mathematics expression for parameter
How can I define a mathematical function as a LaTeX macro?
$S := \{x \in \Bbb R^3: ||x||_2 = 1 \}$ and $T: S^2 \to \Bbb R$ is a continuous function. Is $T$ injective? $S := \{x \in \Bbb R^3: ||x||_2 = 1 \} \subset (\Bbb R^3, || \cdot ||_2 )$ and $T:S^2 \to (\Bbb R, |\cdot |)$ is  a continuous function.  I've already shown that $$T_{\mathrm{max}} := \mathrm{sup}\{ T(x) : x \in S^2\}  \: \: \mathrm{ and }\: \: T_{\mathrm{min}}:= \mathrm{inf}\{ T(x) : x \in S^2\}$$ are reached by $T$ with the argument that $S^2$ is closed and bounded and therefore after the Heine-Borel-Theorem compact, and therefore since T is continuous it assumes its max and min in $S^2$.   Now i have answer the following:     1) Is there a point $x_0 \in S^2$ with $T(x_0)=T(-x_0)$ ?      2) Is there a value $T \in(T_{\mathrm{min}}, T_{\mathrm{max}})$ which is only assumed at exactly one point of $S^2$ ?   I wanted to show that $T$ is injective to get that 1) is false and 2) is true but i didn't get anywhere. Any ideas? Thanks in advance! How would you represent "religion" in an icon, neutrally? How to create this curved 90-degree-angle quickly Do iOS app visits count to your consecutive visits $\lfloor x\rfloor + \lfloor y\rfloor \leq \lfloor x+y\rfloor$ for every pair of numbers of $x$ and $y$ Spatial query to return multiple values I have a woodland scheme advice layer that has a polygon for each area of advice (brown layer in the picture). I also have a polygon layer of UK local authorities (red boundary layer in the picture). I would like to perform a spatial query which returns the local authority name for each woodland advice scheme. This would work fine if each woodland advice scheme was found in just one local authority boundary. However many are found across a boundary of two different authorities. For instance, one scheme sits across the boundaries of the authorities of Fife and Angus. The scheme is the yellow selected polygon in the picture and the red line is the authority boundary. How do I get the spatial join to return a value of Fife; Angus (or something similar to identify two values) instead of just Fife or Angus. Solve differential equation, $x'=x^2-2t^{-2}$ Solve differential equation:  $x'=x^2- \frac{2}{t^2}$  Maybe is it sth connected with homogeneous equation? I have no idea how to solve it. I have two tables, movies and categories, and I get an ordered list by categoryID first and then by Name.  The movie table has three columns ID, Name and CategoryID. The category table has two columns ID and Name.  I tried something like the following, but it didn't work.  var movies = _db.Movies.OrderBy( m =&gt; { m.CategoryID, m.Name }) Given $|z|&lt;1$ with $z \in \mathbb{C}$ prove $\lim_{n \to \infty} z^n=0$.  My attempt given $\epsilon &gt;0$ we wish to show that there exists $\delta &gt;0$ such that $n&gt; \delta$ means $|z^n-0| &lt; \epsilon$.  For $n \in \mathbb{N}$. We have $|z^n|=|z|^n$. We wish $|z|^n &lt; \epsilon$. This means,  $$n \ln |z| &lt; \ln \epsilon$$  But because $\ln |z|&lt;\ln 1=0$ dividing both sides by $\ln |z|$ gives:  $n&gt; \frac{\ln \epsilon}{\ln |z|}$  So that's what we choose for $\delta$.   My question is is how can we get away from the assumption that $n \in \mathbb{N}$. Or is this correct even with the assumption? Why is the mol a fundamental physical quantity? I am starting to study physics in detail and as I read about physical quantities, I was puzzled why mol (amount of substance) is taken as a physical quantity.  A physical quantity is any quantity which we can measure and has a unit associated with it. But a mol represents the amount of substance by telling the number of particles (atoms, molecules, ions, etc.) present. So it is a pure number and numbers are dimensionless. So mol should not be considered a physical quantity.  Also, fundamental physical quantities should be independent of each other. I am wondering whether mass and mol are independent. This is so as they surely affect each other as we can evidently see while calculating the number of moles and using the mass of that sample for calculation.  So how is the mol a fundamental physical quantity and independent of mass? Free, open-source substitutes for Mendeley? is (mainly) a proprietary social network to share basic citation data and research papers.   Are there any free and open source substitutes for this service?   A technical possibility: users could collect BibTeX data and a hash and share this information.  It would be very useful to have such a free service because every journal provides the citation data in a different format. That includes false field entries, broken files and hidden download buttons on the website. The most common way to refer to non-immediate family? Cascading Drop down in SharePoint Online I'm trying to implement cascading drop down for list in SharePoint online  I have 3 different list with name Countries , States, cities.  and the master list Address in which I have Fields Country , State, City.  I'm trying to display Sates for selected country. and City for selected States in SharePoint Online list How can I find local IP addresses (i.e. 192.168.x.x or 10.0.x.x) in Python platform independently and using only the standard library? Can I use Beetel 450TC1 as wireless router? Find the limit $$\lim_{x\to 0^{+}}\dfrac{\sin{(\tan{x})}-\tan{(\sin{x})}}{x^7}$$  My attempt: Since $$\sin{x}=x-\dfrac{1}{3!}x^3+\dfrac{1}{5!}x^5-\dfrac{1}{7!}x^7+o(x^7)$$ $$\tan{x}=x+\dfrac{1}{3}x^3+\dfrac{2}{15}x^5+\dfrac{1}{63}x^7+o(x^3)$$ So $$\sin{(\tan{x})}=\tan{x}-\dfrac{1}{3!}(\tan{x})^3+\dfrac{1}{5!}(\tan{x})^5-\dfrac{1}{7!}(\tan{x})^7+o(x^7)$$ Though this method might solve, I think this problem has nicer methods. Thanks. Move object in inches instead of feet? Whenever I move an object in blender in unit mode(Imperial), it always make it in feet. For example, if I type G Y 12, it will move 12 feet. How can I make it move 12 inches instead? Intuition for groups This is quite a non-standard question, certainly for mathematics, though I believe it is no less important (for me and my peers, i.e. grads).  The course I am reading so far introduced us to Groups, rings, fields, etc. in the first year, progressing to characters, reps etc in the latter years.  However, even with representation theory I still don't feel like I have a good intuition.  How do you gain intuition in such an area. I know there will be a response of doing examples, but this doesn't really help - I've done lots already.  Am I doomed. How to create a virtual audio device and stream audio input with it Here's my concrete problem:  I am broadcasting a game via XFire and it uses the Windows audio device to capture any audio I receive. As I am broadcasting, other users who watch the video stream are communicating with me over Skype, and they hear themselves back within the video stream and it is entirely logical since I am broadcasting the audio I hear.  What I want to do is create another audio device within Windows and redirect (pipe) ONLY the audio input from that game and not the input reveived from Skype. I would then tell XFire to use that newly created "virtual" audio device to broadcast and therefore my partners won't hear themselves back.  Is it possible to create another audio device and redirect only wanted input streams to it? Disk full, du tells different. How to further investigate? I have a SCSI disk in a server (hardware Raid 1), 32G, ext3 filesytem. df tells me that the disk is 100% full. If I delete 1G this is correctly shown.  However, if I run a du -h -x / then du tells me that only 12G are used (I use -x because of some Samba mounts).  So my question is not about subtle differences between the du and df commands but about how I can find out what causes this huge difference?  I rebooted the machine for a fsck that went w/out errors. Should I run badblocks? lsof shows me no open deleted files, lost+found is empty and there is no obvious warn/err/fail statement in the messages file.  Feel free to ask for further details of the setup.