What does variable |= 0; do?
What's the function of the "|" (pipe) operator?
Find the value of $\binom{2000}{2} + \binom{2000}{5} + \binom{2000}{8} + \cdots \binom{2000}{2000}$ Find the value of $\binom{2000}{2} + \binom{2000}{5} + \binom{2000}{8} + \cdots \binom{2000}{2000}$     I've seen many complex proofs. I am looking for an elementary proof. I know the fact that $\binom{2000}{0} + \binom{2000}{1} + \binom{2000}{2} + \cdots \binom{2000}{2000} = 2^{2000}$. This may help here. Is RSA key size the size of private key exponent? I have implemented a key pair generation scheme for RSA algorithm. I have taken the length of private key exponent as RSA key size, but then I've got to know that RSA key size is the size of the modulus $n$. Is it wrong to take RSA key size as the size of the private key exponent?  To generate a private key of 1024 bits, I increase the size of private key exponent regardless the modulus $n$. Rendering low and res/high res images Is it possible to render in Blender with reduced low res files and then ask Blender to render the final render with 4k high res files? Low and high res files would be numbered identically but would be located in different folders.  Many thanks How much does it cost to have a custom ASIC made? Equivalence Relation problem Let $S$ be the relation on $\mathbb{N} \times \mathbb{N}$ defined by $(a,b)S(c,d)$ if and only if $ad=bc$. Prove this is an equivalence relation on $\mathbb{N} \times \mathbb{N}$.  I think I've found this to be reflexive and symmetric, but I'm stuck on transitivity. Can someone check my work so far and assist with testing transitivity?  Reflexive: Let $(x,y)S(x,y)$. Then $xy = yx$. So $S$ is reflexive. Symmetric: Suppose $(a,b)S(c,d)$. Then, $ad = bc$. Therefore, $da = cb$ and $cb = da$. Therefore, $(c,d)S(a,b)$. Thus, $S$ is symmetric.  Transitive: Suppose $(a,b)S(b,c)$. Then $ac = bb$.   This is where I'm stuck. Any ideas? Let $L$ be a finitely generated free module over a commutative ring $A$. Set $n=\operatorname{rank} L$. Let $x_1,\dots,x_m$ be generators of $L$. Then $m \ge n$? If $m = n$, then is $x_1,\dots,x_m$ a basis of $L$? Exponential of the differential operator I am not sure whether this question is even well-posed. But today I learnt that $e^Df(x) = f(x+1)$ where $D$ is differential operator and  $$e^D \triangleq \sum_{i=0}^{\infty} \frac{D^i}{i!}.$$  (ref. )  So I was curious as to whether the differential equation  $$\frac{df(t)}{dt} = e^Df(t) = f(t+1)$$  has any solutions apart from $f = 0$? What species of insect is this? I saw this insect near a beach in Corsica France: Finding Zero Divisors and Units of Rings $\mathbb{Z}_n \times \mathbb{Z}_m$ I have used the package nomencl. In my code I have incorporated the \nomenclature as well. But when I compile, it is not showing me any results. My code is :  \documentclass[pdftex,12pt,a4paper]{report} \usepackage{nomencl} \makenomenclature \begin{document} ---page numbering and other things included----- \include{symbols} -----others--- \end{document}   symbols.tex  \chapter*{NOTATIONS} \addcontentsline{toc}{chapter}{Notations} \nomenclature{$G_e$}{Equivalent Shear Modulus} \printnomenclature   Shall I use \nomenclature while explaining the chapter. Will It give me the same results? like  \documentclass[pdftex,12pt,a4paper]{report} \usepackage{nomencl} \makenomenclature \begin{document} ---page numbering and other things included----- \include{symbols} \include{chap3-calculate_w} -----others--- \end{document}   symbols.tex  \chapter*{NOTATIONS} \addcontentsline{toc}{chapter}{Notations} \printnomenclature   chap3-calculate_w.tex  \chapter{SETTLEMENT RESPONSE OF GEOSYNTHETIC REINFORCED GRANULAR FILL SOFT SOIL} \label{chap:W} \section{INTRODUCTION} In this thesis, the methodology to determine an  Equivalent single layer shear modulus \nomenclature{$G_e$}{Equivalent Shear Modulus},$G_e$ value for a...etc   I am not getting the output in both these. Can someone help me with this? Object selection box in addon In my addon panel, how can I have an object selection box?  And what data type would I use to store the the selected object?  It should look like this: Prove that Gauss map on M is surjective How to identify predatory publishers/journals What are the best strategies for assessing if a journal is a &quot;vanity&quot; or &quot;predatory&quot; journal that should be avoided (both for publishing in and reviewing for)? For example, how would one go about determining if a journal/publisher belongs on ? How does one decide whether to use who or that? In the title example:   The correct pronoun is who. The correct pronoun is that. Either is correct. They're interchangeable. It depends on the context. Repeated error  I have been repeatedly encountering the following error when my laptop freezes. It may occur multiple times during the day, and should take several minutes to recover.  Source Windows  Summary Hardware error  Date ‎11/‎1/‎2018 10:11 AM  Status Not reported  Description A problem with your hardware caused Windows to stop working correctly.  Problem signature Problem Event Name: LiveKernelEvent Code:   144 Parameter 1:    3003 Parameter 2:    ffffad8c0b1a75c8 Parameter 3:    40010002 Parameter 4:    0 OS version: 10_0_17134 Service Pack:   0_0 Product:    256_1 OS Version: 10.0.17134.2.0.0.256.48 Locale ID:  1033   More details  Per my best reflection, the hardware error occurs when I am using the laptop on its docking station. Here is what I observe:   While using the laptop as normal, the "freeze" will occur and I can hardly operate the laptop: mouse cursor won't move continuously and Alt+Tab won't be able to switch across two windows. Shortly after (usually several minutes), things will go back to normal. Upon inspecting through the Performance tab in Task Manager, there had been no spike in anything, be it CPU, Memory or Disk. When I open the View reliability history tool, I will find another entry of Windows Hardware Error   Machine specification:   Thinkpad T480, i7-8650U (newly replaced module thanks to the manufacture)   With the same pair of docking stations, an older unit of T480 has had exactly the same problem.  Docking station: . Same issue has occurred on two identical docks with  slightly different monitors and accessories. OS specification: OS build 17134.376. Most recent collection of "reliability report":    Possible solutions   First, I would like to learn what exactly went wrong, hardware-wise: the manufacturer's support team did not bother too much with the precise error messages, and have ordered me to reinstall the Windows OS up to 5 times in total. [This is what they called "debugging".]   Please advise if you can decipher the recurrent "Hardware error".   Secondly, I will try to push the manufacturer to replace the docking stations, and see if the same "freezing problem' persists. I'm doing some web scraping and sites frequently use HTML entities to represent non ascii characters. Does Python have a utility that takes a string with HTML entities and returns a unicode type?  For example:  I get back:  &amp;#x01ce;   which represents an "ǎ" with a tone mark.  In binary, this is represented as the 16 bit 01ce.  I want to convert the html entity into the value  u'\u01ce' First, this is about beta sites that might not even exist later, but imagine they will for the sake of this question. Taking the latest 10 questions on , I see 3 questions related to Mac OS X:    From my point of view, all 3 could also fit on , the 1st one could also be on SO, the 2nd and 3rd could also be on SU. How we will deal with this type of situation? When should we close a question as &quot;Belongs on [...]&quot;? I already see identical questions asked on 2 or 3 different sites (by the same or different users). Is there anything planned to link these questions or something like that? I mean something else than the already existing &quot;Linked&quot; feature. Something to include answers from site X in site Y or something like this. How do I delete all my mail from my Gmail account? Why crontab scripts are not working? Non superscript citations for figures Is it possible to reference figures like so   taken from \emph{ref. \cite{reference}}   with non superscript numbers, while leaving the rest of the references in the bulk of the document untouched? This is using BibTeX and the .