It was a top 6. First match a team lost 3:2 on penalties after 0:0 score at regulation time. The other one lost 4:3 and the other one lost 4:3 on penalties after 2:2 draw on regulation time Who is the best loser?
We have played a 7-team competition, all seven teams met, we eliminated the team with the lowest points. After that the remaining six teams played knock out games and the three winning teams will go direct to semi-finals.  We decided that 4 teams should be picked to go for semi-final and that the best loser will join the other three to make it four teams to battle at the semi-final and final.  During the quarter final games, one team is beaten 2-0, another is beaten 3-0 and the third team is beaten in a penalty shoot-out by 4-0.  Now should you want a best loser to join the other three teams making four teams to play in the semi-final, who among these three teams would be in the best losers position.
Node.js on multi-core machines Create ColorRamp How can I create a ColorRamp for a Cycles material through Python?  I am able to create a new shader output node with nodes.new(type='ShaderNodeOutputMaterial'). Searching online, I see that it is only possible through C code (however, these pages dates back several years).  Is there a way to fake a ColorRamp (maybe with multiple Math nodes in greater/less than mode) or create a ColorRamp for a Cycles material through Python? How to compute the determinant of a tridiagonal matrix with constant diagonals? How to show that the determinant of the following $(n\times n)$ matrix  $$\begin{pmatrix} 5 &amp; 2 &amp; 0 &amp; 0 &amp; 0 &amp; \cdots &amp; 0 \\ 2 &amp; 5 &amp; 2 &amp; 0 &amp; 0 &amp; \cdots &amp; 0 \\ 0 &amp; 2 &amp; 5 &amp; 2 &amp; 0 &amp; \cdots &amp; 0 \\ \vdots &amp; \vdots&amp; \vdots&amp; \vdots &amp; \vdots &amp; \vdots &amp; \vdots \\ 0 &amp; \cdots &amp; \cdots &amp; 0 &amp; 2 &amp; 5 &amp; 2 \\ 0 &amp; \cdots &amp; \cdots &amp; \cdots &amp; \cdots &amp; 2 &amp; 5 \end{pmatrix}$$  is equal to $\frac13(4^{n+1}-1)$?    More generally:     How does one compute the determinant of the following  (where the three diagonals are constant)?      $$M_n(a,b,c) = \begin{pmatrix} a &amp; b &amp; 0 &amp; 0 &amp; 0 &amp; \cdots &amp; 0 \\ c &amp; a &amp; b &amp; 0 &amp; 0 &amp; \cdots &amp; 0 \\ 0 &amp; c &amp; a &amp; b &amp; 0 &amp; \cdots &amp; 0 \\ \vdots &amp; \vdots&amp; \vdots&amp; \vdots &amp; \vdots&amp; \vdots &amp; \vdots \\ 0 &amp; \cdots &amp; \cdots &amp; 0 &amp; c &amp; a &amp; b \\ 0 &amp; \cdots &amp; \cdots &amp; \cdots &amp; \cdots &amp; c &amp; a \end{pmatrix}$$      Here $a,b,c$ can be taken to be real numbers, or complex numbers.   In other words, the matrix $M_n(a,b,c) = (m_{ij})_{1 \le i,j \le n}$ is such that $$m_{ij} = \begin{cases} a &amp; i = j, \\ b &amp; i = j - 1, \\ c &amp; i = j + 1, \\ 0 &amp; \text{otherwise.} \end{cases}$$  There does not seem to be an easy pattern to use induction: the matrix is not a diagonal  of the type $M = \bigl(\begin{smallmatrix} A &amp; C \\ 0 &amp; B \end{smallmatrix}\bigr)$ (where we could use $\det(M) = \det(A) \det(B)$ for the induction step), and there are no lines or columns with only one nonzero entry, so  gets complicated quickly.  Is there a general pattern that one could use? Or is the answer only known on a case-by-case basis? It's possible to compute the determinant by hand for small $n$:  $$\begin{align} \det(M_1(a,b,c)) &amp; = \begin{vmatrix} a \end{vmatrix} = a  \\ \det(M_2(a,b,c)) &amp; = \begin{vmatrix} a &amp; b \\ c &amp; a \end{vmatrix} = a^2 - bc \\ \det(M_3(a,b,c)) &amp; = \begin{vmatrix} a &amp; b &amp; 0 \\ c &amp; a &amp; b \\ 0 &amp; c &amp; a \end{vmatrix} = a^3 - 2abc \end{align}$$  But there is no readily apparent pattern and the computation becomes very difficult when $n$ gets large. I'm trying to prove the following statements.  Let $G$ be a finite abelian group $G = \{a_{1}, a_{2}, ..., a_{n}\}$.   If there is no element $x \neq e$ in $G$ such that $x = x^{-1}$, then $a_{1}a_{2} \cdot \cdot \cdot a_{n} = e$.   Since the only element in $G$ that is an inverse of itself is the identity element $e$, for every other element $k$, it must have an inverse $a_{k}^{-1} = a_{j}$ where $k \neq j$. Thus $a_{1}a_{1}^{-1}a_{2}a_{2}^{-1} \cdot \cdot \cdot a_{n}a_{n}^{-1} = e$.   If there is exactly one $x \neq e$ in $G$ such that $x = x^{-1}$, then $a_{1}a_{2} \cdot \cdot \cdot a_{n} = x$.   This is stating that $x$ is not the identity element but is its own inverse. Then every other element $p$ must also have an inverse $a_{p}^{-1} = a_{w}$ where $p \neq w$. Similarly to the first question, a rearrangement can be done: $a_{1}a_{1}^{-1}a_{2}a_{2}^{-1} \cdot \cdot \cdot xx^{-1} \cdot \cdot \cdot  a_{n}a_{n}^{-1} = xx^{-1} = e$. And this is where I am stuck since I proved another statement.    Any comments would be appreciated for both problems. Is there a keyboard shortcut to "unselect" in Windows Explorer? Option + click in iTerm In Terminal, Option + click allows me to place my cursor at any point on the current line-- great for editing really long commands.  How can I do this in iTerm2?  Update: According to , the feature has been around since 0.7.0 (2003). I've also noticed that if I hold Command+Option, the cursor changes into the familiar cross like in Terminal...  but clicking still does not reposition my cursor. LaTeX templates for writing a thesis Aside from CTAN, what are good resources/repositories for Latex templates? In particular, I'm looking for some pretty Thesis templates (I'm familiar with classicthesis) Credit score just tanked ~50 points and I'm not sure why I check my credit score monthly using a credit tracking app provided by my credit card company. I use a reminder to check on the same day each month, and have been tracking my credit score for almost two years.  In the last year, the app switched from a TransUnion model to VantageScore 3.0. I have TransUnion data for the entire duration, but only about a year's worth of VantageScore data.  I have good credit, and my TransUnion score has increased steadily from ~740 to ~780 in the last two years.  My VantageScore started around ~760 and increased to ~800 last month. This morning when I checked my credit score, it had dropped just over 50 points to ~750.  My TransUnion score has stayed solid at ~780.  My questions are:  Should I be concerned about this sudden change in score?  Is there any way I can determine what changed to account for the sudden change in credit score? Is there a problem with Beamer in the latest MiKTeX update? How to process all products defined in a project (ConTeXt) If I have  project.tex  \startproject project \product one \product two \endproject   and the corresponding product files, is there an easy way to process all product files, so I get  one.pdf  two.pdf  at once? Position of "only" in “We 𝒐𝒏𝒍𝒚 have two possibilities” Is there any difference in meaning between the three following sentences?   We only have two possibilities. We have only two possibilities. We have two possibilities only.   I have already read  and I know that the position of only and the emphasis (written or oral) may change the meaning of some sentences, but I am not sure if it does here.  Note: It is clear that “only we have two possibilities” has a different meaning. My question is restricted to the sentences I’ve listed above. Index notation Lorentz transfromation matrix Let $A= a(i,j)$ and $B= b(i,j)$ be ($n\times n$) matrices that are positive definite such that $a(i,j) &lt; b(i,j)$ .Let $C= c(i,j)= a(i,j) - b(i,j)$, then $C$ is also positive definite. Why or why not?  what i know is $x^T(C)x= x^T (A-B) x = x^T(A)x - x^T(B)x$. Now both terms on right-hand side are greater than $0$ as $A$ and $B$ are positive definite. Also  $a(i,j) &lt; b(i,j)$. But in such case the sign of $x^T(C)x$ being positive or negative will also depend on the value of elements of vector $x$. So, we can't assure that $x^T(C)x &lt; 0$ always or that $C$ is not positive definite.  Am i right about this? Induction proof: $S$ contains powers of 2 and predecessors implies $S={\bf N}$ Let $S$ be a subset of $\mathbb{N}$ such that $$2^k\in S\quad\forall\ k\in\mathbb N$$ and, $$k\in S\Rightarrow k-1\in S\quad\forall\ k\ge2,k\in\mathbb N$$  Show by induction that $S=\mathbb N$. Cascading Drop down in SharePoint Online OL3 How to select features of a WMS layer Addition is to Integration as Multiplication is to ______ Addition is to Integration as Multiplication is to ______ ? Everyone knows that definite integration  is "a way to sum continuum-many terms" in a rough sense. Can we "multiply continuum-many factors"  in a similar sense? Is there a place to see release dates for previous releases? Failed upgrade from Ubuntu 19.10 to 20.04 "Please update the microcode" While installing the Ubuntu 20.04 updates during upgrade from 19.10 to 20.04, my screen suddenly went completely white and displayed a message saying  Please logout and try again.  But I could not locate my cursor and so I switched off my laptop. After restarting it again, it was showing an error message saying  Please update the microcode  Why did this happen? How can I solve this problem? What is the mechanism behind the slowdown of light/photons in a transparent medium? So light travels slower in glass (for example) than in a vacuum. What causes light to slow down? Or: How does it slow down? If light passes through the medium, is it not essentially traveling in the "vacuum" between the atoms?