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

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2
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2answers
57 views

Conceptual definition: Injection, surjection and Bijection.

I was wondering if this conceptualisation is correct: Injection means that we don't have two arrows come from an element of the domain towards the range. Hence we don't have one archer standing in ...
1
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2answers
85 views

Amann & Escher Integral vs. Lebesgue Integral

In the textbook the authors define the integral via cauchy sequences of simple functions: $$S_n\to F:\quad\int F\mathrm{d}\mu:=\lim_n\int ...
2
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1answer
44 views

Is the symmetric definition of the derivative equivalent?

Is the symmetric definition of the derivative (below) equivalent to the usual one? \begin{equation} \lim_{h\to0}\frac{f(x+h)-f(x-h)}{2h} \end{equation} I've seen it used before in my computational ...
4
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1answer
165 views

Equality of positive rational numbers, Part-2

I am reading this answer. I have some doubts which I want to clarify. Question 1. The author defines a rational number $\dfrac ab$ as, $$b\times\left(\dfrac{a}{b}\right) = a$$ He presumes that ...
0
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1answer
34 views

How the branch cut make a multi-valued function several branches of single valued function?

In the wikipedia article, it describe the branch points and branch cuts: A branch cut is a curve in the complex plane such that it is possible to define a single analytic branch of a multi-valued ...
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1answer
22 views

proving that a subset of a set has a functional mapping that is a subset of another

i wanted to prove: Let $f:X\to Y$ be a mapping from $X$ into $Y$. Show that if $A$ and $B$ are subsets of $X$, then: $$(A \subset B) \implies \left(f(A) \subset f(B)\right)$$ but i thought ...
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1answer
28 views

what is a fiber for function mapping?

I have read that a fiber is the pre image of a mapping.. Does this mean that I can think of a fiber as a line that connects x to y where the line is a function? So for example with $f(x)=2x$ there ...
3
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1answer
57 views

Is this “the winding number”?

Note that $p:\mathbb{C}\rightarrow \mathbb{C}\setminus\{0\}:z\mapsto e^z$ is a covering map. Let $\alpha:[0,1]\rightarrow \mathbb{C}\setminus\{0\}$ be a closed curve. Let $\gamma$ be any ...
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2answers
29 views

True false about direct sum and their bases of vector spaces

I am not entirely sure about the following true/false questions For all the following : $V$ a vector space and $W_1$ and $W_2$ two subspaces such that $V = W_1 ⊕ W_2$ 1) for all subspaces U of V : ...
8
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1answer
238 views

Meaning of the word “conjugate” across mathematics?

Clearly, the word conjugate or conjugation is used for a myriad of different concepts across mathematics and even in science (see the Wikipedia page). Its meaning can range from the fraction used to ...
2
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1answer
114 views

Do these arithmetic rules work? They extend the number system by a zero not based on the empty set that is a divisor with unique quotients.

These rules are part of an attempt to define an additive identity in terms of division in basic standard arithmetic. The difficulties with defining division by $0$ are well known. In order to ...
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0answers
43 views

Definition of $\sum_{a \in A} I_a$

If $(I_a)_{a \in A}$ a family of ideal of $K[x_1,x_2, \dots, x_n]$, I have the following definition in my notes: $$\sum_{a \in A} I_a=\{ a_{i1}+a_{i2}+ \dots+ a_{ij} | a_{ij} \in I_{a_j} \}$$ Is ...
4
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1answer
52 views

Definiton of No Tear and No Paste

Topologists often mention an example beginning by "If there is no tear and no paste, then ...". As a student, I am confused with this "term", and I want to know the exact mean of it. First of all, ...
0
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1answer
29 views

Two different definitions of “scale invariance”

I found the following definition of "scale invariance" in the book Critical Phenomena in Natural Sciences by Didier Sornette. A function $f$ is scale invariant if there is a number $\mu(\lambda)$ ...
3
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2answers
47 views

what is the basic difference between a mapping and a function? [duplicate]

what is the basic difference between a mapping and a function? many say they are same but the opposite views are also seen. is mapping a restricted version of a function?
0
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1answer
43 views

How is the area of a set of points in $\Bbb R^2$ defined?

Let $S$ be a subset of $\Bbb R^2$. If no vertical slice of $S$ contains gaps, we could define the area of $S$ through the following. $$A(S) = \int_{-\infty}^\infty\left(\sup\{y\in\Bbb R\mid (x,y)\in ...
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0answers
18 views

Does this matrix have a name: $x=(x_0, \ldots, x_n)$, $S(x)_{i,j} = x_{i+j \mod(n)}$. Can you help me generalise the idea?

Apologies if this question is vague/unclear (especially the title). Given a vector $x=(x_0, \ldots, x_n)$ define the matrix $S(x)_{i,j} = x_{i+j \mod(n)}$. $$ S(x) = \left( \begin{array}{rrrr} ...
2
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1answer
21 views

What is the definition of “sheet”?

Here is a definition for evenly covered set. Definition Let $C,X$ be topological spaces and $p:C\rightarrow X$ is a continuous function. Let $U$ be an open subset of $X$. Then $U$ ...
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0answers
25 views

Composite residuosity statement.

Consider the following definition. A number $z$ is said to be $n$-th residue modulo $n^2$ , if there exists a number $y \in \mathbb{Z}_{n^2}^*$ such that $$z\equiv y^n \mod n^2$$ Let us take $n=6$ ...
5
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1answer
68 views

Why $R^{op}$ used in the definition of “category of $R$-modules”?

Earlier today, I was thinking: "Oh, an $R$-module is just an additive functor $R \rightarrow \mathbf{Ab}.$" Anyway, I had a bit of a read over at nLab, and it says: For any small ...
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0answers
27 views

different representations of strong induction

I've seen 2 forms of strong induction; just wondering how one follows from the other. $1) f(n_0)\wedge f(n_1)\wedge\cdots \wedge f(n_{k-1})\wedge f(n_k)\wedge \forall_n[f(n-k)\wedge ...
2
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1answer
92 views

Functorial cofibrant replacement does not have to be fibration?

I'm new to model category theory, and I find myself confused about the different meanings of cofibrant replacement in literature. The usual definition is that we assign to every object $X$ in our ...
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1answer
38 views

Caratheodory: Alternative Definition

Idea My idea is to facilitate Caratheodory's construction by composing it with Hahn-Kolmogorov. Problem Given a premeasure on a ring $\mu:\mathcal{R}\to\overline{\mathbb{R}}_+$. Do the ...
2
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2answers
72 views

What does it mean for a function to be holomorphic?

I am trying to wrap my head around the definition of holomorphic. Wikipedia tells me that: A holomorphic function is a complex-valued function of one or more complex variables that is complex ...
3
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1answer
37 views

Why Not Define Connectedness to Mean Path Connected?

All spaces I have seen which are connected are also path connected (apart from examples to show that the two are not equivalent). Is there a reason for using the weaker definition of connectedness ...
2
votes
1answer
43 views

What is a linear equation?

How do we define the linear equation? I mean, it looks like a polynomials with degree one but I'm not sure if $ax+by+c=0$ is a linear equation if $a=b=0$?
2
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3answers
61 views

Is modular arithmetic defined for all rational numbers (with denominators coprime to modulus)?

In the expression $\frac{1}{b}\pmod m$, where $(b,m)=1$, is $\frac{1}{b}$: a) a rational number (and so rational numbers are defined in modulo arithmetic using multiplicative inverses)? b) just ...
2
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0answers
42 views

What is the formal definition of a structure?

I can't seem to find one anywhere, and I've looked in several books and pages. I don't even know how to tag this.
0
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1answer
20 views

Cauchy - Sequences. Different definitons

Is $$\forall \epsilon > 0 \exists N \in \mathbb{N}: \forall j \ge N |a_j - a_N| < \epsilon $$ equivalent to $$\forall \epsilon > 0 \exists N \in \mathbb{N}: \forall m,n \ge N |a_m - a_n| ...
0
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0answers
16 views

anisotropic vector valued function definition

Does anyone have insight into the significance of the following statement which defines an anistropic function $$a:\Omega \times \mathbb{R} \times \mathbb{R}^{N} \rightarrow \mathbb{R}^{N}$$ The ...
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0answers
12 views

Help with understanding why a particular set was chosen in theorem and corollary involving the limit of a function

Here's a theorem from Ross' Elementary Analysis, he gives the limit definition of a function in terms of $\delta$-$\epsilon$ And the corollary that follows: Why is it that in the corollary, the ...
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2answers
37 views

What is effectively continuous?

In Soare's book Recursively Enumerable Sets and Degrees I saw a sentence: $\Phi_e$ is an effectively continuous functional from the Cantor space $2^\omega$ to itself. What does it mean for a ...
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0answers
18 views

Clarification of Definition: Free Algebra

I need some clarification on the definition of free algebra. Here is an extract from Lie Algebras and Lie Groups by Jean-Pierre Serre: I am somewhat confused about the definition of free algebra. ...
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0answers
20 views

trail vs path in Graph Theory v/s Graphical Models

In my course on probabilistic graphical models, I learnt (quoting from page 36 of the book Probabilistic Graphical Models: Principles and Techniques by the same author) Path: We say that X1 , . . . ...
2
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5answers
374 views

Seeking elegant proof why 0 divided by 0 does not equal 1

Several years ago I was bored and so for amusement I wrote out a proof that $\dfrac00$ does not equal $1$. I began by assuming that $\dfrac00$ does equal $1$ and then was eventually able to deduce ...
2
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0answers
32 views

Synonyms for “Theorem”

Some mathematical results, despite being formally proven, are not actually called "theorem". Examples include: Bertrand's postulate Pigeonhole principle Law of large numbers Do these names imply ...
2
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1answer
35 views

Why is an open interval needed in this definition? (definition of a limit of a function)

Here's a part of the definition Ross' Elementary Analysis states for limits of a function: 20.3 Definition (a) For $a\in\mathbb R$ and a function $f$ we write $\lim_{x\to a} f(x)=L$ provided ...
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1answer
24 views

On the definition of transition maps

When defining a manifold the domain and codomain of the transition maps is usually denote like this: $$\varphi_\eta \circ \varphi_\lambda^{-1}: \varphi_\lambda(U_\lambda \cap U_\eta) \to ...
3
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0answers
29 views

Anti-Diagonal Matrix

Hi just a quick question. Can a anti-diagonal have zero in its diagonal? I have this definition in my paper. "An anti-diagonal matrix is a square matrix where all the values are zeroes except those ...
0
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1answer
68 views

What is Bourbaki's definition of subfield? or categorical definition of subfield?

Let $F$ be a field. Let $K$ be a subset of $F$ which is closed under two binary operations $+,\cdot$ Assume $(K,+,\cdot)$ is a field. Is $K$ called a subfield of $F$ in Bourbaki's definition? Or, ...
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1answer
31 views

What is a sparse subset?

In a work about fully homomorphic encryption I found usage of the expression: "sparse subset", as in: Our hint will consist of a set of vectors that has a (secret) sparse subset of vectors whose ...
1
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1answer
45 views

Examples of Free Lie Algebra

In wikipedia, free Lie algebras are defined using the universal property. Can anyone give some concrete examples of free Lie algebras? "In mathematics, a free Lie algebra, over a given field $K$, is ...
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8answers
982 views

Why is the absolute sign needed in the definition of a bounded function

A function $f$ is bounded if there exists a real number $M$ such that $|f(x)| \le M$ for all $x \in \operatorname{dom}(f)$. Why is the absolute sign needed?
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0answers
42 views

Analytic Functions: Notation? [duplicate]

Analytic functions are usually denoted by $\mathcal{C}^\omega$. What does the $\omega$ stand for? (The infinity symbols of a colleague of mine really look like omegas...)
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0answers
65 views

Definition of orientation preserving linear map: is this welldefined?

I am reading a definition in this document here about what it means for a linear map to be orientation preserving: It seems to me though that this is not well-defined: take the unit $x$-vector ...
2
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1answer
58 views

On the definitions of $n$-manifold etc.

I'm doing 3rd year undergraduate geometry (an introductory subject), and we've been given formal definitions of terms like "$n$-manifold" and "smooth $n$-manifold." However, I tend to think about ...
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3answers
56 views

When finding the derivative using its definition, why do we plug $0$ for $h$?

If $\lim h\to 0$, when finding the derivative of the function, why do you plug in the limit that is being approached. Like why would you plug in $0$ in the function $4x+2h$ (which is the derivative of ...
0
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2answers
46 views

What is the polarization identity?

Hi I am studying stochastic calculus and my professor often mentions "Polarization Identity" but I do not know how it is defined. I tried googling it but could not find the right definition and ...
2
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0answers
20 views

Two definitions of “affine stratification”

I see two different definitions of "affine stratification" in the literature: A stratification where each stratum is isomorphic to $\mathbb{A}^n$ for some $n$. A stratification where each stratum is ...
2
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2answers
49 views

Is the angle between a vector and a line defined?

Is the angle between a vector and a line defined? The angle between two lines $a,b$ is defined as the smallest of the angles created. The angle between two vectors $\vec{a},\vec{b}$ is the ...