I'm super new to this Stack and even newer to physics. To be honest, I am currently reading an introductory book on physics called  by Carlo Rovelli (the very first book on physics that I've ever read) and I'm finding extremely interesting.  As you can imagine I have many questions, but one in particular.  In the chapter on particles, he explains briefly that there are "Elementary particles", and from what I understand they are the ones that compose all things. He then mentions that in some galaxies, astronomers observe a halo of matter, but that it is unknown what these halos are made of. He says it is not elementary particles.  Forgive me if my question is dumb (which it probably is); but if everything is made of elementary particles, why can't these halos be made of elementary particles as well? In other words, how do they know they are not made of elementary particles?
There's more gravitational force in our galaxy (and others) than can be explained by counting stars made of ordinary matter. So why not lots of dark planetary systems (i.e., without stars) made of ordinary matter? Why must we assume some undiscovered and unexplained form of matter?
Explain $\iint \mathrm dx\,\mathrm dy = \iint r \,\mathrm \,d\alpha\,\mathrm dr$ It is changing the coordinate from one coordinate to another. There is an angle and radius on the right side. What is it? And why?  I got:  $2\,\mathrm dy\,\mathrm dx = r(\cos^2\alpha-\sin^2\alpha)\,\mathrm d\alpha \,\mathrm dr$,  where $x = r \cos(\alpha)$ and $y = r \sin(\alpha)$.  but cannot understand and get the right side. The problem emerged when trying to integrate $\displaystyle \int_0^\infty e^{\frac{-x^2}{n^2}}\,\mathrm dx$ where I tried to change the problem knowing $r^2=x^2+y^2$ but stuck to this part. What is the change in the title called and why is it so? if $f^{(n)}(x_0),g^{(n)}(x_0)$ exist and $\lim_{x\to x_0} \dfrac{f(x)-g(x)}{(x-x_0)^n}=0$ then $f^{(r)}(x_0)=g^{(r)}(x_0 )$ for all $0\leq r\leq n.$ Show:   if $f^{(n)}(x_0),g^{(n)}(x_0)$ exist and   $$\lim_{x\to x_0} \dfrac{f(x)-g(x)}{(x-x_0)^n}=0$$   then $f^{(r)}(x_0)=g^{(r)}(x_0 )$ for all $0\leq r\leq n.$   My attempt:  let n = 1,  since $f'(x_0),g'(x_0)$ exists then The taylor polynomial exists: $$T_{f,1}(x)=f(x_0)+f'(x_0)(x-x_0), \lim_{x\to x_0} \dfrac{f(x)-T_{f,1}(x)}{(x-x_0)}=0$$  $$T_{g,1}(x)=g(x_0)+g'(x_0)(x-x_0), \lim_{x\to x_0} \dfrac{g(x)-T_{g,1}(x)}{(x-x_0)}=0$$  Then, \begin{align} \lim_{x\to x_0} \dfrac{f(x)-g(x)}{(x-x_0)^n} &amp; =\lim_{x\to x_0} \dfrac{f(x)-g(x)+T_{f,1}(x)-T_{f,1}(x)+T_{g,1}(x)-T_{g,1}(x)}{(x-x_0)^n}\\ &amp;=\lim_{x\to x_0} \dfrac{f(x)-g(x)+T_{f,1}(x)-T_{f,1}(x)+T_{g,1}(x)-T_{g,1}(x)}{(x-x_0)^n}\\ &amp;=\lim_{x\to x_0} \dfrac{f(x)-T_{f,1}(x)}{x-x_0}-\lim_{x\to x_0} \dfrac{g(x)-T_{g,1}(x)}{x-x_0}+\lim_{x\to x_0} \dfrac{T_{f,1}(x)-T_{g,1}(x)}{x-x_0}\\ &amp;=\lim_{x\to x_0}\dfrac{T_{f,1}(x)-T_{g,1}(x)}{x-x_0}\\ &amp;=0 \end{align}  Then, \begin{align} \lim_{x\to x_0}\dfrac{T_{f,1}(x)-T_{g,1}(x)}{x-x_0} &amp;=\lim_{x\to x_0}\dfrac{f(x_0)+f'(x_0)(x-x_0)-g(x_0)-g'(x_0)(x-x_0)}{x-x_0}\\ &amp;=\left(f(x_0)-g(x_0)\right)\lim_{x\to x_0}\dfrac{1}{x-x_0}+f'(x_0)-g'(x_0) \\&amp;=0 \end{align} I am confused because  $\lim_{x\to x_0}\dfrac{1}{x-x_0}$ does not exist. Banach spaces and their unit sphere Let $X$ be a normed vector space.   Show that if a subsequence of a Cauchy sequence converges, then the whole sequence converges. Use the part 1 to show that $S = \{x\in X : \|x\| = 1\}$ is complete if and only if $X$ is Banach.     I have found the first part in a book, but I do not how to prove part 2. Thanx regards Multiplying two dimensional data by a constant List user's folder access permissions What nutrients can humans absorb in the mouth? For instance, I realise being able to absorb simple sugars in the mouth is pivotal in the rapid action of oral glucose gel. Thus I was wondering what nutrients in general can be absorbed directly within the mouth, and at what speed? How to hyperlink only the year part when using natbib and hyperref I am using the natbib and hyperref packages:  \usepackage{natbib}  \usepackage[pagebackref=true, plainpages=false, bookmarks, bookmarksnumbered, colorlinks, linktocpage=true, linkcolor=blue, citecolor=blue, filecolor=black, urlcolor=blue]{hyperref}   With my current settings (above) both the name and the year are hyperlinked.  My question is, whether there is a way to hyperlink (or color) only the year-not the author(s).   I have the impression reading this post  that biblatex does it by default. However changing to biblatex is not an option for me at this point.  Any helpful comments will be greatly appreciated. Projection on the hyperplane $H: \sum x_i=0$ Consider the hyperplane $H: \sum_{i=1}^n x_i=0$. Let $P$ denote the orthogonal projection onto $H$. What exactly is the matrix $P$ and how do we find it? I plan on building a computer with  which has no GPU:     Graphics Specifications      Processor Graphics:   None   on a  motherboard which has an onboard integrated graphics processor:     Integrated Graphics Processor:         1 x D-Sub port, supporting a maximum resolution of 1920x1200@60Hz   1 x DVI-D port, supporting a maximum resolution of 1920x1200@60Hz   1 x HDMI port, supporting a maximum resolution of 4096x2160@24Hz or 2560x1600@60Hz   Support for up to 3 displays at the same time   Maximum shared memory of 512 MB      I have asked  and know that even if the CPU has no GPU, I can still get video as long as the motherboard has onboard graphics. However, when I went to buy the parts at a computer center yesterday, the technician said motherboards nowadays use CPU's GPU to display, including this Gigabyte GA-Z97X-UD5H motherboard (which he meant this board has no real onboard graphics).  So for Xeon E3 1231v3 + Gigabyte GA-Z97X-UD5H to display, do I need a video card or not?  (I don't really care about the graphics quality. I use the computer mostly to write programs, web browsing, and Photoshop, and seldom play games or watch videos.) Compute the period of a decimal number a priori I noticed that , given an operation like $\frac{n}{m},\;n,m \in N$ that result in a periodic decimal number, computes really fast the length of the period.  E.g. $\frac{3923}{6173}$ has a period of 3086: .  I was wondering how this computation is done: is there some method to do this (except the trivial one of executing the division and looking for a sequence repetition) ? Can anybody please suggest how I can compute the moment generating function of the inner product of two Gaussian random vectors, each distributed as $\mathcal N(0,\sigma^2)$, independent of each other? Is there some standard result available for this? Any pointer is highly appreciated. \xymatrix undefined $a^2b^2+a^2c^2+b^2c^2\leq 3$ Let $a, b, c\geq 0$ s.t.$$(a^2-a+1)(b^2-b+1)(c^2-c+1)=1$$  Show that $ a^2b^2+a^2c^2+b^2c^2\leq 3$.  My idea:  I denote $a+b+c=x$, $ab+bc+ac=y$ and $abc=z$.  Then I have $$x^2+y^2+z^2-xy-xz-yz+2z-y-x=0$$ I have to show that $$y^2-2xz\leq 3$$ I tried to prove it with the sign of trinom, but it doesn't work. I have read the Rules of a competition. The text of the Rules include a sentence as follows:     As per stated in the Rules the entrants will be notified by May 30th 2010.   Does the sentence above mean that May 30th is included in the term of notification or not? I have a script that I need to execute on an NTFS partition. The script's permission is set to 600.  I attempted to modify the permissions by running chmod 755 script.sh, which doesn't report a failure or anything - but it also doesn't change the permissions on the file:  $ stat script.sh    File: `script.sh'   Size: 297070      Blocks: 584        IO Block: 4096   regular file Device: 811h/2065d  Inode: 35515       Links: 1 Access: (0600/-rw-------)  Uid: ( 1000/  xxxxxx)   Gid: ( 1000/  xxxxxx) Access: 2010-09-30 14:05:16.041621000 -0700 Modify: 2010-09-30 14:05:05.070157000 -0700 Change: 2010-09-30 14:05:05.070475000 -0700  $ chmod 755 script.sh $ stat script.sh    File: `script.sh'   Size: 297070      Blocks: 584        IO Block: 4096   regular file Device: 811h/2065d  Inode: 35515       Links: 1 Access: (0600/-rw-------)  Uid: ( 1000/  xxxxxx)   Gid: ( 1000/  xxxxxx) Access: 2010-09-30 14:05:16.041621000 -0700 Modify: 2010-09-30 14:05:05.070157000 -0700 Change: 2010-09-30 14:05:05.070475000 -0700   As you can see, it remains unchanged. How to write an RSS feed with Java? I'm using Java, and need to generate a simple, standards-compliant RSS feed. How can I go about this? A few months ago I was introduced to the new  type and was glad 's flaws with regard to time zones were finally taken care of.  However, I was left wondering if there were any overhead or problems that could occur from using this new type.  I work on a multi-locale web application. Does anyone know of anything that could sway me from just using it for all my date/time work? Is there a window for abuse here?  Reference: Force a program to run *without* administrator privileges or UAC? Is there any way that I can force a program that normally requires administrator privileges (via UAC) to run without them? (ie: no UAC prompt and no system-wide access.)  Added: Without modifying the executable itself.    In spite of James's answer, I have found a few ways that it can almost be done:   By modifying the executable I can remove the trustInfo entry from the manifest (or the manifest entirely, so I can use an external one), allowing the program to start without UAC. Unfortunately this modifies the executable, so it exits shortly after due to an internal checksum test. By using Process Explorer I can launch it as a Limited User. However this seems to limit it significantly more than I would like (it runs like Protected Mode IE and so can access significantly less than what my standard un-elevated user can). Is there an idiom for people who boast too much? I am looking for idioms or informal/slang/colloquial expression for some people that make you think that they are able of building a skyscraper, constructing a spaceship, playing the piano better than Mozart or something very fancy that requires remarkable skills, but when it comes to practice they prove to be completely inadequate. Proof of uniform limit of Continuous Functions The uniform limit of Continuous Functions is continuous.   Proof Let $\varepsilon &gt; 0$. There exists $N$ in natural numbers such that $n&gt;N$ implies  $|f_n(x)-f(x)| &lt; \frac{\varepsilon}{3}$ for all $x$ in $S$. In particular $|f_{N+1}(x)-f(x)| &lt; \frac{\varepsilon}{3}$ for all $x$ in $S$. Since $f_{N+1}$ is continuous at $x_0$ there is a $\delta&gt;0$ such that, $x$ in $S$ and $|x-x_0| &lt; \delta$ imply $|f_{N+1}(x)-f_{N+1}(x_0)|&lt; \frac{\varepsilon}{3}$ Now we conclude $x$ in $S$ and $|x-x_0| &lt; \delta$ imply $|f(x)-f(x_0)|&lt; 3 \cdot \frac{\varepsilon}{3} = \varepsilon$.  Can someone explain what is happening here? I don't follow it all.