Monoids as a category and product structures I'm currently going through  and I'm struggling to answer the question posed on page 26 (directly before Lecture 9).  Question:  Characterize what it means for a monoid, regarded as a category, to have products. Can a finite monoid have products? I'm using Definition 3 for products, on the bottom of page 22. Suppose $(M, \times)$ is the monoid that defines the single-object category, I tend to call the object $\cdot$. A product structure will be the $4$-tuple $(\cdot, \pi_1, \pi_2, &lt;-,-&gt;)$ where $\pi_1, \pi_2 \in M$ and for every $f_1,f_2 \in M$ I have some element $&lt;f_1,f_2&gt; \in M$ so that $\pi_i \times &lt;f_1,f_2&gt; = f_i$. I don't see how I could pick such $\pi_i$ unless $M=\{e\}$, in which case $\pi_i=&lt;-,-&gt;=e.$ What other $M$ could this work for?  Any help would be greatly appreciated.
Name for categorical product inside a monoid If a monoid is a category with a single object, is there a "monoid-theoretical" concept that the categorical product translates to? As an analogue, in a poset the product translates to the notion of meet (and the coproduct to its dual, the join).  Just by unwinding the definition one can see that this boils down to the existence of two monoid elements $p_1$ and $p_2$ such that for every $x$ and $y$ in the monoid there's a unique element $\langle x,y \rangle$ that factors as $x = p_1\langle x,y \rangle$ and $y = p_2\langle x,y \rangle$; the question is whether the concept is a known and/or useful one.  Incidentally, if the monoid in question is a group then the existence of a product makes the whole thing collapse into the trivial group; this might or not hint that the concept I'm looking for is "too boring" to have a name. In fact it suffices for one of the projections to be right cancellable, or for the pair $\langle x,x\rangle$ to be left cancellable for some $x$, to make the whole thing collapse.
Proving an alternating Euler sum: $\sum_{k=1}^{\infty} \frac{(-1)^{k+1} H_k}{k} = \frac{1}{2} \zeta(2) - \frac{1}{2} \log^2 2$ Let $$A(p,q) = \sum_{k=1}^{\infty} \frac{(-1)^{k+1}H^{(p)}_k}{k^q},$$ where $H^{(p)}_n = \sum_{i=1}^n i^{-p}$, the $n$th $p$-harmonic number.  The $A(p,q)$'s are known as alternating .     Can someone provide a nice proof that    $$A(1,1) = \sum_{k=1}^{\infty} \frac{(-1)^{k+1} H_k}{k} = \frac{1}{2} \zeta(2) - \frac{1}{2} \log^2 2?$$   I worked for a while on this today but was unsuccessful.  Summation by parts, swapping the order of summation, and approximating $H_k$ by $\log k$ were my best ideas, but I could not get any of them to work.  (Perhaps someone else can?)  I would like a nice proof in order to complete .  Bonus points for proving $A(1,2) = \frac{5}{8} \zeta(3)$ and $A(2,1) =  \zeta(3) - \frac{1}{2}\zeta(2) \log 2$, as those are the other two alternating Euler sums needed to complete my answer.   Added: I'm going to change the accepted answer to robjohn's $A(1,1)$ calculation as a proxy for the three answers he gave here.  Notwithstanding the other great answers (especially the currently most-upvoted one, the one I first accepted), robjohn's approach is the one I was originally trying.  I am pleased to see that it can be used to do the $A(1,1)$, $A(1,2)$, and $A(2,1)$ derivations. I am firmly convinced he is innocent and his accusation is the result of the police conspiracy.     I am firmly convinced he is innocent and his accusation is the result of the police's conspiracy.    Which sentence do you think is a better choice? How to check if a string contains a substring in Bash I have a string in Bash: string=&quot;My string&quot;  How can I test if it contains another string? if [ $string ?? 'foo' ]; then   echo &quot;It's there!&quot; fi  Where ?? is my unknown operator. Do I use echo and grep? if echo &quot;$string&quot; | grep 'foo'; then   echo &quot;It's there!&quot; fi  That looks a bit clumsy. Value of $\frac{\sqrt{10+\sqrt{1}}+\sqrt{10+\sqrt{2}}+\cdots+\sqrt{10+\sqrt{99}} }{\sqrt{10-\sqrt{1}}+\sqrt{10-\sqrt{2}}+\cdots+\sqrt{10-\sqrt{99}}}$ Here is the question:     $$\frac{\sqrt{10+\sqrt{1}}+\sqrt{10+\sqrt{2}}+\cdots+\sqrt{10+\sqrt{99}} }{\sqrt{10-\sqrt{1}}+\sqrt{10-\sqrt{2}}+\cdots+\sqrt{10-\sqrt{99}}} = \;?$$   ()  I think we need to simplify it writing it in summation sign as you can see here:  $$\frac{\sum\limits_{n=1}^{99} \sqrt{10 + \sqrt{n}}}{\sum\limits_{n=1}^{99} \sqrt{10 - \sqrt{n}}}$$  or in Wolfram Alpha input in comments.  I can compute it too! It's easy to write a script for this kind of question. I need a way to solve it. How would you solve it on a piece of paper? Possible to get type of an aliased type in c#? Type.GetType("System.String")   Is there a lookup for the aliases available somewhere?  Type.GetType("string")   returns null. Why does shrinkage work? In order to solve problems of model selection, a number of methods (LASSO, ridge regression, etc.) will shrink the coefficients of predictor variables towards zero. I am looking for an intuitive explanation of why this improves predictive ability. If the true effect of the variable was actually very large, why doesn't shrinking the parameter result in a worse prediction? automatically execute an Excel macro on a cell change Compute the moment generating function of $Y = X_1X_2 + X_1X_3 + X_2X_3$ Is it possible for a system to have negative potential energy? With respect to both gravity and electromagnetism, to the best of my understanding potential energy is added or subtracted from a system based on the distance between two objects such as charged or massive objects.  Would this then imply that when oppositely charged objects or two massive objects are brought together within a small enough radius that the drop in potential energy would be greater than the combined mass of the two objects making the net mass/energy of the system negative? Let $P$ be a prime. Show that $\exists$ $x \in \mathbb{N}$ such that $f(x) = p$ then $\exists$ $y \in \mathbb{N}$ such that $g(y) = p$ What is given?  $$\text{Let P be a prime}$$ $$\text{Let} \space f(x)= 3x+1$$ $$\text{Let} \space g(x)= 6x+1$$  Show that: If there exists $x \in \mathbb{N}$ such that $f(x) = P $ , then there exists a $y \in \mathbb{N}$ , such that $g(y) = P$  What have I tried?   Given the question says 'show that', that leads me to believe I need to do a proof of sorts. I'm guessing it might be a proof by either strong induction, or just induction (or maybe just a direct proof?).  I don't even know how to set this up If $f(x) =$ $3x + 1 = $ prime...for any integer $x$, then there exists an integer $y$ such that $g(y) = 6y + 1 = $ prime?   Any hints or starting points would be greatly appreciated.   $$\text{Thanks in advance!}$$ What are the rules and limits of Accio spell? We know that Accio spell (aka ) has some limits:   It can be blocked (e.g. Harry couldn't Accio the Diadem in RoR in Deathly Hallows). Many more failures are listed on Wikia.   But what are the exact full rules and limitations governing the spell in Harry Potter world?   Distance?  Line-of-sight?  Knowing exact or approximate location?  Specificity of item (e.g. "Accio Broom" seemed to work for Weasleys in OotP to bring their locked-up brooms - why not just a random broom?) Types of Accio-able stuff? Can you Accio water? Lump of earth? Gases/liquids/etc...? Can you Accio based on jQuery selector (or an SQL where clause), for lack of a better description? E.g. "Accio the book that has the word 'Flammel'"? "Accio biggest green ball from this room"? Can you Accio ALL items of a kind? (the reason I thought of this whole question was seeing "Accio all guns" comment in ). Can you Accio vague stuff? ("Accio meal"). Other rules that can be inferred.   NOTE: While Wikia article has some of the info, it is clearly very lacking in details on specific rules based on specific instances in canon. I'm looking for a fully comprehensive, organized ruleset for the charm, not merely 1-2 tidbits in Wikia article. Need help to interpret output of free command I would appreciate help in interpreting the output of 'free -m' below.    -bash-4.2$ free -m               total        used        free      shared  buff/cache   available Mem:          15793        7112          88        7840        8591         611 Swap:         16891        5289       11602    If I have nearly 11GB free then why am I seeing swap usage.  Shouldn't that be 0?   Also, the 'shared' value is 7840.  Is this memory available for a new application? Is only 'cache' memory available for new applications?  In other words, 'shared' and 'buffer' memory are not available for new applications? How should I interpret the 'available' column with a value of 611?  How is this value arrived at? "Is far from..." - How do I connote the simile, without the literal connotation? How can I cancel an edit I made on another user's question or answer? Edits made by users who do not have edit privileges are placed in a queue until it is peer reviewed.  E.g., I edited someone else's answer. but changed my mind. If I'd edited my own question/answer, there would be a "Rollback" button, but there's no such button in the scenario I'm describing (at least none that I can see).  This is more relevant today because users can only have at most five edits waiting in the queue, and allowing one to cancel an edit would permit them to free up one of their five pending edits for something else.  Is there a way to remove one's edit from the peer review queue? Find highest level of a hierarchical field: with vs without CTEs Two correlated marginally Gaussian RV, but not Jointly Gaussian We know if two RVs are marginally Gaussian and uncorrelated, they can be not jointly Gaussian. How about two correlated RVs which are marginally Gaussian? Are they Jointly Gaussian or no, there are occasions when they are not Jointly Gaussian? MacOS dmg file question Not entirely sure if this is where to ask this, but I can't find another place to...  I've recently switched from Windows to Mac and I am trying to understand more in depth what I'm working with.  My question is: are .dmg files the equivalent of .zip files in Windows? I came to this theory when downloading a .dmg and when I opened it, it contained a .app and and text file read me. To me, this seems like a compressed file.   Is this correct? Post to facebook fan page as a user Is it possible for an encryption method to exist which is impossible to crack, even using quantum computers? Quantum computers are known to be able to crack  a broad range of cryptographic algorithms which were previously thought to be solvable only by resources increasing exponentially with the bit size of the key. An example for that is .  But, as far I know, not all problems fall into this category. On , we can read     Researchers have developed a computer algorithm that doesn’t solve problems but instead creates them for the purpose of evaluating quantum computers.   Can we still expect a new cryptographic algorithm which will be hard to crack using even a quantum computer? For clarity: the question refers to specifically to the design of new algorithms. Error Code: 1215. Cannot add foreign key constraint (foreign keys)