How to optimize three numbers such that their sum is always equal?

I know the real numbers $ a,b,c$ and $ d$ and I am trying to find three more numbers – $ x, y, z$ – such that their average is equal to $ d$ and the sum $ |a-x| + |b-y| + |c-z|$ is minimal.

How would I do that? I can use a computer to compute the numbers, but I have no idea how to approach the problem. Any help is appreaciated.

Do I need an entry stamp when returning the the Schengen after a three month absence?

I stayed in Germany for three months, summer of 2018 but my Schengen visa was soon to expire and I needed to exit for a minimum of three months. I went to the UK and was stamped in for 6 months. Then I went to Ireland and was stamped in for 3 months. When I left Ireland and returned to England none of us on the plane went through any passport control leaving Ireland nor entering England. I asked a fellow traveller who lives there and she said it was abnormal. I asked an airport employee and he laughed and said “We let anybody in these days.” I stayed for Christmas and returned to Germany in the Schengen zone after being absent the required three months. Again when I departed England there was no one inspecting or stamping passports and when I arrived in Germany, in a small airport, I was warmly greeted and welcomed by the customs officer but he also did not stamp my passport. Is this usual? Will I run into any problems down the road? Do I need a stamp?

Littlewood’s three precepts of refereeing in mathematics: is it (1) new, (2) correct, (3) interesting?

I have a question regarding Littlewood’s three precepts of refereeing a mathematical paper, namely whether it is (1) new, (2) correct, and (3) interesting.

I have found these mentioned in the literature on refereeing, e.g.:

  • “you should address Littlewoods’s three precepts: (1) Is it new? (2) Is it correct? Is it surprising?” (Krantz, 1997, p. 125); or
  • “the fundamental precepts ‘Is it true?’, ‘Is it new?’, and ‘Is it interesting?’ to which, Littlewood believed, a referee should always respond.” (Moslehian, 2010: 1245)

Unfortunately, I haven’t been able to track down the original source. Does anyone know where Littlewood might have formulated these three precepts?

Thank you!


Krantz, S. G. (1997). A Primer of Mathematical Writing: Being a Disquisition on Having Your Ideas Recorded, Typeset, Published, Read, and Appreciated. Providence, RI: American Mathematical Society.

Moslehian, M. S. (2010). Attributes of an ideal referee. Notices of the American Mathematical Society, 57 (10), 1245.

What is the significance of the three dots “…” on menus and buttons and how to use them right?

Adding three dots after the title of items in a dropdown menu seems to be a common practice (as you can see on the picture of a drop down menu in Google Chrome). They generaly mean that there is “something” after clicking on it.

Google chrome dropdown menu (french version)

These dots are also sometimes presents in the text of action links and buttons.

I am wondering about their utility and relevancy…

In your opinion :

  • What kind of information should be conveyed by these dots ?
  • How and when to use them ?
  • Is it realy relevant to the user, and easily understood by them ?

Different import statement for the “three” 3d lib inside Angular 7 App? (Theoretical Question)

I was wondering if there is any elegant way (best practice-ish) to import/export the Three functions inside of my lazy loaded module so they can get exposed to my components inside the lazy module?

I installed three and the @types/three and currently i have to write something like this in every component that uses three functionalities:

/* Inside a component from a lazy module that uses Three */ import * as THREE from 'three'; 

I’d like to have it something like this (Not sure if this code is valid):

/* Inside a component from a lazy module that uses Three */ import THREE from 'MyLazyModule';  /* Inside my lazy module where the component is declared */ import * as THREE from 'three';     @NgModule({   ...   exports : [ THREE ]   ... }) 

Do i just have to import * as three inside my lazy module and export the namespace? Or is there even a better way so i don’t have to write the import statement in each component that uses three, but only once in my lazy module?

Kind Regards

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How does Vax from Critical Role get three attacks per round?

I am getting ready to play D&D for the first time and really excited! I have been watching Critical Role to learn a little more. I am going to be playing a rogue and on Critical Role Vax often attacks with 3 daggers. When you’re duel wielding do both attacks count as one then using a bonus action to get to third? It has been confusing me.

Calculated Formula for Finding three diferent results

I am working on a calculated formula based on my [Tracking Number] column to get the following results. Find “Andrews” return the result “ANDREWS” Find “Bolling” return the result “BOLLING” Find “Pentagon” return the result “PENTAGON”

The formula that I have come up with returns “ANDREWS” when it finds “Andrews”, and returns #VALUE! for all the others.

This is the formula that I am using. Please help me resolve this issue. =IF(FIND(“Andrews”,[Tracking Number]),”ANDREWS”,IF(FIND(“Bolling”,[Tracking Number]),”BOLLING”,IF(FIND(“Pentagon”,[Tracking Number]),”PENTAGON”)))

Thanks in advance!

Constructing a bivariate normal from three univariate normals

I’m trying to construct correlated bivariate normal random variables from three univariate normal random variables. I realize there is a formula for constructing a bivariate normal random variable from two univariate random normal variables, but I have reasons for wanting to adjust two previously sampled variables by a third in order to give them a correlation $ \rho$ .

Based on the approaches for constructing bivariate normals from two univariate normals, I came up with the following approach and need help verifying its correctness.

First, imagine that we have three univariate normal variables. For simplicity here, we just assume they all have $ \sigma=1$ .

$ $ X_0 \sim Normal(0, 1)$ $ $ $ Y_0 \sim Normal(0, 1)$ $ $ $ Z \sim Normal(0, 1)$ $

Given these three univariate random variables, I construct two new random variables using the following linear combinations:

$ $ X = |\rho| * Z + \sqrt{1-\rho^2} * X_0$ $ $ $ Y = \rho * Z + \sqrt{1-\rho^2} * Y_0$ $

where $ \rho \in [-1, 1]$ represents the correlation coefficient between the two univariate normals.

Can someone help me formally verify that $ X$ and $ Y$ are now correlated random variables with correlation $ \rho$ ?

I’ve convinced my self through empirical simulation. Here are plots of values sampled from $ X$ and $ Y$ for the cases where $ \rho=0$ , $ \rho=1$ , and $ \rho=-1$ .

Plot of X vs. Y for rho of 0

Plot of X vs. Y for rho of 1

Plot of X vs. Y for rho of -1

These plots are exactly as I would expect, but it would be nice to have a formal proof based on my construction. Thanks in advance!