Non-linear function such that $f(ax)=af(x)$ for every $a \in \mathbb R, x \in \mathbb R^n$
Are all multiplicative functions additive?
I've imported several shapefiles which where exported from a Mapinfo .tab. Several  .tab-files are imported resulting in 20+ layers. Afterwards I want to export it to GeoJSON; but I'm reluctant to select each layer and export it manually. Is there a way to merge all the layers into one using QGIS? I've got a huge list of date-times like this as strings:  Jun 1 2005  1:33PM Aug 28 1999 12:00AM   I'm going to be shoving these back into proper datetime fields in a database so I need to magic them into real datetime objects.   This is going through Django's ORM so I can't use SQL to do the conversion on insert. where do I use Database. Methods? I m trying to understand where we use following Database.Methods. I know if we want to insert a record to an object we use DML Statement. What i want to know is why/where we use Database. methods. It might be silly question. Database.saveResults Database.insert Database.query Database.Query Database.error   really appreciate the inputs. Thanks a ton in advance. Countable axiom of choice: why you can't prove it from just ZF This is a follow-up question to the discussion about the finite axiom of choice .  Suppose we have a countable collection of non-empty sets $\{A_1, A_2, A_3,\cdots\}$ Reasoning as indicated in that discussion, we can prove the existence of a choice function $c_n$ such that $c_n(i)$ belongs to $A_i$ for $i = 1,\cdots,n$  Again reasoning as suggested and using induction, we can prove the existence of a function $C$ defined over $\{1, 2, \cdots\}$ such that $C(n)$ is a choice function defined for $i=1,\cdots,n$ with $C(n)(i)$ an element of A_i. We can require in addition that $C(n)(i)$ and $C(n-1)(i)$ are equal for $i=1,\cdots,n-1$.  Now it would it seem that we can prove the countable axiom of choice by taking the union of $C(1)$, $C(2)$, $\cdots$  The question is as follows: why is that proof wrong? I suspect the answer may have something to do with the axiom of union. Is it acceptable to begin a declarative sentence with "Am"? Tail sum for expectation In , the tail sum formula for expectation is introduced for a nonnegative (0,1,...) discrete random variable $X$:  $$E(X) = \sum_{i=0}^\infty P(X &gt; i).$$   I wonder if there is a similar formula for nonnegative continuous random variable $X$:  $$E(X) = \int_0^\infty P(X &gt; x) dx?$$  If no, are there some conditions for it to hold? And how can it be proved?  Here is my thought:  If the cdf $F$ of $X$ is bijective, then $X=F^{-1}(U)$ for some random variable $U$ uniformly distributed over $[0,1)$. So $$E(X) =     \int_0^1 F^{-1}(u) du.$$  To prove the tail sum formula, it suffices to prove $$\int_0^1     F^{-1}(u) du = \int_0^\infty P(X &gt; x) dx.$$ But I am stuck here.   What's more, is the condition that the cdf $F$ of $X$ is bijective really necessary for tail sum formula to hold? Can tail sum formula be generalized to a random variable that is not necessarily nonnegative?   Thanks! How are we able to calculate specific numbers in the Fibonacci Sequence? I was reading up on the Fibonacci Sequence,  $\text {{1,1,2,3,5,8,13,....}}$ when I've noticed some were able to calculate specific numbers. So far I've only figured out creating an array and counting to the value, which is incredibly simple, but I reckon I can't find any formula for calculating a Fibonacci number based on it's position.  Is there a way to do this? If so, how are we able to apply these formulas to arrays? Let $f$ and $g$ be entire functions and $g(z)\neq 0$ for all $z\in \mathbb{C}$. If $|f(z)|\le |g(z)|$, can we say $f$ is constant?  Liouville theorem says that an entire bounded function is constant, but g is not given bounded here. So I think it should be false. What else can we say about $f$? Are spells learned from feats considered to be associated with your class? The multiclassing spellcasting rules (PHB p. 164, or ) state:  Each spell you know and prepare is associated with one of your classes, and you use the spellcasting ability of that class when you cast the spell.  According to the above, every spell you know or prepare should be associated with a class.  However, for spells learned from feats like Drow High Magic or Wood Elf Magic are these actually associated with a class? If so, what class? If not, is the PHB statement above wrong/outdated? Are the spells learned in these feats (and others like it) associated with something else (like your race, in these cases)?  For reference, the  racial feat (XGtE p. 74) states:  You learn more of the magic typical of dark elves. You learn the detect magic spell and can cast it at will, without expending a spell slot. You also learn levitate and dispel magic, each of which you can cast once without expending a spell slot. You regain the ability to cast those two spells in this way when you finish a long rest. Charisma is your spellcasting ability for all three spells.  The  racial feat (XGtE p. 75) states:  You learn the magic of the primeval woods, which are revered and protected by your people. You learn one druid cantrip of your choice. You also learn the longstrider and pass without trace spells, each of which you can cast once without expending a spell slot. You regain the ability to cast these two spells in this way when you finish a long rest. Wisdom is your spellcasting ability for all three spells.  In regards to the third and fourth questions, the fact that the PHB quote says &quot;use the spellcasting ability of that class when you cast the spell&quot; and the fact that the feats specify what ability the spell uses in these examples (independent of class) suggests to me that they are not associated with class. Why is the Sun approximated as a black body at ~ 5800 K? Windows 7 msconfig lets users select the options No GUI boot and OS boot information to have a list of drivers being loaded during boot. Is it possible to have similar detailed messages during shutting down/restarting/logging off? Here's the screen I want disable or give me more details: What can you say about $f$ and $g$ in the case that $fg$ is 1) Injective, 2) Surjective - Cohn - Classic Algebra Page 15 What does "...from nonmagical attacks not made with silvered weapons" mean? Compiling C code in the terminal I've been having a problem with compiling.   I used the following test C code below and saved it as test.c in Home folder.  #include &lt;stdio.h&gt; int main() {   printf("Bruh!");   return 0; }   When I run in terminal the code   gcc test.c -o test   I get  nothing.  Thank you I need to extrude a circular shape from a surface that's basically just a grid of quads. What's the best practice to do this without screwing up the topology too much?  I basically want to have a circular face I can extrude, in order to have a cylinder like protuberance on the object, something like this: If my data is overdispersed when binomial but not when Poisson does that alone mean I should use Poisson GLM? Top is Poisson, bottom is Binomial Source of variation is different people observed over a week long period. 118 people to be exact.  want to make predictions about the proportion of days/week one exercises if they wake up at a given time.  &gt; summary(FMR7)  Call: glm(formula = No.Days.Ex ~ WaketimeStand, family = poisson, data = PSD)  Deviance Residuals:       Min        1Q    Median        3Q       Max   -2.99043  -0.64677  -0.03492   0.67393   1.51943    Coefficients:               Estimate Std. Error z value Pr(&gt;|z|)     (Intercept)    1.36991    0.04689  29.218   &lt;2e-16 *** WaketimeStand -0.08309    0.04806  -1.729   0.0839 .   --- Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1  (Dispersion parameter for poisson family taken to be 1)      Null deviance: 124.55  on 115  degrees of freedom Residual deviance: 121.51  on 114  degrees of freedom AIC: 479.71  Number of Fisher Scoring iterations: 4  &gt; summary(FMR6)  Call: glm(formula = cbind(No.Days.Ex, No.Days.NotEx) ~ WaketimeStand,      family = quasibinomial(link = logit), data = PSD)  Deviance Residuals:      Min       1Q   Median       3Q      Max   -3.7483  -0.9585  -0.0625   1.0428   2.9739    Coefficients:               Estimate Std. Error t value Pr(&gt;|t|)   (Intercept)     0.2595     0.1009   2.571   0.0114 * WaketimeStand  -0.1869     0.1020  -1.832   0.0695 . --- Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1  (Dispersion parameter for quasibinomial family taken to be 2.015388)      Null deviance: 288.42  on 115  degrees of freedom Residual deviance: 281.57  on 114  degrees of freedom AIC: NA  Number of Fisher Scoring iterations: 4  Quite clear that there is much more overdispersion on the quasibinomial than the poisson. The response variable is no.days exercise per week and so my original thought was to analyse with binomial GLM as it is the proportion of days in a week someone exercises. However, when modelled with a Poisson GLM it is much less overdispersed. Is this a sign I am using the wrong distribution. No updates since a long time Using Ubuntu, I have a habit of running: sudo apt update sudo apt upgrade   on every startup. But for a month, I guess, there have been no updates whatsoever. The message shows: All packages are up to date  Now I am starting to worry. Can there be a fault in my update manager? As far as I know, I've made no changes to the package manager settings. But no updates for almost a month seems kind of weird. Any feedback would be helpful. Thanks. System.LimitException: Too many query rows: 50001 What I am trying to do is to set a date in my date field and also update the object. I am using the class below to do so which works just fine in my sanbox but in production I am getting this error:      System.LimitException: Too many query rows: 50001   public with sharing class updateObject{     public  List&lt;NRProducts__c&gt;  values;     public PageReference updateObject() {         values=[SELECT Refreshed_Date__c FROM NRProducts__c ];         for (NRProducts__c s :values){            s.Refreshed_Date__c =Date.today();          }          update values;         return null;                       }     }   VF Page:  &lt;apex:page showheader="false" controller="updateObject" sidebar="false"&gt; &lt;apex:form&gt;  &lt;apex:commandButton action="{!updateObject}" value="Update"/&gt; &lt;/apex:form&gt; &lt;/apex:page&gt; AUC and class imbalance in training/test dataset I just start to learn the Area under the ROC curve (AUC). I am told that AUC is not reflected by data imbalance. I think it means that AUC is insensitive to imbalance in test data, rather than imbalance in training data.  In other words, only changing the distribution of positive and negative classes in the test data, the AUC value may not change much. But if we change the distribution in the training data, the AUC value may largely change. The reason is that the classifier cannot be learned well. In this case, we have to use undersampling and oversampling. Am I right? I just want to make sure my understanding on AUC is correct. Adding Edge Loops Around Airplane Window I would like to add an edge loop around each one of these windows and are not 100% sure on the best way to do it even after watching some videos.  If someone could point me in the right direction that would be great. Thanks.