For example, $\mathbb N$ is equipotent to $\mathbb Q$ which is a field.  $\mathbb R$ is equipotent to itself, which is a field.  But what about $\mathbb R^{\mathbb R}$, $P(\mathbb R^{\mathbb R})$ etc.?
So I took an introductory abstract algebra course a few semesters ago, and we were shown that groups and rings can both be made into products, i.e. if I have some group $G$ (resp. ring $R$) and some indexing set $I$, then I can make a group $G^{I}$ (resp. ring $R^{I}$) with the appropriate cardinality. However, it was also demonstrated that for fields, you cannot keep the multiplicative inverse under a product structure, i.e. if I have fields $F_{1}, F_{2}$, then $F_{1} \times F_{2}$ will still be a ring, but you will not (not just "not in general", but won't) have multiplicative inverses for all non-zero (i.e. not $(0, 0)$) elements, as you could pick $(0, x)$, where $x \neq 0$. According to Wikipedia, an ultraproduct will preserve field structure, but I'm not sure what the cardinality of $\prod_{i \in I} F_{i} / \mathscr{U}$ would be in general.  I saw  that said you could make a field of arbitrarily large cardinality by extending a given field through throwing in a whole bunch of "transcendental" elements. The example given was that if I had $\mathbb{Q}$, I could start by dumping in the complexes, but then I suppose after that I'd just start throwing in dogs and cats or something; I'm really not sure what that responder meant, and ceteris paribus, my dogs tend to stay inside, so I'd rather keep them out of my fields, particularly the large ones the thread was interested in.  Moreover, the thread mentioned earlier seemed more concerned with just making the fields big. My question is a bit more nuanced: Given any cardinality $\kappa &gt; 1$, can I generally construct a field $F$ of a cardinality $\kappa$? Moreover, how vague do I have to be about it (i.e. do I have to use some really choicy methods, or can I make it a bit more straightforward)?  Thanks.  EDIT: Sorry. I am aware that for finite fields, you are limited to powers of primes. I meant for infinite cardinals.
How to animate only the camera? So  i have a scene with a few objects, and a water simulation in the scene. However I want to animate the camera turning around in the scene, while the water simulation is frozen in time. So something like a time freeze effect. But when ever I try to render an animation it renders the water simulation too. I just need one frame of the water simulation. Is there a way to do this? TSA lock reset button stuck and now lock button is also stuck MySQL - varchar length and performance If I have an IEnumerable like:  string[] items = new string[] { "a", "b", "c", "d" };   I would like to loop thru all the pairs of consecutive items (sliding window of size 2). Which would be   ("a","b"), ("b", "c"), ("c", "d")   My solution was is this       public static IEnumerable&lt;Pair&lt;T, T&gt;&gt; Pairs(IEnumerable&lt;T&gt; enumerable) {         IEnumerator&lt;T&gt; e = enumerable.GetEnumerator(); e.MoveNext();         T current = e.Current;         while ( e.MoveNext() ) {             T next = e.Current;             yield return new Pair&lt;T, T&gt;(current, next);             current = next;         }     }   // used like this :  foreach (Pair&lt;String,String&gt; pair in IterTools&lt;String&gt;.Pairs(items)) {     System.Out.PrintLine("{0}, {1}", pair.First, pair.Second)  }   When I wrote this code, I wondered if there are already functions in the .NET framework that do the same thing and do it not just for pairs but for any size tuples.  IMHO there should be a nice way to do this kind of sliding window operations.  I use C# 2.0 and I can imagine that with C# 3.0 (w/ LINQ) there are more (and nicer) ways to do this, but I'm primarily interested in C# 2.0 solutions. Though, I will also appreciate C# 3.0 solutions. Please add close flags, duplicate links and review data to the public data dump I am trying to master the homemade Neapolitan pizza, from dough preparation to cooking it.  After following the recommendations at Seriouseats.com for the dough preparation, I came across some doubts regarding the oven use.  I understand that ideally I would make it in a 900F wood oven, but at home we don't have such equipment!  What is the best practice to make the Neapolitan Pizza in a regular gas oven that heats up to 530F? (i.e. temperature, time, tray positioning...) Are there changes in the way the dough has to be prepared? I want to make a plane for my supply on my PCB and a plane for my MCU to isolate the MCU from the power supply noise.  I found this article in Altium designer :  But it isn't clear how the split plane tool work. Can someone explain it in a different manner?  [EDIT]  This is a picture how I see the planes: Are there cases where a period is placed inside parentheses? In which cases the period (or a comma) should be placed inside the parentheses? Why AUC =1 even classifier has misclassified half of the samples? In my script, I have for loop over many cube objects (~1000) and the treatment is very slow. Looking more in details, I notice that in the same amount of loops:   if I use python operation or simple blender operation like  obj = bpy.data.objects[obj_name]   or   obj.select = True   it takes less than 0.08s (for all 1000 objects) but as soon as I start using blender operators like:  bpy.ops.object.select_pattern() # or bpy.ops.object.duplicate() # or bpy.ops.object.location_clear() # or bpy.ops.object.transform_apply() # etc.   then, the performance drops down deeply with more than 6s for the same amount of objects.   And, a last information, if I reduce the amount of objects from 1000 to 50, the same set of operations with bpy.ops is 0.03s - so extrapolating, it would be 0.6s for the 1000 objects and not 6s. It is as is with 1000 objects, we loose a factor 10 in speed in comparison than with 50 objects.  I tried reducing the complexity of my mesh changing the cubes in planes but it had no effect at whole on performance.  Is there something particular to know to improve these performance or a way to use the bpy.ops methods with better performances? Obviously, I am missing something important probably. Let $a,b,c$ be $3$ distinct integers, and let $P$ be a polynomial with integer coefficients.Show that in this case the conditions $$P(a)=b,P(b)=c,P(c)=a$$ cannot be satisfied simultaneously.  Any hint would be appreciated. How does RAID 1 determine which drive should act as the source for a rebuild? Here's my scenario -- both drives were working perfectly fine. I removed one of the drives for a while (building a new PC, didn't think I was going to use RAID anymore but changed my mind). I actually switch the SATA type to AHCI and was operating for about a week on the new system before deciding to re-enable RAID and reinstall the drive I had removed.   It's currently rebuilding, but I'm curious how it knows which one is the source and which one is out-of-date? How can I randomise the output of a redstone circuit? I have 5 buttons and only one of these can be correct, all others must be false. These 5 outputs will go into 2 outputs, in between there needs to be a circuit which changes which single button out of the 5 will go down the &quot;correct&quot; output of the final 2. The 4 remaining buttons must go to the second of the final 2 outputs. Both final outputs must give out a true signal for a brief period the same as a button on its own would produce. There should also be a way of activating the circuit. Not sure if that is explained well enough so I have made a basic diagram of what I need.  The diagram shows how I need the signal to be split. However in between the buttons and the final two wires there needs to be a circuit (marked by the grey box). This circuit changes the single button connected to the left wire. The right input should start the circuit and randomise the 5 buttons. There are no limits to size, amount of materials used or method used. This question is similar to the one asking about , but I am specifically working with Redstone circuitry - the commands solutions do not apply here How to install gail and atk-bridge? How to know that your machine learning problem is hopeless? Imagine a standard machine-learning scenario:     You are confronted with a large multivariate dataset and you have a   pretty blurry understanding of it. What you need to do is to make   predictions about some variable based on what you have. As usual, you   clean the data, look at descriptive statistics, run some models,   cross-validate them etc., but after several attempts, going back and   forth and trying multiple models nothing seems to work and your   results are miserable. You can spend hours, days, or weeks on such a   problem...   The question is: when to stop? How do you know that your data actually is hopeless and all the fancy models wouldn't do you any more good than predicting the average outcome for all cases or some other trivial solution?  Of course, this is a forecastability issue, but as far as I know, it is hard to assess forecastability for multivariate data before trying something on it. Or am I wrong?   Disclaimer: this question was inspired by this one  that did not attract much attention. It would be nice to have detailed answer to such question for reference. What is the resolution I need to make a picture to make curves look smooth and not pixelated? Recently, I have answered a few questions that were upvoted by factors more than the accepted answer, but not accepted. , at the time of my writing, has an accepted answer that the asker said was not what he wanted. I don't know why. In the future, when people view this, people might only look at the accepted answer and move on. Would it be possible to have a "community accepted" feature? Maybe only the people at the top can vote to accept it or something. How much rep, if any, should go to that person? Can someone use DC generator along with electric motor to make electricity along with mechanical energy? I know this might sound stupid and maybe stupid totally, but I need some clarification....  I came through electric motor principle.  and then I came across the following link on DC generators    Now my question is, can't we combine them both to provide DC current with little DC used in motor side? By "combine," I mean using a DC generator coil only at the other side of the coil of the motor in the motor itself. The condition is that the 2 coils are connected via some apparatus that causes the 2nd coil to move along with the 1st coil.   In this way we can use both the mechanical energy produced by the 1st coil and the electrical energy produced by the 2nd coil, right?   This is a great concept if it works, right? Please throw some light on this. I've often been told by people that I shouldn't refreeze meat (particularly hamburger meat) once it has been thawed.  However, this seems a little silly to me.  I can't imagine how meat that hasn't been bought fresh and local could find its way to my kitchen with out thawing and being refrozen a couple of times.  How much damage can one more thawing and refreezing really do?  Can it really be that dangerous disease wise? Or is this just one of those urban myths? What's wrong with this .desktop file?