Articles of soft question

What is matrix inequality such as $A>0$ or $A\succ 0$?

I am trying to gather here different meanings of the same symbol, inequality symbol or the succ symbol. I find many other use them so many different ways. Sometimes, $A>0$ means $\bar x^T A \bar x >0$. Sometimes, $A>0$ means element-wise i.e. $a_{i,j}>0$ for all $i,j$. How do I know which definition of inequality people […]

Encyclopedia of Mathematics? (non-alphabetical)

Do you know any encyclopedia of mathematics which is in non-alphabetical order, like it starts from basic mathematics and then goes up to very advanced level? And what’s the difference between say, if I am studying calculus from a mathematical encyclopedia and if I am studying calculus from a university based calculus textbook?

Is the notion of “affineness” more general than “linearity”, or vice versa?

This is an incredibly dumb question, so please bear with me. An affine transformation $T$ is equal to a linear transformation $L$ plus a translation $t$. This suggests that affine transformations are more general than linear transformations, because for the former $t$ can be non-zero, but for the latter $t$ must be zero. Likewise, an […]

Differentiation and integration

Which came first : Differentiation or Integration? If one of them was developed to solve certain types of problems, was the other developed for backward compatibility, or was it an independent development and later discovered that they were inverses? Also, how were they discovered to be linked? It was just something that I was thinking, […]

Proof that group is commutative if every element is its inverse (feedback wanted)

This is one of my first proofs about groups. Please feed back and criticise in every way (including style & language). Axiom names (see Wikipedia) are italicised. $e$ denotes the identity element. Let $(G, \cdot)$ be a group. We assume that every element is its inverse. It remains to prove that our group is commutative. […]

Why is recursion theory suffering from terminological bloat?

Several questions on MSE in recent months and most recently this one have made me feel that recursion theory is suffering from terminology bloat. Why have so many synonyms for “recursive” and “recursively enumerable” been introduced? I can just about see why people might want to use “computable” for recursive or use “computably enumerable”, “Turing-acceptable” […]

The set of all things. A thing itself?

If the universe is the set of all things. Does it contain itself? In other words is it a thing itself? I know its a stupid question, but it really grinds my gears. Thanks! Edit 8.12 Okey, someone here said that it cant exist. So what if it would be a proper class does, that […]

What are Goldbach conjecture for other algebra structures, matrix, polynomial, algebraic number, etc?

All: what are Goldbach conjectures for other algebraic structures, such as: matrix, polynomial ring, algebraic number, vectors, and other algebraic structures ? In other word, for other algebraic structures, such as: matrix, polynomial ring, algebraic number, vectors, etc. What does “even” element mean ? What does “prime” element mean ? Can every “even” element for […]

I think I found an error in a OEIS-sequence. What is the proper site to post it?

I checked the link given to this OEIS-sequence : https://oeis.org/A081121 and apparantly the numbers $3136$ and $6789$ appear in the sequence. However, we have $$4192^2=260^3-3136$$ and $$94^2=25^3-6789$$ so the two numbers should not appear in the sequence. $1)$ Did I miss something, or is this actually an error ? $2)$ What is the proper site […]

The complement of a torus is a torus.

Take $S^3$ to be the three-sphere, that is, $S^3=\lbrace (x_1,x_2,x_3,x_4):x_1^2+x_2^2+x_3^2+x_4^4=1\rbrace$. Using the stereographic projection, $S^3=\mathbb{R}^3\cup \lbrace \infty \rbrace.$ Can someone explain how the complement of the solid torus (centered at the origin) $S^1\times D^2$, where $D^2$ is a 2-disk, is also a torus? I am reading Milnor’s paper “On Manifolds Homeomorphic to the 7-Sphere,” and […]