In general we know $A^{bc} = A^{cb}$ for integer $A$.  I want to extend this to the case $A=-1$.   For integers $a,b$ I guess the above relation holds,  \begin{align} (-1)^{2\cdot3} = ((-1)^2)^3 = 1 = ((-1)^3)^2 .\end{align}  But if we include the fraction  \begin{align} (-1)^{-\frac{1}{2}} &amp;= ((-1)^{-1})^{\frac{1}{2}} = (-1)^{\frac{1}{2}} = i \\ &amp;= ((-1)^{\frac{1}{2}})^{-1}=\frac{1}{i} = -i \end{align} this does not hold any more.   Is something wrong with my computation?
This is a silly question but under what conditions is $a^{xy}=(a^x)^y$ true, given all are complex numbers?
I tried to follow the new Blender 2.8 tutorial from Blender Guru.   When entering "Sculpting" with the icing there were suddenly alot of holes in it. Why is this? Moving a doorbell transformer Can a doorbell transformer that is on the outside of a wall be place inside the wall behind the drywall. I’m concerned about it getting too hot. Mine is in the kitchen by the ceiling lights Reading the question posted  left me with a more general question:  Given the professional title and name: Dr. (First Name)(Last Name), is there some way to differentiate between the holder of a philosophical doctorate and a medical doctor? Wouldn't it be more appropriate for a PhD holder to have the title (First Name)(Last Name), PhD? I am trying to get this trigger to work but it is giving me an error and I am not able to figure out why. The error I am getting is "attempt to de-reference a null object)  Why trigger? We have a Destinations object which is linked to Supplier products object via a lookup relationship. There are two types of supplier products(marketplace and curated) We want to count the number of live Supplier products and put them in a separate fields(live counts for each type).  I have gotten so far to actually count the number of live products and it is giving me an error. Can any one help? I am posting the code below  Trigger code :   trigger NoOfLiveProducts on Supplier_Product__c (after insert, after Delete, after update, after undelete) {       set&lt;string&gt; destinationids = new set&lt;string&gt;();      if(trigger.isInsert) {         for(Supplier_product__c products : Trigger.new) {             destinationids.add(products.Destination__c);         }     } else if(trigger.isDelete) {         for(Supplier_product__c products : Trigger.old) {             destinationids.add(products.Destination__c);         }     }else if(trigger.isUnDelete) {         for(Supplier_product__c products : Trigger.new){             destinationids.add(products.Destination__c);         }      } else if(trigger.isUpdate) {         for(Supplier_product__c products : Trigger.new) {             destinationids.add(products.Destination__c);         }     }      if(destinationids.size() &gt; 0){         Productcounter pc = new Productcounter();         pc.updateproductCount(destinationids);     }    }   Class code :  public class Productcounter {      public void updateproductcount(set&lt;string&gt; productIDs) {           if(productIDs == null){                return;         }          if(productIDs.size() == 0){             return;         }           list&lt;Supplier_Product__c&gt; ProductList = new list&lt;Supplier_Product__c&gt;();         ProductList = [Select ID,Destination__c                     From Supplier_Product__c                     Where Destination__c = :productIDs];          integer totalCount;         map&lt;string, integer&gt; DestinationIDToProductCountMap = new map&lt;string, integer&gt;();                for(Supplier_product__c ProductItem : ProductList){             if(DestinationIDToProductCountMap.containsKey(ProductItem.Destination__c)){                 totalCount = DestinationIDToProductCountMap.get(ProductItem.Destination__c);             }else{                 totalCount = 0;             }             totalCount += 1;             DestinationIDToProductCountMap.put(ProductItem.Destination__c, totalCount);         }          list&lt;Destination__c&gt; DestinationUpdateList = new list&lt;Destination__c&gt;();         Destination__c DestinationTemp;         for(string ProductID : DestinationIDToProductCountMap.keySet()){             DestinationTemp.ID = ProductID;             DestinationTemp.No_of_Live_Curated_Products__c = DestinationIDToProductCountMap.get(ProductID);             DestinationUpdateList.add(DestinationTemp);         }         if(DestinationUpdateList.size() &gt; 0){             update DestinationUpdateList;         }           }           } Does there exist a smooth approximation of $x \bmod y$? I'm looking for a function $m(x,y)$ that smoothly approximates $x \bmod y$, and I'm assuming there would be some $n$ or $\varepsilon$ in the body of $m(x,y)$ that defines the degree of approximation such that as $n$ goes to infinity or $\varepsilon$ goes to zero,  $$ m(x,y)= x \bmod y. $$  I've looked at, for example, $\sin(x/y)$ and have been trying to figure out ways to make $\sin(x)$ more "steep" so that I can apply such techniques to $\sin(x/y)$ in order to perhaps more closely approximate $x \bmod y$.  The analogy I'm looking at is the approximation of $\mid x\mid$ as $\sqrt{x^2 + \varepsilon)}$ where epsilon is some small quantity.  Does a similarly smooth function exist for $x \bmod y$? The Complete Rate-Limiting Guide Undefined reference to static class member Square root of 1 is (not) -1 Possible Duplicate:          I was thinking on the following line of thoughts: $1 =  \sqrt{1} = \sqrt{-1 \cdot -1} = \sqrt{-1} \cdot \sqrt{-1} = i^2 = -1$  Of course this is not true, but I was wondering which step in this 'line of thoughts' is forbidden to make?   Thanks for the explanation. Halo particles don't render in Cycles I'm looking for some insight into how Stack Overflow releases are deployed and managed.   What software/tooling is used? What time are releases generally performed? How many and what steps are involved?   I'm assuming it's not too stringent given the light team working on the codebase. But considering the high rate of change, I could see some kind of tooling and automation becoming increasingly relevant, so I'm curious to learn what kind of decisions are being made to deal with the changes in scale and feature set over time.  Is this information available somewhere? Selenium python drop down menu select In a 2D top-down game, how can I create projectiles that have a height? T SQL Table Valued Function to Split a Column on commas I wrote a Table Valued Function in Microsoft SQL Server 2008 to take a comma delimited column in a database to spit out separate rows for each value.  Ex: "one,two,three,four" would return a new table with only one column containing the following values:  one two three four   Does this code look error prone to you guys? When I test it with   SELECT * FROM utvf_Split('one,two,three,four',',')    it just runs forever and never returns anything. This is getting really disheartening especially since there are no built in split functions on MSSQL server (WHY WHY WHY?!) and all the similar functions I've found on the web are absolute trash or simply irrelevant to what I'm trying to do.  Here is the function:  USE *myDBname* GO SET ANSI_NULLS ON GO SET QUOTED_IDENTIFIER ON GO ALTER FUNCTION [dbo].[utvf_SPlit] (@String VARCHAR(MAX), @delimiter CHAR)  RETURNS @SplitValues TABLE (     Asset_ID VARCHAR(MAX) NOT NULL )  AS BEGIN             DECLARE @FoundIndex INT             DECLARE @ReturnValue VARCHAR(MAX)              SET @FoundIndex = CHARINDEX(@delimiter, @String)              WHILE (@FoundIndex &lt;&gt; 0)             BEGIN                   DECLARE @NextFoundIndex INT                   SET @NextFoundIndex = CHARINDEX(@delimiter, @String, @FoundIndex+1)                   SET @ReturnValue = SUBSTRING(@String, @FoundIndex,@NextFoundIndex-@FoundIndex)                   SET @FoundIndex = CHARINDEX(@delimiter, @String)                   INSERT @SplitValues (Asset_ID) VALUES (@ReturnValue)             END              RETURN END Do I choose the target or the ally for the Help action? When a sentence uses a parenthetical plural, should the rest of the sentence treat it as singular or plural? Android adware lets me opt out: is it safe? Recently my mom's phone running Android 5.0 has been infected with a piece of adware:    At first we thought this was caused by applications, but as you can see here, only the Google/METROPCS Internet browser is running.  In the photo, there is a button that reads 'opt out'. So I clicked on it and it took me to this link at http://tracker.tpc.re/optout/mobile/:    BUT both of these options smelly fishy. For starters, the manual app is not a download link in the play store but an APK, making it seem like a virus. The second (manual) as for sensitive information, which could harm me even more.   Which one of these options do I choose? Are they legit or do I have to do get rid of it another way? Showing a tensor is not a simple tensor Let $(e_1,e_2)$ be the standard basis of $\mathbb{R}^2$. Why can the element $e_1\otimes e_2 + e_2\otimes e_1$ in $ \mathbb{R}^2\otimes_{\mathbb{R}} \mathbb{R}^2 $ not be written as a simple tensor $v\otimes w$ for $v,w\in \mathbb{R}^2$?  If we suppose $e_1\otimes e_2 + e_2\otimes e_1=v\otimes w$, then can we conclude that $e_1=\lambda e_2$ for some $\lambda \in \mathbb{R}$? This would give a contradiction since $e_1$ and $e_2$ are linearly independent. However, I am not sure if the argument is valid. When can I be sure a questionable check has cleared? Entropy increases when two substances mix with each other.  For example, the entropy of mixing of two different gases are given by $$\Delta S= 2Nk\ln\frac{V_f}{V_i}\;.$$  But, the entropy doesn't increase when the two gases mixing are same.  This is pointed by Daniel V Schroeder:     It's important to note that this result applies only if the two gases are different like helium and argon. If you start with the same gas on both sides, the entropy doesn't increase at all when you remove the partition.   Now, why is this so? Both the gases, though the same, increase their individual entropies when they expand, don't they?  So, why did Schroeder say there is no entropy of mixing? What actually happens such that there is no change in entropy when the gases are the same? I want to do something like String.Format("[{0}, {1}, {2}]", 1, 2, 3) which returns:  [1, 2, 3]   How do I do this in Python?