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

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5
votes
2answers
181 views

Function of two variables as a function of a function

Consider a function $f : \mathcal X \times \mathcal Y \mapsto \mathbb R$. I want to define $g_x(y) = f(x,y) : \mathcal Y \mapsto \mathbb R$. I want to say that $g_x$ is a ___ of function $f$. ...
2
votes
2answers
177 views

polynomial of $x$?

I want to refer to functions of the form $f(x) = \sum_{i=1}^n a_i x^{\alpha_i}$ where $\alpha_i < 1$. This is not a polynomial, because $\alpha_i$ could be just real arbitrary numbers (though ...
20
votes
4answers
34k views

Difference between axioms, theorems, postulates, corollaries, and hypotheses

I've heard all these terms thrown about in proofs and in geometry, but what are the differences and relationships between them? Examples would be awesome! :)
1
vote
1answer
91 views

Does the term Row-Complete have any synonyms?

I'm wondering if there is other terminology that describes row-completeness, outside the context of a latin square, or if row-complete is actually a general term.
4
votes
1answer
2k views

What are valent vertices?

Page 13 of Tropical Algebraic Geometry by Itenberg, Mikhalkin, and Shustin mentions 1-valent vertices, but I haven't been able to find a source that actually defines this term or managed to guess the ...
3
votes
1answer
307 views

Is there a name for this kind of number?

A perfect number is a number that is the sum of its proper divisors: 28 = 1 + 2 + 4 + 7 + 14 Is there a name for numbers whose only proper divisors are 1, 2, and ...
8
votes
4answers
838 views

How can I read this mathematical sentence aloud in English?

A map $s : \mathbb{N} \to X$ is a computable sequence in $(X,\nu_X)$ when there exists a computable map $f : \mathbb{N} \to \mathbb{N}$ such that $s(n) = \nu_X(f(n))$ for all $n \in ...
2
votes
1answer
109 views

The name for a subobject(subgroup) which is annihilated by action

I know this question is easy, but for the life of me, I cannot remember what we call this thing. Googling for this has offered no help. Consider an object $A$ and a second object $B$(let them be ...
1
vote
1answer
323 views

What is the difference between constants of proportionality and constants of integration?

I've been doing some maths work using the rate of flow of liquids. I've used various models for the flow and various methods to integrate these models. The one thing that is confusing me is the ...
2
votes
0answers
79 views

is there a name for this construction? (taking a pairing and obtaining a perfect one)

I believe that if we are given two modules $M,N$ over a ring $R$, and a pairing between them $M \otimes_R N \to R$, we can construct a perfect pairing $M'\otimes_R N' \to R$ by taking kernels. Is ...
9
votes
6answers
1k views

How is the codomain for a function defined?

Or, in other words, why aren't all functions surjective? Isn't any subset of the codomain which isn't part of the image rather arbitrary?
2
votes
2answers
580 views

What is the correct way to write product or sum in capital pi or capital sigma notation when you wish to exclude an index?

I have a series of terms $\{t_n : t_n = a_n x_n\}$, and I want to talk about the product of each term except $t_j$. Would any of these be an appropriate way to say that? I like this: $$\prod_{i \ne ...
5
votes
3answers
1k views

Notions of equivalent metrics

Let $X$ be a set, and $d,d'$ two metrics on $X$. Consider the identity map $i : (X,d) \to (X,d')$ as a map of metric spaces. There are (at least) three reasonable notions of equivalence for $d$ and ...
7
votes
4answers
5k views

What Are 4 Sided Shapes Called Again?

I apologise for the really basic question. This didn't really fit on any other StackExchange website so the Maths one was the closest one where I could ask. Really Basic Question- What are 4 sided ...
4
votes
1answer
483 views

What is the name of this kind of polynomial?

I'd like to know the name of this kind of polynomial $p(x)=x^n+a_{1}x^{n-1}+\ldots+a_{n-1}x+1$ where the $a_{i}\in\lbrace0,1\rbrace$. Thanks.
1
vote
2answers
597 views

What is the meaning of “reduced schemes of finite type”?

What does the following term mean? category of geometrically reduced schemes of finite type over some field (I know what a category and a field is but I cannot translate any of the middle bits)
4
votes
2answers
194 views

Relationship between torsion modules and topology

I was reviewing my class notes and found the following: "The name 'torsion' comes from topology and refers to spaces that are twisted, ex. Möbius band" In our notes we used the following definition ...
3
votes
1answer
595 views

What are the names of graphs with 2 in/outbound edges?

I know that many types graphs have unique names, ie. a directed graph where each node has exactly one outbound edge is known as a functional graph. Do the graphs ...
0
votes
1answer
110 views

If $x$ is a $p$-value, what do we call $x\cdot (1-x)$?

What do you call this function? $$f(x) = x(1 - x)$$ Does it have a name? It’s cropped up in at least two statistical formulas I've been asked to program so far. (In both cases, $x$ was something ...
18
votes
5answers
2k views

How fundamental is the fundamental theorem of algebra?

Despite its name, its often claimed that the fundamental theorem of algebra (which shows that the Complex numbers are algebraically closed - this is not to be confused with the claim that a polynomial ...
6
votes
0answers
232 views

Terminology for weighted projective spaces

For a sequence of positive integers $a_1, \ldots, a_n$ and a base ring $R$ there is a graded ring $R[x_1,\ldots, x_n]$ where $x_i$ is in degree $a_i$. We can then apply Proj and get a scheme, and ...
14
votes
4answers
6k views

Wedge product and cross product - any difference?

I'm taking a course in differential geometry, and have here been introduced to the wedge product of to vectors defined (in Differential Geometry of Curves and Surfaces by Manfredo Perdigão do Carmo) ...
3
votes
1answer
355 views

Why are differential equations called differential equations?

Why are differential equations called differential equations?
10
votes
1answer
3k views

Why are quadratic equations called quadratics?

The word "quad" generally means 4. Quadratics don't have 4 of anything. Can anyone explain where the name comes from?
18
votes
14answers
3k views

Mathematical concepts named after mathematicians that have become acceptable to spell in lowercase form (e.g. abelian)?

I would like to collect a list of mathematical concepts that have been named after mathematicians, which are now used in lowercase form (such as "abelian"). This question is partly motivated by my ...
4
votes
1answer
187 views

Terminology for point in dent in surface?

This is a simple terminology question. Let $S$ be a (let's say smooth) surface in $\mathbb{R}^3$, and $p$ a point on $S$. Suppose the principle curvatures $\kappa_1$ and $\kappa_2$ at $p$ are both ...
1
vote
2answers
119 views

Terminology for handling probabilities with partial knowledge

Consider a situation where a person has partial knowledge, but we have a more complete picture. For example, suppose that we want to know the probability that a fish is red. Suppose that the person ...
11
votes
3answers
737 views

What is the $x$ in $\log_b x$ called?

In $b^a = x$, $b$ is the base, a is the exponent and $x$ is the result of the operation. But in its logarithm counterpart, $\log_{b}(x) = a$, $b$ is still the base, and $a$ is now the result. What is ...
4
votes
1answer
482 views

How do you show the ring of formal laurent series is well-defined?

The only place I've encountered well-definition is with proving an operation defined on an equivalence class is independent of the choice of representative. On my homework, it asks us to show that ...
2
votes
4answers
357 views

How to reduce lower one number, while another number increases goes up increments

This is such a basic question I'm sure, but I've been trying to find a robust solution to it for a while. Lets say I have the numbers 0-100 in series. As the number increases, I want another number ...
5
votes
3answers
400 views

A 1-1 function is called injective. What is an n-1 function called?

A 1-1 function is called injective. What is an n-1 function called ? I'm thinking about homomorphisms. So perhaps homojective ? Onto is surjective. 1-1 and onto is bijective. What about n-1 and ...
23
votes
3answers
2k views

Why doesn't $0$ being a prime ideal in $\mathbb Z$ imply that $0$ is a prime number?

I know that $1$ is not a prime number because $1\cdot\mathbb Z=\mathbb Z$ is, by convention, not a prime ideal in the ring $\mathbb Z$. However, since $\mathbb Z$ is a domain, $0\cdot\mathbb Z=0$ is ...
9
votes
2answers
1k views

Permutation groups and symmetric groups

Wikipedia has separate pages for symmetric group and permutation group, but I don't understand what the difference between them is. A symmetric group on a set is the set of all bijections from the set ...
5
votes
2answers
2k views

What is “reform calculus”?

In an answer to another question I asked, Isaac suggested a book that is the standard "reform calculus" book. In a comment, I asked what the phrase "reform calculus" means, and Isaac provided a link ...
12
votes
2answers
2k views

Why the name 'FACTORIAL'?

Factorial is defined as $n! = n(n-1)(n-2)\cdots 1$ But why mathematicians named this thing as FACTORIAL? Has it got something to do with factors?
4
votes
3answers
3k views

What is Modern Mathematics? Is this an exact concept with a clear meaning? [closed]

Motivated by this question I would like to know whether there is an exact definition of modern mathematics. In which point in time (century, decade) does one think, when speaking about modern ...
22
votes
7answers
8k views

Why does “convex function” mean “concave *up*”?

A function $f : \mathbb{R} \to \mathbb{R}$ is convex (or "concave up") provided that for all $x,y \in \mathbb{R}$ and $t \in [0,1]$, $$f(tx + (1-t)y) \le tf(x) + (1-t)f(y).$$ Equivalently, a line ...
15
votes
3answers
1k views

What does “only” mean?

I understand the technical and logical distinction between "if" and "only if" and "if and only if". But I have always been troubled by the phrase "only if" even though I am able to parse and ...
7
votes
2answers
625 views

When did the term “tuple” get its current meaning?

In a recent discussion, someone told me tuples in the modern meaning (in particular, tuples are heterogeneous: that is, different elements of a tuple can belong to different sets/have different ...
4
votes
3answers
934 views

Definition of a countable set

What is the proper definition of a Countable Set?
13
votes
5answers
3k views

Blow up of a solution

What exactly does blow up mean, when people say, for example, that a solution (to a pde (say)) blows up. Thanks.
2
votes
1answer
185 views

Algebra terminology question

Suppose A is some algebraic structure, and x and y are two elements of the underlying set. Is there any more concise way of stating, "there exists an automorphism in A which maps x to y"?. It seems ...
14
votes
1answer
1k views

Common English language mistakes in mathematical writing [closed]

Quoting from this excellent answer: If you read enough math papers you'll find that there are certain linguistic ticks that people pick up from each other So here's a question (primarily for you ...
7
votes
1answer
1k views

Is there a name for $[0,1]$?

When writing software, there are often situations where I need a parameter to be a floating point number $x \in [0,1]$. I don't know of a name for that category, but I think there must be one because ...
2
votes
2answers
207 views

Notation of homeomorphism from B(H) to B(K), corresponding to unitary transformation of Hilbert spaces

Let U be a unitary transformation from Hilbert space H to Hilbert space K. How do you call a *-homomorphism f from B(H) to B(K), defined by $f(a) = U a U^{-1}$? I'm interested both in a symbol, ...
4
votes
1answer
583 views

Collective Term for $XY, YZ$ and $ZX$ Planes

Is there a collective term for the $XY, YZ$ and $ZX$ planes in $3D$ co-ordinate geometry? I was thinking "principal planes" but I'm not sure where I heard that.
7
votes
2answers
893 views

What is the name for a shape that is like a capsule, but with two different radii?

I'm looking for the name of a shape that is like a capsule, but where each circle can have different radii. The shape could be described using two circles (two centers and two radii). Something like ...
10
votes
4answers
6k views

Intuitive Way To Understand Principal Component Analysis

I know that this is meant to explain variance butthe description on Wikpiedia stinks and it is not clear how you can explain variance using this technique Can anyone explain it in a simple way?
10
votes
3answers
1k views

If and only if, which direction is which?

I can never figure out (because the English language is imprecise) which part of "if and only if" means which implication. ($A$ if and only if $B$) = $(A \iff B)$, but is the following correct: ($A$ ...
2
votes
6answers
6k views

What's the difference between open and closed sets?

What's the difference between open and closed sets? Especially with relation to topology - rigorous definitions are appreciated, but just as important is the intuition!