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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204 views

Understanding the roots of homomorphism and homeomorphism

I understand the formal (i.e. mathematical) definitions of the terms "homomorphism" and "homeomorphism" as they relate to functions, but I am curious as to the origin of these terms. I don't know ...
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71 views

Are the words “function”, “map”, and “mapping” synonymous?

Is it correct to say that "A function or a map or a mapping is a binary relation such that ..."
5
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156 views

I think a definition is wrong in “Model Categories” by Hovey.

I am working through the book "Model Categories", by Mark Hovey, and have a doubt about a definition given there. At the beginning of page 50 we read: Define a map $f:X\rightarrow Y$ to be a ...
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83 views

Defining “Penon Infinitesimals”.

In this lecture (which is accompanied by these slides), right near the end (so page 9 in the pdf of the slides; I don't think you have to watch the lecture), P. Johnstone refers to the "Penon ...
3
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49 views

What is the difference between $\propto$ and $\sim$

Suppose I have two physical quantities, lets name them $a$ and $b$. I wonder what the difference exactly is between $$a\propto b,\tag{1}$$ and $$a \sim b.\tag{2}$$ I know for eq. 1, that it means ...
3
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42 views

What is the desirable function identification when setting up arrows in the category of types?

My question is which functions can not be allowed in a statically typed programming language, so that the "canonical" category is less coarse than what you get if you define it's arrows to be ...
3
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0answers
69 views

Why not defining a measure as a function on functions?

A measure $\mu$ is a function to $\left[0,\infty\right]$ on the sets belonging to a $\sigma$-algebra. Then for integrable functions $f$ the integral $\int fd\mu$ comes in, having nice properties ...
3
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90 views

Zariski cotangent space, as defined in Arapura's “Algebraic Geometry over the Complex Numbers”

In "Algebraic Geometry over the Complex Numbers", Arapura gives the following definition: Definition 2.5.8. When $(R, m, k)$ is a local ring satisfying the tangent space conditions, we define its ...
3
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82 views

Definition of differential forms

I am reading the book Algebraic Geometric Codes: Basic Notions by Tsfasman, Vladut and Nogin. The authors introduce some definitions about differential forms at the beginning of Section 2.2. These ...
3
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70 views

Definition of small $o$

In one of my homework assignments (in intro to applied mathematics) there is the following definition: Given two functions $f(\epsilon),g(\epsilon)$ that are defined in $D=(0,\epsilon)$ we say ...
3
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50 views

Problem of understanding

Please , can someone help me to understand this part of text please What it means "... for some $k\in \mathbb{N} \cup \lbrace 0 \rbrace$ and uniformly fore a.e $t\in [0,2\pi]$" and please how to ...
3
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106 views

Infinite set ordered like a “infinite tree”

Obviously I don't need to say that I'm still a newbie, so I'll try to explain my question with some pics too. I have a infinite set $T$ I have a visual idea of how this set is ordered, but I don't ...
3
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67 views

Define composition of small cyles and making a big graph

I am having following sub graphs and wish to compose all and make a one bigger graph (say G). After that, I want to select the closed path where it is passing along the outer vertices of that ...
3
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0answers
123 views

Steps toward making the Feynman integral rigorous

I would like to know if there has been any progress recently toward giving the Feynman integral a rigorous definition. I heard that Richard Borcherds is working on giving quantum field theory a ...
3
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0answers
50 views

Defining a Certain Class of Plane Graphs

I'm having problems finding the right words to formulate the following class of graphs in a definition. I'm defining a class of plane graphs with the following properties: Removing any vertex of ...
3
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0answers
85 views

Closed / embedded surface

Given a closed surface in $\mathbb R^3$, is it necessarily an "embedded surface"? I think it is true, but that is just because I can't think of a closed surface for which we cannot construct a smooth ...
3
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0answers
192 views

Can GCD be called an operator?

Can $\gcd(a,b)$ be called a binary operator which takes operands $a$ and $b$ and returns their greatest common divisor. And if for some operator —say $\bigotimes$ —$(a_1\bigotimes a_2 ...
2
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0answers
19 views

Different definitions of an affine algebraic set

Hartshorne assumes we are working over an algebraically closed field $k$ and we simply define algebraic sets to be any set of the form $V(I)\subset k^n$, where $I\subset k[X_1,\ldots,X_n]$ is an ...
2
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59 views

Loss of derivatives

In many books on pdes the expression "loss of derivatives" is used when some estimates on solution are proved. Can someone clarify to me (maybe with an example) the meaning of this expression? For ...
2
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0answers
48 views

Path-independent contour integrals and how to define them

If a contour $C$ is parameterized by $z(t): [a, b] \to \mathbb{C}$, then we define $$ \int_C f(z) \, dz= \int_a^b f(z(t)) \, z'(t) \, dt.$$ If the contour integral on the left side is equal to some ...
2
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0answers
28 views

Connected sum of surfaces with boundary

The connected sum of closed surfaces (2-manifolds) is defined by removing a disk from each and gluing the exposed edges together. When defining the connected sum of surfaces with boundary, is the ...
2
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0answers
59 views

Question about definition of some classes of bimodules.

Suppose that we have a ring $R$ and a $R$-$R$ bimodule $M$ such that: For every $r\in R$ and $m\in M$ there exists $r'\in R$ such that $m\cdot r=r'\bullet m.$ Examples of this bimodules can be seen ...
2
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49 views

Confused about the definitions of atom

I went though half a dozen books on measure theory, and it occurs to me that the definition of atom is not particularly unified. Version $1$: A set $E$ in a $\sigma$-algebra $\Sigma$ is called ...
2
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0answers
64 views

How is this the definition of equi-integrable?

Let $Q=(0,T)\times\Omega.$ I am completely lost with this: No definition of equi-integrability I have seen looks anything like this. Can someone help me please? Presumably it is a fact that a ...
2
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0answers
110 views

Proper functions

I'm a bit confused with the definition of proper function, i.e.: "$f: X \to Y$ is proper if the inverse image of every compact set in $Y$ is compact in $X$." Can anyone provide a more intuitive ...
2
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37 views

Is the empty matrix $M$ ($m \times n$) with $m$ or $n$ equal to zero in RREF ? Does it posses every matrix property one would wish?

Is the empty matrix $M$ ($m \times n$) with $m$ or $n$ equal to zero in RREF ? Reading a proof in Linear Algebra induction starts off by proving $P(n)$ holds for $n=0$, where $P(n)$ is the ...
2
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65 views

Should $0$ be considered a prime?

Typically, a prime is defined as follows: $p$ is prime iff $(p \mid xy \implies p \mid x$ or $p \mid y)$ and $p$ is not a unit or zero. But for ideals, we say the zero ideal is prime. There is a ...
2
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0answers
37 views

What are defintions of ellipticity?

Ellipticity seems to have many definitions. So far, I'm aware of three. Are there other definitions of ellipticity that you know of, and if so, where did you encounter them? The three definitions ...
2
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0answers
64 views

Clarification of Hölder norm in terms of oscillation

Let $\Omega\subset\mathbb{R}^2$ be an open bounded set, $B(x_0, \rho)=\{x\in\mathbb{R}^2\ |\ |x-x_0|\leq \rho\}$, $\Omega(x_0, \rho)\equiv B(x_0, \rho)\cap \Omega$, $u\in L^{\infty}\big(\Omega(x, ...
2
votes
0answers
74 views

What is a Line Integral?

My question is very simple yet crucial to the understanding of many fields of mathematics. What is a line integral? If I choose some arbitrary line segment $\mathbb{A}$ to integrate a function ...
2
votes
0answers
314 views

What is the principal branch of $f(z)=\sqrt{1-z}$, $z\in\mathbb{C}$?

My question is, essencially, about the definition of the principal branch of a function that is not a Logarithm. In this case, $f(z)=\sqrt{1-z}$.
2
votes
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72 views

Meaning of quasiperiodicity in classical KAM

I'm learning about the classical KAM theorem, and I can't quite infer precisely what the term "quasi-periodic solution" means in the theorem's statement. I'm reading the following introductory note: ...
2
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0answers
1k views

Continuity on open and closed intervals

I will be taking Calculus I soon, and I just want to make sure I understand some concepts correctly. So far, reading my book for Calculus I, I've encountered the definition of continuity as being ...
2
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0answers
64 views

definition of the Fourier transform of function on the sphere

Let $f: S^{n-1}\longrightarrow R^n$ be even continuous function. What is the Fourier transform of $f$?
2
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134 views

Taylor series of a vector field?

Can someone give me the definition of the Taylor Series of a vector field in $\mathbb{C}^2$? Thanks!
2
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144 views

Are edge cuts, vertex cuts, and cut sets all variously called “cuts”?

I've seen "cut" being used to refer to all three, in different places, and sometimes in the same book. Which does "cut" most commonly refer to? p.s. I am aware that "cut" itself can be defined to ...
2
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0answers
227 views

Lemma of Whitehead

this is the lemma of Whitehead And i really don't understand the proof How to see that $k$ is well defined (i.e how to write an element from $X\cup_{\varphi_i}e^{\lambda} , i=0,1$ ) and how to ...
2
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0answers
168 views

Exercise on measure theory

I have this exercise: Let $\mathscr {S}$ be a collection of subsets of the set $X$. Show that for each $A$ in $\sigma(\mathscr S)$ there is a countable subfamily $\mathscr C_0$ of $\mathscr S$ ...
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56 views

Can a sample space depend on the parameter to be estimated (example: # of cabs in a city)

In our intro statistic lecture the following we said that the following components made up an estimation problem an at most countable space $\mathcal{X}$ of all possible samples we can observe a ...
2
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0answers
83 views

Graph Theory: Help with a definition

I need some help to see if the definition I found of Cycles and Cycles Decomposition is right, here it is: A graph is a Cycle if it is isomorph to another graph $G=(V,E)$ with the following ...
2
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0answers
206 views

Explanation of Mixed Strategy Definition in Game Theory

Definition: Let $(N, A, u)$ be normal-form game, and for any set $X$ let $\Pi(X)$ be the set of all probability distributions over $X$. Then the set of fixed strategies for player $S_i=\Pi(A_i)$. ...
2
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0answers
43 views

Name for the initial object of a connected component of a category

I was just wondering whether there is already a name for an object of a category as described below. Recall that an initial object in a category $\mathcal C$ is an object $I\in\mathcal C$ such that ...
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86 views

Where can I find a description of math language symbols?

I am reading math articles. I meet math symbols. For example $\exists$ or $\forall$. For example for "For any a exist e that" can be rewriten as: $\forall a \exists e$ Where can I find full ...
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0answers
210 views

How to prove that $e = \lim_{n \to \infty} (\sqrt[n]{n})^{\pi(n)} = \lim_{n \to \infty} \sqrt[n]{n\#} $?

While reading this post, I stumbled across these definitions (Wiki_german) $$e = \lim_{n \to \infty} \sqrt[n]{n\#}$$ and $$e = \lim_{n \to \infty} (\sqrt[n]{n})^{\pi(n)}.$$ The last one seems ...
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55 views

terminology clarification needed: projection of a measure to an open set

I'm reading in Doobs "Classical Potential Theory and its Probabilistic Counterpart" and I'm having trouble with terminology. Specifically, in Part I chapter 6, he talks about the projection of a ...
2
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100 views

question about definition Silverman's AEC

I am sure many of you have Silverman's book the arithmetic of elliptic curves (2nd edition). On page 417, proposition 1.2, he defines $\delta$. He also defines $\xi: G \rightarrow M:\ ...
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0answers
250 views

why are curl and divergence defined the way they are?

Curl of a given vector field, $F$, at a given point is a vector, $C$, that gives the measure of amount of rotation the vector undergoes at that point. It has been defined using cross products and ...
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27 views

Commutative diagrams with antiparallel arrows

A diagram in category theory is said to commute when for all objects $A$ and $B$ in it, every the composite morphism resulting from a possible path from $A$ to $B$ are the identical. Does that mean ...
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16 views

definition of half-integral weight modular forms

i start reading about modular forms of half-integral weight $k/2$ for $\Gamma' \subset \Gamma_0(4)$. As far as i understand these are holomorphic functions $f\colon \mathbb{H} \rightarrow \mathbb{C}$ ...
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32 views

Definition - Logarithmic Zeros and Algebraic Vector Fields

In Algebraic Geometry, what is the difference between an algebraic vector field and a derivation (are they completely different or synonymous)? What does it mean to say an algebraic vector field has ...