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

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19
votes
13answers
6k views

What is a negative number?

I'm trying to get to an abstract definition of a negative number that would fit in with the basic concept of addition/subtraction. There are questions here about multiplying and dividing negative ...
0
votes
0answers
22 views

More on the $n$-dimensional cross product: Orientation

Wikipedia states: This formula is identical in structure to the determinant formula for the normal cross product in $\mathbb R^3$ except that the row of basis vectors is the last row in the ...
1
vote
2answers
27 views

What is the right hand side in this definition of $n$-dimensional cross product

Let ${\bf e_1}, \dots, {\bf e_n}$ be the standard basis for $\mathbb{R}^n$ and let ${\bf w_1} = (w_{11},\dots,w_{1n}), \dots, {\bf w_{n-1}}=(w_{n-1\;1},\dots,w_{n-1\;n}) \in \mathbb{R}^n$. Then one ...
0
votes
0answers
35 views

Simplification of a definition in Hartshorne's algebraic geometry book

I'm reading Hartshorne's book and on page 53 he defines intersection multiplicity of a projective variety and a hypersurface: I'm wondering if we can simplify this definition if we take $Y$ to be a ...
2
votes
1answer
26 views

Meaning and Underlying idea of a definition or a theorem

What does it mean by 'explain the meaning and underlying idea of a definition or a theorem'? For example, if we are asked to explain the Fundamental Theorem of Algebra, how should we explain its ...
1
vote
1answer
30 views

What is a linear embedding from a simplex $\Delta^n \to \mathbb{R}^n$?

As stated in the title, reading Milnor-Stasheff Characteristic classes, I encountered at page 95 the following sentence: let $\Delta^n$ be an $n$-simplex, linearly embedded in the $n$-dim vector ...
2
votes
2answers
187 views

Is the zero ring a domain?

Is the zero ring usually considered a domain or not? Wikipedia says: The zero ring is not an integral domain; this agrees with the fact that its zero ideal is not prime. Whether the zero ring ...
0
votes
0answers
24 views

Name for measure of non-injectivity of a covering map

Suppose that $p:C\to X$ is a covering map. For $x\in X$, is there a name for the number $Card(p^{-1}(x))$? So that for $p(z)=z^5:\mathbb{C}\setminus\{0\}\to\mathbb{C}\setminus\{0\}$, one might say ...
0
votes
1answer
19 views

What is a differentiable distribution on a manifold

Assume $M$ is a smooth manifold and $d:u\to Q_u<T_uM$ is a distribution on $M$. What does "$d$ is a differentiable distribution" mean? What does it mean for $Q_u$ to depend smoothly on $u$?
0
votes
0answers
9 views

meaning of index of a quadratic form for a field of positive characteristic

In "Classification of Algebraic Semisimple Groups" in "Algebraic Groups and Discontinous Subgroups: Procedings of Symposia in Pure Mathematics, Volume IX", Jacques Tits speaks of the index of a ...
3
votes
2answers
67 views

Why are these representations of e the same? [duplicate]

I heard that $e$ can be defined as the limit as n approaches infinity of $(1 + 1/n)^n$, but I also heard that $e$ is also defined as the sum of the reciprocals of the factorials from $0$ to $\infty$. ...
2
votes
1answer
33 views

Rouches theorem, $|f(z)|\gt |g(z)|$ at each point on $C$. $\color{red}{\text{On or In?}}$

Rouche's theorem from both of my resources say the following: Let $C$ denote a simple closed contour and suppose that: Two functions $f(z)$ and $g(z)$ are analytic inside and on $C$ $|f(z)|\gt ...
2
votes
1answer
30 views

Definition of “vector fields never have opposite direction”

Good day! As in my other question I am referring to the book "Differential Equations and Dynamical Systems" by Lawrence Perko, chapter 3.12. I have a question regarding Lemma 2: Lemma 2. If $v$ ...
2
votes
1answer
64 views

Limit definition of integration

The derivative function has the following definition using the limit: $$f'(x) = \lim_{h\to0} \frac{f(x+h)-f(x)}{h}$$ I was wondering whether I could find a similiar definition for the integral. I ...
3
votes
6answers
133 views

Is there proof show that $\log x$ is undefined and make no sense at $ x=0$?

I was asked by someone: why $\log x$ is undefined at $x=0 $? Is there proof show that $\log x$ is undefined at $x=0$? Note(01):: log is the inverse function of the exponential function. note(02): ...
2
votes
2answers
122 views

Isn't there a better way to put the canonical smooth structure on the $n$-sphere?

My differential geometry notes put a smooth structure on the $n$-sphere (denoted $S_n$) as follows. Firstly, $S_n$ is taken as a subset of $\mathbb{R}^{n+1}.$ Secondly, we define $U_0 = S_n \setminus ...
1
vote
1answer
29 views

Implications of Alternate Definition of the Limit of a Function

In Hubbard's Vector Calculus, I saw an unconventional definition for the limit of a function: Definition: Let $X$ be a subset of $\mathbb{R}^n$. A function ...
0
votes
1answer
16 views

Asymmetry of definition of regular value and critical value

Let $f: U \subseteq \mathbb R^n \to \mathbb R^m$ be a smooth map. We say $y \in \mathbb R^m$ is a regular value of $f$ if and only if all points in the set $f^{-1}(y)$ are regular. (see e.g. the ...
3
votes
2answers
69 views

What is precise definition of “monomial curve” in affine $e$-space?

What is precise definition of monomial curve in affine $e$-space ?
0
votes
2answers
19 views

Question about notation, subsets of a graph and intersection of vertices

I have the following description of a graph: Let $G$ be a graph such that all of its vertices are subsets with two elements of $\{1,2,...,n\} (n\ge 2)$ where two sets $A,B$ are adjacent iff ...
1
vote
1answer
39 views

Set of a matrix

I am working on a homework problem which asks me about the Set of a singular $n\times n$ matrix. specifically whether it is a vector space. I looked in the glossary of the book and searched online and ...
2
votes
2answers
38 views

The Standarization of Matrix by Vector Multiplication

I apologize for the trivialness of my question but it has been bugging me as to why the standard for multiplying a matrix by a vector that will give a column matrix mean that the vector has to be a ...
3
votes
2answers
157 views

Is there a formal definition for antiderivatives?

In the way the derivative can be defined as a limit, specifically $$f'(x):=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}$$ or any of the other possible variants, is there a way to define the antiderivative, as in ...
6
votes
2answers
111 views

How do we know that our definitions/axioms are not contradictory?

Let us assume that we have declared some axioms. Now, we wish to declare a new axiom too. How do we establish that the new axiom is not a consequence of, nor a contradiction to the original set of ...
1
vote
3answers
48 views

About $0!=1$ and $a^0=1$ as cases of empty product.

Some useful ''conventions'' as $0!=1$ or $a^0=1$ are particular cases of an empty product, i.e. a product between elements of the empty set. I know that such product is defined as a convention by: $$ ...
1
vote
2answers
38 views

When can I not use the chain rule?

If $z=f(x,y)$ where $x=g(r,\theta), y=h(r, \theta)$, then can you give me a good reason why $$\frac{\partial^2 z}{\partial r^2} \neq \frac{\partial}{\partial x}\frac{\partial z}{\partial ...
1
vote
1answer
21 views

Question about uncorrelatedness of random variables and distributions

I was wondering, if two random variables are dependent, does that mean that they must be correlated? does one imply on the other or vice versa? Also, if I know that a joint distribution of two ...
0
votes
0answers
64 views

Can someone explain presheaf to a calculus student?

From reading some stuff online, there is the claim that continuous functions are presheafs, so that the toy functions I play with in class such as $f(x) = x^2$ can also be thought of as presheafs. ...
4
votes
1answer
58 views

Can someone explain to a calculus student what “dual space is the space of linear functions” mean?

I ran across this phrase today in a post and I am slightly confused. From my understanding, the dual space is the space of functions that sends a vector to a real number. There are two confusions: ...
2
votes
1answer
22 views

What does “points spanned by powers” mean in the Goffinet dragon definition?

The definition of the Goffinet dragon fractal given by Wolfram Mathworld refers to plotting all points spanned by powers of the complex number p=0.65-0.3i What does it mean for points to be ...
1
vote
2answers
25 views

What does this definition of $C^k$ surface mean?

Reference: Richard Millman - Elements of differential geometry I'm reading this book and there is a really wierd definition in this book: To descrive the theorem, here are some definitions in this ...
3
votes
1answer
88 views

What does it mean when two groups commute?

This is probably an easy question, but I can't find the definition in my book. Let $G$ be a group, and let $H$ and $N$ be subgroups. What does it mean for $H$ and $N$ to commute? I have two ...
-1
votes
1answer
29 views

Is this task defined mathematically correctly?

I wish to make a math model for predicting users clicking on context advertising. Math definition: Let $X$ be the set of users, and $Y$ be the set of adverts. We make a mapping $F: X×Y→P$ that should ...
1
vote
2answers
37 views

Topology definition, finite intersection, infinite union?

In my textbook I find: Definition of a topology $\tau$ as a collection of subsets of $X$ is a topology on $X$ if: $\varnothing, X \in \tau$ If $\mathscr{F}\subseteq \tau$ then ...
0
votes
5answers
79 views

Formal definition of the function $Sin$ [closed]

Which is the formal definition of the function $Sin$, starting from axioms of real numbers ? I never found it in any book. Not its Taylor serie, that is based upon the intuitive definition of $Sin$; ...
14
votes
6answers
1k views

How to think of a set?

I am doing self study for the last two months on functional analysis. As I get a bit used to the terms like space, topology, manifold, etc, etc, I realized that everything is defined in terms of set. ...
3
votes
0answers
39 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 ...
4
votes
1answer
35 views

Definition of a “modular Galois representation”

I am trying to pin down a definition for a $n$-dimensional modular Galois representation $$\rho : \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \rightarrow \text{GL}_n(A).$$ I am just looking for ...
0
votes
1answer
73 views

Product topology - definition

Can someone please give me a detailed explanation of the concept of product topologies? I just can't get it. I have looked in a number of decent textbooks(Munkres, Armstrong, Bredon, Wiki :P, Class ...
0
votes
1answer
32 views

Graph: vertex connectivity and edges connectivity

I know that a graph is $k-$ connected if any two vertex can be connected by $k$ independent path. This is what we call the vertex connectivity. But what is the edges connectivity ?
1
vote
3answers
75 views

Is the zero vector in the definition of linear dependence arbritary?

The definition of linear dependence according to wikipedia is The vectors in a subset $S=(v1,v2,...,vk)$ of a vector space $V$ are said to be linearly dependent, if there exist a finite number of ...
3
votes
1answer
49 views

What is a strictly positive probability distribution?

I'm reading about Markov Random Fields. In the wiki page it's written that When the joint probability distribution of the random variables is strictly positive,... I'm so confused! Because a ...
3
votes
1answer
45 views

Definition of functions in $L^p$ space

I know that if we suppose that $1 \leq p \leq \infty$, and if $f$ is in $L^p$, then this means that $\|f\|_p=[\int_X (f^p) dx]^{\frac{1}{p}}$. But I feel as though I'm missing some important ...
7
votes
5answers
861 views

What do people mean by “finite”?

Many arguments about the foundations or philosophy of mathematics centre on the question of whether or not there exist objects or entities (such as certain sets) which are not "finite". (For ...
0
votes
1answer
25 views

What does it mean for an orbit to “accumulate”?

For example, from Beardon's "A Primer On Riemann Surfaces", ...Show that g is a homeomorphism of D onto itself, and that no orbit accumulates in D. I'm simply looking for a definition of the ...
0
votes
0answers
16 views

Definition of parameterised $n$-manifold

Let $U \subseteq \mathbb R^{n+q}$ and let $W = U \cap \mathbb R^n \times \{0\}$. Let $\phi : U \to \phi (U)$ be a diffeomorphism. Then $M=\phi (W)$ is called a parameterised $n$-manifold in $\mathbb ...
1
vote
1answer
23 views

Is this a typo (parameterised $n$-manifold)

In this book here on page 62 parameterised $n$-manifolds are introduced. The example given is that of a regular curve $\gamma (t) = (X(t),Y(t))$ and a parametrisation $\phi (x,y) := (x, y + Y(x))$. ...
4
votes
1answer
101 views

What is the definition for a fine sheaf/ a partition of the unity on a sheaf?

From Griffiths and Harris, p. 42: A sheaf $\mathcal F$ on $M$ is called fine, if for $U=\bigcup_i U_i\subseteq M$ there is a partition of the unity subordinate to the cover $(U_i\mid i\in I)$. By this ...
1
vote
2answers
55 views

What interpretation of the Lie braket is this?

I was reading exercise 1.96 of Gadea's Analysis and Algebra on Differentiable Manifolds. I don't know what definition of Lie braket the author used, but I'm confused, as far as I know ...
3
votes
1answer
25 views

Strings in a dictionary. A partial order, strict order, and total order?

Question: The domain is the set of all words in the English language (as defined by, say, Webster's dictionary). Word $x$ is related to word $y$ if $x$ appears as a substring of y. For example, "ion" ...