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

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8
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0answers
173 views

Is an étale morphism of algebraic stacks locally quasi-finite?

An étale morphism of schemes is unramified, and an unramified morphism is locally quasi-finite. Does the same hold for étale morphisms of algebraic stacks? Let us recall the definitions, following ...
7
votes
0answers
309 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 ...
6
votes
0answers
38 views

Why do isotropic spaces deserve their name?

Wiki defines a quadratic form to be isotropic if it evaluates to zero at some vector. What does this have to do with isotropy in physics i.e uniformity in all directions? From my experience so far, ...
6
votes
0answers
129 views

Why so many 'multi-part' definitions, as opposed to 'unified' ones?

Many definitions consist of multiple parts: an equivalence relation is symmetric AND reflexive AND transitive; a topology is closed over finite intersections AND over arbitrary unions; etc. However, ...
5
votes
0answers
66 views

Looking for functions “similar” to the $ \Gamma $ function

According to wikipedia: https://en.wikipedia.org/wiki/Functional_equation The $\Gamma$ - function is the only solution that suffices the following three equations: $$ f(x)={f(x+1) \over x}\tag{$*$} ...
5
votes
0answers
182 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
votes
0answers
62 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, ...
5
votes
0answers
254 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
votes
0answers
78 views

Why is the unit circle definition of trig functions not rigorous enough?

It has recently come to my attention that the usual unit circle derivation of the elementary trigonometric functions isn't considered rigorous enough. Apparently, this has to do with problems ...
4
votes
0answers
84 views

Using the IVP definition of $\cos$ and $\sin$, how can we show that $\cos^2(x)+\sin^2(x) = 1$ without any “magic”?

One way to define the exponential, hyperbolic and circular functions is to assert that they're the unique solutions to certain IVP systems: The exponential function: $$\exp'(x) = \exp(x), \qquad \...
4
votes
0answers
87 views

Why is cos at $\pi/2$ not undefined?

If the $\cos$ function is based off of the ratio of the adjacent side of Euclidean, right triangle, with fixed hypotenuse length (such as the unit circle), then how does this correspond to a defined ...
4
votes
0answers
30 views

Prove Limit Addition Thm as x approaches infinity

I am supposed to prove prove $\lim_{x\to \infty} [f(x)+g(x)]= L+M$. Starting to realize I don't really understand the formal definition of a limit, although I do understand the general concept. ...
4
votes
0answers
41 views

What is a cusp neighborhood corresponding to a parabolic Möbius transformation in a Riemann surface?

I am referring to this wikipedia entry. So what I understand is that they are defining it using the Fuchsian model. If $\Gamma$ is a Fuchsian group, its parabolic elements correspond to the cusps of $...
4
votes
0answers
107 views

What does it mean to categorify something?

What does it mean to categorify something? Is there any simple illustrative example of this process? How it is done and what is it useful for? Here is where I heard it first: "Khovanov homology is ...
4
votes
0answers
118 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 ...
4
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0answers
100 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 ...
4
votes
0answers
2k 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 ...
3
votes
0answers
38 views

A Cartesian coordinate system is a mathematical or physical thing?

I'm convinced that if I ask what of the coordinates systems in the figure is a Cartesian system almost all say that it is the system $O_1$. This answer comes immediately from our habit and ...
3
votes
0answers
47 views

С* algebras and projective limits

Let $A_i$ be a family of $C^*$-algebras, and let $\varphi_{ij} = A_i \leftarrow A_{j}$ be $*$-morphisms which form some projective system. How can we define a $C^*$-(pre)norm on a projective algebraic ...
3
votes
0answers
92 views

Definition of $C^{1,2}$?

I just realized that I don't really know what the definition of $C^{1,2}$ (or $C^{m,n}$) means. Two candidates come to mind: 1) For every $y$, the function $x\mapsto f(x,y)$ is $C^1$, and for every $...
3
votes
0answers
36 views

How to write this function in a “well-formed” way

Given an input $0 \lt x \lt 1$, find $x$'s Nearest Integer Continued Fraction with structure $$x = a_0 \pm \cfrac{1}{a_1 \pm \cfrac{1}{a_2 \pm \cdots}}.$$ Then $$f(c) = a_0 + 1 \mp \cfrac{1}{a_1 + ...
3
votes
0answers
40 views

Euclidean Neighbourhoods Retracts and Deformation Retracts

Aim of this question is to clarify the differences between the two concepts written in the title, because it's unclear to me wether one condition implies the other, under which circumstances they are ...
3
votes
0answers
104 views

What's the derivative of a map defined on manifolds?

I'm going through Warner's book on differentiable manifolds. On page 8 he defines what it means for a map $f: U \subset M \to \mathbb R$ to be differentiable: $f$ is differentiable iff $f \circ \psi$...
3
votes
0answers
26 views

Characterize in terms of fibre

I am not familiar with the notion "characterize" in the following context. Does this mean to redefine or?.... Any help would be appreciated. Thank you. For a function $f:X\to Y$, and y an element of ...
3
votes
0answers
181 views

Is There a Common Definition of “Finite Support”?

I thought I understood this term, but when I tried to verify this I found three different and conflicting definitions, none corresponding to mine. Is there a generally agreed definition for this term (...
3
votes
0answers
69 views

Properties of $L^{\infty}$

I'm trying to get a better grasp on the idea of $L^{\infty}$. What are the implications if we are given that $f \in L^{\infty}$? Also, how do we write $\|f\|_{\infty}$ in terms of the inf of a set of ...
3
votes
0answers
59 views

What vector field property means “is the curl of another vector field?”

I know that a vector field $\mathbf{F}$ is called irrotational if $\nabla \times \mathbf{F} = \mathbf{0}$ and conservative if there exists a function $g$ such that $\nabla g = \mathbf{F}$. Under ...
3
votes
0answers
68 views

Gradient vector interpretation

I have trouble understanding the gradient vector $\nabla$. For a surface, when we find it's $\nabla$, why is the resultant vector the normal vector when $\nabla$ means gradient vector?? Thanks
3
votes
0answers
78 views

Largest Singular Value / Singular Value

I was wondering, what if the eigenvalues of a matrix A are all negative. So does that simply mean there is no singular value for this particular matrix?, hence I can't calculate the conditional number ...
3
votes
0answers
42 views

Meaning of the term $X/H$ and orbits

I am trying to find representations of the group $G=GL_2(F(t)/t^2) = (M_2(F_p) , + ) \rtimes GL_n(F)$ So I was trying to do exactly what Serre has explained in this section. I am not quite able to ...
3
votes
0answers
30 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 $...
3
votes
0answers
2k views

The definition of continuously differentiable functions

When we say $f \in C^1$, we mean that f is continuously differentiable. Isn't the continuity a redundant word? I mean, we have a theorem that says if $f$ is differentiable then it is continuous. So ...
3
votes
0answers
184 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 ...
3
votes
0answers
65 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 $F(Y)$ is a ...
3
votes
0answers
23 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} ...
3
votes
0answers
205 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
votes
0answers
331 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 ...
3
votes
0answers
127 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
votes
0answers
90 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
votes
0answers
90 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
votes
0answers
70 views

“Identify” terminology

What does it mean to "identify" two objects in algebra, as distinct from having an isomorphism or automorphism between them? I was led to think about this while reading Stewart's Galois Theory, where ...
3
votes
0answers
92 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 ...
3
votes
0answers
126 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
votes
0answers
949 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, $f(z)=\sqrt{1-z}$.
3
votes
0answers
85 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
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
votes
0answers
202 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
votes
0answers
240 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 ...
3
votes
0answers
219 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 $\mathscr S$ ...
3
votes
0answers
88 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 ...