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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1answer
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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 ...
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7answers
333 views

How is greater than defined for real numbers? [closed]

Is there a formal definition of greater than? I need it to describe how much better I am than my friends at math. EDIT: I would like to clarify this is partially a joke, and partially a serious ...
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1answer
48 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, ...
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5answers
2k views

difference between theorem, lemma and corollary

Can anybody explain what is the basic difference between theorem, lemma and corollary. We have been using it for a long time but I never paid any attention. I am just curious to know.Thanks a lot.
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2answers
30 views

what is the basic difference between a mapping and a function?

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?
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1answer
15 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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1answer
33 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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8answers
966 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
100 views

Borel Measures: Atoms (Summary)

Disclaimer: The question here has been solved, now: Finest Measurable Partition (For jeapardy it is stated below, anyway. Have fun! ;) ) Summary: This is a summary of the discussions: ...
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0answers
46 views
+250

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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0answers
13 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} ...
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0answers
24 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 $n=6$ ...
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1answer
60 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
22 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 ...
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2answers
55 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 ...
2
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1answer
29 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 ...
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3answers
43 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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1answer
42 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 $a=b=0$?
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0answers
34 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.
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1answer
18 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| ...
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5answers
330 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 ...
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2answers
28 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
10 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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5answers
421 views

What is the definition of 'within one' in mathematics

I need help with the definition of "within 1": If $x = 8$ and $y = 7$, then $x$ is "within 1" of $y$. If $x = 8$ and $y = 9$, then $x$ is "within 1" of $y$. If $x = 8$ and $y = 8$, is $x$ still ...
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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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0answers
17 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
16 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 , . . . ...
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0answers
31 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 ...
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0answers
17 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: In both parts of the definition, why are open intervals needed? Would it fail if it were a closed interval ...
2
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1answer
169 views

What's with conditionals in mathematical logic?

Having a bit of difficulty understanding the conditional ($\rightarrow$) in mathematical logic. I read up on the already-existing questions and it did help me understand it better (the 'promise' ...
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4answers
3k views

What is a fraction in which the greatest common factor of the numerator and the denominator is 1?

What is this fraction: A fraction in which the greatest common factor of the numerator and the denominator is 1?
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0answers
33 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 ...
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1answer
19 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 ...
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0answers
21 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 ...
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1answer
66 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
28 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 ...
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1answer
37 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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1answer
85 views

When is it insufficient to treat the Dirac delta as an evaluation map?

The Dirac delta "function" is often introduced as a limit of normal distributions $$\delta_a(x)=\frac{1}{a\sqrt{\pi}}e^{-\frac{x^2}{a^2}}\text{ as }a\to0^+.$$ Obviously, this sequence of functions ...
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0answers
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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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2answers
224 views

Motivation for a new definition of the derivative using the concept of average velocity

As everyone knows, for a function $ f: \mathbb{R} \to \mathbb{R} $ and a point $ a \in \mathbb{R} $, we say that the derivative of $ f $ at $ a $ equals $ L $ if and only if $$ \lim_{x \to a} ...
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1answer
53 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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5answers
366 views

What is maths? “Maths is the study of ______”? [closed]

I can fill in the blank by just listing the different fields of maths but my goal is to define all of mathematics. An answer that I would've accepted a few years ago is "Maths is the study of ...
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2answers
36 views

Concept: The graph component

I have the following definition for a Component of a graph: A subgraph $H$ of a graph $G$ is a component of $G$ if $H$ is a maximal connected subgraph of $G$, ...
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3answers
53 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 ...
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3answers
810 views

If $f$ is differentiable at $x = x_0$ then $f$ is continuous at $x = x_0$.

Claim: if $f$ is differentiable at $x = x_0$ then $f$ is continuous at $x = x_0$. Please, see if I made some mistake in the proof below. I mention some theorems in the proof: The condition to ...
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2answers
41 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 ...
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0answers
18 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 ...
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2answers
42 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 ...
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0answers
39 views

What is the definition of boundary-parallel Dehn twist?

I have not been able to find a working definition for the term: "boundary-parallel Dehn twist ". I know what a boundary-parallel surface is, and what a parallel surface is, but I have not been able to ...
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0answers
23 views

Differences between the definition for direct and inverse images.

I am reading a book and it introduces direct and inverse images with the following definitions: Given a function $f:X \to Y$ and subsets $A \subseteq X$ and $B \subseteq Y$, the direct image of $A$ ...