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

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3
votes
1answer
56 views

What's so special about the grounded poset of cardinality $2$?

By a grounded poset, I mean a poset $P$ with a bottom element $0_P$. Arrows between grounded posets are monotone mappings that preserve the basepoint. This gives a self-enriched category; lets call it ...
4
votes
3answers
166 views

A lot of confusion in the “Polynomial Remainder Theorem”?

Lately I've been reading about Polynomial Remainder Theorem from various sources, mainly from the wikipedea article, this post and some high school books. Wikipedea says that if we divide a polynomial ...
3
votes
1answer
145 views

Positive-definite function and Positive-definite matrix

I am trying to understand Positive-definite function and read the wikipedia link: https://en.wikipedia.org/wiki/Positive-definite_function It has a relation to Positive-definite matrix and I did not ...
0
votes
0answers
17 views

Bruhat-Tits building of a non-split group

Where can one read about the definition of parahoric subgroups for a reductive group over a non-archimedean local field $k$ which is not split over $k$?
0
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0answers
9 views

definition of an affine F-variety where F is not algebraically closed

In the first chapter of his book "Linear Algebraic Groups", Springer considers a situation where $k$ is an algebraically closed field and $F$ is a subfield, and seeks to define the notion of an ...
1
vote
0answers
29 views

Help in understanding Bochner's theorem and Pontryagin duality theorem

I am trying to understand Bochner's theorem through wikipedia link https://en.wikipedia.org/wiki/Bochner's_theorem This refers to dual spaces of locally compact abelian group and leads to ...
1
vote
4answers
343 views

Meaning of finite, countably infinite, infinite?

Even after several attempts I could not find the motivation behind the finite, countable and infinite. Is there a simple way to look them differently? I have read the wikipedia definition several ...
5
votes
5answers
276 views

Theoretical function question

Suppose we have the function $f(x)= x^2 $. This function associates real numbers with real numbers ( $f:\mathbb{R}\rightarrow \mathbb{R}$). Now, what i get confused sometimes is what exactly the ...
0
votes
0answers
33 views

Definition of trigonal curves

I'm reading Fulton's book and I'm trying to understand the concept of trigonal in Hartshorne's book (page 345): On the other hand, Fulton's book define the $g_d^n$ in the following manner: So Can ...
0
votes
1answer
16 views

Functional limits and definition of continuity - difference and implications?

Continuity: A function $f : A → \mathbb{R}$ is continuous at a point $c ∈ A$ if, for all $\epsilon > 0$, there exists a $δ > 0$ such that whenever $|x − c| < δ$ (and $x ∈ A$) it follows that ...
0
votes
1answer
26 views

The definition of a series

Given definition: Given a sequence, $(a_n)$, then the series with terms $(a_1,a_2,...,)$ is a sequence $(s_n)$ of partial sums. Does this mean essentially the definition above can be re-stated to say ...
3
votes
2answers
42 views

What is “approximate compactness”? What is an example of an approximately compact set?

I read this: A property of a set $M$ in a metric space $X$ requiring that for any $x\in X$, every minimizing sequence $y_n\in M$ (i.e. a sequence with the property $\rho(x,y_n)\to\rho(x,M)$) has a ...
2
votes
1answer
24 views

Modules over a monoid: trouble with the definition.

I'm having trouble with a definition. We're working in the category of monoids. Take $A\in \mathfrak{Mon}$ and define a module over $A$ to be a set $M$ with an action: $A\times M \rightarrow M$ such ...
2
votes
2answers
41 views

How can I use a precise definition to find values of delta that correspond with given epsilon values

I have been given this problem: For the limit $$\lim_{x\to 2}({x^3-3x+4})=6$$ illustrate "Definition 2" (I have included this below) by finding values of $\delta$ that correspond to $\varepsilon=0.2$ ...
2
votes
1answer
35 views

Definition of finite-by-nilpotent [closed]

In group theory what are definitions of: finite-by-nilpotent, nilpotent-by-finite, abelian-by-finite, or in general: *-by- *?
0
votes
0answers
37 views

Is this an improper method of averaging grades? If so, what is a simple mathematical way of explaining it?

I have a professor who employs a unique method of averaging grades. On each assessment, the professor assigns a raw numerical score to each student based on performance. He then converts particular ...
1
vote
1answer
37 views

Definition of a sheaf: What is $s\rvert_{V_i}$ if $V_i\not\subseteq U$?

I am reading Hartshorne's book on algebraic geometry, which defines a sheaf to be a presheaf $\mathscr F$ on a topological space $X$ such that: For all open sets $U$ and open coverings $\{V_i\}$ of ...
1
vote
3answers
85 views

Numbers and reality [closed]

I have a question I don't really know how to formulate, so apologies for the cloudy mess. The topic is on the meaning of numbers, that is when I say 3, I am referring to, say, $3$ avocados, or $3$ ...
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
36 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
29 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
191 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
23 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
32 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
68 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
139 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
123 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
30 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
20 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
71 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
21 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
39 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
163 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
116 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
39 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
22 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
63 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
23 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
31 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
90 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 ...