uniqueness of embedded submanifolds proof in Lee ITM
Smooth structure of embedded submanifolds
URL Rewriting of a query string in php Windows using too much RAM, how to diagnose resource hog I have 16GB of system RAM. On startup with no applications open except the task-manager Windows is using about 3gb of RAM. I looked in the processes tab, but nothing seems to be out of the ordinary. How can I find out why my Windows is using so much RAM.    all processes from all users      Reading from the poolmon it seems that my wireless broadcom driver is using about 0.4GB of of RAM. Even if I remove  it would still be using  2.6GB on start-up, which is still too much.      After reinstalling the wireless driver associated with the memory leak. I have a new screenshot and would like to confirm that it was indeed a memory leak. Prove that $\frac{a_1^2}{a_1+a_2}+\frac{a_2^2}{a_2+a_3}+ \cdots \frac{a_n^2}{a_n+a_1} \geq \frac12$ Let $a_1, a_2, a_3, \dots , a_n$ be positive real numbers whose sum is $1$. Prove that $$\frac{a_1^2}{a_1+a_2}+\frac{a_2^2}{a_2+a_3}+ \ldots +\frac{a_n^2}{a_n+a_1} \geq \frac12\,.$$  I thought maybe the Cauchy and QM inequalities would be helpful. But I can't see how to apply it. Another thought (might be unhelpful) is that the sum of the denominators on the left hand side is $2$ (the denominator on the right hand side). I would really appreciate any hints. What was the meaning of the Night King's message? In S08E01 of Game of Thrones, we see this message:    We have seen this several times, the symbol drawn with horse heads, the stones around the Weirwood where White Walkers are created and in some caves, drawn by Children of the Forest.  Do we know the meaning behind this symbol? Does it or something similar occur in the books as well? How to handle relative urls correctly with a nginx reverse proxy Reference sObject Fields dynamically from a string value Is it possible to reference the fields of an sObject (opportunity in this example) dynamically from a string value? For example, take the following code:  for(Opportunity o: oppList) {       for(String s : strList) {            // I'd like to set the opportunity field values based on the string             // value that contains the API name of the opportunity field       }  }  So, the variable strList contains the opportunity API field names. I'm trying to clear some fields on the opportunity in a before update trigger, but I have to access the field names dynamically in the strList. Induction Proof: Fibonacci Numbers Identity with Sum of Two Squares Using induction, how can I show the following identity about the fibonacci numbers? I'm having trouble with simplification when doing the induction step.  Identity: $$f_n^2 + f_{n+1}^2 = f_{2n+1}$$  I get to:  $$f_{n+1}^2 + f_{n+2}^2$$  Should I replace $f_{n+2}$ using the recursion? When I do that, I end up with the product of terms, and that just doesn't seem right. Any guidance on how to get manipulate during the induction step?  Thanks! According to HTML specs, the select tag in HTML doesn't have a readonly attribute, only a disabled attribute. So if you want to keep the user from changing the dropdown, you have to use disabled.  The only problem is that disabled HTML form inputs don't get included in the POST / GET data.  What's the best way to emulate the readonly attribute for a select tag, and still get the POST data? What is the definition of $2.5!$? (2.5 factorial) I was messing around with my TI-84 Plus Silver Edition calculator and discovered that it will actually give me values when taking the factorial of any number $n/2$ where $n$ is any integer greater than $-2$. Why does this happen? I thought factorials were only defined for positive integers and $0$, so what is my calculator doing to get the answer $3.32335097$ when I enter $2.5!$? Is there actually a definition of $2.5!$ or is my calculator just being weird? How is the factorial function implemented?  I understand the binary implications of $2.5$, so that could possibly have something to do with it. I get a domain error when trying to take the factorial of $-1, 2.3, e$, and any number that is not of the form $n/2$ where $n$ is any integer greater than $-2$. How specific does "beast you have seen" have to be for Wild Shape? Starting at 2nd level, you can use your action to magically assume the   shape of a beast that you have seen before.   This doesn't say anything about the gender or age of the beast, so can a druid change into a female bear if he has only seen male bears? Can he change into a bear cub if he has only seen adult bears? What about hens/roosters/chickens, which look quite a bit different? Or does the gender of the animal shape just match the gender of druid? I would be happy to get some hints on the following integral: $$ \int_0^1\frac{x^{19}-1}{\ln x}\,dx $$ When I search for my address on Google Maps it cannot be found which means I can't order takeaway etc.  Is there a way of adding my address to Google Maps or even add a street address?  I want the address to be publicly available so others can search it. How do I disable the profiles button in Chrome? Since Chrome version 38, there's a pretty ugly button in the title bar of the Chrome window that allows you to switch profiles quickly.  Since I'm the only one that uses my computer, this is not really useful to me, and I'd rather reclaim the space for more open tabs.  I've looked in the settings but don't see a way to disable it.  How can I get rid of the button? Proofs involving limsup and liminf I've been working with proofs involving $\limsup$ and $\liminf$, and I'm a bit confused regarding their general methodology. More specifically, I'm unsure about whether my approach to the following problem makes sense.  Problem. Let $(s_n)$ and $(t_n$) be sequences and suppose that there exists $N_0$ such that $s_n \leq t_n$ for all $n &gt; N_0$. Show that $\liminf s_n \leq \liminf t_n$ and that $\limsup s_n \leq \limsup t_n$.   The way I approached it was as follows:   Let $N &gt; N_0$. Then $\limsup_{N \rightarrow \infty} \{ s_n : n &gt; N \} \leq t_n$ as $s_n \leq t_n$, and $\limsup s_n $ is the largest possible limit of a subsequence of $s_n$. As $t_n : n &gt; N$ is (by definition) less than $\limsup_{N \rightarrow \infty} \{ t_n : n &gt; N \}$, the proof is complete.  I'm pretty sure this is incorrect, however, and I'm generally unclear about the method behind such a proof. Any help is appreciated! What do parentheses surrounding an object/function/class declaration mean? I'm new to both JavaScript and . In YUI library examples, you can find many uses of this construct:  (function() {     var Dom = YAHOO.util.Dom,     Event = YAHOO.util.Event,     layout = null,         ... })();   I think the last couple of parentheses are to execute the function just after the declaration.  ... But what about the previous set of parentheses surrounding the function declaration?  I think it is a matter of scope; that's to hide inside variables to outside functions and possibly global objects. Is it? More generally, what are the mechanics of those parentheses? How to install Skype 4.3? Note: The legacy Skype for Linux application (v4.*) is now discontinued and was removed from Canonical’s partner repositories, so this question and its answers are obsolete. Please refer to  of our .  Skype 4.3 is just released with considerable improvements. As a regular Skype user I want to use the latest Skype on my Ubuntu 14.04, but there is no updates from Update Manager. Even in Skype website only Skype 4.3 is available only for Ubuntu 12.04. Is there anyway to use this latest Skype on Ubuntu 14.04 or how much we have to wait for the update? Self adjoint operators and trace class property This is a variation of the  questioned some time ago.  For a complex Hilbert space $H$ let $T: H \rightarrow H$ be a bounded operator. We call that $T$ is a trace-class operator if the following sum  $$\sum_{i}{\langle |T|e_{i}, e_{i} \rangle} &lt; \infty$$  converges, where $|T| = (T T^{*})^{\frac{1}{2}}$ is the absolute value of the operator.  Assume that  $$\sum_{i}{\langle Te_{i}, e_{i} \rangle}$$  converges for any basis in the space $H$. How to prove that if the aformentioned property holds then the operator is a trace class operator?  The progress on the problem is the following: Given an arbitrary bounded operator $T: H \rightarrow H$, one can use the following decomposition $$T = \big( \frac{T + T^{*}}{2} \big) ^{*} + i \big( \frac{T^{*} - T}{2i} \big) ^{*}$$  The latter line gives the decomposition $$T = A + i B$$ where $A, B$ are normal operators.  For the normal operators we can apply the spectral theorem that proposes that $T$ is unitary equivalent to  $$(UT U^{-1})(f(x)) = g(x) f(x)$$  where  $$U: H \rightarrow L^{2}(X, \mu)$$ $$g \in L^{\infty}(X, \mu)$$  Though this decomposition classify the operator in a broad sence, i see no direct way to conclude the statement. Are the any hints that may extend the previous argument? If not, are there any ways to conclude the statement? What is LookDev/Material Preview mode for in 2.8? How to keep leading zeros in PHP integer Which shells don't support dollar-parenthesis expansion, and demand backticks?