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

learn more… | top users | synonyms

1
vote
1answer
86 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.
1
vote
2answers
139 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 ...
1
vote
2answers
112 views

Meaning of $\{ a,b \}$, and comparison with $(a,b)$

What does $\{a,b\}$ mean in real analysis? I'm also little bit confused about set definition Can you tell me the main difference between $(a,b)$ and $\{a,b\}$? Thank you.
1
vote
1answer
117 views

Can we say that each monotonic sequence is a bitonic?

Is it mathematically correct to state that every monotonic is actually a bitonic with 0 number of elements in other half? I.e., if it is increasing then we can say it has 0 elements in decreasing ...
2
votes
1answer
123 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 ...
1
vote
0answers
37 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?
0
votes
0answers
87 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 ...
6
votes
3answers
360 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 ...
1
vote
1answer
151 views

$\Sigma_k^\text{P}$−SAT definition is not clear to me

I don't understand if by saying there are $k$ alternating quantifiers on the variables $x_1$,$x_2$...$x_k$, It means we quantify ALL variables (there are only $k$ variables in the SAT formula) or just ...
1
vote
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, ...
8
votes
4answers
885 views

Congruent Modulo $n$: definition

In an Introduction to Abstract Algebra by Thomas Whitelaw, he gives examples of the congruence mod operation, such as $13 \equiv5 \pmod4$, and $9 \equiv -1 \pmod 5$. But when I first learned about ...
4
votes
1answer
131 views

Multiple-valued analytic functions

Although our definition requires all analytic functions to be single-valued, it is possible to consider such multiple-valued functions as $\sqrt{z}$, $\log z$, or $\arccos z$, provided that they ...
0
votes
1answer
52 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 ...
0
votes
2answers
67 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
votes
1answer
26 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 + ...
6
votes
6answers
2k views

Definition of an Ordered Pair

"The ordered pair $(a,b)$ is defined to be the set $\{\{a\},\{a,b\}\}$." ~ Hungerford's Algebra (p.6) I think this is the first time that i've seen this definition. I've read the wiki page. Is it ...
2
votes
2answers
133 views

In a groupoid, do any two objects have a morphism between them?

This is a question about the definition of a groupoid (in category theory). I've read the Wikipedia article, and it says that a groupoid is a small category in which every morphism is an isomorphism. ...
1
vote
1answer
420 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?
0
votes
1answer
138 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.
8
votes
6answers
6k views

What is the Direction of a Zero (Null) Vector?

To be more precise, I am interested in knowing if the intuition that a Euclidean zero vector does not have a particular direction is actually correct, and if there is a rigorous formulation that would ...
0
votes
2answers
123 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
vote
1answer
58 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
102 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 ...
1
vote
1answer
47 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
vote
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: ...
1
vote
1answer
802 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
votes
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
votes
1answer
29 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 $[x,y]$ ...
1
vote
2answers
101 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 ...
0
votes
2answers
67 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
votes
1answer
102 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 $v$ ...
2
votes
1answer
70 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 ...
2
votes
4answers
692 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.
0
votes
1answer
77 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
votes
0answers
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 ...
1
vote
0answers
35 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 ...
5
votes
1answer
191 views

Can someone explain definition of number 1?

I've found the next piece of text: As an example of the second failing, Poincaré recalled the definition of the number 1 offered by another of the logicists, Burali-Forti: $$1 = ...
1
vote
1answer
407 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 ...
3
votes
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 ...
3
votes
1answer
700 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 ...
0
votes
2answers
43 views

Question about $(A - \lambda I_A)\vec{x} = 0$.

Finding a solution to $C\vec{x} = (A - \lambda I_A)\vec{x} = 0$ is the equivalent of considering the determinant of $C$ when it is zero. This means the matrix is linearly dependent and has infinite ...
1
vote
1answer
121 views

A function “extends” to the cone on X

I have the following statement: A map $f : X \rightarrow Y$ is nullhomotopic if and only if it extends to the cone on $X.$ My problem is that I have no idea what "extends" means in this statement (I ...
3
votes
2answers
128 views

Should axioms be viewed as part of the signature?

I included category theory in the tags in order to get feedback from the categorial logic community. It goes without saying that this isn't really category theory. A semigroup can be defined as a ...
0
votes
1answer
47 views

Definition for the action of a category on a set.

I'm trying to understand the definition of the action of a category on a set which is given in nLab, more particularly the first one. If one has a functor $\rho: C \to Set$, one takes the set S as the ...
3
votes
1answer
74 views

If a “group” has two identities then is not a group

The story goes like this: A friend and I found this old exercise: Let $G=\Bbb R-\{-1\}$ and $a*b:=a+b+ab$, is $(G,*)$ a group? I say that $(G,*)$ is not a group because for any $a\in G$ follows ...
0
votes
2answers
105 views

What is the meaning of | operator?

For example, 12|0 = 0 Interesting I can't find this on google anywhere. Which logical connective is this? as in this article http://www.mathblog.dk/strong-induction/
1
vote
1answer
86 views

Mathematics and Origami

I am reading through this paper about the math behind origami: http://www.math.washington.edu/~morrow/336_09/papers/Sheri.pdf However, I am getting confused with definitions 3.3 and 3.4. I am not sure ...
3
votes
2answers
177 views

Math and Origami

I am working on a project for class about the mathematics behind origami and write now I am looking into what is and is not constructible. I've gotten to the definition of origami constructible points ...
0
votes
1answer
112 views

How to tell if relation on set is a partial order when relation is defined as a set of ordered pairs?

Determine whether $R$ is a partial order on set the $S$ and justify your answer. $S=\{1,2,3\}$ and $R=\{(1,1),(2,3),(1,3)\}$. So the job is to consider if $R$ is reflexive, antisymmetric and ...