I am really stuck at calculating $$\frac{d}{dC} \operatorname{trace} \left( X C P (X C)^T \right)$$ where $P \in R^{r\times r}$, $X \in R^{m\times n}$ and $C \in R^{n\times r}$ . Do I need to recall $A=XC$ and then apply chain rule such that $$\frac{d \operatorname{trace}(XCP(XC)^T)}{dC}=\frac{d\operatorname{trace}(APA^T)}{dA}\cdot \frac{dA}{dC}$$ where first factor equals $A (P+P^T) $ (following ) and the second is $X$. Am I right ? More correctly, by a quick dimensionality check, it would make more sense to have $X^T (XC) (P+P^T)$ at the end. Is it right ? Could someone help me prove it ? Moreover, is $\operatorname{trace}(XCP(XC)^T)$ convex w.r.t. $C$ ?
Given three matrices $A$, $B$ and $C$ such that $ABA^T C$ is a square matrix, the derivative of the trace with respect to $A$ is:  $$ \nabla_A \operatorname{trace}( ABA^{T}C )  = CAB + C^T AB^T $$  There is a proof . However, I was not able to understand the second line of the proof. it says something like this:  $$ \nabla_A \operatorname{trace}( ABA^T C )= \nabla_\bullet \operatorname{trace}(f(\bullet)A^{T}C) + \nabla_\bullet \operatorname{trace}(f(A) \bullet^T C) $$  Where  $f(A) = AB$  I tried searching the proof or any hint using google, but I couldn't fine anything. Could anyone please help me with this second line. I cannot sleep. I am obsessed with this proof.
Get full name twice in Bibliography I use biblatex with style=authoryear and when there are two references of the same author in the bibliography there is just a dash. How can I get the full name in the second entry too? How can I cite a range of papers, the output is a range of numbers, instead of a list of numbers? In other words, when I type  ... some dummy text here is due to me \cite{me1, me2, me3, me4} ...   I want the output to be     ... some dummy text here is due to me [3-6] ...   instead of     ... some dummy text here is due to me [3.4.5.6] ...   (I use LaTeX, and am currently using the amsart class and whatever it includes. But I am open to other suggestions.) How to download packages from a command-line given repository? Does borrowing from my 401(k) make sense in my specific circumstance? I know the standard answer and understand that I will be losing out on that compound interest, but I am young and only need to borrow a few thousand. here are some more details:   Need to borrow $6500 to pay off a car and two student loans ($300/month in payments combined) I am 22 and work full time, making about $70K annually I contribute 8% of my income with a 4% company match I increase that by 2% every year The loan would be over 48 months ($140/month) I live very frugally and have almost no discretionary spending that can be cut (don't eat out all the time, no gym membership/ Netflix/ Apple music etc.) I'm cash-poor: 90% of my salary is going to mortgage, groceries, students loans, car payment, newborn baby medical expenses, and commute. These are not my only debts, just the ones I am seeking to pay off in the short term. I work two jobs, one full and one part time. I started a business, and my wife has two jobs. Working more is not a good answer to the original question.   It seems to me that I would be reducing middle to high interest debt while also saving myself $150 per month.   Loans:     Student loan 1: $1200 @ 5%   Student loan 2: $1100 @ 4.25%   Car Loan: $4000 @ 6% Editing photo dimensions in large batches If I had the time, I would go through each and every photo to make sure that the dimensions and quality are pixel perfect. However, I'm in a time crunch, and need to make the couple hundred photos that are 4320 x 3240 into 950x630.   now I could- 1. Upload them into WordPress, uses native image reducing components and J query go from there. 2. Go through each and every photo, reduce file size, and proportion and bring it to Yahoo's PNG processing script.  However, maybe I'm missing a choice or alternative workflow that would allow for processing all of these photos in a clear manner without losing quality or perspectives. Any workflow suggestions? Duplicate flag lists "you have already raised this flag" after flagging as "Unclear what you're asking" I'm attempting to do this in a procedure:  DECLARE @a bit = 1; BEGIN     SELECT * INTO #aTemp FROM OPENROWSET( ... );      IF @a = 0     BEGIN         SELECT ... INTO #bTemp FROM #aTemp;     END     ELSE     BEGIN         SELECT ... INTO #bTemp FROM #aTemp;     END END   I get the error:  Msg 2714, Level 16, State 1, Line 10 There is already an object named '#bTemp' in the database.   Why is this happening and is there a work around?  Update  I've attempted to add a DROP statement as suggested , but it still does not work:  DECLARE @a bit = 1; BEGIN     SELECT * INTO #aTemp FROM OPENROWSET( ... );      IF @a = 0     BEGIN         IF OBJECT_ID('[tempdb]..#bTemp') IS NOT NULL         BEGIN             DROP TABLE #bTemp;         END          SELECT ... INTO #bTemp FROM #aTemp;     END     ELSE     BEGIN         IF OBJECT_ID('[tempdb]..#bTemp') IS NOT NULL         BEGIN             DROP TABLE #bTemp;         END          SELECT ... INTO #bTemp FROM #aTemp;     END END Does the English language have an official Academy? For some languages, there are academies that decide topics such as grammar and spelling of things, for example, for the Spanish language, there are 22 academies in 22 different countries, all making decisions on spanish grammar in their country. I would like to know if there is something that serves the same purpose for the English language. Which is the best way to install new hplip versions? Hplip is the software that drives most hp printers. Usually new printers are only supported in newer versions of the software. Canonical doesn't backport new versions packages to older Ubuntu versions, that means that newer drivers are not going to be available in the Software Center of older Ubuntu versions.  As an example yesterday I needed to hook a hp 1005p to a computer running 8.04. I was forced to install a newer hplip than the version available in the repositories. I installed it using the script from the hp page, but I'd like to know other alternatives.  Which is the best (technically, not the easiest) way to install it and why?. biblatex: Tighter integration of \textcite "in the flow of text" This question led to a new feature in a package:      Many biblatex citation styles feature the \textcite command which is "intended for use in the flow of text, replacing the subject of a sentence" (manual, section 3.6.2). At the moment, this works well for citations of single works -- e.g. (assuming the authoryear style), one may write in the source  The reference work is \textcite{knuth:ct:a}.   which will be typeset as     The reference work is Knuth (1984).   Citing multiple works, on the other hand, is somewhat tedious. If one simply writes (1):  Important works are \textcite{knuth:ct:a,knuth:ct:b,markey}.   the output is rather a list than a correct sentence (2):     Important works are Knuth (1984); Knuth (1986); Markey (2005).   One has to write (3):  Important works are \textcite{knuth:ct:a}, \textcite{knuth:ct:b}, and \textcite{markey}.   to get (4):     Important works are Knuth (1984), Knuth (1986), and Markey (2005).   Can \textcite be modified so that input (1) will yield output (4), hereby achieving a tighter integration "in the flow of text"?  \documentclass{article}  \usepackage[style=authoryear]{biblatex}  \addbibresource{biblatex-examples.bib}  \begin{document}  This is how \texttt{\textbackslash textcite} currently works:\\ Important works are \textcite{knuth:ct:a,knuth:ct:b,markey}.  \bigskip  But I'd like \texttt{\textbackslash textcite} to output the following:\\ Important works are \textcite{knuth:ct:a}, \textcite{knuth:ct:b}, and \textcite{markey}.  \printbibliography  \end{document} The best free and most up to date phylogenetic tree on the internet? I found phylogenetic tree in wikipedia is lacks of accuracy. It actually confusing with versions. Some terms was not scientifically accurate with dna analysis anymore  So are there anyone know where is the most updated datasource on the internet? Create instance of generic type in Java? Is it possible to create an instance of a generic type in Java? I'm thinking based on what I've seen that the answer is no (due to type erasure), but I'd be interested if anyone can see something I'm missing:  class SomeContainer&lt;E&gt; {     E createContents()     {         return what???     } }   EDIT: It turns out that  could be used to resolve my issue, but it requires a lot of reflection-based code, as some of the answers below have indicated.  I'll leave this open for a little while to see if anyone comes up with anything dramatically different than Ian Robertson's . Converse of Schur's Lemma in finite dimensional vector spaces I am trying to prove (or disprove) the converse of Schur's Lemma in finite dimensional vector spaces. I am not sure if it holds in this case, but I have tried to apply the idea that proves it in representation theory (see for example Theorem 4.3  that uses Maschke's theorem or in questions  and ).  The converse of Schur's Lemma in finite dimensional vector spaces is:     Let $V$ be a finite dimensional vector space over the complex numbers. Let $S$ be a set of endomorphisms of $V$ and assume that every endomorphism $A$ of $V$ such that $$AB=BA\text{ for all }B\in S$$ is of the form $\lambda I, \ \lambda\in\mathbb{C}$. Then $V$ is a simple $S$-space.   (*an endomorphism of $V$ is a linear operator from $V$ to $V$  *$V$ is a simple $S$-space if the only $S$-invariant subspaces of $V$ are $V$ itself and the zero subspace)  What I've tried so far:  Assume to the contrary that $V$ is not a simple $S-$space. Then, there exists a subspace $W$ of $V$ such that $W\not =\{0\}$, $W\not= V$ and $W$ is $S$-invariant. Let $W'=V\setminus W$. Then, since $V$ is finite, we can easily prove that $V=W\oplus W'$ (this is my attempt to translate Maschke's theorem in vector spaces). Consequently, for every $v\in V$, there exists unique $w\in W$ and $w'\in W'$, such that $v=w+w'$. Define the projection $P:V\rightarrow V$ by $Pv=w$ for every $v\in V$.   What is left to prove is that $PB=BP$ for all $B\in S$. Then $P$ is clearly not a scalar and thus we have the contradiction we are looking for.  There is a difficulty in showing that $PB=BP$ for all $B\in S$, because $W'$ might be $S$-invariant or not. More precisely:   Let $v\in V=W\oplus W'$ and $v\not =0_V$.  If $v\in W$ then $Bv\in W$ because $W$ is $S$-invariant, therefore: $PBv=Bv$ and $BPv=Bv$.  If $v\in W'$ then $Pv=0$, hence $BPv=B\cdot 0=0$, and  (i) if $Bv\in W'$ then $PBv=0$.  (ii) if $Bv\in W$ then $PBv=Bv$ and this is where the problem occurs.  Any hints, ideas or counterexamples would be very helpful. Make Windows 10 stop installing driver software automatically Roman numerals become “too wide” in table of contents Following minimal document:  \documentclass{report}  \begin{document} \renewcommand*\thechapter{\Roman{chapter}} \tableofcontents \setcounter{chapter}{6} % Problem starts at chapter 7 \chapter{Seven} \chapter{Eight} \section{Eight-one} \section{Eight-two} \end{document}   yields this output (cut to size for your convenience):    Clearly, the chapter (and section) headings hang into the chapter (section) numbers. For the previous chapter numbers, this isn’t a problem since all chapter titles are aligned to the same column and the numbers I–VI are sufficiently narrow.  (The same happens regardless of document type, I’m actually using scrreprt with the same result.)  How can I prevent this? I have seen both usages.    Stay tuned on our Facebook page to know more  Stay tuned to our Facebook page to know more    But don’t know which one is more appropriate. Please help me figure it out. Thanks There are 'n' candies and 't' boxes. Find the number of ways to place the candies in the boxes for each of the conditions (given in the problem). There are 'n' candies and 't' boxes. Find the number of ways to place the candies in the boxes for each of the conditions (all the candies must be spread out) : (a) candies and boxes are different; (b) candies of the same diiferent boxes should not be empty cartons: (c) candies equally new, boxes are different; Edit :(d) candies are different, the boxes are the same, there should be no empty boxes; (e) candies of different boxes alike.(Edit : candies are different, boxes are equal) Specify the display type that matches the placement, if possible. My Answers : (a) Each layout is encoded with a word of $n$ letters from the alphabet of 't' letters $\implies$ possible $n^t$ variants. (b) Write, $n$ candies in the form of balls ina line, we need to put $(t-1)$ partition in $(n-1)$ place, but we can't put two partitions on one place, so we get : $^{t-1}C_{n-1}$. (c) First we need to choose a candy in a box (in the first box $n$ ways, in the second box $(n-1)$, $\cdots$ in the $t-th$ : $(n-t+1)$ ways $\implies$ total $n!/(n-t)!$ methods, and then we distribute the remaining $(n-t)$ candies into $t$ boxes, this is encoded with owrds from the alphabets (for each candy $t$ variants) $\implies$ $t^{n-t}$. The Answer is : $\frac {n!}{(n-t)!}*t^(n-t)$ Are my answers (a), (b) &amp; (c) are correct? For (d) &amp; (e) I don't know how to proceed? Please help me. So I installed vlc daily built snap package. It's working great, no problems there. However I noticed that after upgrade the system has 2 versions of vlc (7 and 9). That can be seen when browsing /snap/vlc/ directory and using system monitor. This is also the case with other snaps. Is there a way to automatically delete older versions of snaps? Or do I just delete them using plain &quot;rm&quot; command? Is it safe to delete them that way? Give an example of a continuous function $f : [0, ∞) \mapsto [0, ∞)$ such that $\int_{0}^{\infty}f(x)dx$ exists but $f$ is unbounded. Give an example of a continuous function $f : [0,  ∞) \mapsto [0, ∞)$ such that  $\int_{0}^{\infty}f(x)dx$ exists but  $f$ is unbounded.   I have been thinking about this. And I have come to the conclusion that I will need to construct a function, $f$, such that $f$ is a sequence of triangles of increasing height, but decreasing base. I obviously need $f$ such that both the height of the triangles, and sum of the bases tends to infinity. But I also need that the $\sum (\text{height} \times \text{base}) \leq \infty $ Choosing C++ debugger on linux