Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [definition]

For requesting, clarifying, and comparing definitions of mathematical terms.

Definitions are at the core of mathematical precision; they answer the question "What is X?" in mathematics. Into this category fit questions regarding equivalence of definitions, clarification of complicated definitions, or proposed new definitions for mathematical notions, with requests for improvements or comments.

7799 questions
3
votes
0 answers

How to "well define" this statement?

For simplifying things, suppose we are working on $\mathbb{R}^3$. We have a surface $S$, a path $C$, and a path $D$ in the $xy$ plane where a line integral is going to be taken. I would start defining the variables I will use: "Let $x,y,z,t \in…
Alan
  • 503
3
votes
1 answer

Why is $X^- = -\min\{X, 0\}$?

Why is $X^- = -\min\{X, 0\}$ ? This is how it is defined in a probability book I am self-studying. For the function $X:\Omega \to \mathcal{R}$, The positive part of $X$ is the function $X^+ = \max \{X, 0\}$. The negative part of $X$ is the function…
Joe St Amand
  • 167
3
votes
2 answers

Characteristic time?

Could somebody tell me the definition of a "characteristic time"? For example, what is the characteristic time for a function $f(t)=\operatorname{tanh}(t)$ to reach 1? I tried looking up a definition, but  there seems not to be a universal…
fred sinclair
  • 33
3
votes
2 answers

Name of the mathematical term $\frac{a-b}{a+b}$

I calculated a kind of proportion between $a$ and $b$, not $\frac{a}{b}$ but $$\frac{a-b}{a+b}$$ Do this mathematical term has a name? If so, it would help me to explain my calculations...
clemtoy
  • 139
3
votes
3 answers

Injective or one-to-one? What is the difference?

What is the difference between the terms 'injective' and 'one-to-one', 'surjective' and 'onto', and 'bijective' and 'isomorphic'?
ahorn
  • 2,236
3
votes
8 answers

How can expressions like $x^2+y^2 = 4$ be defined?

I'm wondering how to define the expression(?) $x^2+y^2 = 4$, because I realised it's not a function because it cannot be expressed in terms of $x$ or $y$ alone. Is it even called an expression? Of course I then thought that you could split it into…
Frank Vel
  • 5,339
3
votes
1 answer

n-tuple function definition

I've read this definition for an hour now and I cannot piece it together abstractly. To define an n-tuple as a function $F$, where $X$ is the index set and domain, and $Y$ is the set containing the elements of the tuple: $$(a_1,a_2, ...,a_n) \equiv…
user222071
  • 31
3
votes
1 answer

What is a Gauss sign?

I am reading the paper "A Method for Extraction of Bronchus Regions from 3D Chest X-ray CT Images by Analyzing Structural Features of the Bronchus" by Takayuki KITASAKA, Kensaku MORI, Jun-ichi HASEGAWA and Jun-ichiro TORIWAKI and I run into a term…
Karl
  • 875
2
votes
3 answers

2+2=4; Not in the Z3 algebraic group

I was reading the article/wiki here When I came across this quote ObviousFact?: examples: 2+2=4 for most people Those with higher mathematical knowledge may disagree - not in the Z3 algebraic group. No, 2+2 is still 4 in Z3, it just also happens…
AnotherUser
  • 349
2
votes
1 answer

$\epsilon$ rules for uniform and pointwise convergence.

Could someone please provide with the $\epsilon$ definitions of uniform and pointwise convergence. I'm trying to really get my head around the differences between them (I do know the differences, but it would help to see them written formally in…
Ellya
  • 11,783
2
votes
2 answers

Does the definition of "mean"/"average" require the result to be in the domain set?

If I have a function that calculates the mean value of a set of elements that is an arbitrary subset of some set $X$, does the mean, by definition, have to also be in $X$? (In other words, if the mean can't be in $X$, does that imply that "mean"…
Jason C
  • 210
2
votes
1 answer

$A_{i,j}B_{i,j}$ is matrix dot-product in Einstein Notation?

Skim-read an engineering book stopped to this assertion "matrix dot product is $\sum_{i,j \in I} A_{i,j} B_{i,j} := A_{i,j} B_{i,j}$ in Einstein Notation". Sorry but what does it really mean? I have never read about matrix dot-product, rather about…
hhh
  • 5,469
2
votes
1 answer

Axiom or Postulate?

In wikipedia we see that the words “axiom” and “postulate” are synonyms: “An axiom, or postulate, is a premise or starting point of reasoning”. But in A Friendly Introduction to Numerical Analysis we read: “The axioms and postulates of the…
Dante
  • 1,908
2
votes
0 answers

Difference between "marginal variance" and "true variance"?

[This question is not about solving a mathematical problem, but about definitions; please tell me if there is a better forum to ask it.] In a paper I am reading, the author writes: ...the sole reason for minimizing the marginal variance instead…
wip
  • 183
1 2 3
…
15 16