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

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0answers
43 views

What is a separable closure? And why the standard definition of algebraic closure is defind as so?

Instead of defining it as a subfield of algebraic closure, what would be the natural way to define it? Here's what I figured out: Let $F$ be a field and $E$ be an extension field of $F$. ...
3
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3answers
487 views

Intuition behind independence & conditional probability

This is a basic question. I have a good intuition that $A$ is independent of $B$ if $P(A \vert B) = P(A)$, and see how you can easily derive from this that it must hold that $P(A,B) = P(A)P(B)$. ...
3
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1answer
26 views

Is normal extension algebraic?

Let $E/F$ be a field extension. Then $E/F$ is called normal iff there exists $\mathscr{A}\subset F[X]\setminus\{0\}$ such that $\forall f\in\mathscr{A}$, $f$ splits over $E$ and $E=F(\{\alpha\in ...
5
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4answers
291 views

Constructing the natural numbers without set theory.

As I understand it the idea of defining everything as sets is a relatively new idea in mathematics. Does that mean there's a non-set theoretic definition of the natural numbers? Could there be?
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1answer
44 views

In an augmented matrix representing a system of equations, why is it a contradiction when the LHS isn't zero and RHS is zero but not when flipped?

In an augmented matrix representing a system of equations, say a $1\times 3$ matrix: $(a,b \mid c)$, why is it a contradiction when $a=b=0, c\neq 0$ but not when $a,b\neq 0, c=0$ ?
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2answers
52 views

Vector bundle and its definition

Take this definition of vector bundle: its a triple $(V,\pi,M)$ where $V$ is a set, $\pi:V \rightarrow M$ a sutjective map such that for every $m \in M$ the set $\pi^{-1}(m)$ has the structure of real ...
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3answers
1k views

What is meant by a stopping time?

TL;DR: is a stopping time some sort of event, or is it a point in discrete time, or something else entirely what is an example of something which is not a stopping time? is my understanding of the ...
3
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1answer
122 views

If $C=M \times [0,1]$, what is an integral over $\partial C$?

Let $M$ be a compact Riemannian manifold. Define $C=M \times [0,1]$ so that $\partial C = M \times \{0\} \cup M \times \{1\}$. If $f:\partial C \to \mathbb{R}$, is then the integral over $\partial C$ ...
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1answer
47 views

Clarification on some definitions in Operator Theory

I'm trying to read this paper http://arxiv.org/abs/1206.3325 , but I'm having a lot of difficulty making sense of two phrases. The setting is $L_2(\mathbb R^d)$. i) He mentions that for a function $...
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0answers
56 views

On definition and usefulness of “Cayley table”

Dummit-foote p.21 Let $G=\{g_1,\cdots,g_n\}$ be a finite group with $g_1=1$. The Cayley table of $G$ is the $n\times n$ matrix whose (i,j)-entry is $g_ig_j$. This definition is based on an ...
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1answer
70 views

How to define a b-open set relative to a larger set?

In the paper of D. Andrijevic entitled "On b-open Sets", it is defined that A subset $S$ of a topological space $(X,\tau)$ is $b$-open if $$ S\subseteq\bar{\operatorname{int} S}\cup \...
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4answers
1k views

Vector Spaces: Redundant Axiom?

Question Why are the axioms for vector space independent? More precisely $1x=x$ seems redundant... (I take the axioms from: Wikipedia) Explanation One has for zero vector: $$\lambda0+\lambda0=\...
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3answers
2k views

What is a point?

In geometry, what is a point? I have seen Euclid's definition and definitions in some text books. Nowhere have I found a complete notion. And then I made a definition out from everything that I know ...
2
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3answers
124 views

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'?
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1answer
62 views

On the definition of sphere in analytic geometry…

Last year, when I was teaching mathematics (analytic geometry) for one of my clever freands, I arrived to the definition of sphere. I said Fix $r>0$, An sphere is the set of all triples $(x,y,z)...
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1answer
37 views

Definition of linear chains

I am reading Hatcher books in algebraic topology and he defined by $LC_n(Y)$ as the subgroup of $C_n(Y)$ generated by linear maps $\Delta^n$ $\to Y$ where $C_n(Y)$ is the abelian group of the $n$ ...
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4answers
853 views

What is the exact definition of an Injective Function

Am I right to believe that a function is injective, if some elements of the first set are mapped to some elements of the second set? It is also possible to 4 elements of the first set, are mapped to ...
3
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2answers
105 views

Holomorphic function definition. Am I missing something very obvious?

I'm reading a book of complex analysis in which the definition of holomorphic function is given as follows: Definition: If $V$ is an open set of complex numbers, a function $f:V \to \mathbb C$ is ...
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1answer
233 views

Number of orbits in a graph.

I am confused with this concept. Consider for instance the graph $G$ with $V=\{v_1,\dots, v_{10} \}$, $E=\{12, 15, 16, 23, 27, 34, 38, 45, 49, 67, 78, 89, 910, 510, 610 \}$. This is a 3-regular graph....
2
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1answer
212 views

Tetration and its inverse to various exponents

I've recently seen in my studies tetration, or the next operation in the addition, multiplication, exponentation... series. I've also heard much discussion about how to extend this operation to "...
3
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0answers
59 views

What vector field property means “is the curl of another vector field?”

I know that a vector field $\mathbf{F}$ is called irrotational if $\nabla \times \mathbf{F} = \mathbf{0}$ and conservative if there exists a function $g$ such that $\nabla g = \mathbf{F}$. Under ...
3
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0answers
68 views

Gradient vector interpretation

I have trouble understanding the gradient vector $\nabla$. For a surface, when we find it's $\nabla$, why is the resultant vector the normal vector when $\nabla$ means gradient vector?? Thanks
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3answers
9k views

Difference between a theorem and a law

There are plenty of theorems out there as well as laws within mathematics. For example, in Boolean algebra: Theorems Idempotent Involution Theorem of Complementarity Laws Commutative ...
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1answer
187 views

Definition of Lebesgue measure on the Circle $X=\mathbb{R}/\mathbb{Z}$.

I was given the following definition: I feel this is not adequate and would only define Lebesgue on $[0,1]$? (Wikipedia uses a pushforard of the complex exponential) Also later I am given a proof ...
3
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0answers
78 views

Largest Singular Value / Singular Value

I was wondering, what if the eigenvalues of a matrix A are all negative. So does that simply mean there is no singular value for this particular matrix?, hence I can't calculate the conditional number ...
2
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0answers
47 views

Quartic operator definition

What is a quartic operator? I googled it but found only some articles which use that term whitout giving a definition (I found that term while studying 2D Ising model, and the use of some ...
2
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1answer
331 views

Well-posed vs Well-conditioned

What's the difference between a well-posed (ill-posed) and well-conditioned (ill-conditioned) problem ?` Here is my finding up to now: "Even if a problem is well-posed, it may still be ill-...
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2answers
64 views

Ordered tuples of proper classes

From time to time I encounter notation like this: A triple $\langle \mathbf{No}, \mathrm{<}, b \rangle$ is a surreal number system if and only if ... The confusing part is that a proper class ...
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2answers
77 views

Two definitions of Jacobson Radical

I have in my notes that the Jacobson radical of a ring $R$ is: $J(R) = \cap${$I$ | $I$ primitive ideal of $R$} $= \cap$ {$Ann_R M$ | $M$ simple $R$-module}. I have now seen elsewhere that $J(R) = \{...
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0answers
83 views

Mathematical structures with name reffering to a country

I am looking for a list of mathematical structures (not theorems) that refer to a country or nationality. I only know of Polish spaces and Polish groups. Does anyone have other examples? Note: many ...
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5answers
583 views

Is there an intuitive, not-too-mathematical way of thinking about limit points? [duplicate]

so I know this question has been asked sooo many times. But I just have a few questions in particular, which despite searching, I haven't found an answer to. I appreciate any help. Book's definition: ...
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2answers
38 views

Comparing Open Bases and Covers

In Topology, I see a resemblance and similarity between open bases and open covers. Although this is a short question, what is the defining difference between the two that sets them apart? ...
2
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1answer
59 views

What's the correct definition of generated ideal in a pseudo-ring?

Given a ring (with $1$) $R$, one defines what, say, a left ideal is. There's also a natural definition of ideal generated by a subset Definition A: $_R(S):=\bigcap\{I\supseteq S:I\text{ is a left ...
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2answers
51 views

Different two definitions for separable extension

Let $E/F$ be an algebraic field extension and $\bar F$ be an algebraic closure of $F$. Define $[E:F]_{\text{sep}}$ as the cardinality of $$\{\sigma\in \operatorname{Mono}(E,\bar F): \sigma \text{ ...
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1answer
57 views

When is it allowed to “take apart” a limit (multiplication of limits)?

When is it allowed to "take apart" a limit? Here's an example to show what I mean: $\displaystyle\lim_{n\to\infty}\frac{\frac 1 ne^{\frac 1n}(e-1)}{e^{\frac 1 n}-1}=e-1$ since we can "take apart" ...
15
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3answers
935 views

True Definition of the Real Numbers

I've found lots of resources that say this is a real number if it's not rational, but what is a real number, really? I mean what is the definition of a real number? If nothing else, anyone know of a ...
17
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3answers
1k views

What is category theory?

I have read the page about category theory in wikipedia carefully, but i don't really get what this theory is. Is category theory a content in ZFC-set theory? (Just like measure theory, group theory ...
2
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1answer
39 views

The domain of a function as a function: the “domain-function”

The domain of a function $f:X\to Y$ is normally defined as $\operatorname{dom}f\equiv X$, but I would like the domain-function $\operatorname{dom}$ to be a funtion itself, i.e. I would like to define ...
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2answers
303 views

Motive for the definition of inner product

Mathematicians pride themselves on writing proofs of propositions in an elegant way, but frequently (maybe even usually?) neglect to formally write motivations of definitions with the same elegance, ...
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2answers
316 views

Application of Picard-Lindelöf to determine uniqueness of a solution to an IVP

I am still struggling quite a lot with the Picard-Lindelöf-Theorem (also known as the Cauchy-Lipschitz-Theorem). Problem: Consider the following IVP with $\alpha \neq 1$ $$\begin{cases} y'&= ...
2
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1answer
137 views

How to understand cocategories

$\newcommand\CC{\mathsf{C}}$The notion of a category is well-known. There are multiple equivalent definitions; small categories can be seen as an internal category in $\mathsf{Set}$, that is a ...
2
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3answers
400 views

Alternative definition of hyperbolic cosine without relying on exponential function

Ordinary trigonometric functions are defined independently of exponential function, and then shown to be related to it by Euler's formula. Can one define hyperbolic cosine so that the formula $$\...
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2answers
267 views

Can someone explain definition of number 1?

I've found the next piece of text: As an example of the second failing, Poincaré recalled the definition of the number 1 offered by another of the logicists, Burali-Forti: $$1 = \imath\,T'\{Ko\...
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0answers
85 views

Correct understanding of col and diag operators

In a scientific paper I am currently working with, a definition of $Col$ and $Diag$ operator is introduced: We use the operator $Col_{k\in K}(x_k)$ which stacks up its vector (or matrix) ...
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6answers
612 views

Removable discontinuity question

A quick and easy question on terminology: If we have a function $$f(x) = \frac{x(x-2)}{(x-2)}$$ then clearly the function is not defined at $x = 2$. If we cancel out "$(x-2)$" then the function is ...
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3answers
39 views

Continuity definition of a functional

I'm having a hard time understanding the formal definition of continuity of a functional. I'm not sure if such questions are appreciated on this site; so let me know. Definition: The functional $J[y]$...
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2answers
22 views

Properties of exponents when dealing with induction.

This will most likely be a simple question for most of you. While watching my lecture today the white board cut out and the instructor didn't explain the final step in an example. He went from $(3^{2^...
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2answers
101 views

What is the definition of a finite set $S$? [closed]

This question is intended mainly for beginners. We can say "$S$ is not infinite" or "counting elements of $S$ is a procedure that (theoretically) terminates", a little of maths appears in “$S$ is ...
3
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0answers
41 views

Meaning of the term $X/H$ and orbits

I am trying to find representations of the group $G=GL_2(F(t)/t^2) = (M_2(F_p) , + ) \rtimes GL_n(F)$ So I was trying to do exactly what Serre has explained in this section. I am not quite able to ...
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1answer
29 views

Are there terminologies distinguishing these two ranks?

Definition 1. Let $R$ be a ring with invariant basis number and $M$ be a free $R$-module. Then, the rank of $M$ is the cardinality of an $R$-basis of $M$. ${}$ Definition 2. Let $...