For a function $g: X \times Y \rightarrow \mathbb R$ show that: $$\sup_{y\in Y}\inf_{x\in X} g(x,y) \leq \inf_{x\in X}\sup_{y\in Y} g(x,y)$$ My attempt: By definition: $\inf_{\bar x\in X} g(\bar x,y) \leq \sup_{\bar y\in Y} g(x,\bar y)$ for each $x,y$. In particular, this is true for the pair $(x,y)$ such that $y$ maximizes the LHS and  $x$ minimizes the RHS. So the claim holds. My difficulty In my proof, I've assumed that the infimum and supremum are attained. This isn't necessarily the case if $X,Y$ are not closed or if the infimum/supremum is at $\infty$. What is a way to clean up this argument? My thought is to instead consider sequences that approach infimum and supremum, respectively. Is there some more direct way?
Came across this little practice exercise, and I couldn't properly convince myself of this relation:  Let $X,Y \subset \mathbb{R}^n$ and $g:X\times Y \rightarrow \mathbb{R}$. Show that $$\sup_{y \in Y} \inf_{x \in X}g(x,y) \leq \inf_{x \in X} \sup_{y \in Y} g(x,y).$$  My thinking was starting with $\inf_{x \in X}g(x,\overline{y}) \leq\sup_{y \in Y} g(\overline{x},y)$, but does that even hold for all $\overline{x} \in X$ and $\overline{y} \in Y$?
We know  speedup brute-force attacks two times faster in block ciphers (e.g brute-forcing 128-bit keys take $2^{64}$ operations, not $2^{128}$). That explains why we are using 256-bit keys to encrypt top secrets. But  on AES shows brute-forcing AES-256 take $2^{100}$ operations. Does this attack work with Grover's search to make AES cipher quantum unresistant? Infectious chicken pox My 21 year old son has chicken pox and has been home for a few days now. The spots are just starting to crust over.  I work with elderly people but have had chicken pox as a child - should I carry on working while he is here (hes usually away at Uni) and infectious? How can I get the Cartesian product (every possible combination of values) from a group of lists?  Input:   somelists = [    [1, 2, 3],    ['a', 'b'],    [4, 5] ]   Desired output:  [(1, 'a', 4), (1, 'a', 5), (1, 'b', 4), (1, 'b', 5), (2, 'a', 4), (2, 'a', 5) ...] Why is 4 not an instance of Number? Just curious:    4 instanceof Number => false new Number(4) instanceof Number => true?   Why is this? Same with strings:    'some string' instanceof String returns false new String('some string') instanceof String => true String('some string') instanceof String also returns false ('some string').toString instanceof String also returns false   For object, array or function types the instanceof operator works as expected. I just don't know how to understand this.  [new insights]  Object.prototype.is = function() {         var test = arguments.length ? [].slice.call(arguments) : null            ,self = this.constructor;         return test ? !!(test.filter(function(a){return a === self}).length)                : (this.constructor.name ||                   (String(self).match ( /^function\s*([^\s(]+)/im )                     || [0,'ANONYMOUS_CONSTRUCTOR']) [1] ); } // usage var Newclass = function(){};  // anonymous Constructor function var Some = function Some(){}; // named Constructor function (5).is();                     //=&gt; Number 'hello world'.is();           //=&gt; String (new Newclass()).is();        //=&gt; ANONYMOUS_CONSTRUCTOR (new Some()).is();            //=&gt; Some /[a-z]/.is();                 //=&gt; RegExp '5'.is(Number);               //=&gt; false '5'.is(String);               //=&gt; true Reproject GeoJSON from EPSG:3857 to EPSG:4326 I am using this  to convert a GeoJSON point from EPSG:3857 to EPSG:4326.   My command fails. What am I doing wrong?   $ echo '{"type":"Point","coordinates":[-13720479.997764934,5688254.733325758]}' | reproject --from=EPSG:3857 --to=EPSG:4326   /usr/local/lib/node_modules/reproject/cli.js:98             throw new Error("Could not find definition for CRS \"" + crsName + "\".");                   ^ Error: Could not find definition for CRS "EPSG:3857".     at lookupCrs (/usr/local/lib/node_modules/reproject/cli.js:98:10)     at Object.&lt;anonymous&gt; (/usr/local/lib/node_modules/reproject/cli.js:25:1)     at Module._compile (module.js:456:26)     at Object.Module._extensions..js (module.js:474:10)     at Module.load (module.js:356:32)     at Function.Module._load (module.js:312:12)     at Function.Module.runMain (module.js:497:10)     at startup (node.js:119:16)     at node.js:901:3 Is it normal for focal length to change while focusing? When I change the focus distance on my Pentax 55-300mm lens, the focal length appears to change.  More specifically, the lens appears to zoom in as I focus towards the minimum focusing distance.  This is most noticeable at the 55mm end of the lens.  Is this normal? How do I print an IFrame from javascript in Safari/Chrome Can someone please help me out with printing the contents of an IFrame via a javascript call in Safari/Chrome.  This works in firefox:  $('#' + id)[0].focus(); $('#' + id)[0].contentWindow.print();   this works in IE:  window.frames[id].focus(); window.frames[id].print();   But I can't get anything to work in Safari/Chrome.  Thanks  Andrew The following definition is needed for my actual question:  For $A, B \in \mathcal{M}_{n \times n}$ define $$  \langle A, B \rangle = \text{Trace}(B^{T}A) = \sum_{j=1}^{n}\sum_{i=1}^{n}b_{ij} \, a_{ij} $$  And here is the actual question:  For $B \in \mathcal{M}_{n \times n}$ show that $$ \text{Trace}(B)^{2} \:\leq\: n \,\text{Trace}(B^{T}B) $$  I know I have to use the Cauchy-Schwarz Inequality, but I wasn't exactly sure how to.  Here is my attempt:  Since I only have one matrix I decided to use the identity matrix $$ \begin{align} \text{Trace}(B)^{2} \:&amp;\leq\: n \,\text{Trace}(B^{T}B) \\ \text{Trace}(I^{T}B)^{2} \:&amp;\leq\: n \, \langle B, B \rangle \\ \langle I_{n}, B \rangle^{2} \:&amp;\leq\: n \, \Vert B \Vert^{2} \\ \langle I_{n}, B \rangle \:&amp;\leq\: \sqrt{n} \, \Vert B \Vert \end{align} $$  I feel like I either did it incorrect, or that I'm close but am missing something. I know that python has a len() function that is used to determine the size of a string, but I was wondering why it's not a method of the string object.  Update  Ok, I realized I was embarrassingly mistaken. __len__() is actually a method of a string object. It just seems weird to see object oriented code in Python using the len function on string objects. Furthermore, it's also weird to see __len__ as the name instead of just len. Custom Adapter for List View Why is it that Android still can't be installed as regular OS? Are there any technical or others reasons why Android is still a firmware and not an OS which can be just installed without a lot of hassle?  Why there is no universal image which could detect appropriate drivers (kernel modules), partition NAND in some standard way and configure boot loader to boot updated OS version?  What stops Google from making a real (less firmware-like) OS (an OS for small ARM, x86 etc. devices)? How to load JAR files dynamically at Runtime? Why is it so hard to do this in Java? If you want to have any kind of module system you need to be able to load JAR files dynamically. I'm told there's a way of doing it by writing your own ClassLoader, but that's a lot of work for something that should (in my mind at least) be as easy as calling a method with a JAR file as its argument.  Any suggestions for simple code that does this? How to resolve merge conflicts in Git repository? An irreducible $f\in \mathbb{Z}[x]$, whose image in every $(\mathbb{Z}/p\mathbb{Z})[x]$ has a root? Problem: Is there an irreducible $f\in \mathbb{Z}[x]$, whose image in every $(\mathbb{Z}/p\mathbb{Z})[x]$ has a root for $p$ prime? If there is, what is the minimal degree possible?   I can only prove that $x^2-c$ is impossible, by quadratic reciprocity and Chinese remainder theorem. Even the case of $a x^2 - c$ is unknown to me.  Meanwhile, if $f$ is not required  to be irreducible, but only have no root in $\mathbb{Z}$, then $(x^2-a)(x^2-b)(x^2-ab)$ is a solution for non-sqaure $a,b,ab$, since if both $a,b$ are not squares in $\mathbb{Z}/p\mathbb{Z}$, then $ab$ is a square. Yet I still don't know if this is of minimal degree.  Also, it is natural to pose generalizations   When $p$ is not necessarily prime (equivalently for all prime powers). When $p$ is odd prime. When $p$ represents sufficiently large primes (equivalently all but a finite number left). Schengen visa permit I am a student with a Singapore passport currently studying in the Czech Republic for a semester. My permit is valid until 2nd of February, I plan to fly to the UK to travel for 3 days before heading back to Singapore from there.  Would there be any problems, such as exit immigration from the Czech Republic to the UK? I searched online that the UK is not part of Schengen Area Agreement states, and my passport has a visa free entry to the UK. Wireless USB Adapter 0bda:c811 Realtek Semiconductor Corp From Euler we've learned that $z=re^{i\theta}$.  And it's easy to see that $|z|^2=r^2$, since $re^{i\theta}\times re^{-i\theta}=r^2$.  Why must we use e to represent these numbers correctly? It seems that I could arbitrarily choose a different exponent $z=r\pi^{i\theta}$ and get the same size for $z$ as I did before: $|z|^2=r\pi^{i\theta}\times r\pi^{-i\theta}=r^2$  What did I miss? What is the simplest way of proving (to a non-mathematician) that the power set of the set of natural numbers has the same cardinality as the set of the real numbers, i.e. how to construct a bijection from $\mathcal{P}(\mathbb{N})$ to $\mathbb{R}$? Google Search Console domain property not in the list of properties I can add to Google Analytics Is Cursed Child canon to the original story? From what I am hearing the stage-play called The Cursed Child is based on the characters from the Harry Potter series. It is set 19 years from the end of Deathly Hallows (i.e. When Harry's children are starting their time at Hogwarts).   I have also seen the J.K. Rowling has had at least some involvement and been on the set and a part of rehearsals, etc.   My question is has there been a statement given relating to the canon of the stage play to original series? If so, would any conflicting information be considered a retcon?  Related:  - but not dupe because this asking about in-universe information.