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

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X - Y in a finite set

Question: The domain is the power set of a finite set. $X$ is related to $Y$ if $X - Y$ is not empty. Is it a partial or strict order? If it is a partial order or strict order, is it also a total ...
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1answer
20 views

Partial order or strict order

Question: The domain is the set of all positive integers. $a$ is related to $b$ if $b = a⋅ 3n$, for some positive integer $n$. Because of the equal sign, isn't this relation symmetric, transitive, ...
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2answers
35 views

How to define a specific ring using a homomorphism

If we have a ring $R$ then I can form a ring of matrices isomorphic to $R$ by setting $r \overset{\phi}{\mapsto} \left( \begin{array}{ccc} r & 0 \\ 0 & 0 \end{array} \right) $ and defining ...
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0answers
10 views

About equal substitutions

When we say that two substitutions, say $\theta,\sigma$ agree on the variables of a term $t$ what exactly do we mean ? Is it that both substitution act on the same variables and substitute them with ...
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21 views

Transitive relation vs. transitive action

A transitive relation $\circ$ is a relation with the property that: $$ (a\circ b \wedge b \circ c) \Rightarrow a \circ c.$$ A transitive group action is a group action $$\phi : G \times X \rightarrow ...
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1answer
25 views

What is the definition of ``2nd-order quasilinear parabolic'' ? for partial differential systems?

I have to know why the mean curvature flows are 2nd-order quasilinear parabolic. Let $\Omega\subset\mathbb{R}^n$ be a bonded domain (or a smooth manifold of $n$ dimensional) and $N\geq 2$. When the ...
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1answer
24 views

Are these partial order or total orders?

I just want to know if the following are considered partially ordered sets or total set. Here is the definition: $\mathbb{R}^2$ : (a,b) R (c,d) iff $a\leqslant c$ and $b\leqslant d$ $\mathbb{R}^2$ ...
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1answer
7 views

Terminology for probability matrix.

I have two related questions about terminology. If a matrix contains probabilities such that each column (or row or both) sums to $1$ , is this matrix always called a stochastic matrix i.e. even if ...
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1answer
43 views

Is my understanding of a limit correct?

When i first learned about limit i was taught that the limit $\lim \limits_{x \to a}$ f(x) = L exists if f(x) $\rightarrow$ L when x $\rightarrow$ a. Or that f(x) gets arbitrarily close to L when x ...
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21 views

It is correct this definition of limit of a function?

I have a definition of the limit of a function in some point $\alpha$ for metric spaces on this manner: We have two metric spaces $(E,d)$ and $(F,p)$; $A\subset E$, $f:A\to F$. Then ...
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1answer
44 views

Rephrasing the definition of a limit

Is $\large{\lim_{x \to a}f(x)=L}$ The same as saying "one can get $\large{f(x)}$ as close as imaginable to $\large{L}$, by setting $\large{x}$ close enough to $\large{a}$" and vice versa "one can ...
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What is a projective rational function in Fulton's book?

I'm reading Fulton's algebraic curves and I don't know if I understood correctly this definition on the page 46: What does the author mean by $f,g\in \Gamma_h(V)$ being of the same degree? Since ...
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5answers
47 views

How is “smaller than” defined on $\mathbb{R}$?

According to http://en.wikipedia.org/wiki/Binary_relation Binary relations are used in many branches of mathematics to model concepts like "is greater than", "is equal to", and "divides" in ...
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1answer
23 views

What is the definition of a binary relation?

According to http://en.wikipedia.org/wiki/Binary_relation it is first defined as "a collection of ordered pairs of elements of A" and then as "an ordered triple (X, Y, G) where X and Y are arbitrary ...
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1answer
32 views

Epsilon Delta Limit Intuition

I am having a hard time grasping the definition of a limit. I initially learned this loose definition of a limit: $\lim\limits_{x \to a}f(x)=L$ iff $f(x)$ approaches L when $x$ approaches $a$ from ...
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0answers
12 views

Definition of curvilinear coordinates?

Please can someone give me a formal definition of curvilinear coordinates, preferably with as source. The once that I have found don't seem to be very formal.
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1answer
20 views

What is a Galois closure and Galois group?

I was reading wikipedia for Galois groups and this term suddenly appears and there is no definition for it. What is a Galois closure of a field $F$? Does this mean a maximal Galois extension of $F$ ...
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2answers
36 views

Is this definition correct for the inverse of a function?

Is this definition correct for the inverse of a function? Let $f:X\to Y$ be a function. The inverse of $f$ is the function $g:Y\to X$ such that $g\circ f=i_X$ and $f\circ g=i_Y$. We denote the ...
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1answer
43 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
15 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
19 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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+50

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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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
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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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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
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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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2answers
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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
828 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 ...
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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
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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3answers
59 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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4answers
723 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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0answers
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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
34 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
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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
40 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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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 ...
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1answer
42 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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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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0answers
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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0answers
24 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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2answers
40 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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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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37 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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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
30 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
21 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?