Articles of terminology

Divisible abelian $q$-group of finite rank

What does “finite rank” mean in the context of divisible abelian $q$-group? A divisible abelian $q$-group of finite rank is always a Prüfer $q$-group or it can be also a finite product of Prüfer $q$-group? Thanks for all future answers to my questions.

Is an unit-cube polyhedron? What about other platonic solids?

Definitions According to my linear programming course and screenshot here (Finnish), a polyhedron is such that it can be constrained by a finite amount of inequalities such that $$P=\{\bar x\in \mathbb R^n | A \bar x\geq \bar b\}, A\in \mathbb R^{m\times n},\bar b\in \mathbb R^m$$ and a convex function $f(x)$ must satisfy $$f(\lambda \bar x+(1-\lambda)\bar […]

Relative merits, in ZF(C), of definitions of “topological basis”.

Typical Terminology: A basis $\mathcal{B}$ for a topology on a set $X$ is a set of subsets of $X$ such that (i) for all $x\in X$ there is some $U\in\mathcal{B}$ such that $x\in U$, and (ii) if $x\in U\cap V$ for some $U,V\in\mathcal{B}$, then there is some $W\in\mathcal{B}$ such that $x\in W\subseteq U\cap V$. The […]

What is the meaning of equilibrium solution?

What are the equilibrium solutions for the differential equation $\dfrac{\mathrm{d}y}{\mathrm{d}t} = 0.2\left(y-3\right)\left(y+2\right)$ My Question: What does equilibrium solution mean in this context of this problem?

Why do we call Functional Analysis like this?

Functional analysis is ‘a kind of mathematical analysis’ where the object of study are functions. The tool for studying functions are the operators. A specific type of operators are the functionals. My question is why do we call this subject as functional analysis, while the main role here is the operator? And if someone has […]

What does it mean to “identify” points of a topological space?

I was recently reading about circle rotations (a basic example in dynamical systems) and got confused by some notation. It said consider the unit circle $S^{1} = [0,1]/{\sim}$, where $\sim$ indicates that $0$ and $1$ are identified. What does “identify” mean? and how is the set $[0,1]/{\sim}$ different from the set $[0,1]$? Thanks!

Good way to describe “converging parallel lines”?

Not sure if this question is on topic. I am looking for a nice correct and succinct way to describe “2 lines are limiting parallel ” that is: Understandable for newbies to hyperbolic geometry. Geometrically correct not mentions “ideal points”. (because they don’t really exist. “going in the same direction” is also not allowable, lines […]

Number of variables and dimension of a function

Why is a function $f(x)$ called a single-variable function if it has coordinates represented by $x$ and $y$? Can it be called a 1D function if its plot is 2D? Subsequently, can two-variable functions $f(x,y)$ (that have coordinates $x,y,z$) be called a 2D function if its plot is 3D? This is confusing and potentially ambiguous. […]

Is there a name for a topological space $X$ in which very closed set is contained in a countable union of compact sets?

Is there a name for a topological space $X$ which satisfies the following condition: Every closed set in $X$ is contained in a countable union of compact sets Does Baire space satisfy this condition? Thank you!

When does intersection of measure 0 implies interior-disjointness?

If there are two “nice” shapes in $R^2$, such as circles or polygons, whose intersection has area 0, then they must be interior-disjoint, as their intersection can only contain pieces of their boundary. My question is: what is a simple term for those “nice” subsets of $R^2$ for which intersection of area 0 implies interior-disjointness? […]