I am trying to understand the proof that uses a maximal-lexicographic ordering.  For an infinite ordinal $\kappa,$ the canonical well-ordering of $\kappa \times \kappa,$ denoted by $&lt;_{cw}$ is defined as follows: $(\alpha_1, \beta_1)&lt;_{cw} &lt; (\alpha_2, \beta_2)$ iff either one of the following holds.  $(1): \max\{\alpha_1,\beta_1\} &lt; \max\{\alpha_2,\beta_2\}$  $(2): \max\{\alpha_1,\beta_1\} =\max\{\alpha_2,\beta_2\}$ and $\alpha_1&lt;\alpha_2$   $(3): \max\{\alpha_1,\beta_1\} =\max\{\alpha_2,\beta_2\}$ and $\alpha_1 = \alpha_2$ and $\beta_1 &lt; \beta_2.$  How would I prove that this ordering is well ordered?  EDIT: I'm not looking for a proof of $\kappa\cdot\kappa= \kappa,$ for infinite cardinals, I'm looking for a proof that the ordering as it's defined is well ordered on $\kappa\times\kappa$.
I don't understand the proof to the a/m claim. How we know that $\eta &lt; \kappa$ and $(\alpha,\beta)&lt;_{cw} (0, \eta) $ and hence $h: \mu \to \eta \times \eta$ is injective? Appreciate if anyone could advise.  Here is an excerpt of the proof:  For an infinite ordinal $\kappa,$ the canonical well-ordering of $\kappa \times \kappa,$ denoted by $&lt;_{cw}$ is defined as follows: $(\alpha_1, \beta_1)&lt;_{cw} &lt; (\alpha_2, \beta_2)$ iff either one of the following holds.  $(1): \max\{\alpha_1,\beta_1\} &lt; \max\{\alpha_2,\beta_2\}$  $(2): \max\{\alpha_1,\beta_1\} =\max\{\alpha_2,\beta_2\}$ and $\alpha_1&lt;\alpha_2$   $(3): \max\{\alpha_1,\beta_1\} =\max\{\alpha_2,\beta_2\}$ and $\alpha_1 = \alpha_2$ and $\beta_1 &lt; \beta_2.$  Claim: For all infinite cardinals $\kappa, (\kappa \times \kappa, &lt;_{cw}) \cong (\kappa, \in).$  Proof: (It has been show that it is true for $\kappa = \omega$) Assume there is a counterexample  $\kappa.$ Let $A= \{\lambda \in \kappa + 1| \lambda\geq \omega \ is \ a \ cardinal \ and \ (\lambda \times \lambda, &lt;_{cw}) \not\cong (\lambda, \in)\}.$ Then, $\kappa \in A.$ Let $\mu$ be least element of $A.$ Thus, $\mu &gt; \omega$ and for all $\lambda,$ if $\omega \leq \lambda &lt; \mu$ and $\lambda$ is a cardinal, then $(\lambda \times \lambda, &lt;_{cw}) \cong (\lambda, \in).$  By Comparability theorem, $(\mu, \in)$ is isomorphic to initial segment of $(\mu \times \mu, &lt;_{cw})$ since $|\mu \times \mu| \geq \mu$ and $\mu$ is cardinal. Let $(\alpha, \beta) \in \mu \times \mu$ such that $(\mu, \in)$ is isomorphic to initial segment of $(\mu \times \mu, &lt;_{cw})$ given by $(\alpha, \beta).$ Let $h$ be the isomorphism. Let $\eta= \max\{\alpha, \beta\}+ \omega.$ Then $\eta &lt; \kappa$ and $(\alpha,\beta)&lt;_{cw} (0, \eta).$ HENCE, $h: \mu \to \eta \times \eta$ is injective.
I'm not that into dynamic programming languages but I've written my fair share of JavaScript code. I never really got my head around this prototype-based programming, does any one know how this works? var obj = new Object(); obj.prototype.test = function() { alert('Hello?'); }; var obj2 = new obj(); obj2.test();  I remember a lot discussion I had with people a while back (I'm not exactly sure what I'm doing) but as I understand it, there's no concept of a class. It's just an object, and instances of those objects are clones of the original, right? But what is the exact purpose of this &quot;.prototype&quot; property in JavaScript? How does it relate to instantiating objects? Update: correct way var obj = new Object(); // not a functional object obj.prototype.test = function() { alert('Hello?'); }; // this is wrong!  function MyObject() {} // a first class functional object MyObject.prototype.test = function() { alert('OK'); } // OK  Also these  really helped a lot. We had a rule check tonight that we couldn't answer. On the Pathfinder wiki I could only find info on natural 20s and 1s in regards to attack rolls and saves in combat.  Do they also apply to skill checks? I have a data set for a predictive model(predicting survival rate with certain acute medical condition on some animals) with 25 predictors where around 30% of the predictors are complete, 3 predictors are sitting at 30%,25% and 20% missing, and the others are missing at around 5% level. Around 50% of my data are complete cases. I'm new to dealing with missing values so I have a couple of questions regarding how to deal with it:   What can I do with the variable with 30% missing assuming it's MAR? Is 30% too high for imputation? What kinds of metrics can I use to make the decision between removal of the predictor, listwise removal, imputation or some other options? How should I deal with a predictor with 20% - 25% missing when I have reason to believe that it's MNAR? I'm thinking of using imputation on the remaining predictors with around 5% missingness. How do I make the decision on which imputation methods to use? Are they chosen on a case by case basis based on the individual predictors? How does one impute categorical values?  How does(should?) the imputation carry out in practice? Should it use complete cases or compute it iteratively somehow?  Does feature selection comes in before or after dealing with missing values? if at all? Can a Raspberry Pi run Ubuntu? I was thinking about buying a , and running Ubuntu on it. I have looked around the web, and a lot of stuff says that it's difficult for it to work on ARM processors. Has anyone figured out how to get a Raspberry pi to run Ubuntu? introduces  which will allow the introduction of new literal syntax based on existing literals (int, hex, string, float) so that any type will be able to have a literal presentation.  Examples:  // imaginary numbers std::complex&lt;long double&gt; operator "" _i(long double d) // cooked form {      return std::complex&lt;long double&gt;(0, d);  } auto val = 3.14_i; // val = complex&lt;long double&gt;(0, 3.14)  // binary values int operator "" _B(const char*); // raw form int answer = 101010_B; // answer = 42  // std::string std::string operator "" _s(const char* str, size_t /*length*/)  {      return std::string(str);  }  auto hi = "hello"_s + " world"; // + works, "hello"_s is a string not a pointer  // units assert(1_kg == 2.2_lb); // give or take 0.00462262 pounds   At first glance this looks very cool but I'm wondering how applicable it really is, when I tried to think of having the suffixes _AD and _BC create dates I found that it's problematic due to operator order. 1974/01/06_AD would first evaluate 1974/01 (as plain ints) and only later the 06_AD (to say nothing of August and September having to be written without the 0 for octal reasons). This can be worked around by having the syntax be 1974-1/6_AD so that the operator evaluation order works but it's clunky.   So what my question boils down to is this, do you feel this feature will justify itself? What other literals would you like to define that will make your C++ code more readable?    Updated syntax to fit the final draft on June 2011 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: What does "$$" mean in the shell? What does the output of the command below mean? (result is positive integer)  echo $$ How do I cd into a directory in the home folder? How do I change directories to downloads, from the home folder, in the latest version Ubuntu 12.10? I tried cd /downloads, but it gave an error saying that there is no such directory. Complete induction proof that every $n > 1$ can be written as a product of primes I try to find a good proof for invertibility of strictly diagonally dominant matrices (defined by $|m_{ii}|&gt;\sum_{j\ne i}|m_{ij}|$).  There is a proof of this in  but I'm wondering whether there are are better proof such as using determinant, etc to show that the matrix is non singular. How to organically merge nested associations? How to stop an animation in C# / WPF? I have something like this:   barProgress.BeginAnimation(RangeBase.ValueProperty, new DoubleAnimation(     barProgress.Value, dNextProgressValue,     new Duration(TimeSpan.FromSeconds(dDuration)));   Now, how would you stop that animation (the DoubleAnimation)? The reason I want to do this, is because I would like to start new animations (this seems to work, but it's hard to tell) and eventually stop the last animation... Firefox - Websites with custom fonts not displaying How to change kernel at boot I installed the latest kernel (4.1) and want to boot into it instead of 3.9. How can i do this on ubuntu 15.04? How would I convert a char array to int? Equivalence of conditions for Convergence of Non-negative Random Series Question  Let $X_n\geq 0$ be independent for $n\geq 1$. The following are equivalent.  $\sum_{n=1}^\infty X_n&lt;\infty$ a.s. $\sum_{n=1}^\infty[P(X_n&gt;1)+E(X_nI(X_n\leq 1)]&lt;\infty$ $\sum_{n=1}^\infty E(X_n/(1+X_n))&lt;\infty$   My attempt I was able to prove $(1)$ implies $(2)$ and $(2)$ implies $(3)$ but not $(3)$ implies $(1)$. For $(1)$ implies $(2)$, observe that by Borel-Cantelli $\sum_{n=1}^\infty P(X_n&gt;1)&lt;\infty$, else $X_n&gt;1$ infinitely often w.p.1 so that $(1)$ is violated. To see that $\sum_{n=1}^\infty E(X_nI(X_n\leq 1)]&lt;\infty$, apply Kolmogorov three-series theorem (I think). For $(2)$ implies $(3)$ we can write that $$ \begin{align} \frac{X_n}{1+X_n}&amp; =\frac{X_n}{1+X_n}I(X_n\leq 1)+\frac{X_n}{1+X_n}I(X_n&gt;1)\\ &amp;\leq X_nI(X_n\leq 1)+ I(X_n&gt;1). \end{align} $$ Now take the expectation of both sides and sum on $n$ using $(2)$ to conclude $(3)$. For the last implication $(3)$ implies $(1)$, I tried to use Kolmogorov three series theorem. To this end put $Y_n=X_nI(X_n\leq 1)$. First note that $\sum_{n=1}^\infty P(X_n&gt;1)&lt;\infty$ since $$ P(X_n&gt;1)\leq P\left(\frac{X_n}{1+X_n}&gt;\frac{1}{2}\right)\leq 2E\frac{X_n}{1+X_n} $$ by Markov's inequality. But I am unable to show that $\sum EY_n&lt;\infty$ and $\sum\text{Var}(Y_n)&lt;\infty$. Any help is appreciated. How to run a fresh version of AIX in a Virtual Machine with a Linux host? I'm looking for the noun form of "person with intermediate skill". For example, in the context of a particular activity, "person with no skill" might be designated a novice, and "person with much skill" might be designated an expert. However I know of no such word in between these two extremes.   My only thought is amateur, which has a distinct meaning from describing level of skill, although skill level can often be inferred. I'm looking for a better solution.  EDIT: due to the attention this question has received, I'll try to offer some clarification in order to reduce repeat questions.   "Journeyman" would certainly not work. I am a software developer by profession. However the word I'm looking for should hopefully be profession-agnostic. I am trying to designate the skill level of the person that some learning material might be appropriate for. The word that I'm looking for follows the preposition "for".   For Beginners  For [word for intermediate-skilled person] For Experts  Sorry, but even after all the great answers, I'm still learning towards "amateur" even though it is technically incorrect. Also, the word should not be so esoteric that only people with English degrees know what it means. Keep the ideas coming! How to edit `postgresql.conf` with pgAdmin 3? How does one edit the 'postgresql.conf' file with the pgAdmin app?  With pgAdmin 1.18.1 on Postgres 9.3.x, when I choose File > Open postgresql.conf I get a file picker dialog. And, yes, I'm connected in pgAdmin as the superuser 'postgres'.  I know I've done this in older versions of Postgres. Found an error in my thesis I've found a error in my thesis recently, after its submission. I made a mistake that wrongly accepted the hypothesis which should be rejected. What should I do??