Mathematics Stack Exchange Community Digest

Top new questions this week:

Collatz conjecture but with $n^2-1$ instead of $3n+1.$ Does the sequence starting with $13$ go to infinity?

Let's consider the following variant of Collatz $(3n+1) : $ If $n$ is odd then $n \to n^2-1.$ $1\to 0.$ $3\to 8\to 1\to 0.$ $5\to 24\to 3\to 0.$ $7\to 48\to 3\to 0.$ $9\to 80\to 5\to 0.$ $11\to 120\to ...

number-theory recreational-mathematics collatz-conjecture  
asked by Adam Rubinson Score of 17
answered by Will Jagy Score of 1

"Multiply everything so far, plug into polynomial" - can these always yield primes?

EDIT: I forgot how open number theory is! (I think that gets me put on mathematician probation or something.) For this question, I will accept any answer which assumes "standard conjectures" ...

number-theory prime-numbers  
asked by Noah Schweber Score of 15

Why does it seem more natural to think of $r$ as a function of $\theta$, rather than the other way around?

When teaching functions in polar coordinates, the nearly universal practice is to consider functions of the form $r = f(\theta)$. I think I have never seen any examples in which $\theta$ is expressed ...

algebra-precalculus functions polar-coordinates  
asked by mweiss Score of 12
answered by user2661923 Score of 2

Are these equations "properly" defined differential equations?

Are these equations properly defined differential equations? Modifications were made to a deleted question to re-focus it Intro Recently, in this answer I figure out that the following autonomous ...

ordinary-differential-equations partial-differential-equations physics mathematical-modeling finite-duration  
asked by Joako Score of 12

The coupon collector's most collected coupon

Suppose a coupon collector is collecting a set of $n$ coupons that he receives one-by-one uniformly at random. If the collector stops exactly when the collection is complete, we know the expected ...

probability combinatorics expected-value coupon-collector  
asked by Brocellous Score of 11
answered by Sangchul Lee Score of 8

Prove $\sum_{i,j:i<j} μ(E_{i} \cap E_{j} )=\infty$.

The problem is: Assume that $(X,\mathcal{A} , μ)$ is a finite measure space (i.e., $μ(X) < \infty $ ) and the sequence of sets $E_{j}\in \mathcal{A}$ satisfies $\sum^\infty_{j=1} μ(E_{j} )=\infty$....

real-analysis measure-theory  
asked by rose Score of 11
answered by Robert Israel Score of 8

Understanding Legendre transform (and convex conjugate)

I am learning convex analysis on my own and I would appreciate some help. I know that convex conjugate is the generalization of the Legendre transform. I also know the formula known for Legendre ...

real-analysis convex-analysis legendre-transformation  
asked by Pegi Score of 11

Greatest hits from previous weeks:

A loss and gain problem

This is a very simple but confusing puzzle. A customer buys goods worth $200$ rupees from a shop. The shopkeeper selling these goods makes zero profit from this purchase. The lady gives him a $1000$ ...

puzzle  
asked by constantlearner Score of 7
answered by fgp Score of 8

How many even numbers of four digits can be formed with the digits 0,1,2,3,4,5 and 6 no digit being used more?

My attempt to solve this problem is: First digit cannot be zero, so the number of choices only $6 (1,2,3,4,5,6)$ The last digit can be pick from $0,2,4,6$, so the number of choices only 4 Second ...

combinatorics  
asked by akusaja Score of 4
answered by akusaja Score of 2

How to calculate the area of a 3D triangle?

I have coordinates of 3d triangle and I need to calculate its area. I know how to do it in 2D, but don't know how to calculate area in 3d. I have developed data as follows. ...

triangles 3d area  
asked by iamgopal Score of 51
answered by Vafa Khalighi Score of 45

Proof of triangle inequality

I understand intuitively that this is true, but I'm embarrassed to say I'm having a hard time constructing a rigorous proof that $|a+b| \leq |a|+|b|$. Any help would be appreciated :)

inequality absolute-value  
asked by ivan Score of 67
answered by k.stm Score of 122

Which way is length and which way is width?

I hear people refer to the dimensions of things as "$2$ by $4$" etc. and I know its length by width, but I can't tell if the length dimension is vertical (up and down) or horizontal (side to side). ...

geometry terminology  
asked by David Score of 15

Inverse of the sum of matrices

I have two square matrices: $A$ and $B$. $A^{-1}$ is known and I want to calculate $(A+B)^{-1}$. Are there theorems that help with calculating the inverse of the sum of matrices? In general case $B^{-...

linear-algebra matrices inverse  
asked by Tomek Tarczynski Score of 179

Is a matrix multiplied with its transpose something special?

In my math lectures, we talked about the Gram-Determinant where a matrix times its transpose are multiplied together. Is $A A^\mathrm T$ something special for any matrix $A$?

matrices  
asked by Martin Ueding Score of 285
answered by Simon Markett Score of 190

Can you answer these questions?

Max edit distance between binary strings of length n

Given a binary string s of length $n>2$.An edit operation is a single character insert, delete or substitution. The edit distance between two strings is the minimum number of edit operations needed ...

binary dynamic-programming  
asked by Hosho Kana Score of 1

Is a certain map $\mathbb{R}\to \widehat{\mathbb{R}}$ a homeomorphism?

I'm studying harmonic analysis, and I'm trying to understand the fact that $$\phi: \mathbb{R}\to \widehat{\mathbb{R}}: s \mapsto (t \mapsto \exp(2\pi i st))$$ is an isomorphism of topological groups (...

complex-analysis analysis harmonic-analysis topological-groups  
asked by Andromeda Score of 1
answered by QuantumSpace Score of 0

How many ways are there to arrange $3$ men, $3$ women, $3$ children in a circle such that they alternate clockwise in this order: man, woman, child?

Problem How many ways are there to arrange $3$ men, $3$ women, $3$ children in a circle such that they alternate clockwise in this order: man, woman, child? The seats are not numbered. My Opinion I ...

combinatorics permutations  
asked by WS Wee Score of 1
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