According to scientists, if an object approaches the speed of light, a stage is reached where the object's speed is responsible for the increase in mass of the object. Beyond that stage, all the acceleration will increase the mass of the object. So where does this mass come from and how this mass created?
Some people say that mass increases with speed, some people say that the mass of an object is independent of its speed.  I understand how some (though not many) things in physics are a matter of interpretation based on one's definitions. But I can't get my head around how both can be 'true' is any sense of the word.  Either mass increases or it doesn't, right?  Can't we just measure it, and find out which 'interpretation' is right? e.g.: (in some sophisticated way) heat up some particles in a box and measure their weight?  ------------------------------  UPDATE  Right, so I've got two identical containers, each with identical amounts of water, each on identical weighing scales, and each in the same g field. If one container has hotter water, will the reading on its scale be larger than the other? If the answer is yes, and g is constant, does this mean that the m in w=mg has got bigger?
How to relate areas of circle, square, rectangle and triangle if they have same perimeter?? I was given a question which was like:     Suppose that a circle, square, rectangle and triangle have same perimeter. How area their areas related??   My work:  I broke the question in parts and tried to prove it seperately:  STEP $1$. Suppose the given perimeter is $P$.So the radius of circle will be ${r=P/2\pi}$.  And hence area of the circle will be:   ${P^2/4\pi}$.  Side of square will be $P/4$ and hence area of square will be $P^2/16$.   It is obvious that ${P^2/4\pi}$  >  $P^2/16$.   So Area of circle is more than area of square.  STEP $2$: Whether by using $AM&gt;GM$ inequality or by using a bit of differentiation, we can get that product off two quantities is maximum when they are equal.  So, Area of square is more than Area of rectangle.  Here, the hardest part come (at least for me). I can't relate the area of triangle to any of Circle, Square or Rectangle. It would have been easy if the triangle was Equilateral, Right angle but the question is talking about a general triangle.  However, when I considered the situation with different examples (Assuming the perimeter and calculating the area, I got the relation:  Circle > Square > Rectangle > Triangle.  I shall be highly thankful if you guys can establish the relation in purely mathematical way.  EDIT: This question does not ask that whether circle have the largest area among all 2-D figures or not. The OP just want to relate the areas of Circle, Square, Rectangle, Triangle in a pure mathematical way. I have a computer with a video card that has no Linux support.  I discovered this issue while trying to use a persistent usb to boot from.   I found that I can boot like that if I change options and edit so as to add nomodeset there (see ).  Also, editing the syslinux.cfg file from my live persistent USB and adding nomodeset in a line there made this setting automatic for booting from the persistent USB.  Can I make a similar edit for proper install from the same live USB?  Is it possible to edit something on the live USB in order to install the SO from it,  -- that is: having automatically the nomodeset setting already there when the installation starts?  My problem is that in order to install I think I already have to have added nomodeset or I'll see nothing on my display. I need a way to be sure that nomodeset it's permanent from the start. Is it true that a dihedral group is nonabelian?  I'm not sure if the result is true. I checked it for some lower order and I think the result may correct.   But I failed to prove/disprove the result. Unable to resume after suspend I'm using Ubuntu 12.04 &amp; everything about it is great except that it just won't resume after suspend. I've tried the suggestions listed in this thread  but none worked. I'm using Lenovo Y510P. I'm new to Ubuntu, or unix for that matter, so please let me know if there's anymore  information that I should be posting. Thanks.  update  I tried the suggestion by Dima on this post . The machine wakes up from hibernation but immediately the screen gives a screen like this     Wondering if this is an issue with the graphics card or something. When can we speak of collinearity In linear models we need to check if a relationship exists among the explanatory variables. If they correlate too much then there is collinearity (i.e., the variables partly explain each other). I am currently just looking at the pairwise correlation between each of the explanatory variables.  Question 1: What classifies as too much correlation? For example, is a Pearson correlation of 0.5 too much?  Question 2: Can we fully determine whether there is collinearity between two variables based on the correlation coefficient or does it depend on other factors?  Question 3: Does a graphical check of the scatterplot of the two variables add anything to what the correlation coefficient indicates? example.com is hosted on IP address c.c.c.c at host1, no access any windows server admin at all blog.example.com is  on IP a.a.a.a at host2, full access to Linux server admin, using WordPress   So basically, I can fiddle with subdomain all I want.  I can't 301 redirect because the content on host 1 is an eCommerce site that I have no server control over. Can't put WordPress on it, can't run scripts, can't do anything on the server period.  Is there a method to "mask" all the URLs on root blog.example.com so they appear as example.com/blog/ folder?  They can't actually be sent to example.com/blog/ as that doesn't exist. It just needs to look to the surfer as if they are there.  Can this be done with some kind of masking or something?  Will this be better for SEO as hopefully search engines will see the blog in a directory on the main domain? I have several std::vector, all of the same length. I want to sort one of these vectors, and apply the same transformation to all of the other vectors.  Is there a neat way of doing this? (preferably using the STL or Boost)?  Some of the vectors hold ints and some of them std::strings.  Pseudo code:  std::vector&lt;int&gt; Index = { 3, 1, 2 }; std::vector&lt;std::string&gt; Values = { "Third", "First", "Second" };  Transformation = sort(Index); Index is now { 1, 2, 3};  ... magic happens as Transformation is applied to Values ... Values are now { "First", "Second", "Third" }; Preview: PDF Highlighting & Search Stop Working until Restart A problem has recently become much more prevalent for me when using Preview to search PDFs for text (using Command + F or the search box in the upper right).  At first, everything generally seems to go fine:    (as it should be) a list of thumbnails appears in the sidebar, indicating all of the pages that contain the search term. Generally, though not always, the term I search for will be highlighted on the first page that it occurs on.   But, after this, things quickly start malfunctioning:   Generally speaking, if I click on thumbnails lower on the list, Preview will take me to that page in the PDF (like it used to).  But, it will no longer highlight the searched for terms. Instead, Preview will make a low-pitched beeping noise of the sort that occurs when one tries to perform some un-permitted action in a variety of mac situations. Furthermore, once this occurs, I cease being able to make any highlights at all in the document I am viewing with Preview.  In fact, I can't even select any text any more.   Below is a list of unsuccessful efforts to resolve this:   Closing the side bar does not work to resolve this. x-ing out the search box in the upper right does not resolve this. Selecting the "Tools" menu and then "Text Selection" does not work either, nor does clicking on the text selection icon in the edit panel. If I quit preview and then re-open the PDF, I am able to highlight again.  But, the problem will re-surface as soon as I try performing another search. Working with the same files in other PDF programs (e.g. Skim, Acrobat, etc.) poses no problems.  I'm quite certain it isn't a problem with the underlying PDFs. None of the responses to  resolve this.  (indeed, I supplied one such response myself a while ago, but that trick and the others no longer work).   I've used Preview for many years without problem.  This issue started cropping up relatively recently for me, I believe within the past 3-4 months.  I had hoped that upgrading to Catalina might resolve it, but it did not.    I'd like to think I am fairly sophisticated with computers (I have a reputation of close to 8,000 on the main stackoverflow page), but I am baffled by this.  I also don't have a good handle on how I can get diagnostic information or how I can create a reproducible example.    Any leads on how to tackle this would be greatly appreciated. Let $p,q,r$ be integers such that the symmetric sum of fractions $\dfrac{pq}{r}+\dfrac{qr}{p}+\dfrac{pr}{q}$ is an integer. Prove that each of the numbers: $\dfrac{pq}{r},\dfrac{qr}{p},\dfrac{pr}{q}$ is an integer. How to do this? Someone posted but deleted soon. Has the below question been answered here before?  Prove: For a given enclosed volume, a sphere has minimum surface area.  Please provide link or ways to solve it. I know it is a problem of Minima and involves finding derivative and second derivatives. However, how to to frame the leading equations which has to be differentiated?  I am looking for a way to solve only using High School level Calculus. I think there must be an answer. Product of Non-Disjoint Cycles I am trying to learn how to find the product of non-disjoint cycles, as you may have guessed from the title. I have the basic idea, but I do not understand it entirely.  I am trying to find $(1352)(256)$ and $(1634)(1352)$, multiplying from right to left.  Please give me some hints/suggestions.  Thank you. Exterior power "commutes" with direct sum I know that for vector spaces $V, W$ over a field $K$, we have the following identity :  $$ \bigoplus_{k=0}^n \left[ \Lambda^k(V) \otimes_K \Lambda^{n-k}(W) \right] \simeq \Lambda^n(V \oplus W) $$ which holds in a canonical way : for $0 \le k \le n$, the maps  $$ \Lambda^k(V) \times \Lambda^{n-k}(W) \to \Lambda^n(V \oplus W), \\ (v_1 \wedge\cdots \wedge v_k,w_{k+1} \wedge \cdots \wedge w_n) \mapsto (v_1 \wedge \cdots \wedge v_k) \wedge (w_{k+1} \wedge \cdots \wedge w_n) $$ factors through the tensor product's universal properties, which gives us a map $\varphi_k : \Lambda^k(V) \otimes_K \Lambda^{n-k}(W) \to \Lambda^n(V \oplus W)$, and since the category of $K$-modules has biproducts, we get a canonical map  $$ \varphi = \bigoplus_{k=0}^n \varphi_k : \bigoplus_{k=0}^n \left[ \Lambda^k(V) \otimes_K \Lambda^{n-k}(W) \right] \longrightarrow \Lambda^n(V \oplus W) $$  What if I replace $K$ by a (commutative ring with $1$) ring $A$? Does this map have a kernel? I know it is surjective (it suffices to "look at it", i.e generators of the codomain are obviously all hit by $\varphi$). I don't know if it helps, but I am working in integral domains. In the field case, this is obvious if $V$ and $W$ are finitely generated, since one can use dimension arguments. I wouldn't mind if the above identity held only in the finitely generated case either. If it worked in general it would be cool though.   Note that when trying to compute the inverse map with the universal property, there seems to be a problem... my guess would have been $$ (v_1 + w_1,\cdots,v_n + w_n) \mapsto \bigoplus_{k=0}^n (v_1 \wedge \cdots \wedge v_k) \otimes (w_{k+1} \wedge \cdots \wedge w_n) $$ and this map is obviously $A$-multilinear, but there is a problem with the alternating property ; I get some terms left. I'm guessing this is the wrong map... if there is one. How to list files installed by a snap package? After a snap has been installed with sudo snap install [package] how can I list the files which have been installed by the snap? How to remove all comment tags from XmlDocument I know how to detect when resistors are arranged in parallel or series arrangement and I can also find their equivalent resistance in simple circuits or when resistances are connected in form of triangle but what happens when the arrangement is complex like this :   Which resistors are parallel and which are in series ? How can I find the equivalent resistance in such cases ? Is there rule or method for figuring this out ? Evaluating and proving $\lim\limits_{x\to\infty}\frac{\sin x}x$ I've just started learning about limits. Why can we say $$ \lim_{x\rightarrow \infty} \frac{\sin x}{x} = 0 $$ even though $\lim_{x\rightarrow \infty} \sin x$ does not exist?   It seems like the fact that sin is bounded could cause this, but I'd like to see it algebraically.   $$ \lim_{x\rightarrow \infty} \frac{\sin x}{x} =  \frac{\lim_{x\rightarrow \infty} \sin x} {\lim_{x\rightarrow \infty} x}   = ? $$  L'Hopital's rule gives a fraction whose numerator doesn't converge. What is a simple way to proceed here? Which is correct?     How much is the two fares?      How much are the two fares? Can I charge computers by inducing 5-volt into their USB connectors? Is this possible? I noticed that when my EeePC 900 is off but plugged in charger, there is power coming out of the USB port. Can this be a sustainable way of charging a PC, or a flaw in design? How can we communicate short messages long-distance? How to install "vanilla" TeXLive on Debian or Ubuntu?