Questions on the usage and meaning of words in mathematics, the names for mathematical entities, and other such questions.

learn more… | top users | synonyms (2)

4
votes
1answer
142 views

Terminology for blow-ups in algebraic geometry

Definition (from Eisenbud-Harris' Geometry of Schemes): Let $X$ be any scheme, $Y \subset X$ a subscheme. We say that $Y$ is a Cartier subscheme in $X$ if it is locally the zero locus of a single ...
5
votes
1answer
133 views

Languages with context-free grammar having only one non-terminal symbol

As seen in this question, the class of languages that can be generated by a context-free grammar having only one non-terminal symbol (i.e. the start symbol) is a proper subclass of the class of ...
-1
votes
3answers
250 views

Is there a term for any number that is $2^n$?

I am looking for a term for numbers that have a base of $2$ with any power so for example, $2,4,8,16,32,\cdots$. I would say a base $2$ number but am under the assumption that that refers to binary ...
0
votes
3answers
160 views

Two Parabolic Mirrors Opposite of Each other

Suppose we have two parabolic mirrors opposite of each other (e.g. $x=y^2$ and $x= -y^2+10$). Also suppose the first mirror is smaller than the second mirror. If a light ray enters into the opening ...
5
votes
1answer
102 views

Terminology for metric space with “anti-symmetric” distance

I'm interested in spaces that have a two-place function $d$ with non-negative real values, satisfying the following three conditions (for all $x$, $y$, $z$): $d(x, x) = 0$ $d(x, y) + d(y, z) \geq ...
4
votes
1answer
157 views

Does this theorem have a name?

Let P(x) be a polynomial of degree n. Let H(i) represent the number of 1's in the binary expansion of the integer i. Although reasonably easy to prove, it may seem surprising that the following ...
6
votes
1answer
60 views

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..
7
votes
0answers
121 views

Urysohn's Lemma needn't hold in the absence of choice. Alternate terminology for inequivalent definitions of “normal” spaces?

A topological space $\langle X,\tau\rangle$ is said to be normal if any two disjoint closed subsets are separated by open sets, meaning that for disjoint $E,F\subseteq X$ with $X\setminus E,X\setminus ...
1
vote
2answers
106 views

$G$-set terminology

When a group action $G \times X \rightarrow X$ is defined with a group $G$ and a set $X$, why is there not a special name for the set $X$? I know that this is referred to as a $G$-set, but the set $X$ ...
4
votes
1answer
139 views

Why they use the word “abstract” for naming mathematical fields of study?

I've found some books with titles such as abstract analysis - but I don't understand why they choose such word. For me it seems quite vague and perhaps misleading - consider the examples: Real ...
5
votes
2answers
280 views

$\omega$-consistency and related terms

We know that a theory $T$ is $\omega$-inconsistent if there is a formula $\psi$ such that $T$ proves $(\exists x)\psi(x)$, and $T$ also proves $\lnot \psi(n)$ separately for each standard natural ...
1
vote
2answers
280 views

Difference between equilibrium point and Unique Equilibrium Point?

Is there any difference between unique equilibrium point and equilibrium point? If yes, please tell me what it is and How is it used to solve a dynamical system. You can consider a dynamical system ...
8
votes
5answers
327 views

Motivation for the concept of “open set” in topology

I am looking at the section "Motivation" for the Wikipedia entry on "open sets": https://en.wikipedia.org/wiki/Open_set#Motivation and I am not sure it is doing such a good object of motivating open ...
3
votes
2answers
83 views

Categories without identities

What's the name of "categories without identities", i.e. of digraphs with just an associative binary operation on its "matching" arrows (disregarding identities)?
6
votes
1answer
116 views

Who introduced the term “norm” into mathematics?

I've always been curious about the motivation behind the use of the word norm, as used in linear algebra and functional analysis, for a function that assigns a positive number to a vector. Who ...
3
votes
2answers
114 views

What does it mean “adjoin A to B”?

What does it mean that we can obtain $\mathbb{C}$ from $\mathbb{R}$ by adjoining $i$? Or that we can also adjoin $\sqrt{2}$ to $\mathbb{Q}$ to get $\mathbb{Q}(\sqrt{2})=\{a+b \sqrt{2}\mid a,b \in ...
1
vote
1answer
67 views

Unit length tangent vectors on a Riemannian manifold

Let $X$ be a Riemannian manifold and $TX$ its tangent bundle. Is there a name for the $S^1$-bundle given by the unit length tangent vectors?
6
votes
3answers
985 views

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.
1
vote
2answers
65 views

Why isnt area the same as surface area? [closed]

Doesn't a 2D shape have a surface? In that case, is it incorrect to call the area of it surface area?
0
votes
0answers
35 views

The relation of GCD and capacity?

I saw some books call GCD(Greatest Common Divisor) capacity. Why call it this name? or it's just a name without further learning value? capacity in wiki, related with measure, however there is no ...
1
vote
0answers
25 views

Joint discrete and continuous optimization problem

What is an optimization problem which involves joint discrete and continuous variables called? I hope here be a good place to ask this. If it is not, please tell me where to ask it.
1
vote
1answer
83 views

Why are regular $p$-groups called “regular?”

In the concept of regular $p$-Groups, what does "regularity" stand for? What is "regular" in such groups? I would like to know idea behind defining these groups, and naming these groups "regular." ...
1
vote
1answer
55 views

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] = ...
1
vote
0answers
38 views

Condition for being 'injective in each variable separately'

Is there an established term for a function on a product set that is not injective but is injective with respect to each argument individually? The motivating example is the canonical projection $\pi: ...
12
votes
2answers
620 views

What is the mathematical intuition behind àl-jàbrà?

The term algebra comes from the arabic term àl-jàbrà that means "to force", "to restore". Over centuries mathematicians, in east and west, celebrate by this term mathematical disciplines. What is ...
0
votes
5answers
328 views

Why do we say a real line is an open set, but the complex plane is not an open set?

Why do we say a real line is an open set, but the complex plane is not an open set? I don't understand that? I'm fully confused between these differences?
0
votes
3answers
93 views

Is there a word for the classification of a set as continuous or discrete?

For example, in computer science, there can be zero, one, two, etc. parameters to a computer program, and this is called its "arity". Sets can be countable or uncountable. Is there some word I can use ...
3
votes
3answers
92 views

Is there a word for the classification of a set as infinite or finite?

For example, in computer science, there can be zero, one, two, etc. parameters to a computer program, and this is called its "arity". Sets can be countable or uncountable. Is there some word I can ...
3
votes
0answers
68 views

Is there a name for the following property in the theory of topological spaces?

I worked out the following lines. Probably it is totally trivial, but maybe this is interesting in the theory of topological/metric spaces. Let $\Omega$ be an open subset of the Euclidean $n$-space. ...
2
votes
1answer
64 views

Monoid with inversion

Is there a name for monoid with operation $a\mapsto a^{-1}$ conforming the equations $(a^{-1})^{-1}=a$ and $(b\cdot a)^{-1} = a^{-1}\cdot b^{-1}$? (with no requirement that $a^{-1}\cdot a$ would be ...
1
vote
1answer
53 views

What do we call a continuous function that induces a homeomorphism onto its image?

I know that an order-homomorphism that induces an isomorphism onto its image is called an order-embedding. So, I was expecting that a continuous function that induces an homeomorphism onto its image ...
2
votes
1answer
81 views

Terminology for $\phi(xy)=\phi(x)\phi(y)$

I have a model which contains a function $\phi:{\mathbb R}_+ \rightarrow {\mathbb R}_+$ that satifies: $$\tag{*}\phi(xy)=\phi(x)\phi(y)$$ for all $x,y\in{\mathbb R}_+$. In Number Theory there is a ...
0
votes
2answers
101 views

Is LinearTransformation's matrix a square matrix?

Linear Transformation’s confusions. Is linear transformation's matrix a square matrix? Some books define linear transformation is $V\to W$, and say linear operator is $V\to V$. Some books define ...
1
vote
0answers
51 views

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: ...
0
votes
0answers
35 views

The difference of change of bases and change of basis

The difference of change of bases and change of basis is? A little question. Are they the same? At the bottom of the wiki's page, External links, you can see change of bases.
2
votes
2answers
74 views

Please help me check my metric definition of isolated point

I translated the word definitions into the more symbolic form below, but as they aren't mere negations of each other, it was a little tricky. Is there any mistake below (especially for 'isolated ...
2
votes
1answer
49 views

Term or factor?

I was reading a book on probability and encountered a summation expression like $$P(Y\mid X, Z) = \sum_{W}P(Y\mid X, Z, W)P(W\mid X,Z)$$ followed by the author referring to "terms of the summation". ...
0
votes
2answers
95 views

Changing the index of a summation - what is it called?

About $\sum$ (summation), what is it called when you change the index of a summation? My teacher does it all the time and I just don't get it! Please send some links so I can learn it.
10
votes
0answers
99 views

Smallest $n$ such that $G$ is a subgroup of the symmetric group $S_n$ [duplicate]

A well-known result, Cayley's theorem, says that any group is isomorphic to $S_n$ for some $n$. Given a (finite) group $G$, is there a standard name for the smallest such $n$? This seems like a very ...
3
votes
0answers
42 views

Is there a name for subtracting a set of values from their max?

I hope this question is appropriate here - if it isn't let me know and I will remove it. I am wondering if there is a verb for the following operation: given a set of non-negative numbers, I take ...
0
votes
1answer
59 views

What is the name of two commutative squares?

How do you refer two commutative squares sharing one side as follows?
1
vote
0answers
26 views

Terminology for multilinear functionals the sum of whose cyclic shifts is zero?

Let $\varphi$ be an $n$-linear functional on a vector space $X$. Suppose that $\varphi$ has the property that, for all $(x_1,\ldots,x_n) \in X^n$, we have $$ \varphi(x_1,\ldots,x_n) + ...
7
votes
6answers
917 views

What is the meaning of infinitesimal?

I have read that an infinitesimal is very small, it is unthinkably small but I am not quite comfortable with with its applications. My first question is that is an infinitesimal a stationary value? It ...
0
votes
2answers
579 views

Definition of accumulation point

I have here a definition of accumulation point: A point $x$ in a metric space $M$ is called an accumulation point of $A \subset M$ if every neighbourhood of $x$ contains some point of $A$ distinct ...
9
votes
1answer
165 views

A category where maps are factorizations - what is this called?

Let $\mathcal C$ be a category, and define $\mathcal D$ to be the category whose objects are maps in $\mathcal C$, and where a map $f\to g$ is a factorization $pfq=g$. Composition of $(p_1,q_1):f\to ...
0
votes
2answers
99 views

How to define an interior point in terms of $\epsilon$-balls?

Which is the technically correct definition? I) An interior point of a set $B$ is a point that is the centre of some $\epsilon$-ball in $B$. II) An interior point of a set $B$ is a point that is in ...
2
votes
2answers
411 views

Precise definition of epsilon-ball

My textbook gives the following definition: "For each $\epsilon>0$, the $\epsilon$-ball about a point $x$ in a metric space $M$ is the set $\{y\in M:d(x,y)<\epsilon\}$." Is this correct? ...
3
votes
1answer
75 views

Does a neighbourhood need to be a *connected* set?

I have in my topology/ real analysis textbook the definition of neighbourhood of a point as an open set containing that point. But isn't a neighbourhood necessarily a connected set? Wikipedia also ...
1
vote
0answers
27 views

Standard nomenclature for certain embedded tori

The standard embedded torus, parametrized by longitude $\alpha$ and latitude $\beta$ is given by $$ \begin{align} x & = (R+\cos\beta)\cos\alpha, \\ y & = (R+\cos\beta)\sin\alpha, \\ z & = ...
1
vote
1answer
55 views

Recursive application of a function : Symbol of [duplicate]

I need to apply a function $f(x)$ recursively/repeatedly for n times; how do I express it (mathematically) ? Is their a mathematical symbol which denotes $f(x)$ applied n times ie $g(x,n)$ ...