How the cost function of logistic regression given below is derived ?  Here Sigmoid function is used for logistic regression. After heavy search on internet i have come up here to put on my question , no other question on this portal talks about it so please don't mark it as duplicate.
There is a statement that maximizing the likelihood is equivalent to minimizing the cross-entropy.  Are there any proof for this statement?
Complex numbers equation: $z^4 = -16$ Can I gift games from my library? I understand that once a game is in the library it cannot be removed, only hidden.  My question is, can I gift an uninstalled copy of a game to a friend?  I understand that this may be impossible due to legal issues, and that once a game is in the library it stays. Using shell's read command with live editing functionality (readline like) Is there a standard (POSIX) way of asking the user some data from within a shell script, with  for example, while allowing live edition of the text being typed (what readline does)?   I know bash has read -e varname that allows for the person launching the script to use keyboard arrows for exemple, to edit or correct what has just been typed without deleting the last entered characters with backspace.   However, read -e is . And still, it is pretty cumbersome to delete all that has been wrote if you realise you made a mistake at the beginning of your long sentence... Where to or how to ask "Best of" Questions for what software tools to use I am wondering where is a good place or how is a good way to ask a question on what software tools to use for a specific task.  A lot of the time when searching for information on these types of problems stackoverflow comes up with the top and actually useful discussions and answers, but the question is locked.  I completely understand the point of discouraging "best of" or list questions as discussed in this question () and agree they often aren't constructive enough to justify the effort in moderating them.  What I am wondering is where is the right place to have a discussion like this? Stackoverflow has a good and active community so perhaps there is another section or related site I can ask less specific questions. And I do find some value in reading people's opinions, even if subjective on issues like this.  Here are some examples of the kind of questions I mean:    and   First results in google when searching for those subjects and some useful discussion.  Perhaps asking a question like this but with a strict and detailed specification it might be considered a valid and reasonable question to ask on stackoverflow; perhaps it's just a case of re-thinking how to ask a question. Or perhaps there's a more relevant place for this kind of discussion. Please let me know what you think. How to put small numbers outside the pie slices without making a mess I'm new to LaTeX and need some help with my figures. I would like to to put small numbers outside the pie slices without making a mess. Having found this thread , I tried it out, but didn't get all the way through with it. Right now, my pie looks like this:    I used the following code:  \usepackage{tikz,stackengine} \makeatletter  \tikzstyle{chart}= [legend label/.style={font={\scriptsize},anchor=west,align=left}, legend box/.style={rectangle, draw, minimum size=5pt}, axis/.style={black,thin,-&gt;}, axis label/.style={anchor=east,font={\tiny}}]  \tikzstyle{bar chart}=[ chart, bar width/.code={     \pgfmathparse{##1/2}     \global\let\bar@w\pgfmathresult}, bar/.style={thin, draw=black}, bar label/.style={font={\bf\small},anchor=north}, bar value/.style={font={\footnotesize}}, bar width=.75,]  \tikzstyle{pie chart}= [chart, slice/.style={line cap=round, line join=round, thin, draw=black}, pie title/.style={font={}}, slice type/.style 2 args={     ##1/.style={fill=##2},     values of ##1/.style={}}]  \pgfdeclarelayer{background} \pgfdeclarelayer{foreground} \pgfsetlayers{background,main,foreground}  \newcommand{\pye}[3][]{     \begin{scope}[#1]         \pgfmathsetmacro{\curA}{90}         \pgfmathsetmacro{\r}{1}         \def\c{(0,0)}         \node[pie title] at (270:1.3) {#2};         \foreach \v/\s in{#3}{             \pgfmathsetmacro{\deltaA}{\v/100*360}             \pgfmathsetmacro{\nextA}{\curA + \deltaA}             \pgfmathsetmacro{\midA}{(\curA+\nextA)/2}             \path[slice,\s] \c             -- +(\curA:\r)             arc (\curA:\nextA:\r)             -- cycle;             \pgfmathsetmacro{\d}{max((\deltaA * -(.5/50) + 1) , .5)} %%%              %\pgfmathprintnumberto{#3}{\fnum}  %\ifthenelse{\lengthtest{#3 pt &lt; 10pt}} %{\path (O) -- ++ (\midangle:\radius) %node[inner sep=0,align=left,anchor=west,rotate=\midangle](nums#3)    {~~\fnum};} %{\path (O) -- ++(\midangle:\innerpos)  %node {\scalefont{#3}\beforenumber\fnum\afternumber};}           %%%                         \begin{pgfonlayer}{foreground}                 \path \c -- node[pos=\d,pie values,values of \s]{$\v\%$} +    (\midA:\r);             \end{pgfonlayer}             \global\let\curA\nextA}     \end{scope}}  \newcommand{\legend}[2][]{     \begin{scope}[#1]         \path         \foreach \n/\s in {#2}         {++(0,-5pt) node[\s,legend box] {} +(2pt,0) node[legend label]     {\n}};     \end{scope}}      \begin{document}         \begin{center}      \begin{figure}[H]     \centering \begin{tikzpicture} [pie chart, slice type={A}{black!0}, slice type={B}{black!10}, slice type={C}{black!20}, slice type={D}{black!30}, slice type={E}{black!40}, slice type={F}{black!50}, slice type={G}{black!60}, slice type={H}{black!70}, pie values/.style={font={\small}}, scale=2]  \pye{1994}{6.1/A, 2.9/B, 0.2/C, 54.3/D, 25.2/E, 1.4/F, 6.7/G, 3.2/H} \pye[xshift=2.2cm,values of coltello/.style={pos=1.1}]% {2005}{6.8/A, 3.0/B, 0.5/C, 52.3/D, 25.6/E, 1.8/F, 7.1/G, 2.8/H} \pye[xshift=4.4cm,values of caffe/.style={pos=1.1}]% {2015}{7.2/A, 3.1/B, 0.6/C, 50.6/D, 26.5/E, 2.3/F, 7.5/G, 2.2/H}  \legend[shift={(0cm,-1.5cm)}]{{A}/A, {B}/B}  \legend[shift={(1.4cm,-1.5cm)}]{{C}/C, {D}/D}  \legend[shift={(2.8cm,-1.5cm)}]{{E}/E, {F}/F} \legend[shift={(4.2cm,-1.5cm)}]{{G}/G, {H}/H} \end{tikzpicture} \end{figure}  \end{document}   This looks quite the way I want it to, only the small numbers should be placed outside the pies.  Thanks in advance :) Let $\displaystyle \sum a_n$ be  convergent series of positive terms and set  $\displaystyle b_n=\sum_{k=n}^{\infty}a_k$ , then prove that $\displaystyle\sum \frac{a_n}{b_n}$ diverges.   I could see that $\{b_n\}$ is monotonically decreasing sequence converging to $0$ and I can write $\displaystyle\sum \frac{a_n}{b_n}=\sum\frac{b_n-b_{n+1}}{b_n}$, how shall I proceed further? If Steve Rogers wasn't so weak, could Cap be even stronger? Proving by induction that $ \sum_{k=0}^n{n \choose k} = 2^n$ I'm planning to buy a new Android phone and I'd like to be able to transfer the progress I have made at several games to the new device, so I won't have to play all the levels again from scratch.  I currently have an HTC Desire and will buy an HTC Evo 3D.  There's probably no simple/general solution to this.  But I was wondering if anyone did something similar and could provide some advice (e.g. where to look for files that might hold game progress information). $\mathbb Z[1/2]$ is not finitely generated ?   Maybe I misunderstood, what finitely generated means.  it says, we need finitely many elements and I think $1$ and $1/2$ suffices as generators.  But , at the end of Proposition $5.1.4$, from the fact that $\mathbb Z[1/2]$ is not finitely generated it is concluded that $1/2$ is not integral. Broken URL for data.stackexchange.com help page Proving $\sup B=\inf A$ Let $A$ be a nonempty and bounded below, and define $B= \{b\in \mathbb{R}: b$ is a lower bound for $A\}$. Show sup$B$ = inf$A$.   So far I have: Let $A$ be nonempty and bounded below. This implies $\exists l \in A, \forall a \in A, l\leq a$ and $l =$ inf$A$. Let $B = \{b\in \mathbb{R}: b$ is a lower bound for $A\}$ and let $M=$ sup$B$. This implies $\exists b\in B, \forall b \in B, b\leq M$. By definition of $B$, we know inf$A \in B$. Since $b\leq$ sup$B$, with sup$B$ the largest element in $B$, we have inf$A \leq M =$ sup$B$, so inf$A \leq$ sup$B$.  I realize I have to prove that sup$B \leq$ inf$A$ to get that sup$B$=inf$A$. I am having trouble doing so.   Part B: Use (a) to explain why there is no need to assert that greatest lower bounds exist as part of the Axiom of Completeness. Dividing polynomial $f(x)$ by $x-3$ and $x+6$ leaves respective remainders $7$ and $22$. What's the remainder upon dividing by $(x-3)(x+6)$? I want to setup a new virtual machine with some specified packages (name and version), that are provided.  For example apache2 in version 2.2.20-1ubuntu1 with all dependencies. Even if there is a new version of this package on the servers this one should be installed.  The solution has to work/scale with multiple (n) "setups". Another virtual machine might need an older version of apache2.  I currently know of some possibilities that install the exact packages, but do not scale that good:   Copy all required *.deb to every virtual machine manually and enter: dpkg -i ... -> Could work, but it is very error prone. (Manual scripts etc.) Create and use a new Ubuntu repository for each setup. -> Does not work because I would need n repositories. Setup the machine once and copy the VM / create a snapshot. -> Does not work because I would need to store n VMs.   My problem could be labeled as patch management, but I do not want to update my packages to the current version. My goal is to install old packages. Oracle SQL Syntax: Quoted identifier Money transfer from India to USA My daughter is a US Citizen. We (parents)are also planning to permanently settle in US. We will go their as Permanent Resident,as per US Law. We have a house here in India worth Rs.2 Crores.  We want to sell it and take money with us. We have ordinary savings Bank account here in India.  My US Citizen Daughter is a Income Tax payer.  Now your advises are required as under:   What are the routes to take money from India to US and our tax liabilities Can I gift this whole amount to my US Citizen Daughter or what is the maximum limit of Gift amount.  What will be the tax liability on me and on my Daughter in case of Gift   Whether I have to show it in my Income Tax Return or in my Daughter's Tax Return. What US Income Tax Laws says. What will be the procedure to send money as Gift to my Daughter. Will the money will go directly from my Bank Act.to my Daughter's Bank Account. Will there will be wire transfer from bank to bank Can I send money through other money sender Certified Companies also.    Please advise. If there is something more on the subject to make me understand, please advise. Thanks, An equality involving the mean value property I am trying to establish that  $$\frac2{\pi}\int_0^{2\pi} f(e^{i\theta})\cos^2\left(\frac{\theta}{2}\right)\;\mathrm d\theta=2f(0)+f^\prime(0)$$  for $f$ analytic on $|z| &lt; R,\; R &gt; 1$  I think I am making a mistake somewhere, please point it out to me.Here's what I've done unsing the mean value property and Cauchy's integral formula:  $$2f(0)+f&#39;(0)=2(\frac1{2\pi}\int_0^{2\pi} f(e^{i\theta})d\theta + \frac1{2\pi i}\int_0^{2\pi}\frac{f(e^{i\theta})ie^{i\theta}}{e^{i2\theta}}=\frac{1}{\pi}\int_0^{2\pi} f(e^{i\theta})d\theta\frac1{2\pi}\int_0^{2\pi}\frac{f(e^{i\theta})}{e^{i\theta}}\quad\text{(1)}$$  But when I develop the expression $\frac2{\pi}\int_0^{2\pi} f(e^{i\theta})\cos^2(\frac{\theta}{2}) d\theta$ I get:  $$\frac1{\pi}\int_0^{2\pi} f(e^{i\theta})d\theta+\frac1{2\pi}\int_0^{2\pi}f(e^{i\theta})[{e^{i\theta}}+e^{i-\theta}]d\theta \quad\text{(2)}$$  The second integral in (1) differs from the second integral in (2). Thanks. Algebra of LDA. Fisher discrimination power of a variable and Linear Discriminant Analysis Apparently,     the Fisher analysis aims at simultaneously maximising the   between-class separation, while minimising the within-class   dispersion. A useful measure of the discrimination power of a variable   is hence given by the diagonal quantity: $B_{ii}/W_{ii}$.     I understand that the size (p x p) of the Between (B) and Within-Class (W) matrices are given by the number of input variables, p. Given this, how can $B_{ii}/W_{ii}$ be a "useful measure of the discrimination power" of a single variable? At least two variables are required to construct the matrices B and W, so the respective traces would represent more than one variable.  Update: Am I right in thinking that $B_{ii}/W_{ii}$ is not a trace over a trace, where the sum is implied, but the matrix element $B_{ii}$ divided by $W_{ii}$? Currently that is the only way I can reconcile the expression with the concept. Specifying geography with cardinal directions So here in America we are in "the West".  Would this become "the Western world" or "the western world? "  I'll assume the same applies to "the East" (Asia). Prove by induction $\sum_{i=1}^ni^3=\frac{n^2(n+1)^2}{4}$ for $n\ge1$ Prove the following statement $S(n)$ for $n\ge1$:  $$\sum_{i=1}^ni^3=\frac{n^2(n+1)^2}{4}$$  To prove the basis, I substitute $1$ for $n$ in $S(n)$:  $$\sum_{i=1}^11^3=1=\frac{1^2(2)^2}{4}$$  Great. For the inductive step, I assume $S(n)$ to be true and prove $S(n+1)$:  $$\sum_{i=1}^{n+1}i^3=\frac{(n+1)^2(n+2)^2}{4}$$  Considering the sum on the left side:  $$\sum_{i=1}^{n+1}i^3=\sum_{i=1}^ni^3+(n+1)^3$$  I make use of $S(n)$ by substituting its right side for $\sum_{i=1}^ni^3$:  $$\sum_{i=1}^{n+1}i^3=\frac{n^2(n+1)^2}{4}+(n+1)^3$$  This is where I get a little lost. I think I expand the equation to be  $$=\frac{(n^4+2n^3+n^2)}{4}+(n+1)^3$$  but I'm not totally confident about that. Can anyone provide some guidance?