Let $(G,\cdot)$ be a group and define the comutator as $[a,b] := aba^{-1}b^{-1}$ for every $a,b\in G$. About to prove that $[G,G] \leqslant_n G$, first i need to prove that is in fact a subgroup, wich is were i failed. My question is:     Given $a,b,c,d\in G$, find $x,y\in G$ such $[a,b]\cdot[c,d] = [x,y]$.   This is for proving that $[\,\cdot\,,\,\cdot\,]$ are closed at $[G,G]$. The otter statments wich defines a group i've allready check it, but they doesn't defines a group if the operation $\cdot$ wouldn't give me an element of $[G,G]$. Thanks for the help.  Mistakes. In fact, $[G,G]$ isn't the comutator as well, so $[G,G] {\require{cancel}\cancel{\leqslant}} G$, so, the product of the comutators isn't a comutator as well
It is suggested as an exercise in Serge Lang's book "Algebra" to show that the commutator subgroup $G^c$ of a group $G$ is a normal subgroup.   I'd like to do that but I am afraid I need help,   I think the first thing I need to figure out is how a general element in the commutator subgroup looks like, so that I can check that the defining condition for normality is satisfied.  That is, supposing for a moment that a general element in $G^c$ is denoted by $g$, I need to show that $aga^{-1} \in G^c,$ for all $a \in G$.  But here I get stuck, first because I am unsure how to write a general element in $G^c$ - a simple product in $G^c$ is of the form $xyx^{-1}y^{-1}aba^{-1}b^{-1}$ where $a,b \in G$. I cannot see a way to simplify this - I am sure there is one, but somehow I am blind today.  The second thing then is, even if one tries out the conjugation of a simple element like $xyx^{-1}y^{-1}$ in $G^c$, again not simplification offers itself easily I think .. what am I missing ?  An alternative would be to find a homomorphism of $G$ whose kernel is precisely $G^c$ - here I tried to think of this as a map $G \times G \to G$ but whatever I cook up is not a homomorphism.  Thanks for your hints !!
Deleting your own closed questions fails silently on the mobile site There is some sort of delay for deletion of your own closed questions (2 days, I think) for non-mods. Now, when you try deleting a closed question on the desktop site, it nicely tells you "please wait xyz to allow for possible reopening" (in a box similar to the "please consider adding a comment explaining why you downvoted" thingy).  On mobile, it just reloads the page—there is no indicator that deletion is not allowed.  Either completely disable the delete button on mobile for closed questions, or give it a popup as well. Not letting the user know what's wrong is confusing. Sharepoint 2013. Add JS to NewForm.aspx of a list What is the difference between a PDU and a power strip (both 120V, 15A)? I just chatted with an APC rep about upgrading the UPSes at our office. She recommended a single higher-capacity 6-outlet Smart-UPS to replace the four Back-UPS units we currently have. When I asked how she recommended plugging in all the current devices, she recommended using a APC's  PDU, but said not to use a power strip. At first she said I had to use an APC brand PDU, but after I inquired about using a Tripp-Lite PDU, she said any brand PDU would be fine.  The APC PDU previously referenced looks like a standard 120V power strip with overload protection but no surge protection. Other than overload protection (which seems redundant if plugging into the UPS), is there something else I'm missing, or should any power strip (without surge protection) be fine?  Edit: I didn't mention it earlier, but we don't have a proper rack--though I did still plan to mount the PDU or power strip to something. I guess I'm wondering if there's any special reason I should pay as much as $180 for the low-end APC PDU (which just looks like a power strip to me) vs. $20-$30 for a . Vacuum sealing vs. water displacement method when doing longer sous vide cooking Modernist Cuisine recommends that for longer sous vide cooking (times greater than 36 hours) that bags are vacuum sealed rather than just using the water displacement method to remove the air from a ziploc bag.  What is the reason for this? Will I be losing something essential if I just use a regular ziploc bag without vacuum sealing? Performance differences between equal (=) and IN with one literal value find and rsync? I want to be able to search for files over 14 days and over 10k and than rsync those found files to a destination.  Is there a way to combine these two commands?  find ./ -mtime +14 -size +10k  rsync --remove-sent-files -avz /src /dest How do I decrease first byte time with GoDaddy and GitHub pages? Uniform Scattering with vector I'm attempting to create a random spread of dots to create a gradient look. I'm getting really close to the effect I want by blending two scatter brushes. The issue I'm having is that the blended brushes creates unwanted lines, and the dots run into each other. I keep adjusting the settings with no luck.  Here's the effect I'm wanting to achieved.    Smooth transition  On Circular path Dots don't converge Scattered at random-ish     Here's how close I'm able to get:   Blended scatters create unwanted lines Dots converge Spacing isn't gradual from edge to center How do I get query string value from script path? Why is everybody so concerned about etc/passwd? Here is the content of my vagrant machine of this particular file:  root:x:0:0:root:/root:/bin/bash daemon:x:1:1:daemon:/usr/sbin:/usr/sbin/nologin bin:x:2:2:bin:/bin:/usr/sbin/nologin sys:x:3:3:sys:/dev:/usr/sbin/nologin sync:x:4:65534:sync:/bin:/bin/sync games:x:5:60:games:/usr/games:/usr/sbin/nologin man:x:6:12:man:/var/cache/man:/usr/sbin/nologin lp:x:7:7:lp:/var/spool/lpd:/usr/sbin/nologin mail:x:8:8:mail:/var/mail:/usr/sbin/nologin news:x:9:9:news:/var/spool/news:/usr/sbin/nologin uucp:x:10:10:uucp:/var/spool/uucp:/usr/sbin/nologin proxy:x:13:13:proxy:/bin:/usr/sbin/nologin www-data:x:33:33:www-data:/var/www:/usr/sbin/nologin backup:x:34:34:backup:/var/backups:/usr/sbin/nologin list:x:38:38:Mailing List Manager:/var/list:/usr/sbin/nologin irc:x:39:39:ircd:/var/run/ircd:/usr/sbin/nologin gnats:x:41:41:Gnats Bug-Reporting System (admin):/var/lib/gnats:/us$ nobody:x:65534:65534:nobody:/nonexistent:/usr/sbin/nologin syslog:x:100:103::/home/syslog:/bin/false   Could anybody explain me why it is bad if some evil guy could get this file of my production server? Types in MySQL: BigInt(20) vs Int(20) How many solutions has the equation $\sin x= \frac{x}{100}$ ? How did Enrico Fermi calculate the classical Fermi Problem? From Wikipedia: Fermi was known for his ability to make good approximate calculations with little or no actual data, hence the name. One example is his estimate of the strength of the atomic bomb detonated at the Trinity test, based on the distance travelled by pieces of paper dropped from his hand during the blast. Fermi's estimate of 10 kilotons of TNT was remarkably close to the now-accepted value of around 20 kilotons, a difference of less than one order of magnitude.  I have not been able to find any references explaining how he made this calculation. Please provide a reference or an example calculation.  See also: Changing parameter values according to other parameter choice in Processing script using QGIS 3? I couldn't find anything about this problem.  Suppose to have the first parameters in a Processing script defined as an Enum:     self.addParameter(         QgsProcessingParameterEnum(             self.CHOICE,             self.tr('Choose the parameter'),             ['param1', 'param2']         ) )   While the second one as a number:  self.addParameter(     QgsProcessingParameterNumber(         name=self.VALUE,         description=self.tr('My value'),         type=QgsProcessingParameterNumber.Double,         defaultValue=40,         optional=False,         minValue=0,         maxValue=100     ) )   Is it possible to change the minValue, maxValue, description, ecc.. of the second parameter depending on the first or second choice of the first parameter already in the GUI?  e.g. if the user choose param1 then the second parameter should become:  self.addParameter(     QgsProcessingParameterNumber(         name=self.VALUE,         description=self.tr('My value'),         type=QgsProcessingParameterNumber.Double,         defaultValue=40,         optional=False,         minValue=0,         maxValue=100     ) )   while if he choose param2 it should became:  self.addParameter(     QgsProcessingParameterNumber(         name=self.VALUE,         description=self.tr('My AWESOME value'),         type=QgsProcessingParameterNumber.Double,         defaultValue=0,         optional=False,         minValue=50,         maxValue=500     ) ) Shapefile: "table1.shp"   Field name: "Name" Field name: "Result"   Flat file: "referenceTable"   Field name: "STREET" Field name: "AKA"   I need to compare the value "Name" in "table1.shp" to the value "STREET" in "referenceTable"  If the values are the same then put the "AKA" value into the "Result" in "table1.shp"     Here is my code.  I got stuck on accessing the elements from the array  import arcpy, os, sys, string from array import *   table1 = "G:\\xStreetNew\\table1.shp" table1_column = ['Name', 'Result'] referenceTable = "G:\\xStreetNew\\Test.gdb\\referenceTable" referenceTable_column = ['STREET', 'AKA']  referenceTable_List = [] with arcpy.da.SearchCursor(referenceTable, referenceTable_column) as rCur: for r_row in rCur:     reference_row = r_row[0].split(",")     referenceTable_List.insert(0,reference_row) ##print sorted(referenceTable_List)  Table1_Update_Cursor = arcpy.da.UpdateCursor(table1, "Name") for row in Table1_Update_Cursor:   if row[0] == None:   row[0] = row[0] else:   row[0] = row[0].replace(";",",")   Table1_Update_Cursor.updateRow(row)  table1_List = [] with arcpy.da.SearchCursor(table1, table1_column) as tCur: for t_row in tCur:     row = t_row[0].split(",")     table1_List.insert(0,row) ##print sorted(table1_List)  Table1_row = len(table1_List) Table1_column = len([table1_List]) ref_row = len(referenceTable_List) for i in range(0,Table1_row):  #Table Row   for ii in range(0,Table1_column):   #Table Column     print table1_List[i][ii]   Below was my old post.  Sorry for the confusion.  I have a shapefile named "table1.shp".     I would like to get all values from fielname "Name" from this table and put them into a list. Able to output each element from the list     So far, I'm able to create a list but not too sure how to get to value "AVENUE 33" or "CHAPMAN ST"  This is my code:  import arcpy, os, sys, string from array import *   table1 = "G:\\xStreetNew\\table1.shp" referenceTable = "G:\\xStreetNew\\Test.gdb\\referenceTable" cursor = arcpy.da.UpdateCursor(table1, "Name") for row in cursor: if row[0] == None:     row[0] = row[0] else:     row[0] = row[0].replace(";",",") cursor.updateRow(row) table1_column = ['Name'] table1_curObj = arcpy.da.SearchCursor(table1, table1_column) table1_List = [] for row in table1_curObj:    table1_row = row[0].split(",")    table1_List.insert(0,table1_row)    print table1_List[0] Same gravatar is displayed for different users in stackexchange.com filters Same gravatar is displayed for different users in the stackexchange.com , under the Top Network Askers in the right side bar.   The red circled image's source (&lt;img src=) are from   https://www.gravatar.com/avatar/?s=18&amp;d=identicon&amp;r=PG&amp;s=30                                ^   Seems the gravatar id is missing here in the image source.  For an example from the list, the user  gravatar id should be    https://www.gravatar.com/avatar/66c66c4d97a7bc3859a85253f003e1fb?s=18&amp;d=identicon&amp;r=PG&amp;s=30   but it is mapped with   https://www.gravatar.com/avatar/?s=18&amp;d=identicon&amp;r=PG&amp;s=30   which caused the display issue. Showing that Y has a uniform distribution if Y=F(X) where F is the cdf of continuous X Let $X$ be a random variable with a continuous and strictly increasing c.d.f. function $F$ (so that the quantile function $F^{−1}$ is well-deﬁned). Deﬁne a new random variable $Y$ by $Y = F(X)$. Show that $Y$ has a uniform distribution on the interval $[0, 1]$.  My initial thought is that $Y$ is distributed on the interval $[0,1]$ because this is the range of $F$.  But how do you show that it is uniform? Error with square bracket in table What is the maximum possible length of a .NET string? What is the longest string that can be created in .NET? The docs for the String class are silent on this question as far as I can see, so an authoritative answer might require some knowledge of internals. Would the maximum change on a 64-bit system?  [This is asked more for curiosity than for practical use - I don't intend to create any code that uses gigantic strings!] Let $X_1$ and $X_2$ be two independent exponential random variables with the PDFs $f_{X_1}(x_1)=\lambda_1 \exp(-\lambda_1 x_1)$ and $f_{X_2}(x_2)=\lambda_2 \exp(-\lambda_2 x_2)$.   Let $X= \frac{X_1X_2}{X_1+X_2+a}$ be a r.v. for which I want to derive the CDF, where $a$ is a positive constant.  In other words, I need to calculate $\mathbb{P}\{X &lt;x \}$.  Any ideas or hints?  Edit (attempt):  Using a similar approach to that used , we have $X_1 \in [0, \infty)$ and  $X_2 \in [0, \frac{x(x_1+a)}{x_1-x})$. The latter interval results from: $\frac{x_1x_2}{x_1+x_2+a} &lt; x$, thus $x_2(x_1-x)&lt;x(x_1+a)$ which implies $x_2 &lt; \frac{x(x_1+a)}{x_1-x}$; but I am not sure if this is correct since $x_1-x$ can sometimes be negative (?). Based on the above, we can write: \begin{align}  \mathbb{P} \{ \frac{X_1X_2}{X_1+X_2+a} &lt; x \}  &amp; = \int_{x_1=0}^\infty \int_{x_2=0}^{ \frac{x(x_1+a)}{x_1-x}} \lambda_1 \exp(-\lambda_1 x_1) \lambda_2 \exp(-\lambda_2 x_2)  dx_2 dx_1  \\ &amp;= \int_{x_1=0}^\infty \left(1-\exp(-\lambda_2\frac{x(x_1+a)}{x_1-x}) \right) \lambda_1 \exp(-\lambda_1 x_1) dx_1  \\ &amp; = 1-\lambda_1 \int_{x_1=x}^\infty \exp(-\lambda_2\frac{x(x_1+a)}{x_1-x} ) \exp(-\lambda_1 x_1)   dx_1 \end{align} In the last equality: for the integral of $\lambda_1 \exp(-\lambda_1 x_1) dx_1$, I suppose that $x_1 \in [0,\infty)$, whereas for the second term I suppose that $x_1 \ge x$ (since $x_2$ should be positive) (?).  Is my approach correct ?