Shading / smoothing - how does it work with edges, faces, vertices?
What is vertex and edge shading?
I can prove the converse of it, but I cannot do this one. Here is the problem:     Prove that the implicit function theorem implies the inverse function theorem. Steps to solve this system of equations: $\sqrt{x}+y=7$, $\sqrt{y}+x=11$ I want to solve this system of equations, I have been out of Maths for a long!!  $$\sqrt{x}+y=7$$  $$\sqrt{y}+x=11$$  Just wondering easiest step to find values for $x$ and $y$ from the above equations? Suppose $A$ is a general $n \times n$ matrix and $B$ is obtained by interchanging two rows of $A$. Prove that $\det(B) = -\det(A)$ Suppose that $A$ is a general $n \times n$ matrix and $B$ is obtained by interchanging the first two rows of $A$. Prove that $\det(B) = -\det(A)$.  By general $n \times n$ matrix, I mean $$\left[\begin{matrix} a_{11} &amp; a_{12} &amp; \cdots &amp; a_{1n} \\ a_{21} &amp; a_{22} &amp; \cdots &amp; a_{2n} \\ \vdots &amp; \vdots &amp; \ddots &amp; \vdots \\ a_{n1} &amp; a_{n2} &amp; \cdots &amp; a_{nn} \end{matrix} \right]$$ I'm thinking of using the cofactor expansion of the 2nd row of the first matrix, yielding $$ \det(A) = c_{21} a_{21} + c_{22} a_{22} + \cdots+ c_{2n} a_{2n}  $$ which will be positive or negative depending on $n$ and the values of $a_{21}, \ldots,a_{2n}$, and using the same on the first row in matrix $B$ to get $c_{11} a_{21}$ etc. and then use the fact that the power of the $c$ subscripts will alter $\det(B)$ to be the opposite sign, but I'm not sure how to express all of this mathematically in a proof directly. Help please? How to create a page like this? I'm a LaTeX newbie. I want to create a page layout like in the picture below on every page.  Can someone help please? Thanks. Can I have two Schengen visas in one passport? Is Innate Spellcasting counterspellable? On p.85 of Xanathar’s Guide to Everything, there’s a section clarifying that you need to perceive a spell being cast (by either a verbal, somatic or material component) in order to counterspell it.  It then goes on saying this :     If the need for a spell’s components has been removed by a special ability, such as the sorcerer’s Subtle Spell feature or the Innate Spellcasting trait possessed by many creatures, the casting of the spell is imperceptible.   The thing is that many monsters’ Innate spellcasting specify that only material components are discarded (not verbal or somatic). So I am not sure whether you can actually counterspell innate spellcasting when the stat block only mentions removing material components.  There seems to be  supporting that such spells could be counterspelled (although it is confusing a bit).  It basically comes down to whether the above quote’s “many creatures” refers to all creatures with innate spellcasting, or only those that forgo all spell components to innanely cast. So, which is it ? Proving that a normal, abelian subgroup of G is in the center of G if |G/N| and |Aut(N)| are relatively prime. SQL Servers has Traces and XEvents. These are used to capture and analyze what is going on with our SQL Server Instances. Events are stored in the stack for later analysis. For example,  If I decide to monitor any dead lock in the database, I just query the trace file to see the history of deadlock for a period of time. Here is my question:  While events occur, in our example deadlock event, is there a way to get an email notification using msdb.dbo.xp_send_dbmail? Why does shell Command Substitution gobble up a trailing newline char? Endpoint of a line knowing slope, start and distance In a Cartesian system, I've got the slope, start point and distance of a line segment.  What's the formula to find the endpoint? FME - Calculate distance of point along a line I have created an FME workspace which inputs a pipeline as well as other linear features which cross it (waterways, roads, railways, fences, other pipelines, etc).  The output is a point file which contains the crossing type.  What I would like to do is calculate the distance along the pipe from the start of the pipeline for each intersection.  So, as you travel along the pipeline, the distance attribute will increase.   I'm thinking that I'll need to use some of the linear referencing tools like MeasureGenerator, MeasureExtractor, or something like that, but I'm not sure how to do so.  Does anyone have an idea of how to do this? I have the below list of files  aro_tty-mIF-45875564pmo_opt aro_tty-mIF-45875664pmo_opt aro_tty-mIF-45875964pmo_opt aro_tty-mIF-45875514pmo_opt aro_tty-mIF-45875524pmo_opt   that I need to rename to  aro_tty-mImpFRA-45875564pmo_opt aro_tty-mImpFRA-45875664pmo_opt aro_tty-mImpFRA-45875964pmo_opt aro_tty-mImpFRA-45875514pmo_opt aro_tty-mImpFRA-45875524pmo_opt Why is $\omega$ the smallest $\infty$? I am comfortable with the different sizes of infinities and Cantor's "diagonal argument" to prove that the set of all subsets of an infinite set has cardinality strictly greater than the set itself. So if we have a set $\Omega$ and $|\Omega| = \aleph_i$, then (assuming continuum hypothesis) the cardinality of $2^{\Omega}$ is $|2^{\Omega}| = \aleph_{i+1} &gt; \aleph_i$ and we have $\aleph_{i+1} = 2^{\aleph_i}$.  I am fine with these argument.  What I don't understand is why should there be a smallest $\infty$? Is there a proof (or) an axiom that there exists a smallest $\infty$ and that this is what we address as "countably infinite"? or to rephrase the question "why can't I find a set whose power set gives us $\mathbb{N}$"?  The reason why I am asking this question is in "some sense" $\aleph_i = \log_2(\aleph_{i+1})$. I do not completely understand why this process should stop when $\aleph_i = \aleph_0$.  (Though coming to think about it I can feel that if I take an infinite set with $\log_2 \aleph_0$ elements I can still put it in one-to-one correspondence with the Natural number set. So Is $\log_2 \aleph_0 = \aleph_0$ (or) am I just confused? If $n \rightarrow \infty$, then $2^n \rightarrow \infty$ faster while $\log (n) \rightarrow \infty$ slower and $\log (\log (n)) \rightarrow \infty$ even slower and $\log(\log(\log (n))) \rightarrow \infty$ even "more" slower and so on).  Is there a clean (and relatively elementary) way to explain this to me?  (I am totally fine if you direct me to some paper (or) webpage. I tried googling but could not find an answer to my question) Most of the recipes I'm using (mainly Modernist; stocks, caramelized vegetables, confit) suggest cooking at 15psi. My pressure cooker has two settings 12psi and 7psi. I've always had success just using 12psi in recipes where it says 15psi. However recently the 12psi mode has broken on my PC. I can now only cook at 7psi. How can I convert my recipes to use this lower pressure?  For example I would cook a chicken stock at 12psi for 90mins, I presume at 7psi the cooking time would need to be longer but by how much? Are film capacitors affected by DC bias? I added this to /etc/crontab on a few different linux &amp; freebsd systems:  # monthly reboot: 3rd Tuesday of every month 56 07 15-21 * 2 root /sbin/shutdown -r now   I want a reboot on the 3rd TUESDAY of every month.  However, all the systems rebooted on the 3rd Wednesday of this month (the 19th).  What am I doing wrong?  Update: Thanks to Ranon's answer below, looks like the below revision will work, can anyone confirm or is there an even better way of doing it?  # monthly reboot: 3rd Tuesday of every month 56 07 15-21 * * root test $(date +\%u) -eq 2 &amp;&amp; /sbin/shutdown -r now How do I say what is the current UTC in eastern zone Can I nohup/screen an already-started process? I am currently thinking about open-sourcing a project of mine and am in the process of preparing the source code and project structure to be released to the public. Now I got one question: how should I handle the signature key for my assemblies? Should I create a new key for the open-source version and publish it along with the other files to the SVN repository? Should I leave the key out and everyone who wants to compile the code should generate his own key?  How do you handle this? I feel a little bit uncomfortable with releasing a signature key to the public. Incompatibility between etextools and etoolbox command \dolistloop (\forlistloop)?