Often in geology &amp; archaeology we listen that they calculated the age of things (bones, pots) using radio active dating. Radioactive dating works using half life of unstable isotopes. And what geologists/archaeologists do is use the fraction of remaining isotopes. But this makes no sense to me. I mean should not it be that those isotopes start decaying right after they were created (probably inside star cores or explosions 💥) instead of when that pot was created?
So I have this word problem and I’m a bit confused about it. I have the answer and explanation but I still don’t understand: The half-life of carbon-14 is approximately 5730 years, while the half-life of carbon-12 is essentially infinite. If the ratio of carbon-14 to carbon-12 in a certain sample is 25% less than the normal ratio in nature, how old is the sample?     A. Less than 5730 years      B. Approximately 5730 years      C. Significantly greater than 5730 years, but less than 11460 years      D. Approximately 11460 years  Correct Answer: A Explanation: Because the half-life of carbon-12 is essentially infinite, a 25 percent decrease in the ratio of carbon-14 to carbon-12 means the same as a 25 percent decrease in the amount of carbon-14. If less than half of the carbon-14 has deteriorated, then less than one half-life has elapsed. Therefore, the sample is less than 5730 years old. Be careful with the wording here—the question states that the ratio is 25% less than the ratio in nature, not 25% of the ratio in nature, which would correspond to choice (D). How is the ratio of the carbon isotopes relevant to half-lives? What is the purpose of saying the half-life of an isotope is infinite? What is meant by “the normal ratio in nature”? Just based on the answer, it seems like the question said “25% of a carbon sample decayed, how old is this sample?” Obviously it’s younger than 5,730 years (the time for its first half life) because if only 25% decayed that means half a half-life has passed. I don’t see how isotopes and ratios would change this problem at all. Edit: Every answer here was very informative and helpful. It was tough to pick a best answer. I picked Cosma’s because it clicked with me. If you’re reading this and want to understand radiocarbon dating, look at every answer because there is some useful information in every answer
What is the best way to "carve" a terrain created from a heightmap? I have a 3d landscape created from a heightmap. I'd like to "carve" some holes in that terrain. That will allow me to create bridges, caverns and tunnels inside it.    That operation will be done in the game editor so it doesn't need to be realtime. In the end, rendering is done using traditional polygons.  What would be the best/easiest way to do that ? I already think about several solutions :  Solution 1  1) Create voxels from the heightmap (very easy). In other words, fill a 3D array like this : voxels[32][32][32] from the heightmap values.  2) Carve holes in the voxels as i want (easy too).  3) Convert voxels to polygons using some iso-surface extraction technique (like marching cubes).  4) Reduce (decimate) polygons created in 3).  This technique seems to be the most promising for giving good results (untested). However the problem with marching cubes is that they tends to produce lots of polygons thus reducing them is mandatory. Implementing 4) also seems not trivial, i have read several papers on the web and it seems pretty complex. I was also unable to find an example, code snippet or something to start writing an algorithm for triangle mesh decimation. Maybe there is a special decimation algorithm (simpler) for meshes created from marching cubes ?  Solution 2  1) Create some triangle mesh from the heighmap (easy).  2) Apply severals 3D boolean operation (eg: subtraction with a sphere) to carve the mesh.  3) apply some procedure to reduce polygons (optional).  Operation 2) seems to be very complex and to be honest i have no idea how to do that. Also applying many boolean operation seems to be slow and will maybe degrade the triangle mesh  every time a boolean operation is applied. At the moment I'm working with the . For this I read the following paper: .  But I have some mathematical problems with this formalism. I don't really see why this is a variation principle since it is opartor-valued.   The action principle is defined by: $\delta S  = G_1 -G_2$, where $S$ is the action operator $S = \int_{t_1}^{t_2} dt ~L$ and $G$ the generator operators of the transformations.  What does it mean that the action principle is operator-valued? For the classical action where $S$ is real we search for a stationary point but for operators? I also have a problem that $\delta S \neq 0$. What interpretation has this additional term $G_1 - G_2$? Book set in contemporary times about a travel writer who is inducted into the world of the supernatural why are the best questions the ones which have been closed? Is "Logan" the last movie with Hugh Jackman as Wolverine? has played the role of Wolverine as a major character in various X-Men films since 2000 and contributed highly to the brand of the X-Men movie series. But is he playing the Wolverine character for the last time in Logan?  I am getting to know from my friends that  is the last movie in which he is playing Wolverine and they all say that this is his last contracted film.  Are these rumors true? Is there any official information that confirms that this is Hugh Jackman's last movie as Wolverine? preventDefault() on an  tag Obscure fantasy movie involving flying sea-galleons and a time weapon Been trawling the internet for hours, can't find any trace or reference of this movie. Not really sure what station I saw it on (only once), but it was probably Sci-Fi Channel (mind you, a very long time ago:  Movie opens with flying ships (sailing galleons), that proceed to siege a towering city. The city fires back with its own cannons, and drives the invaders away. The main protagonist is a young boy who longs to be a knight... or this movie's equivalent thereof. He has an older brother who is already recognized as a warrior. The boy lives in the city that successfully repelled this siege.  The king of this city has some sort of wizard or wise-man as a councilor, who conceives an "ultimate weapon" against their enemies- an hourglass of sorts that accelerates time against a target. However, during the course of the movie this weapon is stolen by the city's enemies (led by an evil queen or sorceress), who proceeds to use it against the city itself and all its inhabitants.  So the king sends out his armies to retrieve the artifact, all the while everyone and everything begins aging rapidly. Our boy-hero watches his older brother (now much older) set out with other knights on the quest, and there's a token romance between the boy and the king's daughter. The knights do not return, however, and the time-curse continues to wreak havoc on the city.  Ultimately, the boy (now a man) sets out to find his brother, the hourglass, and save the day. He finds his brother at a wreckage-site (I believe of a flying galleon), and they share a brief moment before the elder brother finally succumbs to old age. I'm sure I'm forgetting some other poignant details, but that's essentially the crux of it. Ends... if I remember correctly, with  the artifact being disabled and the evil queen/sorceress defeated.  Question is: what is this movie titled? Show inbox updates in the page title Background  Recently  was implemented. With this feature the amount of questions with new activity is shown in parenthesis at the beginning of the page title, e.g.     (2) Meta Stack Overflow   means that MSO has two questions with new activity since the user last viewed it.   this feature is unrelated to inbox updates. Here is an example to illustrate this:    Even though there are two inbox messages in this screenshot this is not reflected by the page title. The page title is currently only sensitive to questions with new activity.  Suggestion  I suggest that the page title also reflects inbox updates. For the screenshot above this would mean that the page title shows that there are two inbox updates and two questions with new activity.  I am not sure what format is best for showing this. For chat * in the page title is used to show that there are messages directed at you which is similar in meaning to inbox updates. However, * if is placed after the number in parenthesis to denote inbox updates there might be confusion as there is no connection between the number in parenthesis and * like in chat. Thus, one might prefer another format such as one of the following (all are meant to reflect the status of the screenshot above):      * (2) Meta Stack Overflow     2* (2) Meta Stack Overflow     [2] (2) Meta Stack Overflow     2i 2q  Meta Stack Overflow     (2/2)  Meta Stack Overflow     (2|2)  Meta Stack Overflow    For the alternatives without an explicit hint of which number refers to what it might be best to place the number for questions with new activity first as it will in most cases be greater. This will to a greater degree avoid combinations such as "(0/2)" which may be confused zero of two.  An advantage of the suggested feature is that if the user has a browser tab with a Stack Exchange site then they will be instantly notified of inbox updates. They do not need to focus a tab with a Stack Exchange site to know if there are inbox updates, they merely have to check the relevant page titles among their tabs.  A disadvantage of the suggested feature is that it might be experienced as verbose. I suggest that it is implemented such that it is as unobtrusive as possible while still being informative. Think of a situation where a user has   one Meta Stack Overflow tab with two inbox updates relating to Meta Stack Overflow and two questions with new activity, and one Super User tab with one inbox updates relating to Super User and three questions with new activity.   Then the sum of the inbox notification in the page titles should be local to sites and not global. The page titles for the tabs above should be something like the following pair:      2i 2q  Meta Stack Overflow     1i 3q  Super User My wireless/WiFi connection does not work. What information is needed to diagnose the issue? Your situation   You have successfully installed Ubuntu.  You have just downloaded and booted Ubuntu live media.  The latest LTS (see also ) or latest non-LTS release are preferred. See the list of  that are currently supported.) You upgraded your Ubuntu installation to the latest release that the software updater offered you. WiFi worked before, but not now on the new release. You  to new hardware.   Your problem   The wireless of your laptop or desktop is not working. You tried switching the wireless switch off and on and you tried rebooting several times, but you don't see any WiFi access points. You can see your wireless access point, but you cannot establish a connection. You want to analyze the problem, but you don't know where to start or what information you can provide.   This question and answer was written to give people the wireless script that will provide the information needed in most cases to diagnose their wireless issue's.   Related questions What could cause exposure problems after my camera got wet? I had an unfortunate accident where my DSLR camera (and me) briefly ended up in a river. The camera wasn't fully submerged but it did get very wet. It still works, sort of, although the electrics are doing some random things. I'm hoping that's just a bit of dampness doing that, that will fix itself when it dries out fully.  More concerning is that the documents exposure metering seems to be all wrong when using the viewfinder. Pictures are extremely overexposed and nearly white. But if I use live view, then they are fine.  Why could cause this? I assume the light is routed a different way when using the viewfinder and maybe it is going through something with water/condensation on it? Context: I was looking at . I didn't know the result referred to in Brian M. Scott's answer is hard enough to require a separate proof   ― .  ― .  (1)  ― What's the intuition? Is Bezout Identity supposed to be intuitive?  (2) — Why doesn't this try work? Let $c$ be any common divisor of $a, b$. Therefore $c\mid a, c\mid b \iff cj_1 = a, cj_2 = b$ for some integers $j_1, j_2$.     Call $g = \gcd$. By definition $\gcd(a,b)\mid a$ and $\gcd(a,b)\mid b$ and $c \le \gcd(a,b)$. Therefore $g\mid a, g\mid b \iff gk_1 = a, gk_2 = b$ for some integers $k_1, k_2$.   The two paragraphs together precipitate $cj_1 = a = gk_1,  cj_2 = b = gk_2$ but this only proves $c\mid gk_1, c\mid gk_2$?    Origin - Elementary Number Theory, Jones,  p9, Exercise 1.8 Bibliography with BibLaTeX: Do not abbreviate first given (first) name and abbreviate all further given names The first given name of an author or editor in my bibliography should be written out, but all further first names should be abbreviated (initials with a period -- and space if there are multiple first names/initials, see below). I am using style=authoryear with BibLaTeX (v. 2.7) and Biber (v. 1.7).  My bibliography looks like this:     Doe, John Arthur Kyle (2013). The secret life of John Doe.   It should look like this:     Doe, John A. K. (2013). The secret life of John Doe.   Minimal example:  example.tex  \documentclass{scrartcl}  \usepackage[ngerman]{babel} \usepackage[babel,german=quotes]{csquotes} \usepackage[backend=biber,style=authoryear]{biblatex} \bibliography{bibliography}  \begin{document}  \cite[123]{Doe} \printbibliography  \end{document}   bibliography.bib  @book{Doe,     title = {The secret life of John Doe},     author = {Doe, John Arthur Kyle},     year = {2013} }   (Worked with said versions of BibLaTeX and Biber, Texmaker (v. 4.0.3) and MiKTeX (v. 2.9.4813).)  I could not find a solution in biblatex.def and firstinits=true abbreviates all first names, including the first one. Using JQuery and Prototype in the same page Several of my pages use both JQuery and Protoype. Since I upgraded to version 1.3 of JQuery this appears to be causing problems, because both libraries define a function named '$'.  JQuery provides a function noConflict() which relinquishes control of $ to other libraries that may be using it. So it seems like I need to go through all my pages that look like this:  &lt;head&gt;           &lt;script type="text/javascript" src="/obp/js/prototype.js"&gt;&lt;/script&gt;     &lt;script type="text/javascript" src="/obp/js/jquery.js"&gt;&lt;/script&gt; &lt;/head&gt;   and change them to look like this:  &lt;head&gt;           &lt;script type="text/javascript" src="/obp/js/prototype.js"&gt;&lt;/script&gt;     &lt;script type="text/javascript" src="/obp/js/jquery.js"&gt;&lt;/script&gt;     &lt;script type="text/javascript"&gt;         jQuery.noConflict();         var $j = jQuery;     &lt;/script&gt; &lt;/head&gt;   I should then be able to use '$' for Prototype and '$j' (or 'jQuery') for JQuery. I'm not entirely happy about duplicating these 2 lines of code in every relevant page, and expect that at some point somebody is likely to forget to add them to a new page. I'd prefer to be able to do the following    Create a file jquery-noconflict.js which "includes" jquery.js and the 2 lines of code shown above Import jquery-noconflict.js (instead of jquery.js) in all my JSP/HTML pages   However, I'm not sure if it's possible to include one JS file in another, in the manner I've described? Of course an alternate solution is simply to add the 2 lines of code above to jquery.js directly, but if I do that I'll need to remember to do it every time I upgrade JQuery. Can't upgrade due to low disk space on /boot I try to do do-release-upgrade but then I get:       Not enough free disk space      The upgrade has aborted. The upgrade needs a total of 25.7 M free   space on disk '/boot'. Please free at least an additional 25.7 M of   disk space on '/boot'. Empty your trash and remove temporary packages   of former installations using 'sudo apt-get clean'.   Output of df:                      237251272 214797108  10402504  96% / udev                    488120         4    488116   1% /dev tmpfs                   198676       668    198008   1% /run none                      5120         0      5120   0% /run/lock none                    496684         0    496684   0% /run/shm /dev/sda1               233191    225867         0 100% /boot   How come there is no space left on boot? Here's the output of ls -as /boot:  total 221839     4 .     4 ..   645 abi-2.6.32-34-generic-pae   698 abi-2.6.35-30-generic-pae   727 abi-2.6.38-12-generic-pae   727 abi-3.0.0-12-generic-pae   727 abi-3.0.0-13-generic-pae   727 abi-3.0.0-14-generic-pae   727 abi-3.0.0-15-generic-pae   727 abi-3.0.0-16-generic-pae   727 abi-3.0.0-17-generic-pae   727 abi-3.0.0-19-generic-pae   761 abi-3.0.0-20-generic-pae   115 config-2.6.32-34-generic-pae   128 config-2.6.35-30-generic-pae   136 config-2.6.38-12-generic-pae   140 config-3.0.0-12-generic-pae   140 config-3.0.0-13-generic-pae   140 config-3.0.0-14-generic-pae   140 config-3.0.0-15-generic-pae   140 config-3.0.0-16-generic-pae   140 config-3.0.0-17-generic-pae   140 config-3.0.0-19-generic-pae   140 config-3.0.0-20-generic-pae     5 grub 10773 initrd.img-2.6.32-34-generic-pae 13619 initrd.img-2.6.35-30-generic-pae 15365 initrd.img-2.6.38-12-generic-pae 16481 initrd.img-3.0.0-12-generic-pae 16487 initrd.img-3.0.0-13-generic-pae 16501 initrd.img-3.0.0-14-generic-pae 16476 initrd.img-3.0.0-15-generic-pae 16481 initrd.img-3.0.0-16-generic-pae 16478 initrd.img-3.0.0-17-generic-pae    12 lost+found   174 memtest86+.bin   176 memtest86+_multiboot.bin  1700 System.map-2.6.32-34-generic-pae  1841 System.map-2.6.35-30-generic-pae  2115 System.map-2.6.38-12-generic-pae  2141 System.map-3.0.0-12-generic-pae  2141 System.map-3.0.0-13-generic-pae  2143 System.map-3.0.0-14-generic-pae  2146 System.map-3.0.0-15-generic-pae  2147 System.map-3.0.0-16-generic-pae  2147 System.map-3.0.0-17-generic-pae  2148 System.map-3.0.0-19-generic-pae  2149 System.map-3.0.0-20-generic-pae     2 vmcoreinfo-2.6.32-34-generic-pae     2 vmcoreinfo-2.6.35-30-generic-pae     2 vmcoreinfo-2.6.38-12-generic-pae     2 vmcoreinfo-3.0.0-12-generic-pae     2 vmcoreinfo-3.0.0-13-generic-pae     2 vmcoreinfo-3.0.0-14-generic-pae     2 vmcoreinfo-3.0.0-15-generic-pae     2 vmcoreinfo-3.0.0-16-generic-pae     2 vmcoreinfo-3.0.0-17-generic-pae     2 vmcoreinfo-3.0.0-19-generic-pae     2 vmcoreinfo-3.0.0-20-generic-pae  4092 vmlinuz-2.6.32-34-generic-pae  4347 vmlinuz-2.6.35-30-generic-pae  4567 vmlinuz-2.6.38-12-generic-pae  4675 vmlinuz-3.0.0-12-generic-pae  4676 vmlinuz-3.0.0-13-generic-pae  4681 vmlinuz-3.0.0-14-generic-pae  4698 vmlinuz-3.0.0-15-generic-pae  4700 vmlinuz-3.0.0-16-generic-pae  4700 vmlinuz-3.0.0-17-generic-pae  4703 vmlinuz-3.0.0-19-generic-pae  4705 vmlinuz-3.0.0-20-generic-pae   Output of uname -a:   Linux kitsch 3.0.0-17-generic-pae #30-Ubuntu SMP Thu Mar 8 17:53:35 UTC 2012 i686 i686 i386 GNU/Linux An instructor has an exam of 5 questions and a question bank of 50 questions. Are there $50\choose 5$ or $(50)_5$ possible exams? Where $(50)_5 = 50! / 45! $ and $50\choose 5$ $= 50!/(5!45!)$   I've narrowed it down to these two possibilities (though even these may be wrong), but I think I'm confusing myself as to which one I use. Can anyone suggest a good way of thinking of this problem that will make it apparent whether I need use use the falling factorial or the binary coefficient?   Edit: thank you for all the answers! Indeed, I meant $50!/45$ as the falling factorial. Also, the question doesn't specify whether it was meant to be ordered or unordered, but I think I understand my confusion now! we all study mathematics , and all of us learn mathematical methods to solve problems , we learn how to prove , how to think mathematically   but the question is, what is mathematics ? how can we define it as a branch of science ? Express $x^8-x$ as a product of irreducibles in $\Bbb Z_2[x]$ Express $x^8-x$ as a product of irreducibles in $\Bbb Z_2[x]$.  Work so far:  $$x^8 - x = x(x^7 - 1) = x(x - 1)(x^6+x^5+x^4+x^3+x^2+x+1)$$  From here, I think the Zeros of an Irreducible over a splitting field Theorem would be the method. I am unsure how to proceed Cannot delete files, nothing seems to work I am on Windows 7.  I have some files on my computer leftover from an old version of Photoshop.  (I think... I don't remember exactly where they came from, but they seem to be Adobe files.)  There are 124 of them in a folder, which I have named "KILL."  (I don't remember what the original name of this folder is, it's been on my comptuer a long time.)  These files seem to be absolutely undeletable, as far as I can tell.  I can move them around and change the name of the folder, but deleting them is out of the question.  I also can't move them onto a different hard drive or flash drive, I've tried that.  So, let me list the things I've tried first:   A whole bunch of different "unlocking programs."  Including: IObit Unlocker, Lock Hunter, FileAssassin, and Project Kill box.  What I use LockHunter I am told: "No processes locking this file or folder have been found."  All of the other programs I've tried give me similar messages, telling me that nothing is locking the files.  Yet when I try to delete the files using LockHunter, I am simply told "Cannot delete x."  No reason is given. Tried using cmd.exe.  Even when I run a command window under administrator, it won't even let me access the KILL folder, much less delete anything.  I also tried killing explorer.exe from the task manager, and then using the cmd window.  Still can't even access that folder. Tried using safe mode in combination with these other options, doesn't change a thing. Tried changing permissions of the files using the security tab under properties.  It simply says "not inherited."  It will not allow me to change a thing. I am definitely the administrator of this computer. Tried Disk Cleanup and Defrag.  I suspected that some files must be damaged or something, but Disk Cleanup wasn't able to find any errors.  Defrag didn't help.   So, let me just say that I am not a computer expert.  But based on all the things I've tried, I suspect that this might be some hardware problem?  It seems strange that these files don't seem to be used by any processes, and yet are absolutely undeletable no matter what I do.  Is there anything else I can try?  Has anyone had any similar experience, and have any idea why this might have happened?  EDIT: Interesting recent development.  As I have toyed with moving the folder around, I moved it to the Program Files folder.  Now that it's there, it suddenly CANNOT be moved nor renamed.  It is stuck there, apparently permanently, as a folder called "KILL."  How irritating. Evaluating logistic regression and interpretation of Hosmer-Lemeshow Goodness of Fit Why particle hole symmetry and chiral symmetry are called symmetries? $PHP^{-1}=-H$ (particle-hole symmetry) and   $\Gamma H \Gamma^{-1}=-H$ (chiral symmetry)   I understand why we get the negative signs but im just a bit confused as to why such equalities mean $H$ is particle hole and chiral symmetric.   When we say $H$ is symmetric under some operation, like time reversal,  don't we usually mean that $H$ is invariant under the said transformation?  $THT^{-1}=H$ if $H$  is time reversal symmetric.  On the contrary, we say $H$ is symmetric under particle hole and chiral operations when $H$ picks up a negative sign. This seems to contradict our usual convention of what we mean by $H$ being symmetric(invariant).