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

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1answer
47 views

Is there any difference between covariant vectors and one-forms?

Just need to have it clarified, are the 2 expressions interchangeable, or is there any difference? I'm trying to learn differential geometry on my own and it is really hard.
7
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1answer
78 views

The category of locally $P$ spaces

Let $P$ be a class of topological spaces (for example, compact spaces). The class of locally $P$ spaces consists of those spaces in which every point has a neighborhood basis consisting of $P$ ...
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21 views

How to define a non Triangle Free Graph

Consider a Regular Connected Graph $G= \bigcup\limits_{i=1}^{m} U_i$ where- $U_i$ is a sub-graph of $r$ vertices $\forall i$.(i.e. $G$ is a $r$ regular graph) $\forall i$, $U_i$, is not a ...
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51 views

Definitions of Weil Algebras

I am confused by several definitions of Weil Algebras and their connection to each other. Kock's book on synthetic differential geometry defines a Weil algebra over a ring $R$ as an $R$-algebra of ...
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1answer
50 views

What is the standard definition of a “periodic” function?

This question is admittedly pedantic, but I like my definitions precise. Tom Apostol, in his calculus book, defines a periodic function as follows. A function f is said to be periodic with ...
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1answer
36 views

How to prove equivalence of definitions for matrix similarity

It seems the most usual way to define matrix similarity is as follows: "Let $A$ and $A'$ be two n-by-n matrices, we say they are similar if there exist some invertible n-by-n matrix $P$ such that ...
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33 views

Symplectic matrix over $\mathbf{H}$ has determinant $1$ as well?

According to this question, symplectic matrices over $\mathbf{R}$ have determinant $1$. Does the equivalence carry over if we go back to the quaternion definition for $Sp(n)$? In the original ...
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1answer
42 views

Definition confirmation: cover of a subset

In lectures we defined: Let $X$ be a topological space, $Y \subset X$ a subset. A collection $\mathcal{A} \subset \mathcal{P}(X)$ is a cover of $Y$ by sets open in X if each element of ...
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2answers
33 views

Definitional question: difference between a correspondence and a function

Is there a difference between a correspondence and a function? For example, in game theory I am told that for a given strategy set, $\Sigma_i$, the best response given by ...
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1answer
22 views

basic understanding of hereditary property in matroid

I'm trying to prove something is a matroid and to do that I must understand what a matroid is. I don't get the hereditary property. A matroid is an ordered pair $(S, I)$. I is a non-empty family ...
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2answers
26 views

Can someone make clear the concept of a “restricted metric”

I never understood this concept of restricted metric. Consider the following theorem. http://www.math.psu.edu/wysocki/M403/Notes403_4.pdf I don't quite understand this concept since under usual ...
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1answer
49 views

Why are “the” morphisms of the category of topological spaces continuous maps?

On the Wikipedia page for "morphism (category theory)" it says that: In the category of topological spaces, morphisms are continuous functions and isomorphisms are called homeomorphisms. In what ...
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39 views

Meaning of “Form” in mathematics

There are a number of occasions in the mathematics where the word "form" is used: Modular form Bilinear form Quadratic form ... It seems that form is a special kind of function. But I cannot ...
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1answer
29 views

Definition of a recursive ordinal

I'm having trouble understanding the definition of reclusive or computable ordinal - Wikipedia defines it as follows: "...an ordinal $\alpha$ is said to be recursive if there is a recursive ...
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2answers
113 views

Precise definition of free group

I have seen the definition of a free group go like this: Let $S = \{s_i : i\in \mathbb{N} \}$ be a countable set. Let $S^{-1}$ be the set $\{s_i^{-1}: i\in \mathbb{N}\}$. Here one is to understand ...
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3answers
80 views

Is there a theoretical (or practical) definition of $n$-gon, for $n < 0$?

Background This is purely a "sate my curiosity" type question. I was thinking of building a piece of software for calculating missing properties of 2D geometric shapes given certain other ...
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1answer
47 views

Why do we need the condition “cocommutative” in the definition of a coPoisson Hopf algebra?

In this paper, page 5, Section 3.6, in the definition of a coPoisson Hopf algebra $H$, it is said that: a coPoisson Hopf algebra is a cocommutative Hopf algebra $A$ with a map $\delta: A \to A \otimes ...
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2answers
49 views

Mutually exclusive/independent events

I am having some conflicts with the basic definitions of mutually exclusive and independent event. I can't seem to understand the difference(or relation) between the two. Here's a statement from my ...
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1answer
55 views

What has a chain homotopy to do with homotopy?

I'm in an intro algebraic topology class. In the textbook Algebraic Topology (Hatcher), a chain homotopy is defined by saying that a map $P$ is a chain homotopy between two maps $f$ and $g$ if $dP + ...
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1answer
20 views

Clarification about the definition for polynomial discriminant?

On the wikipedia page about polynomial discriminants, it shows this definition: $$\Delta = a_n^{2n-2}\prod_{i<j}(r_i-r_j)^2$$ What I'm getting from this is that $\Delta$ is obviously the ...
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1answer
54 views

Definition of absolute value of complex number

The definition I see everywhere for the absolute value of a complex number is: Let $z=x+iy$ then $|z| := \sqrt{x^2+y^2}$. But the square root operation is multivalued in complex analysis. So while ...
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1answer
32 views

T/F: pointwise equicontinuous $\Leftrightarrow$ all $f_n$ continuous, uniform equicontinuous $\Leftrightarrow$ all $f_n$ uniformly continuous

Is it true that: Given a sequence of functions $(f_n)$ on a set not necessarily compact pointwise equicontinuity $\Leftrightarrow$ all $f_n$ are continuous and, uniform ...
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0answers
19 views

Ideal Sheaf: Making sure I got it right

I want to know if I unwrapped the definition of ideal sheaves correctly. Let $\mathscr{O}$ be a sheaf of rings on topological space $X$. The definition of an ideal sheaf is as follows: The ideal ...
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2answers
32 views

What is the meaning of the locus of points P satisfying some conditions?

A rod AB of length 15 cm rests in between two coordinate axes in such a way that the end point A lies on x axis and end point B lies on y axis. A point P(x,y) is taken on the rod in such a way that ...
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2answers
62 views

$\Delta$-complex structure on the singular complex $S(X)$

I am confused with the definition of $S(X)$ so that I can't see why $S(X)$ is a $\Delta$-complex. Here are some material by Allen Hatcher. Though singular homology looks so much more general ...
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27 views

Vector sort operator

I was wondering whether there exists a definition of the sort operator for vector, i.e., the operator that given a vector of $\mathbb{R}^n$ yields a vector of $\mathbb{R}^n$ whose components are ...
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1answer
78 views

Meaning of $[\mathbb K : \mathbb Q]$

What is the meaning of $[\mathbb K : \mathbb Q]$ where $\mathbb K$ and $\mathbb Q$ are fields. This is galois theory, abstract algebra. What does this actually mean?
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2answers
41 views

Definition of uniformly cauchy

In this post, the definition of uniformly Cauchy is defined as: Uniformly cauchy sequences A sequence of functions $f_n$ is said to be uniformly cauchy if $$\forall \varepsilon > 0 \ \exists ...
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1answer
39 views

Polyhedra vs Polytope

I am having a hard time understanding what is the main difference between a polyhedron and a polytope. Could anyone explain me what is the difference between these two structures?
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3answers
52 views

Intuition behind the definition of a surface

From the book Elementary Differential Geometry by Andrew Pressley, this is the definition of a surface: A subset $S$ of $\mathbb{R^3}$ is a surface if, for every point $p \in S$, there is an open ...
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1answer
27 views

Understanding the vector space $\bigoplus_{i\in I} F$

Let $F$ be a field and define for any set $I$ the vector space $F^I = \{f \colon I \to F \}$ with addition $(f+g)(i) = f(i) + f(i)$ and rescaling $(\alpha f)(i) = \alpha f(i), \forall f,g\in F^I, ...
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2answers
23 views

What does bounded in module mean?

I'm trying to understand Liouville's theorem and in the book I'm reading it's stated as Let $f$ be a holomorphic function on the complex plane, which is bounded in module in a neighborhood of infinity ...
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4answers
69 views

Why is this first-order ODE considered non-linear?

Consider the following equation: $$y'=\sin y.$$ What causes this equation to be considered non-linear? $y'$ is not multiplied by a power, and neither is $\sin y$.
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1answer
31 views

Why are matrix norms defined the way they are?

Given $A$ a square matrix Define: $\|A\|_1$ as the max absolute column sum $\|A\|_2$ as the sum of the squares of each element $\|A\|_\infty$ as the max absolute row sum Pray tell, why are ...
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0answers
57 views

Contribution of a loop to the degree of a vertex in a graph

Prehistory I've recently started learning graph theory in my institute (as a part of discrete mathematics course). During a lecture the professor had given a definition for a local degree of a vertex ...
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1answer
110 views

What does “dual statement” mean exactly in category theory?

I have long been confused about this notion. I know that for a statement within a single category, forming the dual statement is just reversing every arrows. But what about a statement concerning ...
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3answers
67 views

Defining Surjection as an Implication

Is the following logical notation valid in describing the condition for a function $f$ being surjective? If $f: A \rightarrow B$ is a function, then $f$ is surjective if $$\forall y, \exists x \in A ...
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4answers
109 views

How to define $x^a$ for arbitrary real numbers $x$ and $a$

Questions like this, which asks to solve $$x^{\frac43} = \frac{16}{81}$$ confuse me. The solution for real $x$ is $\pm \frac8{27}$. The question would presumably be a bit different to solve ...
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1answer
61 views

What is the definition of a local property for a function?

I came across this question while thinking about the question whether the integrability is a local property. I first thought that the MSE question What's the definition of a "local ...
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3answers
48 views

In measure theory $\emptyset$ and $\{a\}$ are considered as intervals . Why?

I recently started self learning measure theory. While reading a note , I came across this , In measure theory $\emptyset$ (empty set) and $\{a\}$ (singleton set) are considered as intervals . Why ...
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0answers
43 views

Is integrability of a function a local property?

I always thought, that integrability of a function is a global property while differentiability is a local property. After reading What's the definition of a "local property"? I doubt ...
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40 views

Conditional probability between two random vectors: input and output of a channel.

Suppose we send a sequence of $n$ bits $X=\{x_1,...,x_n\}$ through a channel. Each $x_k=1$ with probability $p$, and $x_k=0$ w.p. $1-p$. The channel flips one input bit with probability $q$. If ...
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128 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, ...
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1answer
22 views

Understanding the definition of the direct sum of subspaces of a vector space

I have a question regarding the definition of direct sum of a vector space in relation to subspaces. Definition: A vector space $V$ is called the direct sum of $W_1$ and $W_2$ if $W_1$ and $W_2$ ...
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39 views

What is the name of the sub-matrix?

Given a matrix $\mathbf{A}$ of size $n\times n$. Let $I=\{i_1,\ldots,i_k\}\subseteq\{1,\ldots,n\}$ for some $k\leqslant n$. How to call the sub-matrix of $\mathbf{A}$ that has its indices in $I$? (I ...
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1answer
36 views

Are these two definition of boundedness equivalent?

Definition 1: A set $S \subset M$ is bounded if $\forall x \in M, \exists r > 0,$ such that $S \subset B_r(x) = \{y \in M | d(x,y) < r\}$ Definition 2: A set $S \subset M$ is bounded if ...
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27 views

What is $L_{loc}^1$

What is $L_{loc}^1$? I know what $L^1$ is. I saw this on page 15 in the book Variational Analysis in Sobolev and BV Spaces. Let us first consider a function $f\in L_{loc}^1(\Omega)$, where ...
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2answers
65 views

Defining Material Conditional

Defining the conditional statement "$\phi \implies \psi$" so as to capture what is meant by implication requires that we ignore the issue of causation. From Keith Devlin's Introduction to ...
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3answers
91 views

The given limit is a derivative, but of what function and at what point? [closed]

How would I solve for something like this?? $$\lim_{h\to 0} \frac{\sqrt[4]{16+h} - 2}{h}$$ using the definition of derivatives.
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1answer
29 views

Definition of trajectory

I am writing something that involves comparing the solutions of many different differential equations, and I need precise definitions of the terms trajectory and solution curve. Given a dynamical ...