Definitions are the core of mathematical precision; they come to answer "what is X" in mathematics. Into this category fit questions regarding equivalence of definitions; clarifications regarding complicated definitions; as well questions with purposed definitions for mathematical notions with ...

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1k views

real numbers and number line

While reading some articles, I got a bit confused by the definitions of numbers. Specifically, Can the number line contain decimal values? I read that Real numbers = All numbers on the number ...
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5answers
16k views

What is the difference between Singular Value and Eigenvalue?

I am trying to prove some statements about singular value decomposition, but I am not sure what the difference between singular value and eigenvalue is. Is singular value just another name for ...
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1answer
188 views

What set operation is this?

Given two sets $ A = \{\{1\} , \{2 , 6\} \}$ and $ B = \{\{2\} , \{3\} , \{4 , 5\} \}$, what set operation can produce $$ C = \{ \{ 1 , 2 \} , \{ 1 , 3 \} , \{ 1 , 4 , 5 \} , \{ 2 , 6 , 2 \} , \{ 2 , ...
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1answer
79 views

Which distribution can be abbreviated as “LD”?

Which distribution can be abbreviated as LD and which PDF is expressed as a formula with sum of erfc() functions? $$p(o)=\frac{1}{4\ell} ...
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2answers
327 views

the definition of the area of a surface

When we say the area of a rectangle is the product of the length by the width is it a definition based on geometric intuition or is it a result? I know it is a result that we can find after defining ...
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1answer
99 views

Definition of the 1-dimensional $\mathbb{C}GL(V)$ module “$\det ^n$”

I'm reading through my notes on representation theory of $S_n$ and $GL(V)$, and have come unstuck on a definition which I can't understand - furthermore I can't seem to find any information on it ...
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1answer
122 views

Definition of “succession of central extensions of abelian groups”

What is the meaning of the phrase: "A group $G$ can be realized as a succession of central extensions of abelian groups"?
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2answers
1k views

A simple explanation of differential calculus and its link to geometry?

The wikipedia articles on differential calculus and differential geometry are quite long and not so straightforward for a layman like me. Is there a master of math vulgarization out there that could ...
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1answer
51 views

Math vocab: operator on $S$ and into $S =$?

Is there a special name for a binary operation on the set $S$ that is also into $S$, that is unambiguous with other uses. I.e. if it's "operator on $S$", I've heard that in other places meaning the ...
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1answer
117 views

What do mathematicians call the Two's Complement on 8-bits group?

It is isomorphic to $\mathbb{Z}_{2^8},$ only difference is the symbols usually identifying the elements of the set are from $\{-128, \ldots, 127 \}$ and not $\{0, \ldots, 256\}.$ What is an elegant ...
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2answers
288 views

Is 2+2=4 an identity?

I know this seems like a silly question, but someone was trying to debate with me about how 2+2=4 should be called an identity and not an equation. I mentioned how it has no variables and isn't true ...
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5answers
450 views

What's the difference between tuples and sequences?

Both are ordered collections that can have repeated elements. Is there a difference? Are there other terms that are used for similar concepts, and how are these terms different?
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2answers
984 views

$\{0,1\}^n$ and $[0,1]^n$ notations

Can someone please help me clarify the notations/definitions below: Does $\{0,1\}^n$ mean a $n$-length vector consisting of $0$s and/or $1$s? Does $[0,1]^n$ ($(0,1)^n$) mean a $n$-length vector ...
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2answers
132 views

Defining a Differentiable Function in a Non-Pointwise Manner

Is there a way to define a differentiable function in a non-pointwise manner? That is without defining function differentiable at a point first. Just like we define a continuous function through open ...
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0answers
57 views

Are equidimensional ideals and unmixed ideals the same?

Zariski-Samuel define an unmixed/equidimensional ideal to be one whose associated primes have the same dimension. At other places I have seen definitions saying unmixed=all associated primes have same ...
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1answer
149 views

Name of binary relation: if $(x, y)\in R$ then there is no $z$ such that $(y, z)\in R$

Is there a term for a binary relation $R\subset A^2$ on some set $A$ such that if $(x, y)\in R$ then there is no $z$ such that $(y, z)\in R$ ? Are there any examples of it? Are there any related ...
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1answer
371 views

Pairwise non-integral numbers

I have a set of complex numbers a_1 through a_n which are said to be "pairwise non-integral numbers". Could someone explain to me exactly what this means? Thanks. From comment below: I should also ...
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2answers
471 views

What are the primitive notions of real analysis?

My dad introduced my to primitive notions in geometry in high school. It's come back to haunt me as I study real analysis; I find myself wondering, Have we given this a formal definition? ...
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3answers
246 views

A question about a certain way to define mathematical objects

It is common in mathematics to see definitions of the following form: we begin with a certain object $A$. we perform some construction depending on a choice of some parameter $\lambda\in\Lambda$ for ...
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1answer
66 views

Does this function have a “name”, somewhat linked to Euler totient

If $\varphi$ denotes the Euler totient and $n=p_1^{k_1}\cdots p_r^{k_r}$ is the prime factorization of $n>1$ we have $ \varphi(n)= \varphi(p_1^{k_1}) \varphi(p_2^{k_2}) \cdots\varphi(p_r^{k_r})= ...
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2answers
124 views

Elaborate on $A^{c}:=\{p\in\mathbb Q : 0<p<\sqrt{2}\}$ not open and not closed in $\mathbb R$

I know that $\sqrt{2}\not\in\mathbb Q$ and $\sqrt{2}\in\mathbb R$ but it is not obvious to me why $\{p\in\mathbb Q : 0<p<\sqrt{2}\} \subset \mathbb R$ is not open. If it is not open, it means ...
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3answers
72 views

A parametrized surface

If I am given a map $f:U\subset \mathbb R^2 \to \mathbb R^3$ where $(x,y)\to (f(x,y),g(x,y),h(x,y))$. Is this necessarily a "parametrized surface"?  Am I right in thinking that any map of the above ...
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2answers
228 views

Definition of a point and object

Is there any theory in which a point has a definition? What is the definition of "object" as seen in category theory?
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9answers
3k views

What makes elementary functions elementary?

Is there a mathematical reason (or possibly a historical one) that the "elementary" functions are what they are? As I'm learning calculus, I seem to focus most of my attention on trigonometric, ...
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1answer
515 views

Understanding various definitions of TREE($n$) in Friedman's finite form of Kruskal's tree theorem.

I was reading the Wikipedia article on Friedman's finite form of Kruskal's tree theorem, and am interested in the large numbers TREE(n). I would like to verify TREE(2)=3 myself, but find conflicting ...
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3answers
2k views

why do we use 'non-increasing' instead of decreasing?

In english based math language it seems that non-increasing $\Longleftrightarrow$ less or equal (non-strict decreasing) decreasing $\Longleftrightarrow$ strict less ( strict decreasing) ...
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1answer
239 views

What is the definition of an induced cut

I am reading A simple min-cut algorithm (Stoer & Wagner, 1997), and the proof uses some terms I don't understand. Specifically I am unclear on what is meant by "$C_v$ the cut of $A_v \cup \{v\}$ ...
6
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2answers
184 views

What is $\mathbb{N}^{<\mathbb{N}}$?

What is the definition of this symbol $\mathbb{N}^{<\mathbb{N}}$? How is it related to the infinite product $\mathbb{N}^{\mathbb{N}}$?
4
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1answer
183 views

What is the difference between $\mathrm{E}[Y|X = x]$ and $\mathrm{E}[Y|X]$ and between $\mathrm{Var}(Y|X = x)$ and $\mathrm{Var}(Y|X)$?

I am slightly confused about the different between $\mathrm{E}[Y|X = x]$ and $\mathrm{E}[Y|X]$ and similarly for Variance. It seems to me the first should be a scalar, because we first pick a ...
2
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2answers
231 views

What is the product of the empty set?

Give: $fn(S)=\prod_{x\in S}x$ what is: $fn(\emptyset)$ I can see reason that it would be defined as 1 or 0. Note: I thought about restricting the domain of $S$ but that would make the problem ...
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2answers
95 views

Definition clarification on ideals

Suppose $P$ is the set of all subsets of a set $X$ and $P$ is a ring. Let $p$ be an element in $P$ (so that $p$ is a subset of $X$). What does it mean to say "an ideal generated by $p$"? And suppose ...
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1answer
253 views

Is Gödel's completeness theorem a representation theorem?

In general a representation theorem is — according to Wikipedia — a "theorem that states that every abstract structure with certain properties is isomorphic to a concrete structure". ...
2
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1answer
280 views

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 ...
4
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1answer
307 views

The definition of the logarithm.

One usually gets several definitions of the logarithm along his studies. You might be first introduced to the exponential and then told that the logarithm is its inverse. You might be given $$\log ...
2
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1answer
39 views

Terminology clarification: ***exchanges***

I need help with a terminology definition. If we say "R is a reflection that exchanges the sides a and b in some triangle", does it mean sides a and b have the same length and the reflection maps one ...
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0answers
92 views

Definition of stereoprojection and Möbius maps

@WillieWong has kindly pointed out that there are 2 definitions of stereographic projection. One with the unit sphere placed on top of the plane, the other where the plane is at the equator of the ...
2
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1answer
45 views

Reducts of categories

There are several ways to reduce a category. The skeleton of a category is the category with isomorphic objects collapsed into one i.e. the only isomorphisms that remain are the identities. What's ...
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1answer
137 views

Motivation behind the definition of reflections in affine hyperplane

What is the motivation behind the definition of the reflection map in affine hyperplanes? $R: x \to x-2(x\cdot u-c)u$ where $u\cdot x=c$ defines the affine plane. Of course one requirement is for it ...
2
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1answer
72 views

Definition clarifications : on adjectives of functions

Could somebody please explain what are the differences between a differentiable function and a holomorphic function analytic function and conformal function? (Am I right to think that all analytic ...
2
votes
1answer
281 views

Perfect Hash Function just an Injection?

I just read up on the concept of perfect hash functions on a set $S$. I am quoting: "A perfect hash function for a set S is a hash function that maps distinct elements in S to a set of integers, with ...
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2answers
238 views

Solvability and Simplicity

I have read that Burnside's theorem implies that a group with order $p^aq^b$ cannot be simple. So I looked up Burnside's theorem and saw that it doesn't mention "simple" explicitly, rather it says ...
7
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1answer
468 views

Higher Order Trigonometric Function

Once in a time, I had to work with functions that have the following Taylor series expansion: $$ t_m(x)=1-\frac{x^m}{m!}+\frac{x^{2m}}{(2m)!}+\cdots =\sum_{k=0}^\infty \frac{(-1)^k x^{km}}{(km)!}. $$ ...
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2answers
619 views

How to write “let” in symbolic logic

How do I write let in symbolic logic? For example, if I am in the middle of a proof and there is a variable which I can assign to an arbitrary value, what would I write? My best guess is: $$ x := a ...
5
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1answer
336 views

Understanding the homotopy extension property

I'm reading Chapter 0 of Hatcher right now, and there's something in the definition of homotopy extension property that I don't understand. Suppose one is given a map $f_0:X\to Y$, and on a ...
2
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2answers
588 views

What is the differentiation operator

On Wikipedia the Differential Operator is described as an operator defined as a function of the differentiation operator. The link that underlies the words "differentiation operator" in fact gives ...
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3answers
100 views

Oddities in the Definition of IntegralCosinus ${\rm Ci}(x)$

Reading the defintion of the IntegralCosinus $$ {\rm Ci}(x) = \gamma + \ln x + \int_0^x\frac{\cos t-1}{t}\,dt $$ I wonder what happens, if I to split the function in the integral: $$ ...
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2answers
514 views

How many 'supremum(s)' and 'infimum(s)' can a set have?

I am learning calculus/real analysis with Apostol's Calculus (2nd Edition). I have a doubt about the grammer of this book. Apostol, everywhere, uses a supremum (or a least upper bound) and an infimum ...
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2answers
166 views

On the definition of jets

I have some problems with the definition of jets and it would be great if someone could help me here: In many books it is written, that the $r-th$ order jet $j^r_xf$ of a smooth function $f:M ...
0
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1answer
193 views

definition: dual of a vector field

Let $X:\mathbb{R}^3\rightarrow T\mathbb{R}^3$ be a vector field, what is the definition of its dual ? I know that the set of vector fields on $\mathbb{R}^3$ forms an $\mathbb{R}$-vector space.
15
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2answers
531 views

What is it to be normal?

I'm interested to find out why the word 'normal' crops up so many different areas of mathematics, especially but not exclusively in abstract algebra, and how the definitions are related, if at all. ...