I would like to bring this formula in the form Re(D) + jIm(D):    This is my current progress:    My question is:  How can I get rid of the imaginary part jωC2*R?  When the denominator is completely real I can just split up the formula and have an imaginary and a real part.
I have a mathematical problem with the following formula. The goal is to transform this equation to have a seperated real and imaginary part. I have already done this with several other equasions, but for this particualar one I don't know where to start.  I dont expect you to do the complete work of transforming for me, it would already be very nice if you could give me a hind on how you would start.  Thank you in advance!  C||RLC      This is my current progress:    My question is:  How can I get rid of the imaginary part jωC2*R?  When the denominator is completely real I can just split up the formula and have an imaginary and a real part.
Explain why $\mathbb{Z \times Z}$ and $\mathbb{R \times R}$ is not a field and that why any external direct sum of two fields cannot be a field. I believe it has much to do with the lack of every non-zero element having an inverse, however I am having difficulty seeing it. Petersen graph is not a Cayley graph How can I show that the Petersen graph is not a Cayley graph?  I don't know very much about Cayley graphs, I know that they are vertex-transitive, but so is the Petersen graph. It probably has to do with the group structure of $\Gamma$ in $Cay(\Gamma,S)$ (which is a group of order $10$, i.e., it is either the cyclic group $C_{10}$ or the dihedral group $D_{10}$). Then I am guessing we go for some sort of contradiction(?). My group theory is a bit rusty, I would very much appreciate someone showing me how to do this one! Weird artifacts in my UV mapped texture Any ideas what's causing those chunks of geometric artifacts?  Argh. Propose a privilege to sit in the 500 reputation mark Does the new finding on "reversing a quantum jump mid-flight" rule out any interpretations of QM? new finding by Minev et al. seems to suggest that transitions between atomic states are not instantaneous, but continuous processes wherein a superposition smoothly adjusts from favoring one state to another (if I understand it correctly). The authors also claim to be able to catch a system "mid-jump" and reverse it. Popular articles are  and .  I am curious if this finding rules out any interpretations of QM. It seems to generally go against the Copenhagen attitude, which describes measurements as collapsing physical systems into a definite classical state. The popular articles indeed claim that the founders of QM would have been surprised by the new finding.  The link with the asterisk mentions that something called "quantum trajectories theory" predicts what was observed. Is this an interpretation, or a theory? And are they implying that other interpretations/theories don't work? Problem with package lineno I'm using package lineno for proofreading of my documents.  So I usually have something like the following MWE:  \documentclass{article}  \usepackage{blindtext}  \usepackage{ifthen}           % to use if then else \newboolean{korrektur}        % for proofreading: true/false  \setboolean{korrektur}{true} % choose one, comment the other %\setboolean{korrektur}{false}  % What we need for proofreading \ifthenelse{\boolean{korrektur}}% proofreading wanted? {% YES:   \usepackage{showframe}          % shows type area   \usepackage{lineno}             % numbers lines   \pagewiselinenumbers            % lines per page }% {% NO: % error if lineno was used before:   \typeout{***** Please delete .aux files if you get errors! *****} % What to add here so that I get no errors?  %&lt;================= }  \begin{document} \Blindtext \end{document}   Run it two times. Everything should be well, blindtext is numbered.  Now please move the % sign from line 8 (false) to line 7 (true).    Because we had had the run before there is an .aux-file, including lines with the macro \@LN.  If you compile now you will get the following error message (and some more):   ***** Please delete .aux files if you get errors! ***** (C:\Users\...\LaTeX\_TeX.SX\lineno\test-lineno.aux ! Undefined control sequence. l.2 \@LN         {0}{0}   Question:  Is it possible to add in line 20 (see %&lt;===========) something to get rid of the error messages? Something that resets the meaning of macro \@LN? Proving that $e^{\pi}-{\pi}^e\lt 1$ without using a calculator Prove that $e^{\pi}-{\pi}^e\lt 1$ without using a calculator.    I did in the following way. Are there other ways?  Proof : Let $f(x)=e\pi\frac{\ln x}{x}$. Then,  $$e^{\pi}-{\pi}^e=e^{f(e)}-{e}^{f(\pi)}\tag1$$ Now, $$f'(x)=\frac{e\pi(1-\ln x)}{x^2},\quad f''(x)=\frac{e\pi (2\ln x-3)}{x^3},\quad f'''(x)=\frac{e\pi (11-6\ln x)}{x^4}.$$ Since $f'(x)\lt 0$ for $e\lt x\lt\pi$, one has $f(e)\gt f(\pi)$. By , there exists a point $c$ in $(e,\pi)$ such that  $$\begin{align}f(\pi)&amp;=f(e)+(\pi-e)f'(e)+\frac{(\pi-e)^2}{2}f''(c)\\&amp;\gt f(e)+(\pi-e)\cdot 0+\frac{(\pi-e)^2}{2}\cdot \frac{e\pi(2\ln e-3)}{e^3}\\&amp;=f(e)-\frac{\pi(\pi-e)^2}{2e^2}\tag2\end{align}$$ because $f'''(x)\gt 0\ (e\lt x\lt \pi)$ implies $f''(c)\gt f''(e)$.  By the  and $(2)$,  $$e^{f(e)}-e^{f(\pi)}\lt (f(e)-f(\pi))e^{f(e)}\lt \frac{\pi (\pi-e)^2}{2e^2}\cdot e^{\pi}=\frac{e\pi(\pi-e)^2}{2}\cdot e^{\pi-3}\tag3$$  Since $e^x\lt \frac{1}{1-x}\ (0\lt x\lt 1)$ and $0\lt \pi-3\lt 1$,  $$e^{\pi-3}\lt\frac{1}{4-\pi}\tag4$$  From $(1)(3)(4)$,  $$e^{\pi}-{\pi}^e\lt \frac{e\pi(\pi-e)^2}{2}\cdot\frac{1}{4-\pi}\lt\frac{3\times\frac{22}{7}\left(\frac{22}{7}-2.718\right)^2}{2}\cdot\frac{1}{4-\frac{22}{7}}\lt 0.993\lt 1$$ In one of the concluding scenes of the movie Thor, it's shown that Loki deliberately let go of the Scepter, and falls deep into the vast intergalactic sea sort of thing. How does he survive and return in The Avengers? How to increase ping time/network connection latency on Windows? I need to permanently increase network latency in approximately 100ms, without losing packets.  Reason: On a certain server and internet game my ping is too much lower than the other people playing it. I have 15ms, when average is 150ms which gives me cheater-like advantages.  So also out of curiosity I'm looking for ways to increase my ping while keeping connection stable. What are my options if any?  I thought about VPN services, but it seems like a bad option since games use UDP?  Is there any router/software that would assist me on this?  I'm on windows. Recursive time series forecasting model In a recursive forecasting model, let's say you are trying to predict sales of Target for the next month and you will append that prediction to your input and predict the month after. Basically, your target is Sales quantity, which you lagged. But let's say you have other continuous numerical features like Price which you will lag as well. As your model is making predictions on Sales_QTY, it will feed it back to sales_qty as input so you will never ran out of values for that one. So how do you deal with your other features (other than the target) in order to generate them as next months inputs in a recursive forecasting model (because eventually you will run out of them)? DO you create a sub-model and try to predict them as well? I have an email account that has passed 60GB of emails, and currently I'm having a lot of trouble using an email client to archive emails of last year (2011).  Via terminal, I am trying to use  to locate the files between 2011-01-01 and 2011-12-31, but no avail.  How can I find files between two dates?  If relevant, the end goal will be a batch that will move each file found, matching the date interval, to a folder. What is $\int_0^1\frac{x^7-1}{\log(x)}\mathrm dx$? The code below compiles, but has different behavior for the char type than for the int types.  In particular     cout &lt;&lt; getIsTrue&lt; isX&lt;int8&gt;::ikIsX  &gt;() &lt;&lt; endl;    cout &lt;&lt; getIsTrue&lt; isX&lt;uint8&gt;::ikIsX  &gt;() &lt;&lt; endl;    cout &lt;&lt; getIsTrue&lt; isX&lt;char&gt;::ikIsX  &gt;() &lt;&lt; endl;   result in 3 instantiations of templates for three types: int8, uint8 and char. What gives?  The same is not true for ints: int and uint32 which result in the same template instantiation, and signed int another.  The reason seems to be that C++ sees char, signed char and unsigned char as three different types. Whereas int is the same as a signed int. Is this right or am I missing something?  #include &lt;iostream&gt;  using namespace std;  typedef   signed char       int8; typedef unsigned char      uint8; typedef   signed short      int16; typedef unsigned short     uint16; typedef   signed int        int32; typedef unsigned int       uint32; typedef   signed long long  int64; typedef unsigned long long uint64;  struct TrueType {}; struct FalseType {};  template &lt;typename T&gt; struct isX {    typedef typename T::ikIsX ikIsX; };   // This  int==int32 is ambiguous //template &lt;&gt;            struct isX&lt;int  &gt;    { typedef FalseType ikIsX; };  // Fails template &lt;&gt;            struct isX&lt;int32  &gt;  { typedef FalseType ikIsX; }; template &lt;&gt;            struct isX&lt;uint32 &gt;  { typedef FalseType ikIsX; };   // Whay isn't this ambiguous? char==int8 template &lt;&gt;            struct isX&lt;char  &gt;  { typedef FalseType ikIsX; }; template &lt;&gt;            struct isX&lt;int8  &gt;  { typedef FalseType ikIsX; }; template &lt;&gt;            struct isX&lt;uint8 &gt;  { typedef FalseType ikIsX; };   template &lt;typename T&gt; bool getIsTrue(); template &lt;&gt;           bool getIsTrue&lt;TrueType&gt;() { return true; } template &lt;&gt;           bool getIsTrue&lt;FalseType&gt;() { return false; }  int main(int, char **t ) {    cout &lt;&lt; sizeof(int8) &lt;&lt; endl;  // 1    cout &lt;&lt; sizeof(uint8) &lt;&lt; endl; // 1    cout &lt;&lt; sizeof(char) &lt;&lt; endl;  // 1     cout &lt;&lt; getIsTrue&lt; isX&lt;int8&gt;::ikIsX  &gt;() &lt;&lt; endl;    cout &lt;&lt; getIsTrue&lt; isX&lt;uint8&gt;::ikIsX  &gt;() &lt;&lt; endl;    cout &lt;&lt; getIsTrue&lt; isX&lt;char&gt;::ikIsX  &gt;() &lt;&lt; endl;     cout &lt;&lt; getIsTrue&lt; isX&lt;int32&gt;::ikIsX  &gt;() &lt;&lt; endl;    cout &lt;&lt; getIsTrue&lt; isX&lt;uint32&gt;::ikIsX  &gt;() &lt;&lt; endl;    cout &lt;&lt; getIsTrue&lt; isX&lt;int&gt;::ikIsX  &gt;() &lt;&lt; endl;  }   I'm using g++ 4.something I'm having a problem finding out a way to project from top view and the side view so I texture better. It just replaces the current projection with the top view    So is it possible to project from view twice on the same image? UV mapping a straight image around a ring shaped flat mesh Is it possible to mapping a image around a ring shaped mesh? I would imagine there should be a way to curve/distort the image around the mesh. If not, how can I straighten out the mesh easily instead to fit the image? Setting Objects to Null/Nothing after use in .NET Should you set all the objects to null (Nothing in VB.NET) once you have finished with them?   I understand that in .NET it is essential to dispose of any instances of objects that implement the IDisposable interface to release some resources although the object can still be something after it is disposed (hence the isDisposed property in forms), so I assume it can still reside in memory or at least in part?  I also know that when an object goes out of scope it is then marked for collection ready for the next pass of the garbage collector (although this may take time).  So with this in mind will setting it to null speed up the system releasing the memory as it does not have to work out that it is no longer in scope and are they any bad side effects?  MSDN articles never do this in examples and currently I do this as I cannot  see the harm. However I have come across a mixture of opinions so any comments are useful. Best way to get last identity inserted in a table Clarifying question on meaning of derivative of a linear transformation In my analysis class, my professor (and the textbook) reference the fact that $\mathbf{D}T(\bar{x}) = T$ for any $\bar{x} \in \mathbb{R}^n$ where $T \in \mathcal{L}(\mathbb{R}^n,\mathbb{R}^n) $. I'm wondering about the precise meaning of this statement. Is it that the differential of any linear transformation is also linear? Or is that the differential of any linear transformation is itself (and therefore also linear)? The second seems confusing to me because the derivative of a function is often not itself. For example, $f(x) = 2x$ is a linear transformation but $f'(x) = 2 \ne 2x$ for all $x \in \mathbb{R}$. postgresql - replace all instances of a string within text field Not a problem, but have been seeing questions recently of people not being offered upgrade from 18.04 to 20.04.  I thought these upgrades were offered back in June.  I upgraded to 20.04 from 19.10 and this was offered back in May.  Imagine that different versions paths have different schedules, but google does not help explaining(that I found).  Does someone know the schedules?