## 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!

REFERENCES

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.

## 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”)))

## 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!

## Numbers that can be written as a sum of three cubes in exactly one way (a^3 + b^3 + c^3)

Based on online info, it seems that most of these numbers have many solutions. Are there any that have only 1 known solution or only a few solutions?