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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221 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 ...
5
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64 views

Can we consider a hypergeometric function as a closed-form?

Let's say a calculus problem like an integral or a series has a solution that inevitably involving a hypergeometric function. It turns out that hypergeometric function cannot be expressed in term of ...
5
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74 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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168 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 ...
4
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84 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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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 ...
3
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0answers
42 views

Conceptual question on showing properties of the absolute value function on $\mathbb{Q}$

I have a rather conceptual question about showing certain small lemmas regarding the absolute value function on $\mathbb{Q}$. I want to only give one example: Let $a,b \in \mathbb{Q}$ and $|.|$ ...
3
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53 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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51 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
76 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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112 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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86 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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72 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 ...
3
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0answers
74 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
votes
0answers
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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0answers
114 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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70 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
votes
0answers
128 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
votes
0answers
88 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
200 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
votes
0answers
14 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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55 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 ...
2
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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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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 ...
2
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19 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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0answers
111 views

Functions $h$ such that $h(x*x') = f(x) * g(x').$

Definition 0. Call a magma $X$ surjective iff the distinguished binary operation of $X$ induces a surjective function $X \times X \rightarrow X$. Now for the main idea: Definition 1. Let: ...
2
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0answers
16 views

Integrable Systems-Description

I am relatively new to the topic of integrable systems and I came across the following description: "Integrable systems share a fundamental property which is related to the geometry of the initial ...
2
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0answers
36 views

Definition of “one element normalizes another element”

I understand that someone has asked the definition of "an element of a group normalizes a subgroup". Let $G$ be a group and $a,b\in G$. Then what does "$a$ normalizes $b$" mean? Does it make sense at ...
2
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0answers
26 views

How to define a Holder seminorm of a section

I'm reading "Variational Problems in Geometry",Seiki Nishikawa, in the figure below. Let $(M,g)$ be a compact $m$ dimensional Riemannian manifold with no boundary. $T>0, 0<\alpha<1, ...
2
votes
0answers
19 views

What is $R$-algebra?

(To be clear, I mean a ring by a ring with unity.) Here is a definition in Wikipedia: Let $R$ be a commutative ring. Let $(M,+,\cdot)$ be an $R$-module. Let $\ast$ be a binary operation ...
2
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52 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 ...
2
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26 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
votes
0answers
60 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
votes
0answers
55 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
31 views

Meaning of proper antichain

I saw this sequence on the Online Encyclopedia of Integer Sequences, which describe the number of 3-element proper antichains of an n-element set. What does it mean to be a 3-element proper antichain ...
2
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0answers
35 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
61 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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0answers
58 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
74 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
votes
0answers
210 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
votes
0answers
41 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
votes
0answers
40 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
votes
0answers
71 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
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0answers
78 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
364 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
0answers
75 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
65 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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0answers
150 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!