While studying reflection through a plane mirror, I have been told that when the object is real the image will be virtual and the image will be real while the object is virtual.  What are virtual objects? Do they really exist in nature?
I'm doing magnification and lens in class currently, and I really don't get why virtual and real images are called what they are.  A virtual image occurs the object is less than the focal length of the lens from the lens, and a real image occurs when an object is further than focal length.  By why virtual and real? What's the difference? You can't touch an image no matter what it's called, because it's just light.
Windows update giving error 80070002 and manual update not working how to update? I have been trying to update my pc from 20h1 to 20h2 but the windows update shows error message 80007002 and when I manually install the update using windows 10 update assistant it downloads completely and then even starts install and gets stuck at 87 percent on the screen where it says do not turn off. What to do? This is the log file. Matching Profile found: FindRollbackFailure - 3A43C9B5-05B3-4F7C-A955-88F991BB5A48 SetupDiag version: 1.6.0.0 System Information:     Machine Name =      Manufacturer =      Model =      HostOSArchitecture = 1033     FirmwareType = UEFI     BiosReleaseDate = 20160513000000.000000+000     BiosVendor = A12     BiosVersion = A12     HostOSVersion = 10.0.19041     HostOSBuildString = 19041.1.amd64fre.vb_release.191206-1406     TargetOSBuildString = 10.0.19041.621 (vb_release_svc_prod1.201014-1758)     HostOSLanguageId =      HostOSEdition = CoreSingleLanguage     RegisteredAV = Quick Heal Internet Security     FilterDrivers = FileInfo     UpgradeStartTime = 12/20/2020 2:54:38 PM     UpgradeEndTime = 12/20/2020 5:30:40 PM     UpgradeElapsedTime = 02:36:02     CV = Pi3IdU3SlEqpFx93     ReportId =    Error: SetupDiag reports rollback failure found. Last Phase = Pre OOBE Boot Last Operation = OOBE boot apply Error = 0xC1900101-0x4000D LogEntry:   Refer to &quot;https://docs.microsoft.com/en-us/windows/desktop/Debug/system-error-codes&quot; for error information.  Last Setup Phase: Phase Name: Pre OOBE Boot     Phase Started: 12/20/2020 4:37:50 PM     Phase Ended: 1/1/0001 12:00:00 AM     Phase Time Delta: 00:00:00     Completed Successfully? False   Last Setup Operation: Operation Name: OOBE boot apply     Operation Started: 12/20/2020 4:37:50 PM     Operation Ended: 1/1/0001 12:00:00 AM     Operation Time Delta: 0:00:00:00.0000000     Completed Successfully? False What is the difference between the search() and match() functions in the ?  I've read the  (), but I never seem to remember it.  I keep having to look it up and re-learn it.  I'm hoping that someone will answer it clearly with examples so that (perhaps) it will stick in my head.  Or at least I'll have a better place to return with my question and it will take less time to re-learn it. Install on Second Hard Drive with startup boot option? In theory, what I want to do is very simple, but I've done some google research and read some posts on this site and I'm still confused.   I want to keep win7 on my primary C:\ drive, and install Ubuntu (12.10) on a secondary drive (that I have yet to buy or install. I just want to be certain that I'm successful when I do this). If at all possible I'd also like it if when I start the computer I have the option to boot either Win7 or Ubuntu, like an account log in type thing where I just pick which drive I'm running with that day. And just to reiterate, I do not want to partition my C:\ drive or alter, change, or affect my Win7 system in any way.  The reason I'm still fairly uncertain about all of this, is that a lot of the guides I've read have been for earlier versions of Ubuntu. Also the vast majority of guides are about partitioning one drive to fit both OS'. I want two separate OS running from two separate drives that I can pick at boot. The guides I did find mentioned that there is some issue with the grub being installed on C:\ but the rest of the OS being installed on the other drive. Is this still the case? How do I avoid that?  Also, if it helps, I've been running 12.04lts on my laptop for a few months now in and out of class and I love it.  Finally, if you could please point me to a guide or write your answer in the form of a guide for a two year old that'd be awesome. I'm still an Ubuntu novice and I don't want to mess up my win7. Is there an alternative to the word "coincidence" to describe when two say the same word spontaneously? Is there a word to describe a scenario where two people having a conversation utter the same word/phrase together, simultaneously, and unconsciously? Something else than just a coincidence. Is there a reliable way to determine the OS upgrade history leading to Windows 10 or if it was a fresh install? I have been around on Twitter for awhile and I regularly find people using "Please to retweet this" or "Please to help me", etc. Is this proper English? I do not think prefixing the infinitive form with a 'to' is necessary here at all. In Radian mode, continually pressing the $\cos$ function of a number causes the result to converge to $x=0.739085133$, a fixed point of $\cos(x)$. Repeating this behavior with the $\sin$ function causes the result to converge to $x=0$, a fixed point of $\sin(x)$.   What happens if this is done using the tan function? It seems that $x=0$ is a repelling fixed point with no convergence in this case.  What would be the best way to explain these results mathematically? To make sure that we are talking about the same, I would like to post the relevant definitions I know first.  Definitions:  A pair $(G, +)$ where $G$ is a set and  $+: G \times G \rightarrow G$  is called a commutative group if it has all of the following three characteristics:   (associativity): $\forall x,y,z \in G: (x+y)+z=x+(y+z)$ (identity element): $\exists e \in G \forall x \in G: e + x = x = x +e$ (inverse elements): $\forall x \in G \exists x^{-1} \in G: x^{-1} + x = e = x + x^{-1}$ (commutativity): $\forall x,y \in G: x+y=y+x$.     A triple $(\mathbb{K}, +, \cdot)$, where $\mathbb{K}$ is a set and  $+: \mathbb{K} \times \mathbb{K} \rightarrow \mathbb{K}$  $\cdot: \mathbb{K} \times \mathbb{K} \rightarrow \mathbb{K}$  is called a field if it has all of the following three characteristics:   $(\mathbb{K}, +)$ is a commutative group $(\mathbb{K} \setminus \{0\}, \cdot)$ is a commutative group distributive properties: $\forall x,y,z \in \mathbb{K}: x \cdot (y+z) = (x \cdot y) + (x \cdot z)$ and $\forall x,y,z \in \mathbb{K}: (y+z) \cdot x = (y \cdot x) + (z \cdot x)$     $\mathbb{N^+} := 1, 2, 3, 4, 5 ...$ (all positive integers)  $\mathbb{N}_0 := \{0\} \cup \mathbb{N}$ (the natural numbers with zero)  $\mathbb{N}$ can be both: $\mathbb{N^+}$ or $\mathbb{N}_0$.    My question: Does a field exist that has $\mathbb{N}$ as its set?  I know that $(\mathbb{Q}, +, \cdot)$ is a field and I know that $(\mathbb{N}, + )$ is only a commutative semigroup.   But maybe it's possible to define two mappings $\circ, *$ for $\mathbb{N}$ in such a way that $(\mathbb{N}, \circ, *)$ is a field.    Related:     The smallest field containing the integers is the field of rational   numbers.   Source: en-Wikipedia:   I don't know if this is true. If somebody can prove (or at least scribble the proof of) the quote it would be a good answer, I guess. Minipage: 2 tables side by side isn't working Allowed Wave Functions of System Given a single-particle system with Hamiltonian $H$, what constraints can be put on the wave function at a particular point in time $\psi(x)$? Of course $\psi(x)$ must obey boundary conditions given by $H$. However, in situations where $H$ does not yield strict boundary conditions (e.g. the harmonic oscillator) can $\psi(x)$ be any normalised function? My intuition says no for the following reason: QM textbooks say that any valid wave function can be represented as a linear combination of the eigenstates of $H$. In the case of the harmonic oscillator, the eigenstates of $H$ are a countably infinite set. However the set of all normalised functions is uncountably infinite. It seems to me that one cannot change between one complete set of basis functions to another and reduce the "dimensionality" somehow. This makes me think that the eigenstates of $H$ are not complete, that they imply some constraints on $\psi(x)$. This is not an area of mathematics that I understand well. Can somebody help me out? How to add PlotLegends in a Show function? I'm having trouble getting basic APA bibliographies to work. The same thing works with nejm style, but changing the style to apa gives errors:  ! Undefined control sequence. &lt;argument&gt; \mkbibdateapalongextra  {year}{month}{day}\iffieldundef {endyear}{...  Package biblatex Warning: Macro 'name:apa:last-first' undefined. (biblatex)                Using \newbibmacro.  Package biblatex Warning: Macro 'name:apa:first-last' undefined. (biblatex)                Using \newbibmacro.   I'm using texlive 2012, as you can see I'm using biber, yes, I reran biber, and also tried delting the .bbl file and rerunning tools. My source markup is:  \documentclass[12pt]{article}  \usepackage[hidelinks]{hyperref} \usepackage[backend=biber,style=apa]{biblatex}  \addbibresource{BibResource.bib}  \begin{document}  test\parencite{ormrod2004human}  \clearpage  \printbibliography  \end{document} Newton's Law of Graviation: Why $G$ and not e.g. $\dfrac{1}{4\pi G_0}$? How to perform a true GIS clip of polygons layer using a polygon layer in R? Which site deals with career development questions for programmers? Is there any site in the network where career development questions about the programming profession can be asked ? I thought Programmer's was the site for it, but that doesn't appear to be the case. What are recursive triggers? How can recursive triggers be controlled? What are recursive triggers? How can we control them, or prevent infinite recursion?  Are these any risks inherent in them, and any limits that we should be aware of?  Please explain with an example. Let ($x_s$) be a convergent sequence, where $x_s&gt;0$ for all s, and $y_s$ be a sequence such that   $$\displaystyle \Large y_s=\frac{s}{\frac{1}{x_1}+...+\frac{1}{x_s}}$$  Prove $\displaystyle \lim x_s$=$\lim y_s$  So I know things like $x_s$ is bounded, and that it's Cauchy, but I can't think of anything to solve it.  Any help is appreciated. Find this integral $$\operatorname I=\int\limits_{0}^{1}\dfrac{\arcsin{\sqrt{x}}}{x^4-2x^3+2x^2-x+1}\operatorname d\!x$$  My try: let $$f(x)=x^4-2x^3+2x^2-x+1$$ I found  $$f(1-x)=(1-x)^4-2(1-x)^3+2(1-x)^2-x+1=x^4-2x^3+2x^2-x+1=f(x)$$ so $$I=\int_{0}^{1}\dfrac{\arcsin{\sqrt{(1-x)}}}{x^4-2x^3+2x^2-x+1}dx$$ so $$2I=\int_{0}^{1}\dfrac{\arcsin{\sqrt{x}}+\arcsin{\sqrt{(1-x)}}}{x^4-2x^3+2x^2-x+1}dx$$ then I can't,Thank you very much Does the Black Panther suit give extra speed? So in Civil War, the Black Panther suit seems to give the suit holder some extra speed, because the guy was running as fast as Bucky and Captain America.  So does it give extra speed? If so, how does that work? Is a finite group uniquely determined by the orders of its elements?