the system a few times just to be sure it was not some temporary corruption of the list.  I'm running OS X 10.6.6 on a 2011 17" MacBook Pro with a fresh install of OS X (i.e. I didn't use Migration Assistant.app or a Time Machine backup). I did, however, sync all my preferences and other files using MobileMe like I always do, so I imagine it's feasible that a preference file somewhere is causing problems? I didn't notice whether this was happening before or after my MobileMe sync.  How can I fix this so one app shows? Allow a user to read some other users' home directories I am trying to make a computer monitor. This means I have to have an image texture for the screen. My image texture is currently connected to an Emission BSDF shader. Unfortunately, when the emission is bright enough to emit light, the image is white-washed. I tried adding a Brightness/contrast node to the image before pushing it into the Emission BSDF, but it seems to directly counter the Emission's strength.  The desired light level   The desired image brightness