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does anyone know the following result? If it holds in this form and any source which presents it? Thanks a lot.  Consider metric space $(X,d_{X})$.  Let $f:A \subset (X,d_{X}) \rightarrow \mathbb{R}$ be a Lipschitz continuous function i.e. $\exists K &gt; 0$ such that  \begin{align*} |f(x)-f(y)| \leq Kd_{X}(x,y) ~~~~\forall x,y \in A \end{align*}  then define functions  $$\bar{f_{1}}(x) := \sup\lbrace f(y) - Kd_{X}(x,y): y\in A \rbrace ~~ \forall x  \in X$$ and $$\bar{f_{2}}(x) := \inf\lbrace f(y) + Kd_{X}(x,y): y\in A \rbrace ~~ \forall x  \in X$$ it follows that  \begin{align*} \bar{f_{1}}, \bar{f_{2}}: X \rightarrow \mathbb{R}  \end{align*} are Lipschitz continuous with  \begin{align*} |\bar{f_{i}}(x)-\bar{f_{i}}(z)| \leq Kd_{X}(x,z) ~~~\forall x,z \in X \text{ and } \forall  i = 1,2 \end{align*} and  \begin{align*} \bar{f_{i}}(x) = f(x) ~~~ \forall x \in A \text{ and } \forall i =1,2 \end{align*} Ordered Pair Formal Definition I understand what an ordered pair is, but I'm having trouble the formal Kuratowski definition which says that $\langle a,b \rangle = \{\{a\},\{a,b\}\}$. What about this definition imposes order on the pair? Max dimension of a subspace of singular $n\times n$ matrices I am sure the answer to this is (kind of) well known. I've searched the web and the site for a proof and found nothing, and if this is a duplicate, I'm sorry.  The following question was given in a contest I took part. I had an approach but it didn't solve the problem.     Consider $V$ a linear subspace of the real vector space $\mathcal{M}_n(\Bbb{R})$ ($n\times n$ real entries matrices) such that $V$ contains only singular matrices (i.e matrices with determinant equal to $0$). What is the maximal dimension of $V$?   A quick guess would be $n^2-n$ since if we consider $W$ the set of $n\times n$ real matrices with last line equal to $0$ then this space has dimension $n^2-n$ and it is a linear space of singular matrices.  Now the only thing there is to prove is that if $V$ is a subspace of $\mathcal{M}_n(\Bbb{R})$ of dimension $k &gt; n^2-n$ then $V$ contains a non-singular matrix. The official proof was unsatisfactory for me, because it was a combinatorial one, and seemed to have few things in common with linear algebra. I was hoping for a pure linear algebra proof.  My approach was to search for a permutation matrix in $V$, but I used some 'false theorem' in between, which I am ashamed to post here. Here comes Bob.  Bob is coming here.  Would anyone please explain what is  the difference between these? How to delete duplicate LiDAR points? How can I delete duplicate LiDAR points? I incorrectly read the same las twice and saved as a one file resulting in duplicate XY points.  How should I delete these duplicate points? Where can I get Blender test builds? I know that  provides downloads for official releases. I see on forums that people use newer versions of Blender. Where can I get newer version of Blender (nightly builds, alpha versions... etc). Solving Hardy Weinberg problems I really fail to understand Hardy-Weinberg equilibrium and can't find an easy enough source of information.  Can you help me to understand Hardy-Weinberg equilibrium?  My goal is to be able to solve the following kind of problem     In a population with two alleles for a certain locus, B and b, the allele frequency of B is 0.7. What is the frequency of heterozygotes if the population is in Hardy-Weinberg equilibrium? Delete specific line number(s) from a text file using sed? I want to delete one or more specific line numbers from a file. How would I do this using sed? Bottom of tool shelf grayed out What is the name of the tactic that politicians use to bury people with torrent of words? Some people write bloated books and long essays with skilful use of hooks, e.g. Jared Diamond; some others speak in long-drawn sentences with torrents of words, e.g. Noam Chomsky. It reminds me of a tactic politicians use in public discourse to waste people's time. I came across this word before but I can't retrieve it from either memory or the internet. How to transform negative values to logarithms? I would like to know how to transform negative values to Log(), since I have heteroskedastic data. I read that it works with the formula Log(x+1) but this doesn't work with my database and I continue getting NaNs as result. E.g. I get this Warning message (I didn't put my complete database because I think with one of my negative values is enough to show an example):   &gt; log(-1.27+1) [1] NaN Warning message: In log(-1.27 + 1) : NaNs produced &gt;    Thanks in advance  UPDATE:  Here is an histogram of my data. I'm working with palaeontological time series of chemical measurements, E.g the difference between variables like Ca and Zn is too big, then I need some type of data standardization, that is why I'm testing the log() function.   This is my I have a hard time finding a how-to.  I would like to set up a simple ESD mat and wrist wrap, but I am not sure about is the the 1 MOhm resistor should be, if at all.  Several options:   1 MOhm from ground outlet to the mat, wire from the common earthing on the mat to the wrist wrap. 1 MOhm from ground outlet to the mat, wire from another contact on the mat to the wrist wrap. 1 MOhm from ground outlet to the mat, 1 MOhm from the same mat contact to the wrist wrap. wire from ground outlet to the mat, 1 MOhm from the same mat contact to the wrist wrap. wire from ground outlet to the mat, wire from another mat contact to the wrist wrap.   I am not sure why the 1 MOhm wire is needed since the mat itself already has that kind of resistance (as long as the wrist wrap does not connect to ground directly, but through the mat).  Which solutions above are ok? Which is best? Should I do something else instead? Photoshop To CSS: How do you translate Fill to CSS Usage of Some: Does "some" require a plural or singular verb? In Ruby on Rails, how do I format a date with the "th" suffix, as in, "Sun Oct 5th"? Careers Stack Overflow - Tag "ruby-on-rails" returns jobs that don't contain the tag Prove that for any finite sequence of decimal digits, there exists an $n$ such that the decimal expansion of $2^n$ begins with these digits. What part of speech is there in the sentence “There is a book on the table?”  Also, while typing it out, another question pops up vis-à-vis punctuation. In my complete first sentence above, I ended it with a question mark since my main sentence is a question. The quote is not a question, but it looks like it because of the question mark. Is this the right way to punctuate? Use Stokes's Theorem to show $\oint_{C} y ~dx + z ~dy + x ~dz = \sqrt{3} \pi a^2$ I am a little stuck on the following problem:  Use Stokes's Theorem to show that  $$\oint_{C} y ~dx + z ~dy + x ~dz = \sqrt{3} \pi a^2,$$  where $C$ is the suitably oriented intersection of the surfaces $x^2 + y^2 + z^2 = a^2$ and $x + y + z = 0$.  OK, so Stokes's Theorem tells me that:  $$\oint_{C}\vec{F} \cdot d\vec{r} = \iint_{S}\operatorname{curl} \vec{F} \cdot \vec{N} ~dS$$  I have calculated:  $$\operatorname{curl} \vec{F} = -\vec{i} - \vec{j} - \vec{k}.$$  I then figured that on the surface $S$ we must have:  $$\vec{N}dS = \vec{i} + \vec{j} + \vec{k}dxdy$$  since this follows from the equation of the given plane.  However this will then give me:  $$\operatorname{curl} \vec{F} \cdot \vec{N} = -1 -1 -1 = -3$$  And thus I would get, if project this onto the $xy$-plane:  $$\iint_{S} \operatorname{curl} \vec{F} \cdot \vec{N} ~dS = -3 \iint_{A} dA = -3 \pi a^2$$  which is obviously not correct.  I would greatly appreciate it if someone could help me with this.  I actually had multivariable calculus a few years ago, and I know that I knew this stuff then.  However, now that I need it again I notice that I've become quite rusty.  Thanks in advance :) How to prove that the radial basis function is a kernel? How to prove that the radial basis function $k(x, y) = \exp(-\frac{||x-y||^2)}{2\sigma^2})$ is a kernel? As far as I understand, in order to prove this we have to prove either of the following:   For any set of vectors $x_1, x_2, ..., x_n$ matrix $K(x_1, x_2, ..., x_n)$ = $(k(x_i, x_j))_{n \times n}$ is positive semidefinite. A mapping $\Phi$ can be presented such as $k(x, y)$ = $\langle\Phi(x), \Phi(y)\rangle$.   Any help?