# Tagged Questions

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### Why we use ANY in the definition of a maximal element?

I am confused about the following definition: "a maximal element of a subset S of some partially ordered set is an element of S that is not smaller than any other element in S." I do not understand ...
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### Definition of null space

I have two definitions of null space. One by Serge Lang Suppose that for every element $u$ of $V$ we have $\langle u,u\rangle=O$. The scalar product is then said to be null, and $V$ is called a ...
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### In arbitrary commutative rings, what is the accepted definition of “associates”?

In an integral domain, the following are equivalent: $r \mid s$ and $s \mid r$ $r=us$ for some unit $u$ However in arbitrary commutative rings this is no longer the case; in particular, (2) ...
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### Name of a shape that is intersected once by each ray that starts at a given point

Is there a particular name for a shape that is intersected exactly once by each ray that starts at a given point? To illustrate: I'm looking for a name for shapes like the left one in this image: ...
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### When is a binary operation bipotent?

I learnt that $\max(-,-)$ is a bipotent binary operation but I'm not able to find a definition of bipotent operation. QUESTION A binary operation $*:M\times M \rightarrow M$ is bipotent if ...
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### Name for “relative difference”

If we want to express that two numbers $x, y$ are not so far away from each other absolutely we use the absolute value function $|\cdot|$ with $0 < \epsilon \ll 1$: $$|x-y| < \epsilon$$ ...
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### Has this topology a name?

let $(X,\tau )$ be the topological space where $\tau =\{\emptyset, X, \{x\}, X-\{x\}\}$ , $x \in X$. Does this topology have a name? Thanks in advance!
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### “Identify” terminology

What does it mean to "identify" two objects in algebra, as distinct from having an isomorphism or automorphism between them? I was led to think about this while reading Stewart's Galois Theory, where ...
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### Terminology: Name for general, non-differential equations over $\mathbb{R}^n$

Let $x \in \mathbb{R}^n$ be a variable, and $f_1, \ldots , f_n : \mathbb{R}^n \mapsto \mathbb{R}$ be a sequence of functions, each having a closed form expression. Assume that we want to solve the ...
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### Why to see that $\overline{B}(x;r)$ is closed if it was just defined?

I'm reading Conway's A Course in Point Set Topology. He defines open and closed balls and then he introduces some examples, one of these examples is this: (c) For any $r>0$, $\overline{B}(x;r)$ ...
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### Critical points of a function of absolute value

Say I have the function $f(x) = |x|$ I believe that $x = 0$ is a critical point, although not I'm not positive. As the function is decreasing and increasing each side of $x = 0$ does that alone make ...
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### Definition of homogeneous ODE

In my lecture notes, it gives this following definition of a homogeneous ODE: A differential equation is called homogeneous if it can be written in the form $x′=f(\frac{x}{t})$ Then in one of ...
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### Binary operation (english) terminology

Foreword: I have read R.H. Bruck's A Survey of binary systems, where the notion of halfoperation is given. A halfoperation $\ast$ differs from a (binary) operation since $a\ast b$ may not be defined ...
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### Word for “openness”/“closedness” of an interval

What word properly completes the phrase the radius of convergence does not depend on the $\text{______}$ of the interval to mean that it doesn't matter whether $(a, b)$, $[a, b)$, $(a, b]$, or ...
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### If a series converges does its sequence of partial sums converge?

By Definition, a series $\sum_{n=1}^\infty a_n$ converges if it's sequence of partial sums $S_n = \sum_{k=1}^n a_k$ converges. My question is, is the converse true? If $\sum_{n=1}^\infty a_n$ ...
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### The term $rank$ in methematics

Reading wikipedia's disambiguation page about the "rank" word I see many concept of rank of many different matematical object. I only know about the rank of a graded poset and the rank of a set that ...
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### What is this metric called?

Ahlfors -complex analysis p.20 Consider a stereographic projection between the 2-sphere and $\overline{\mathbb{C}}$ (i.e. one-point compactification of $\mathbb{C}$) Let $z,w$ be complex numbers. ...
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### Embedding and monomorphism

What is the difference between an embedding and a monomorphism? As far as I can see, most introductory abstract algebra texts treat them as if they are the same, i.e. an injective function from one ...
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### A set with a supremum and an infinum inside

What is the name of a set $A$ that has its infimum $i \in A$ and a supremum $s \in A$? Examples: $[0,1]$ has a supremum $1$ and an infimum $0$ which are inside $[0,1]$. $\{0,1,2,3\}$ has a supremum ...
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### How is the word “contains” defined in set theory? (In relation with neighborhoods in topology).

From Wiki: Some basic sets of central importance are the empty set (the unique set containing no elements) Thus, this make me think that "contained" is equivalent to the $\in$, as in: if $a$ is ...
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### What does “characteristic” mean in mathematics?

In Germany, I have heard "ABC is 'characteristic' for XYZ" sometimes from math students. It was used like "if you know ABC, then you know you're talking about XYZ" or "ABC describes XYZ completely". ...
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### Definition of $\Omega$-group and $\Omega$-composition series

What are the definitions of $\Omega$-group and $\Omega$-composition series? No luck searching on the internet..
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### difference between theorem, lemma and corollary

Can anybody explain what is the basic difference between theorem, lemma and corollary. We have been using it for a long time but I never paid any attention. I am just curious to know.Thanks a lot.
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### Analogous to convolution

A convolution of two functions $f$ and $g$ is defined as $$[f*g] = \int_{-\infty}^{\infty} f(\tau) g(t-\tau) d\tau.$$ I am interested on an analogous transformation of the form [f\star g] = ...
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### With infinite size, we can have $P \cdot M = M \cdot D$ (D diagonal) but where $M^{-1}$ does not exist. Can we say “P is diagonalizable”?

(I had this question in mind for longer time, but it is just triggered now by some comments at that recent question in mse) (Background) I was looking at properties of the Pascal-matrix: ...
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### What is a fraction in which the greatest common factor of the numerator and the denominator is 1?

What is this fraction: A fraction in which the greatest common factor of the numerator and the denominator is 1?