Changing `Deleted Recent Questions` to `All Deleted Questions`
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Alternating sum of binomial coefficients: given $n \in \mathbb N$, prove $\sum^n_{k=0}(-1)^k {n \choose k} = 0$ How to add some non-standard font to a website? Is there a way to add some custom font on a website without using images,  or some other graphics?  For example, I was working on a wedding website, and I found a lot of nice fonts for that subject. But I can't find the right way to add that font on the server. And how do I include that font with CSS into the HTML? Is this possible to do without graphics? I've just started to learn about the tensor product and I want to show: $$(\mathbb{Z}/m\mathbb{Z}) \otimes_\mathbb{Z} (\mathbb{Z} / n \mathbb{Z}) \cong \mathbb{Z}/ \gcd(m,n)\mathbb{Z}.$$  Can you tell me if my proof is right:  $\mathbb{Z}/m\mathbb{Z}$ and $\mathbb{Z} / n \mathbb{Z}$ are both finite free $\mathbb{Z}$-modules with the basis consisting of one single element $\{ 1 \}$. So $(\mathbb{Z}/m\mathbb{Z}) \otimes_\mathbb{Z} (\mathbb{Z} / n \mathbb{Z})$ has the basis $\{ 1 \otimes 1 \}$.  Therefore, any element in $(\mathbb{Z}/m\mathbb{Z}) \otimes_\mathbb{Z} (\mathbb{Z} / n \mathbb{Z})$  is of the form $(ab) 1 \otimes 1$ and any element in $\mathbb{Z}/ \gcd(m,n)\mathbb{Z}$ is of the form $k 1 = k$ where $k \in \{ 0, \dots , \gcd(n,m) \}$.  I would like to construct an isomorphism that maps $ab$ to some $k$. Let this map be $ab (1 \otimes 1) \mapsto ab \bmod \gcd(n,m)$.  This is a homomorphism between modules: it maps $0$ to $0$ because it maps the empty sum to the empty sum. It also fulfills $f(a + b) = f(a) + f(b)$ because there is only one element, $a = 1$.   It is surjective. So all I need to show is that it is injective. But that is clear too because if $ab \equiv 0 \bmod \gcd(m,n)$  then both $a \equiv 0 \bmod n$ and $b \equiv 0 \bmod m$ so the kernel is trivial.  Many thanks for your help!! How to use regex in String.contains() method in Java I saw this in a  and am putting it here for posterity.  Problem: Let $A$ and $B$ by square matrices and set $C=AB-BA$. If $AC=CA$, prove $C$ is nilpotent. Exercise on infinite series: pointwise product between convergent serie and unbounded sequence I am not able to solve this exercise. I've tried everything.     Exercise Let be $a_n &gt; 0$ a sequence and suppose that $\sum a_n$ converges. Prove that exists a sequence $c_n &gt; 0$ such that $\lim c_n = \infty$ and $\sum c_na_n &lt; \infty$   I don't even know if I have to find an "always-valid" closed form for $c_n$ (maybe in function of $a_n$?) or prove its existence in another way.  Thanks in advance. Why do some Jedi disappear when they die while others don't? Some Jedi disappear when they die:   In Star Wars Episode IV: A New Hope       At the end of his duel with Vader on the first Death Star, Obi-Wan Kenobi let down his guard and concentrated for a moment, disappearing entirely just as Vader's lightsaber passed through his empty robe.   In Star Wars Episode VI: Return of the Jedi      When Luke returned to Dagobah in the Return of the Jedi, Yoda also disappears when he peacefully dies.  In Star Wars Episode VIII: The Last Jedi      When Luke is exhausted after “being” on Crait he looks into the Horizon and disappears.    And others don't:   Star Wars Episode I: The Phantom Menace      When Qui-Gon Jinn's dies after his duel with Darth Maul, he doesn't  disappear and is cremated.  Star Wars Episode VI: Return of the Jedi      When Vader/Anakin Skywalker dies on the second Death Star, he does not disappear and he is cremated by Luke.      Also, they all became a "ghost" after their death. Why did Dumbledore 'put on' the Resurrection Stone (Ring)? Dumbledore put on Marvolo's ring, knowing that it was the Resurrection Stone which was on the ring. He also knew (even though he &quot;lost his head&quot; in his excitement to see his family again) that the ring was a Horcrux and would &quot;surely carry a curse&quot;:  When I discovered it, after all those years, buried in the abandoned home of the Gaunts - the Hallow I had craved most of all, though in my youth I had wanted it for very different reasons - I lost my head, Harry. I quite forgot that it was now a Horcrux, that the ring was sure to carry a curse. I picked it up, and I put it on...  -Deathly Hallows, Ch 35 - KING'S CROSS Now when Harry realised that the Resurrection Stone was inside the Snitch, all he had to do was turn it three times for it to 'work':  He closed his eyes and turned the stone over in his hand three times. He knew it had happened, because he heard slight movements around him that suggested frail bodies shifting their footing on the earthy, twig-strewn ground that marked the outer edge of the forest.  -Deathly Hallows, Ch 34 - THE FOREST AGAIN I may be missing something, but did Dumbledore incorrectly use the Resurrection Stone? or did Harry just 'think' that he had to turn it thrice? I'm assuming the answer may have been that Dumbledore wanted to have the ring on, and then turn it thrice whenever he felt like conjuring up his family; or that he simply &quot;lost his head&quot;; but he surely didn't need to put it on right? What is the correct way to use the Resurrection Stone? GCD of rationals Disclaimer: I'm an engineer, not a mathematician  Somebody claimed that $\gcd$ only is applicable for integers, but it seems I'm perfectly able to apply it to rationals also:    $$ \gcd\left(\frac{13}{6}, \frac{3}{4} \right) = \frac{1}{12} $$  I can do this for a number of cases on sight, but I need a method. I tried Euclid's algorithm, but I'm not sure about the end condition: a remainder of 0 doesn't seem to work here.    So I tried the following:    $$\gcd\left(\frac{a}{b}, \frac{c}{d} \right) = \frac{\gcd(a\cdot d, c \cdot b)}{b \cdot d}$$  This seems to work, but I'm just following my intuition, and I would like to know if this is a valid equation, maybe the correct method. (It is in any case consistent with $\gcd$ over the natural numbers.)  I'm not a mathematician, so please type slowly :-) Are Variable Operators Possible? Do you use "a" or "an" before acronyms / initialisms? 99% of the time, I'm clear on when I should use "a" versus "an." There's one case, though, where people &amp; references I respect disagree.  Which of the following would you precede with "a" or "an," and why?   FAQ FUBAR SCUBA   [Note: I've read the questions  and , but the rules given there don't necessarily apply here.]    [Edited to add]  Here's a shorter (and hopefully clearer) version of the question…  In written English, which is correct (and why): "a FAQ" or "an FAQ"?  Some references with differing opinions:   an: the  and  a: the Microsoft Manual of Style for Tech Publications, 3e either: the  and What's the difference between the five masses: inertial mass, gravitational mass, rest mass, invariant mass and relativistic mass? Chosing a reference frame in which the Earth is at rest and doesn't rotate Could not open file archive.ubuntu.com_ubuntu_dists_eoan-updates_restricted_cnf_Commands-amd64.xz Greetings from Bangladesh. I am an intermediate user of Ubuntu OS. I am facing problem with apt update. I have searched online but could not find any answer to my problem so I am posting this question please help me.  Err:20 http://archive.ubuntu.com/ubuntu eoan-updates/restricted amd64 c-n-f Metadata   Could not open file /var/lib/apt/lists/partial/archive.ubuntu.com_ubuntu_dists_eoan-updates_restricted_cnf_Commands-amd64.xz - open (13: Permission denied) [IP: 91.189.88.152 80] Ign:35 http://archive.ubuntu.com/ubuntu eoan-updates/multiverse amd64 c-n-f Metadata Err:37 http://archive.ubuntu.com/ubuntu eoan-backports/main amd64 c-n-f Metadata   Could not open file /var/lib/apt/lists/partial/archive.ubuntu.com_ubuntu_dists_eoan-backports_main_cnf_Commands-amd64.xz - open (13: Permission denied) [IP: 91.189.88.152 80]   Reading package lists... Done E: Failed to fetch http://archive.ubuntu.com/ubuntu/dists/eoan-updates/restricted/cnf/Commands-amd64  Could not open file /var/lib/apt/lists/partial/archive.ubuntu.com_ubuntu_dists_eoan-updates_restricted_cnf_Commands-amd64.xz - open (13: Permission denied) [IP: 91.189.88.152 80] E: Failed to fetch http://archive.ubuntu.com/ubuntu/dists/eoan-backports/main/cnf/Commands-amd64  Could not open file /var/lib/apt/lists/partial/archive.ubuntu.com_ubuntu_dists_eoan-backports_main_cnf_Commands-amd64.xz - open (13: Permission denied) [IP: 91.189.88.152 80] E: Some index files failed to download. They have been ignored, or old ones used instead.   I have tried with         removing partial folder's content completely.   Cleaning apt by sudo apt clean   moving the lists to lists.old   then updating the apt      But without luck. Prove that $2730$ divides $n^{13} - n$ for all integers $n$. Prove that $2730$ divides $n^{13} - n$ for all integers $n$.   What I attempted is breaking $2730$ into $2, 3, 5$, and $7, 13$. Thus if I prove each prime factor divides by $n^{13} - n$ for all integers $n$, it will prove the original problem.   I already proved $2, 3$, and $5$ work. Struggling with how to do $91$?  Other solutions are welcome, but they should be associated with mod and solutions of this kind, because my answer should pertain to what was learned in class.  (Edit: Fixed the 91 problem, being slow today...In any case, I've got this problem now, so no answers are necessary. Posting my own soon.) Can I run my CPU at 100% usage for a long time? Integer solutions for $x^2-y^3 = 23$. Identify cartoon about a set of fairy/demon twins in a far future where the demon uses propaganda to take over the world I'm looking for a cartoon from the 70's (I believe) which is about a set of twins; one plump "fairy" and a skeletal "demon". The "fairy" has magic powers and ends up living a life of pleasure, while the "demon" flees and turns evil.  When the "demon" finds his new weapon in the ruins of the ancient world (roughly modern day) he plots to enslave the world; the weapon in question is a set of propaganda movies with Adolf Hitler. Finding the missing digits of $23!$ Difference between StringBuilder and StringBuffer What is the main difference between StringBuffer and StringBuilder? Is there any performance issues when deciding on any one of these?