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

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1answer
33 views

simplicial homology definition

I am revising for my algebraic topology exam based on Hatchers 2nd chapter of algebraic topology and I have this question regarding the definition of simplicial homology on pages 104-105: Hatcher ...
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1answer
48 views

Question about the definition of a supremum

I want to know if the next definition is correct or I should fix it: Lets say that we have set $K$ and $a\in K$. Then, if $a\sup K$, for every $x\in K\Rightarrow x<a$? Or I shold cahge it to $x\le ...
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1answer
14 views

Minimal planar domain

I am recently studing minimal surfaces on my own. I have meet in many places the fallowing statement: The only connected, properly embedded, minimal planar domains in $\mathbb{R}^3$ are a plane, a ...
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0answers
17 views

What is a purely inseparable extension?

There are many different definitions of purely inseparable extension, and below is what I have chosen for my definition. (Since I don't know what is a standard one, if you know please tell me what ...
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0answers
16 views

Characterizing affine subspaces order-theoretically

Let $V$ denote a real vectorspace and $\mathrm{Con}(V)$ denote the poset of convex subsets of $V$. The goal is to identify those elements of $\mathrm{Con}(V)$ that happen to be affine subspaces of $V$ ...
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10 views

On random functions taking values in the space of continous functions.

I upload the image of the passage that is unclear to me (Theoretical statistics by Keener): I do not understand how $W_1, W_2, \dots$ take values in $C(K)$. For every different realization of the ...
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0answers
10 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 $F$. ...
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3answers
252 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 $P(A,B) = P(A)P(B)$. ...
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1answer
17 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 ...
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4answers
83 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
26 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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32 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
816 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
110 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
35 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
42 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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0answers
17 views

(Partial) derivatives exist vs. are finite?

Is there a difference between the following two statements or do they mean the same? The (partial) derivatives of $f$ exist. The (partial) derivatives of $f$ are finite. I believe that it ...
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1answer
50 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
716 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: ...
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4answers
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 ...
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3answers
58 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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0answers
6 views

Definition of a kind of section

Let $L$ a line bundle of a complex algebraic surface. What is a rational function of $L$? (Or if you want you can take the case of the Riemann surfaces)
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1answer
33 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 ...
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1answer
20 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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13 views

Name for maximal number of pre-images under a function

Let $E$ and $F$ be sets and $f:E\to F$ be any function. I want a name for the quantity $$\displaystyle\sup_{y\in F}\operatorname{Card}(f^{-1}(y)).$$ I am currently calling it the "maximal ...
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4answers
39 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 ...
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2answers
67 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
41 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 ...
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4answers
112 views

Are all groups algebraic?

I know the definition of a group as a set with an operation that satisfies certain axioms. I have heard that there is something called an algebraic group and that this is a group with a topology such ...
2
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1answer
158 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 ...
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0answers
24 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 ...
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23 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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5 views

Spherical representation on locally compact group

What is the definition of a spherical representation of the the pair $(G \times G, G)$, where $G$ is a locally compact group?
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2answers
39 views

What does $×$ mean in this context

I have two definitions, from real analysis - Metric Space: Given a set $X$, a function $d:X×X→\mathbb{R}$is a metric on $X$ if for all $x, y∈X \dots $ Function: Let $A,B$ be two sets. A function ...
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3answers
6k 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
38 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 ...
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0answers
16 views

Which is the correct definition of degree of a curve

I found those definitions of what a degree is and they contradict each other so I don't know which is the correct. The first definition it was by the last year professor saying that the degree of an ...
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0answers
36 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 ...
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28 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 ...
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0answers
20 views

Definition of Strongly Absolutely Continuous

I recall seeing a definition of the phrase Strongly Absolutely Continuous in measure theory, but I can't seem to find it anywhere now. Can anyone provide a definition, or am I misremembering?
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34 views

Differentiate between Fourier analysis and Fourier decomposition

I am a beginner. I am confused between two terms i.e. Fourier analysis and Fourier decomposition.I don't understand when to use Fourier analysis term and when to use Fourier decomposition term. It ...
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2answers
42 views

Formalization of the definition square root [closed]

I found it extremely difficult to formalize the following definition: We define the square root of $a \geq 0$, denoted by $\sqrt a$ to be an other number $b \geq 0$ such that $b^2 = a$ Can anybody ...
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1answer
38 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 ...
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0answers
20 views

Function Mapping Definition in Relation

So I have two relations: i) Ordinary relation ii) Disjunctive relation. Definition of Ordinary relation: $\Sigma$ be a finite set of attribute names, where for any attribute name $A\in \Sigma $, ...
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2answers
51 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
55 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
46 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
512 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
20 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? ...
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1answer
35 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 ...