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

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6
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234 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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155 views

From the viewpoint of modern geometry, is there a “best” definition of the term “triangle”?

I can think of at least six different possible definitions of the term "triangle" in Euclidean geometry. a subset of $\mathbb{R}^n$ that can be expressed as the convex hull of three or fewer points. ...
5
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70 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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187 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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86 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 ...
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88 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 ...
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21 views

Definition of Dihedral group via semidirect product

Let $G$ be an abelian group and let $\varphi:Z_2\rightarrow Aut(G)$ be the homomorphism where $\varphi(\overline{1})$ is the inversion automorphism. Define $Dih(G)=G\rtimes_{\varphi} Z_2$. Now set ...
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174 views

What is the “correct” way of making $\mathcal{P}(X)$ into a topological space?

If $X$ is a topological space, I want to know the "correct" way of making the powerset $\mathcal{P}(X)$ into a topological space. So let $\mathsf{SupLat}$ denote the infinitary Lawvere theory whose ...
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58 views

Functors that preserve the subset relation on constructs

Is there a name for functors between concrete categories over Set that preserve '$\subset$'? Example, for a functor $F:\mathbf {Top}\to \mathbf {Grp}$, if $Y$ is a subspace of $X$ then $F(Y)$ ...
3
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44 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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44 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 $|.|$ ...
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64 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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61 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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83 views

Strong Integral

Is there a strong(!) notion of integral that can face all of those issues: Singularities Decay Modes Oscillations Measure Spaces Locally Convex Spaces For example combining decay modes with ...
3
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126 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 ...
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99 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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442 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, $f(z)=\sqrt{1-z}$.
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76 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 ...
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51 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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171 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!
3
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127 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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185 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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71 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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137 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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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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93 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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209 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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27 views

Does the relation $\mid^*$ have any interesting applications for understanding the structure of commutative rings that aren't integral domains?

There is a binary relation $\mid^*$ defined on any commutative ring as follows: $a \mid^* b$ iff $ak=b$ for some $k \in R$ that is not a zero divisor. This is always transitive, and it is reflexive ...
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25 views

Unique Union Problem

Given a the set $a = \{1,...,n\}$ and let $b$ denote a set of subsets of $a$. Find a subset of $c$ of $b$ so that the union of all subsets in $c$ is equal to $a$ and the intersection of any of the ...
2
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33 views

What is the “minimal” structure in which points and lines are defined?

Points, straight lines and planes are fundamental concept of geometry. Usually this entities are defined in a structure. We can easily define points in a vector space, or in a affine or projective ...
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53 views

Relations in a finitely complete category with images

I can't use tikz inside MathStackExchange; so my question is on an image below.
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17 views

What is the standard definition of Torsion element?

Here are two different definitions of torsion element. Let $M$ be an $R$-module and $m\in M$. Wikipedia: $m$ is a torsion element iff there exists a nonzero regular element $r$ (i.e. Not a zero ...
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41 views

Uniform Boundedness: Am I right or my TA?

I am a student, and I disagree with the solutions our TA has prepared. I am seeking verification that I am correct or explanation as to why I am wrong. It seems to be a disagreement or ...
2
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0answers
28 views

Correct way of defining a mathematical object (linguistic)

I am writing my thesis and my advisor made a correction to the sentence below: Transitional Rule commonly denoted by $\phi$ is defined by the map $\Sigma^n \rightarrow \Sigma$. He has changed ...
2
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0answers
51 views

Two definitions for non-singular in codimension 1

I am trying to understand how the following definitions are the same. Shafarevich definition (pg 128) - A variety is non-singular in codimension one if the singular locus has codimension $> 1$. ...
2
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48 views

What abstract structures allows us to describe “nets that converge toward each other”?

Topological spaces are equipped with a bare minimum of structure to allow for a formalization of the statement "the net $a$ converges to the point $x$." Actually this isn't strictly true, but its true ...
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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} ...
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44 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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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 ...
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22 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
118 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: ...
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0answers
21 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 ...
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0answers
45 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 ...
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29 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, ...
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0answers
31 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 ...
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59 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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28 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 ...
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61 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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69 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 ...
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53 views

Types of definitions

My question is about terminology. Consider the following two types of definitions of a circle: A circle is the collection of all points at the same distance from a given point. A circle is the ...