Here is Prob. 19, Sec. 2.3, in the book Topics in Algebra by I.N. Herstein, 2nd edition:  If $S$ is a set closed under an associative operation, prove that no matter how you bracket $a_1a_2 \ldots a_n$, retaining the order of the elements, you get the same element in $S$ (e.g., $\left( a_1 \cdot a_2 \right) \codt \left( a_3 \cdot a_4 \right) = a_1 \cdot \left( a_2 \cdot \left( a_3 \cdot a_4 \right) \right)$; use induction on $n$).  My Attempt:  Of course the assertion holds for $n = 1, 2, 3$. Suppose it holds for every $k = 1, 2, 3, \ldots, n$ for some $n \in \mathbb{N}$ such that $n \geq 3$. Let $a_1, a_2, a_3, \ldots, a_n, a_{n+1}$ be any elements of set $S$.  How to proceed from here? How to prove rigorously that, no matter how we bracket $a_1a_2a_3\ldots a_n a_{n+1}$, retaining the order of the elements, we obtain the same element of $S$.
I've always heard this reasoning, and it makes obvious sense, but how do you actually show it for some arbitrary product? Would it be something like this?  $$(a(b(cd)))e=((ab)(cd))e=(((ab)c)d)e=abcde?$$  Do you just say that the grouping of the parentheses now corresponds to just multiplying straight through? Thanks.
Relativity and Galaxy Rotation Speed If time travels slower nearer gravity wells, why can't the galaxy rotation speeds being faster on the outer edges than the inner areas be explained by relativity? What necessitates dark matter? Has a double slit experiment ever been done using a track chamber or even contemplated? I tried searches and the question has been posed in other fora, but no experiment came up.   Track chambers (cloud chambers, bubble chambers , time projection chambers, solid state detectors like the vertex detectors at LHC)  give the track of the particle as it ionizes the  medium, and could be carried out in geometry after the particle has passed the double slit. The straight track should be pointing back to the slit it came from and its record could be used as the points on the screen in the classical double slit experiment.  The set up as I see it would be the classical setup for single electron through the double slits but instead of a "screen" one has a detector and detects the track.  It should be a long enough detector to get an accuracy less than the slit difference so it could point back to the slit, as the interslit distance is of the order of 100 microns and detectors are giving order of microns accuracies.  This experiment, if possible,  would resolve the controversy whether the detection of the slit destroys the interference pattern or the  at the slits change the boundary conditions and destroy the interference pattern.  An expert's opinion is necessary whether the experiment is possible, whether the energies of the electrons to show interference with a specific d separation is enough to create an accurate track in a solid state detector. If not a cloud chamber would do , but again the energy of the electron would be important because it would have to pass the barrier air/chamber.  It could succeed if the double slits were within a cloud/ bubble chamber; the beam count was low  (10 to twelve per picture) but it was spread in the vertical direction. If the beam could be focused on the slits , it should be doable. If I'm given a complex number (say $9 + 4i$), how do I calculate its square root? Well I thought this is obvious. since $\gcd(a,b)=1$,  then we have that $a\not\mid b$ AND $a\mid bc$. This implies that $a$ divides $c$. But apparently this is wrong. Help explain why this way is wrong please.   The question tells you give me two relatively prime numbers $a$ and $b$ such that $a$ divides the product $bc$, prove that $a$ divides $c$. how is this not obvious? Explain to me how my "proof" is not correct. I'm studying special relativity and saw the 2 postulates of Einstein. The most remarkable one for me is the universal speed of light. Einstein postulated that the speed of light in vacuum is the same for all inertial observers, regardless of the motion of the source. But what if we are accelerating and we are changing from one inertial frame to another, how will we observe the speed of light? Is it still the same? I suggest it's not? Because I alread saw a small introduction to general relativity where the light will bend in a gravitational field and therefore also when you accelerate. So does the speed of light change because it bends? What spell slot count formula can I use to base my homebrew mechanic on? Read Post Data submitted to ASP.Net Form I have a working login form in an asp.net application.  Standard stuff with a username and password text box and a button to process the login.  Works fine.  I have a new requirement to allow the user to input the username and password from a separate plain html page that is not a part of my asp.net application.  I plan on achieving this using standard html - form, input, submit button etc.  The form action will be the URL of my asp.net login page and its method will be POST.    What I want to do in the C# code behind page of the asp.net login form, presumably in the Page_Load event, is to check if the request for the page contains a username and password value being passed in.  If it does then I need to read those values and process the login as if someone had clicked the login button on the asp.net page.  If not then I will display the login form as usual.    How do I check for the existence of, and read, the username and password values in the request for my page. Prove that the ideal $(X_1-a_1,...,X_n-a_n)$ is maximal in $K[X_1,\dots,X_n]$ Let $K$ be a field, and $a_1,\dots,a_n \in K$. Prove that the ideal $$(X_1-a_1,\dots,X_n-a_n)$$ is maximal in $K[X_1,\dots,X_n]$.   I tried proving that the only elements outside the ideal are the invertibles of $K$ (I should still prove that this implies maximality, but it shouldn't be too difficult).  Is there a better strategy, or another stategy? Sum of First $n$ Squares Equals $\frac{n(n+1)(2n+1)}{6}$ Summation of Central Binomial Coefficients divided by even powers of $2$ Whilst working out  the following summation emerged: $$\sum_{m=0}^n\frac 1{2^{2m}}\binom {2m}m$$ The is equivalent to  $$\begin{align} \sum_{m=0}^n \frac {(2m-1)!!}{2m!!}&amp;=\frac 12+\frac {1\cdot3}{2\cdot 4}+\frac{1\cdot 3\cdot 5}{2\cdot 4\cdot 6}+\cdots +\frac{1\cdot 3\cdot 5\cdot \cdots \cdot(2n-1)}{2\cdot 4\cdot 6\cdot \cdots \cdot 2n}\\ &amp;=\frac 12\left(1+\frac 34\left(1+\frac 56\left(1+\cdots \left(1+\frac {2n-1}{2n}\right)\right)\right)\right) \end{align}$$  and terms are the same as coefficients in the expansion of $(1-x)^{-1/2}$.  Once the solution  $$ \frac {n+1}{2^{2n+1}}\binom {2n+2}{n+1}$$ is known, the telescoping sum can be easily derived, i.e.    $$\frac 1{2^{2m}}\binom {2m}m=\frac {m+1}{2^{2(m+1)-1}}\binom {2(m+1)}{m+1}-\frac m{2^{2m-1}}\binom {2m}m$$  However, without knowing this a priori, how would we have approached this problem? I am trying to create an Apex Trigger that performs a Rollup from a custom object to Contacts.   Here is the error that I am receiving:      Didn't understand relationship 'Assignments__r' in FROM part of query   call. If you are attempting to use a custom relationship, be sure to   append the '__r' after the custom relationship name. Please reference   your WSDL or the describe call for the appropriate names.   The child relationship name is Assignments.  Below is the code:      trigger RollupActiveAssigmentLevelsToContact on pse__Assignment__c (after insert, after update, after delete) {      set&lt;Id&gt; set_id = new set&lt;Id&gt;();      List&lt;Contact&gt; con_list = new List&lt;Contact&gt;();      if(trigger.isInsert || trigger.isUpdate){         for(pse__Assignment__c myAssignment : trigger.New) {             set_id.add(myAssignment.pse__Resource__c);         }     }     else if(trigger.isDelete){         for(pse__Assignment__c myAssignment : trigger.Old){             set_id.add(myAssignment.pse__Resource__c);         }     }      if(trigger.isAfter &amp;&amp; (trigger.isInsert || trigger.isUpdate || trigger.isDelete)){         con_list = [SELECT id,Total_Levels__c, (SELECT id,name FROM Assignments__r WHERE Active_Assignment__c = TRUE) FROM Contact WHERE id IN :set_id];          for(Contact con : con_list){             if(con.Assignments__r.size()&gt;0)                 con.Total_Levels__c = con.Assignments__r.size();             else                 con.Total_Levels__c = 0;         }         if(!con_list.isEmpty())         update con_list;     } }   Below is a snap of the Child Relationship name: I am creating a controller for a 12 V system using an Arduino Uno microcrontroller. For the outputs I am using a relay shield to switch the 12 V components. I have a 12 V toggle switch that turns on some 12 V components in the system and I want to use a trigger signal off of this same switch to send to an Arduino digital input. I know that the Arduino can only handle 5 V max. What would be the best way to step down the 12 V coming off of the switch to the 5 V for the input?  EDIT: The system is for use in a car. Would the amperage of the car battery need to be lowered somehow as to not blow up the components? Can players on a PC and an Xbox play together in any way? I.e. can Xbox users connect to a (modfied?) PC server, or vice versa? How to show a binomial random variable dominates another binomial random variable with a smaller success value? Let $X\sim B(n,p_h)$ and $Y\sim B(n,p_\ell)$ be two random variables following a respective binomial distribution, where $p_h&gt;p_\ell$. I want to show that  $$P(X\ge\alpha)\ge P(Y\ge\alpha),$$ for any $\alpha\in\{0,1,\dots,n\}$. In other words, I want to show  $$P(X\ge\alpha)=\sum_{i=\alpha}^{n}\binom{n}{i}p_h^i(1-p_h)^{n-i}\ge P(Y\ge\alpha)=\sum_{i=\alpha}^{n}\binom{n}{i}p_\ell^i(1-p_\ell)^{n-i}.$$  The statement to be proven is obviously true, but proving it is extremely difficult (at least for me). Can anyone give me some guidance on proving it?  It would be nice if someone can give an algebraic proof too, that is, show that  $$\sum_{i=\alpha}^{n}\binom{n}{i}p_h^i(1-p_h)^{n-i}\ge \sum_{i=\alpha}^{n}\binom{n}{i}p_\ell^i(1-p_\ell)^{n-i}.$$  Thank you very much! How to do joins in LINQ on multiple fields in single join Today, when you accept an answer you did not provide, you get a +2 reputation, however, if you accept your own answer, you get +0.  I expect the reason for this is to stop people gaming the system by asking and answering their own questions repeatedly.  However, this has the downside of providing a disincentive of accepting your own answer when it seems the best and instead accepting an inferior one in order to get the reputation bonus.  Therefore, I propose that the accept your own answer is only worth zero when there are no answers provided by anyone other than yourself.  I feel this keeps the intent of preventing gaming the system for reputation, while removing disincentive from picking the best answer, even if it is your own. Displaying two XML listings side by side. I have two XML outputs from a program that I would like to include in a LaTeX document. I would like them to appear side-by-side each in its own box. So that a reader can compare the two outputs while reading the document. Can't seem to find a way to do this, without making images for each output and adding both as images/graphics. Could there be a better option? How do you generate all the permutations of a list in Python, independently of the type of elements in that list?  For example:  permutations([]) []  permutations([1]) [1]  permutations([1, 2]) [1, 2] [2, 1]  permutations([1, 2, 3]) [1, 2, 3] [1, 3, 2] [2, 1, 3] [2, 3, 1] [3, 1, 2] [3, 2, 1] Prove that $\langle a^n \rangle \bigcap \langle a^k \rangle = \langle a^{lcm (n,k)} \rangle$ How can one use multi threading in PHP applications Is there a realistic way of implementing a multi-threaded model in PHP whether truly, or just simulating it. Some time back it was suggested that you could force the operating system to load another instance of the PHP executable and handle other simultaneous processes.  The problem with this is that when the PHP code finished executing the PHP instance remains in memory because there is no way to kill it from within PHP.  So if you are simulating several threads you can imagine whats going to happen.  So I am still looking for a way multi-threading can be done or simulated effectively from within PHP. Any ideas?