The convolution of two function $f$ and $g$ is defined as $$(f*g)(x) = \int f(y) g(x-y) dy.$$   Terry Tao  that the convolution with a bump function can be thought of as "blurring" the function. There is also a .  In these contexts, it appears the bump function used is usually symmetric about the $y$-axis, so it shouldn't matter whether we write $g(x-y)$ or $g(x+y)$. However, I have never seen the convolution defined with a plus sign: $$(f*g)(x) = \int f(y) g(x+y) dy.$$  If the point is to translate a bump function by $x$, then why is it standard to write $g(x-y)$ and not $g(x+y)$? Are there other uses of convolutions where it is important to write $g(x-y)$?
Why do we use $x - y$ rather than $x + y$ in the definition of the convolution? Is it just convention? (If we are thinking of convolutions as weighted averages, for instance against "good kernels," it should make no difference.)  Why $(f * g) (x) = \int f(y) g(x - y) dy$ rather than $(f * g) (x) = \int f(y) g(x + y) dy$?  Edit: I'm finding it really hard to choose a best answer. There are at least three very good ones here.
How can I compile, run and decompile C# code in Ubuntu terminal? Why do we need the trilogy instead of just one site? How should I use a possessive apostrophe when a name is between commas? I have recently noticed that in chat when I type enough and the profile picture expands with my message, the hat stays the same size and it stays in the same place (I have the  app): I need something explained or corrected:  In my number theory class we proved that there are an infinite number of primes using Euler's Phi Totient. It went something like this:  Let $M = p_1 p_2 \dots p_n$ be the product of all primes. Consider $1 &lt; A \le M$:  Some prime must divide $A$, call it $q$. Since $q$ must be one of the primes, $q$ must divide $M.$  So the $gcd(A,M) &gt; 1.$ Thus $\phi(M) = 1$ ...?   Which is not even and contradicts the theorem that $\phi(N)$ is even for $N&gt;2.$ Therefore there exists an infinite number of primes.  I get confused by the statement "Thus $\phi(M) = 1$"....  Did i possibly copy this proof down wrong? or am I missing something? Thank you in advance.   Edit: by consider a such that...i meant consider an integer '$a$' I replaced it with $A$ to hopefully make it more clear.   Edit2: I'm sorry I am not familiar with using the equation editor. This is not a homework assignment, just studying for my exam. I just want to be able to understand this or clearify it. Got 2,919 PB/s in Ubuntu Terminal I have problem (seems to be trivia) about Ubuntu Terminal. Once my network connection slow down then it (sometimes but very rare to happened) showed some information like this.     I shocked because I have been got 2,919 PB/s. Sorry my mouse block some part of letter 'P' (I actually rush to print screen) but I believe you can recognize it.    EDIT    based on answer given to me, and I got similar info then, I'd like to add more clearly new image after.    Can someone explain it to me?  Best Regards! When I try to upload an image, I'm presented with the following dialog box,    The dialog box does not allow uploading either from my computer or by posting a link.  This happens only on Chrome browser (running on Windows). Clearing cache and cookies, and restarting computer do not resolve.  Is this just my computer? Or is this a site-wide bug?  Additional info:  Clicking on 'click' to upload does nothing. Dragging an image onto dialog redirects me to the file location on my computer. Clicking 'Add picture' redirects me to http://[SE-domain].com/upload/image?https=true.  UPDATE: A Norton extension for Chrome was the source of the 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: Suppose $s$ is a complex number with $\Re(s) \in (0,1]$ and $\{a_n\}$ is a complex sequence converging to $a \neq 0$. Must the Dirichlet series $$\sum_{n=1}^\infty\frac{a_n}{n^s}$$ diverge? Matrix reciprocal positive to prove $λ_{max⁡}=n$ Suppose we have $n\times n$ matrix $A$ having only positive elements and satisfying the property $a_{ij}=1/a_{ji}$ (a matrix satisfying this property is called a reciprocal matrix).  If its largest eigenvalue $λ_{max}$ is equal to $n$, then the matrix $A$ satisfies the property (consistency property) $a_{ij}a_{jk}=a_{ik}$ where $i,j,k=1,2,...,n$.    I already found an example of $5\times 5$ reciprocal matrix  to show this condition:  \begin{bmatrix}1&amp;1/2&amp;1&amp;1&amp;1/4\\2&amp;1&amp;2&amp;2&amp;1/2\\1&amp;1/2&amp;1&amp;1&amp;1/4\\1&amp;1/2&amp;1&amp;1&amp;1/4\\4&amp;2&amp;4&amp;4&amp;1\end{bmatrix}  It has simple characteristic equation $λ^5-5λ^4=0$, so the eigenvalues of matrix $A$ are $λ=0$ and $λ=5$ and it's shown that $λ_{max}=n=5$.  Is there anyone can help me to give another positive reciprocal matrix example  that has simple characteristic equation and integer number as its eigenvalues ($4\times 4, 5\times 5$ or $6\times 6$ matrix)? Where to get 2010 Census Block data? I would like to download 2010 Census Block Data for several towns and I can't seem to find it on American Factfinder.   Is it there? And if not, does anyone know where I could find it? Showing directional line symbol in ArcGIS for Desktop? Is this possible in ArcGIS to have variable number of arrows on the line to show direction?  This is an ArcGIS example, where I can have fixed (1,2,3,4,5...) number of arrows and it stays the same doesn't matter what the line length is. First and last always stick to the end, doesn't help with network mapping, where there is usually the node to show. Having just one in the middle is never enough, check line at the right...   No such issues with ArcView 3: AND gate made out of transistors letting current pass despite only one input high I recently built a gate with transistors, however my meter reads 1.6 volts when only input B is high, and 2.3 while both are high. How can I fix this? (I used 3 volts instead of 6 and 2k instead of 10k) What does "sudo apt-get update" do? I am wondering what sudo apt-get update does?  What does it update? Why did I receive a message saying I have been granted a privilege, yet I don't have enough reputation? The site shown a message saying I have been granted a privilege, but the reputation required for that privilege is higher than the one I actually have. What happened? How do I put a backdrop behind the model that I am creating so that I can compare the picture to the model while editing? How can I randomize order of elements in a list? I cheated on the Disciplined badge, how bad is it? I was looking for badges I didn't have yet, and I saw the Disciplined badge. So I deleted an answer of mine with 3+ votes, and waited. Five minutes later, sweet, a new badge! Then I went to my answer again, and undeleted it. No problem at all, and I keep the badge!   What will happen? Will my badge stay here or not? More generally, what happens when the conditions to get a badge are valid at one time, and then not valid?  Other badge in this case:   Unsung hero - "Zero score accepted answers: more than 10 and 25% of total"   What happens if answers get a vote after getting the badge? Are the badge frequently recalculated? How do I run a command before or after every command written in the console? I am trying to override the menu.css file in the default Classy theme through my library file. I have read through the documentation, but anything I tried doesn't work.  What would be the correct markup to override or disable this file?