So i have an answer here, but i need to know how to ask question for that answer:     ans:he is his 3rd son.      que: which son is he to him??   Is that question framing correct? Thanks.
How to ask the position(?) of a president (like 10th) of a country, correctly?
Lets say I have a function defined as $f(x) = \sqrt {x^2}$. Common knowledge of square roots tells you to simplify to $f(x) = x$ (we'll call that $g(x)$) which may be the same problem, but it isn't the same equation. For example, say I put $-1$ into them:  $\begin{align} f(x) &amp;= \sqrt {x^2} \\ f(-1) &amp;= \sqrt {(-1)^2} \\ f(-1) &amp;= \sqrt {1} \\ f(-1) &amp;= 1 \end{align}$  $\begin{align} g(x) &amp;= x \\ g(-1) &amp;= -1 \end{align}$  thereby, we conclude that $f(x)$ and $g(x)$ do not produce the same results even though they are mathematically the same. This is also shown when we try to graph the functions:  :    :    :    From this, we can see that given $f(x) = \sqrt {x^2}$, when simplified is not the same as $f(x) = x$. So, is there any way to prove that $y = \sqrt {x^2}$ is not the same as $y = x$ for negative values, but is infact the same as $y = \mid x \mid$? What software can join videos? What's the best &amp; easiest GUI software that can join a video that I have downloaded in several parts? How to print line numbers to the log. Say when outputting some information to the log, I also want to print the line number where that output is in the source code. As we can see in the stack trace, it displays the line number where the exception has occurred. Stack trace is available on the exception object.  Other alternative could be like manually including the line number when printing to the log. Is there any other way? How to set default magnification for all windows Difference between sentences using 'was' and 'being'? What is the meaning of below sentences using was and being?     He committed suicide after he was severely beaten up by four teachers. He committed suicide after being severely beaten up by four teachers. In addition to the .tex and .dvi or .pdf files, TeX/LaTeX/ConTeXt produces and uses lots of other files. What do they do? Which ones are used to create the document and which ones are automatically created by the compiler. What files can I safely delete? Proof explanation $A\setminus B = A \setminus B \cap A$ $A\setminus B = A \setminus B \cap A$  Proof $A\setminus B \subseteq A \setminus B \cap A$:  $x\in A\setminus B \to x\in A \land x \notin B$  $x\in A \land (x \notin B \color{red}{\lor x\notin A})$  $x \in A \land (x\notin B \cap A)$  $x \in A\setminus B\cap A$  Proof $A \setminus B \cap A \subseteq A\setminus B$:  $x \in A \setminus B \cap A \to x\in A \land x\notin B \cap A$  $x \in A \land (x\notin B \color{red}{\lor x\notin A})$  $x \in A \land x \notin B$  $x \in A \setminus B $  Therefore $A\setminus B = A \setminus B \cap A$  I am confused how the stuff in the red is formed. Can we add an or statement to anything and ignore if its true or not if the first part of the or is true? I can't crack this one.      Prove: If $\gcd(a,b,c)=1$ then there exists $z$ such that $\gcd(az+b,c) = 1$ (the only constraint is that  $a,b,c,z \in \mathbb{Z}$ and $c\neq 0)$ What's the associated matrix of this linear operator? Let $V$ be a complex vector space of dimension $n$ with a scalar product, and let $u$ be an unitary vector in $V$. Let $H_u: V \to V$ be defined as  $$H_u(v) = v - 2 \langle v,u \rangle u$$  for all $v \in V$. I need to find the characteristic polynomial of this linear operator, but the only way to find it that I know of is using the associated matrix of the operator.  I don't know how to find this matrix because I don't know how to deal with the scalar product. Is there some other way to find the characteristic polynomial? If not, how can I find the associated matrix of this linear operator?  Thanks in advance. In the LaTeX document pasted below, I still see page numbers upon compiling, even though I've included a \pagestyle{empty} command below the \maketitle command. Any ideas what I'm doing wrong? Thanks in advance.  \title{Thesis Abstracts} \author{         Kurt Han Peek } \date{\today}  \documentclass[a4paper,11pt]{article} \usepackage{graphicx} \usepackage[margin=2cm]{geometry}  \begin{document} \maketitle \pagestyle{empty}  \section*{An Analysis of the Effects of Digital Phase Errors on the Performance of a FMCW-Doppler Radar} In modern frequency-modulated continuous-wave (FMCW) radars, the transmitter is increasingly being implemented using direct digital chirp synthesis (DDCS), which provides improvements in sweep linearity, stability, precision, agility, and versatility over analog techniques. Its main limitations are errors due to sampling of the modulating signal, phase truncation, and digital-to-analog converter (DAC) quantization, which produce spurious signals due to their deterministic nature. This thesis presents an analysis and simulation of the effect of two sources of digital phase errors -- namely, sampling of the modulating signal and phase truncation -- on the performance of a FMCW-Doppler transceiver employing a DDCS in its transmitter. An upper bound for the amplitude of spurious targets resulting from these digital phase errors is established. Further, it is shown that provided the phase errors are periodic with the sweep period, the spurious targets are not offset from the target in Doppler. An algorithm for selecting the digital chirp parameters of a DDCS so as to ensure periodic and phase-continuous sweep transitions, which guarantee periodic phase errors and reduce `spectral splatter', is devised. Finally, we investigate parallels of FMCW radar in the optical domain, and consider the fundamental question whether range resolution is fundamentally limited by the bandwidth of a transmitted signal, or its carrier frequency.  \section*{Estimation and Compensation of Frequency Sweep Nonlinearity in FMCW radar} One of the main issues limiting the range resolution of linear frequency-modulated continuous-wave (FMCW) radars is nonlinearity of frequency sweep, which results in degradation of contrast and range resolution, especially at long ranges. Two novel, slightly different, methods to correct for frequency sweep nonlinearity by digital post-processing of the deramped signal were introduced independently by Burgos-Garcia et al. (Burgos-Garcia, Castillo et al. 2003) and Meta et al. (Meta, Hoogeboom et al. 2006). In these publications, however, no formal proof of the techniques was given, and no limitations were described. In this thesis, we prove that the algorithm of Meta is exact for temporally infinite chirps, and remains valid for finite chirps with time-bandwidth products which are large compared to unity, provided the maximum frequency component of the phase error function is small compared to the square root of the nominal chirp rate. It is also shown that the algorithm of Meta reduces to that of Burgos-Garcia in this limit. A digital implementation of both methods described. Finally, we propose a novel method to measure the systematic phase errors which are required as input to the compensation algorithm.  \end{document} Make IN clause behave like AND clause I've asked the same question on , but got no answer, so that's why I'm asking it here (I've read that simple sql questions should be asked at SO, but I have no choice).  Suppose I have 3 tables: posts, post_categories and categories. I'm implementing some kind of page with filters where I can filter posts by categories. User can select multiple categories. When he selects more than one, they should be summed. I can't get it working. What I have now is simple IN() SQL clause:  SELECT posts.id, post_categories.category_id FROM posts  JOIN post_categories ON posts.id = post_categories.post_id  WHERE post_categories.category_id IN (1,2,3) LIMIT 10;   But it is not matching all ids, it is OR and I need AND. What I need here is to find all posts that have categories with id=1 AND id=2 AND id=3, but instead this query returns posts that have at least one category from the IN() list. xgboost tayler expansion detail This is the objective function for Xgboost.  I have no idea where $g_{i}$ and $h_{i}$ came from  is some one explain how this two terms came form?  or direct me to the related tutorial page then I will search for it Return single column from a multi-dimensional array IANAL et al. is usually the start of answers to [legal] questions on any SE site (e.g. ). You usually end up with personal opinions which sound sensible and the accepted answer is often the one closest to the OP's hopes. But it's probably often not correct. I know the situation can be rather complicated and even differ by location, but it would be really helpful if the SE team were providing official lawyers (indicated maybe by using a § in the username, forbidden for everyone else, just like the mod ♦) to provide some orientation that one could actually rely on beyond much doubt... Equivalent of "both" when referring to three or more items? What would be the correct word to use when referring to three or more items, in the same manner as the word both?  For example, using two words, with the word both:     "There are several recommendations I   have to further improve the sites —   both to improve their profit, and   decrease their cost."   Using three words, with a blank space in place of the correct word:     "There are several recommendations I   have to further improve the sites —   _ to improve their profit, decrease their cost and improve their   usability."   So, what would be the correct word to use in place of the __? I'm really interested in quantum theory and would like to learn all that I can about it. I've followed a few tutorials and read a few books but none satisfied me completely. I'm looking for introductions for beginners which do not depend heavily on linear algebra or calculus, or which provide a soft introduction for the requisite mathematics as they go along.  What are good introductory guides to QM along these lines? iPad won't turn on after a complete battery drain My iPad mini retina is out of warranty,  I forgot to plug in my iPad after the critical battery warning and it went some minutes following the warning. I was not worried because same thing had happened a few times before and iPad was perfectly okay after charging it. However, this time I remember seeing a flash of blue screen immediately after it went black. Then, I plugged it in an waited for a while and tried starting it. Ever since, it fails to start hanging on the apple logo with white background screen.   I tried to connect to my Mac but nothing showed up in iTunes. I am worried now. Can I troubleshoot this or is it broken and needing repair? So, I was messing around in the Skyrim console and increased my speed a lot. Now I want to reset my speed but the command player.setav speedmult 100 isn't working. Is there any way to reset this? Gay rights – does Star Trek ever address this issue? I am aware of the picture of the man wearing the 'dress' from TNG, which promotes gender equality, but are gay rights ever addressed in Star Trek at all? (All seasons accepted.) The habitual past "would" versus "simple past" tense One can use the habitual past 'would' to describe past events. For example one could say,      Jim would spend weekends at the farm. He would cut weed and water the plants.   One can also use the simple past.       Jim spent weekends at the farm. He cut weed and watered the plants.   Would the use of habitual past be considered "weak" writing or prose, as opposed to the simple past?  How does the simple past in the following sentence "He would cut weed and water the plants." without any preceding time expression e.g. "spend weekends"  make a distinction between a repeated action/habit and a one-time action of cutting weeds and watering plants?