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

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1answer
82 views

confusion about rank and nullity of matrix and rank-nullity theorem

I can see that rank + nullity = number of columns of the matrix. Does that mean the dimension of matrix is the number of columns? Isn't dimension of a $m\times n$ matrix $mn$? Thanks.
20
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5answers
863 views

$\epsilon, \delta$…So what?

Over the course of my studies I often encounter phrases in reference material of the type "and this avoids the need for using $\epsilon$, $\delta$ definitions" or "by this we can omit those ...
2
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1answer
50 views

What is the defenition of $\mathcal{c}$ and $\aleph_1$ if we assume ZFC without CH.

I am reading an intro to a chapter of Andreas Blass called "Combinatorial Cardinal Characteristics of the Continuum" and I am getting a bit confused. When I studied "Discrete Mathematics", it was ...
3
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0answers
152 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 ...
2
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1answer
179 views

What's with conditionals in mathematical logic?

Having a bit of difficulty understanding the conditional ($\rightarrow$) in mathematical logic. I read up on the already-existing questions and it did help me understand it better (the 'promise' ...
3
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2answers
173 views

Radius of Convergence and its application to a Power Series including $x^{2n}$ rather than $x^n$

(Radius of Convergence) Consider the Power Series $f(x)=\sum_{n=0}^{+ \infty}a_n x^n$, the radius of convergence $\rho$ can be found using $$\rho = \displaystyle \lim_{n \to + \infty} \left| ...
8
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1answer
253 views

What is the difference between ❋3.01 and ❋4.5 in Whitehead and Russell's PM?

This baby step from ❋3.01 to ❋4.5 is so tiny that I can barely see the difference. Please kindly explain why it is so important to distinguish the two. What is the philosophical importance of this ...
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1answer
110 views

A question concerning Ulam's Theorem from Oxtoby's “Measure and Category”

I am reading the following theorem from Oxtoby's Measure and Category Theorem 5.6 (Ulam). A finite measure $\mu$ defined for all subsets of a set $X$ of power $\aleph_1$ vanishes identically if ...
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2answers
43 views

In a commutative algebraic theory, do all constant symbols necessarily represent the same value?

Let $T$ denote a commutative algebraic theory with two nullary function symbols $a$ and $b$ (i.e. constants). Is it an automatic consequence of the definitions that $a=b$ is a theorem of $T$? My ...
2
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1answer
57 views

Axiom or Postulate?

In wikipedia we see that the words “axiom” and “postulate” are synonyms: “An axiom, or postulate, is a premise or starting point of reasoning”. But in A Friendly Introduction to Numerical ...
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2answers
118 views

definition of primes for higher hyperoperations

I was reading yesterday when I came across the history of counting. There was some evidence of an early understanding of prime numbers. I thought that I would try changing the definition of primality ...
2
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2answers
182 views

Proving that $f(x) = \cos(x)\implies f'(x) = -\sin(x)$ using the definition of a derivative

I'm having trouble grasping the concept which proves that the derivative of $f(x) = \cos(x)$ is $f'(x) = -\sin(x)$. It needs to be proven using the definition of a derivative--and I can't quite piece ...
2
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5answers
629 views

What is maths? “Maths is the study of ______”? [closed]

I can fill in the blank by just listing the different fields of maths but my goal is to define all of mathematics. An answer that I would've accepted a few years ago is "Maths is the study of ...
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1answer
55 views

Why exclude the constant sequence from this definition

As in the title: I am interested in why it is necessary to exclude the constant sequence from the following definition of a functional limit: Note the $0 < |x-c|$. Why would the definition break ...
4
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2answers
171 views

Why do differential geometry textbooks bother with equivalence classes of smooth structures?

In contemporary textbooks on differential geometry, the definition of smooth manifolds is given in a (IMHO) awkwardly obfuscated way, by saying that a smooth manifold is a topological space endowed ...
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1answer
107 views

Definition of $p$-torsion module

Let $p$ be a prime. What is the definition of a $p$-torsion module ? I have googled but was not convinced.
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2answers
148 views

Infinite sum with 0 terms: comparison to infinite product

Depend on what text you read, an infinite product with an infinite number of terms that are 0 is either divergent, or diverge to 0. Even though, obviously, the partial product is still a convergence ...
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0answers
39 views

What is $(j,\epsilon)$-normality?

In looking at the concept of normality for real numbers I have come across the notion of $(j,\epsilon)$-normality, but cannot find a definition for this. could anyone explain what this term means?
2
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1answer
130 views

Subspace with different vector space operations

Let $A,B$ be vector spaces such that $A\subseteq B$. Is it true that $A$ is a subspace of $B$? I claim that the answer is no, because it is possible that $A$ and $B$ might be equipped with different ...
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0answers
90 views

Maximum measurable collection and Caratheodory extension

Remember Caratheodory extension theorem: we get an outer measure by extending a premeasure over an algebra. We also have Caratheodory theorem, which give us a measure by restricting that outer measure ...
1
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1answer
45 views

Quick question about chain homotopies.

In the definition of a chain homotopy (say $h$) between two chain maps (say $f$ and $g$), are the maps $h_i$ comprising the chain homotopy required to commute with all other maps involved (the $f_i$s, ...
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3answers
389 views

How can people understand complex numbers and similar mathematical concepts?

In mathematics, how does something like complex numbers apply to the real world? Why do complex numbers exist? How can we comprehend addition of complex numbers? For example, addition of natural ...
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1answer
61 views

What is “superficial density” in the context of integration (over surfaces)?

I have read something like this: We have $$\int_{M}f(x)d\sigma(x) = 1$$ where $d\sigma$ is the superficial density. What does this mean? $M$ is a hypersurface/manifold. The author does not ...
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1answer
83 views

Definition of functions on metric spaces.

In the post Definition of functions, it is stated in the accepted answer that one way to define a function is to define it as the triple $(f, X, Y)$ where $f \subset X \times Y$. My question is what ...
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2answers
89 views

Laws of indices

Well, given any real number $x$ and any positive integer $n$, the number $x^{n}$ is defined to be the product $x.x.x. ... x.x $ ($n$ times). But, how do we define $x^{r}$ when $r$ is a negative ...
2
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1answer
28 views

What is a cofactor in this case?

Im looking at a homework problem I have and I am a bit confused. The first part of the question is to show that 8128 is a perfect number. This is simple enough: $1 + 2 + 4 + 8 + 16 + 32 + 64 + 127 + ...
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1answer
480 views

Upside-down triangle symbol on function

I came across the symbol that looks like an upside-down triangle, and coming in front of a function $f(x,y)$. What does that mean?
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1answer
161 views

Beginnings of Topology: Homeomorphisms

Why is a knot and a circle homeomorphic? The general definition of a homeomorphism requires that you be able to deform each to one another.
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5answers
198 views

Definition of determinant [closed]

Determinant is a certain function from the set of all $n\times n$ matrices to the set of scalars. How is the determinant defined? What characterizes the determinant function?
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2answers
127 views

Why do we use “if” in the definitions instead of “if and only if”? [duplicate]

I often write my notes as logical statements and constantly wonder why people use only the "if" direction in the definitions instead of the "if and only if". Consider: "A homomorphism $\phi$ is said ...
1
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1answer
69 views

Why is a graph an ordered pair?

From the source of all knowledge a graph is an ordered pair G = (V, E) comprising a set V of vertices or nodes together with a set E of edges or lines, which are 2-element subsets of V Why ...
4
votes
1answer
109 views

How does this definition capture the intuitive notion of an algebra?

On page 15 of this document, the author writes: Definition 1.1.1. Let $\mathcal{E}$ be any category. Given an endofunctor $\Gamma : \mathcal{E} \rightarrow \mathcal{E}$, a $\Gamma$-algebra ...
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1answer
49 views

Help in this definition of morphism

I need help in this definition of morphism of affine algebraic sets which I found in a book: Let $X$ and $Y$ affine algebraic sets and say $$f:X\to X'\ \text{and}\ g:Y\to Y'$$ isomorphisms with ...
2
votes
2answers
27 views

$M=(a_{ij})$, with $1\leq x\leq3$ and $1\leq x \leq 3$, such that $a_{ij}=4i+2j-6$.

My exercise asks me to write the following matrix $M=(a_{ij})$, with $1\leq x\leq3$ and $1\leq x \leq 3$, such that $a_{ij}=4i+2j-6$. What I did not understand, and tried unsuccessfully was what ...
1
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1answer
37 views

On (absolute) convergence of $f_c:= c + \sum_{n=0}^{+ \infty} \frac{a_n}{n+1}x^{n+1}$

Let $R> 0$ and let $g: (-R,R) \longrightarrow \mathbb{R}$ be given by the convergent power series $$g(x):= \sum_{n=0}^{+ \infty} a_nx^n$$ for $|x| < R$. Let $c \in \mathbb{R}$ and let $f_c: ...
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2answers
1k views

For all but finitely many $n \in \mathbb N$

In my book I have the following theorem: A sequence $\langle a_n \rangle$ converges to a real number $A$ if and only if every neighborhood of $A$ contains $a_n$ for all but finitely many $n \in ...
0
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1answer
37 views

Question on definition of little o

I would like to generalize the definition of little o. The definition from Wikipedia is as such: Let $f$ and $g$ be two real valued functions. We write $f(x) = o(g(x))$ as $x \to \infty$ if for all ...
0
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1answer
30 views

Definition convex sets

$C$ is convex $\Longrightarrow \forall x,y \in C$ and $\forall$ $t\in [0,1], \space(1 − t ) x + t y \in C$ My question how comes this formula $(1 − t ) x + t y$ describes all the elements in $[x,y]$ ...
0
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2answers
74 views

Minnor differences in notation used in definition of graphs

One of book states A graph G consists of two finite sets: a nonempty set V(G) of vertices and a set E(G) of edges, where each edge is associated with a set consisting of either one or two ...
2
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1answer
127 views

Definition of pull-back analogous to push-forward

Let $U\subseteq\mathbb{R}^n, V\subseteq\mathbb{R}^m$ be open subsets, and $f:U\rightarrow V$ a $C^1$ map. Let $u,v$ be vector fields on $U,V$ respectively. Then the push-forward $f_*u$ is equal to $v$ ...
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1answer
83 views

Difference between the concepts of graph and trace

I'm a little confused with the definition of graph and trace. If I have a function (or a curve) $f:\mathbb R\to \mathbb R,\ f(t)=t^2$ and I draw the graph we have a parabola since the graph is the ...
3
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0answers
103 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 ...
2
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1answer
80 views

Ring of rational functions for reducible variety

Let $X$ be some affine algebraic variety over $\mathbb{k}$ (i.e. some closed subset in $\mathbb{A}_\mathbb{k}^n$). First suppose $X$ to be irreducible. Then the algebra $\mathbb{k}[X]$ is a domain and ...
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0answers
36 views

In a translation of a classic math book: is the term *normal* translated correctly? Is there a better term?

I am reading an English translation of the classic book by Gelfond & Linnik: Elementary Methods in the Analytic Theory of Numbers. Here is the definition of "normal" from the book (page 5 in my ...
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1answer
497 views

Definition of “Universe of Discourse” and Definition of “Set” [duplicate]

I want to axiomatize "the concept of set" in my head, but every time I face some circular definition or intuition. In predicate logic, we quantify over some "Universe of Discourse". Intuitively ...
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4answers
106 views

The differentiation of $ \sin, \cos$ through a Taylor Series

This question has been asked quite a lot on math SE, however, please before you mark this as a duplicate carry on reading, I will try to highlight my doubts and concerns as clear as possible. First ...
0
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2answers
153 views

Well-posed problem

In the definition of a well-posed problem it states that a problem is well posed if: 1.A solution exists. 2.The solution is unique. 3.The solution's behaviour changes continuously with the initial ...
3
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1answer
851 views

Understanding Elliptic Operators

I don't have a strong background in partial differential equations so some of these questions might be quite basic. I first want to give some definitions which I am using. Definition: We define a ...
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2answers
102 views

Notation and terminology for functions, interpreting $f(y)$

It seems to me there are two different interpretations of a symbol $f(y)$. I will explain what I mean: Suppose I have a function $f(x) = x$. (I took the identity map to have a simple example). Also ...
2
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4answers
885 views

What is the Riemann Sphere?

Reading from wikipedia I understood that Riemann Sphere is used to represent extended complex plane. But it would be great if someone could explain it in a less technical manner.