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

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4
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1answer
48 views

Definitions of order-dense Riesz spaces

I would like to get an intuitive idea as to why the definitions of order-dense and locally order-dense Riesz subspaces were chosen in the following way: A Riesz subspace $F$ of $E$ is order-dense if ...
0
votes
2answers
35 views

Splitting condition of forcing posets

I was looking at Wikipedia for brief reminders of what I learned in my elementary set theory class, and discovered the forcing page (which I did not learn): A forcing poset is an ordered triple, ...
3
votes
2answers
86 views

Notation for Tautologies

I've been stuck for a while in this question and so far I don't understand the flaw of my reasoning please if you guys could help me out. See, this is my context. From the definition of argument we ...
0
votes
0answers
31 views

$\mathbf{A}$ is unimodular $\Rightarrow$ $\mathbf{A}$ has entry in $\{-1, 0, 1\}$?

Is it true that $$\mathbf{A}\;\text{is unimodular}\;\Rightarrow\mathbf{A}\;\text{has entry in}\; \{-1, 0,1\}?$$ Also can an unimodular matrix $\mathbf{A}$ has entry in $\mathbb{R}$?
0
votes
2answers
60 views

Rudin's Chain Rule

Rudin's chain rule theorem goes like this: Suppose $f$ is continuous on ${[a,b]}$, $f'(x)$ exists at some point $x\in [a,b], g$ is defined on an interval $I$ which contains the range of $f$, and ...
0
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2answers
33 views

Possible definition of the matrix representation of a linear transformation with respect to given bases

Let $E$, $F$ be vector spaces with basis $\{e_1,\dots,e_m\}$, $\{f_1,\dots,f_n\}$. Let $T:E\to F$ be a linear transformation. We say that the matrix $A\in\mathbb{R}^{m\times n}$ represents $T$ with ...
4
votes
4answers
363 views

Why do Topologies get “finer”?

Why are topologies with many elements called "fine" and topologies with few elements called "coarse"? It seems as though the finer a topology is, the more likely it is for a function defined from that ...
1
vote
1answer
49 views

Help in understanding Bochner's theorem

This relates to the Bochner's theorem stated in the link: https://en.wikipedia.org/wiki/Bochner%27s_theorem My question is related to the unique probability measure μ on G. I want to express the ...
0
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1answer
32 views

Ambient Isotopy of Knots

Statement 1: Knots of opposite chirality have ambient isotopy, but not regular isotopy. Statement 2: We can then define two such knots to be equivalent if they are ambient isotopic, meaning that ...
0
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0answers
33 views

Isotopy: Definition

An isotopy is a homotopy from one embedding of a manifold $M$ in $N$ to another such that at every time, it is an embedding. In this definition, I am wondering why $M$ and $N$ are required to be ...
0
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1answer
29 views

Nurbs parametric coordinate span

I am using the Nurbs definition of Wikipedia. I might have missed something in the definition but I cannot understand how to know on which interval does the parametric coordinate span. Particularily ...
2
votes
1answer
34 views

a real number as a point

An element of the domain of a real-valued function of a real variable is often called a point. For example, an element (point) $p$ in the domain of a real-valued function $f$ of a real variable where ...
1
vote
1answer
24 views

Terminology help for a set relation: for sets $X, Y$, not necessarily disjoint, such that neither is a subset of the other.

Is there an existing term for pairs of sets $X, Y$, not necessarily disjoint, such that neither $X \subseteq Y$ nor $Y \subseteq X$? Would it be incorrect (or misleading) to call them something like ...
2
votes
0answers
35 views

Are there non-holomorphic or non-analytic polynomials of several complex variables?

Having no prior exposure to several complex variables, I am trying to read some papers involving this subject. I came upon the terms analytic polynomial and holomorphic polynomial. Do they simply ...
1
vote
1answer
21 views

Definition of quasi-cyclic and full rational groups

In Unit Groups of Classical Rings by Karpilovsky, p.96, we can see this theorem: Let $G$ be a divisible abelian group. Then $G$ is a direct product quasi-cyclic and full rational groups. I want ...
0
votes
1answer
45 views

Unknowledgable of single variabled integral

Browsing Stack-Exchange and other sites, I have noticed this come up quite a few times, an integral with only one letter or number! This would be a great example: $$\int_{a}$$ What in the world does ...
5
votes
0answers
50 views

Looking for functions “similar” to the $ \Gamma $ function

According to wikipedia: https://en.wikipedia.org/wiki/Functional_equation The $\Gamma$ - function is the only solution that suffices the following three equations: $$ f(x)={f(x+1) \over x}\tag{$*$} ...
7
votes
5answers
416 views

Necessity of being Hausdorff in the definition of compactness?

According to R Engelking - General Topology: A topological space $X$ is called a compact space if $X$ is a Hausdorff space and every open cover of $X$ has a finite subcover, i.e., if for every ...
2
votes
3answers
123 views

If $AB\not=BA,$ then do $A$ and $B$ have to be matrices

Is this solvable? Or are there other things that fit the bill for $A$ and $B$?
1
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2answers
30 views

What is “an increasing sequence of step functions”?

I'm reading Alan Weir's "Lebesgue Integration and Measure". In exercise 8 on page 30 he talks about "...an increasing sequence of step functions $\{\phi_n\}$..." and "...an increasing sequence of ...
1
vote
1answer
34 views

how to define a “directed spanning tree”?

In all my books and articles about "graph theory", I didn't find the definition of "directed spanning tree". Could you please give this definition and the reference? How to judge if a directed graph ...
0
votes
1answer
27 views

Filters and filter bases in order theory

Hi I would like to confirm the following ideas regarding filters in order theory: By definition I have that a filter is a subset of a poset $(P, \leq)$ which satisfies: $\mathcal{F}$ is non-empty. ...
5
votes
11answers
2k views

Why every definition is an “iff”-type statement? [duplicate]

Suppose that we are trying to define a mathematical object $M$. The statement of the definition generally takes the form (or some of its equivalent variant), A mathematical object is said to be ...
3
votes
1answer
79 views

Explanation of a joke on abelian groups (grapes).

Q: What's purple and commutes? A: An Abelian grape. Q: What is lavender and commutes? A: An Abelian semigrape. Q: What's purple, commutes, and is worshipped by a limited number of people? A: A ...
0
votes
0answers
18 views

Filter double elements from tuple

Let's say we have a tuple $h = (g_1,...,g_n)$ for example $h = (1,2,3,2,2,4,1)$. Now I have trouble finding a formal specification for the following problem: I want to know how many different ...
3
votes
2answers
45 views

What is the bar symbol over a complex scalar in the expression $\overline{\lambda}$?

I have the following problem from section 1.4 (Vector Spaces) of Peter Peterson's Linear Algebra textbook. I am having trouble with the way multiplication is defined on the given vector space, ...
0
votes
0answers
16 views

Define univalent homomorphism

I'm reading a paper that uses the phrase univalent homomorphism. From context and a cursory internet search, it seems that this just means "injective". I'd like to confirm this - it doesn't seem to ...
0
votes
3answers
70 views

Give a serious explanation of the difference between an equation and a function.

What's the difference between an equation and a function? I mean, I am not seeking for a high school-like answer like "an equation has an equals sign". I want to know what is the fundamental ...
1
vote
1answer
39 views

Is every field a Krull domain?

Background: A Krull domain is an integral domain $A$ with a family $(v_i)$ of valuations on the field of fractions $K$ for $A$ satisfying the following conditions: Each $v_i$ is discrete. The ...
1
vote
1answer
79 views

Bizarre definition of convergent sequences

I've recently begun studying Analysis I, specifically the sequence and series part. I've just come across the definition of convergence: A sequence $(a_n)$ converges to a real number $a$ if, ...
2
votes
2answers
73 views

What is the actual definition of a function?

I am learning precalculus and my book defines the following: A function $f$ from a set $A$ to a set $B$ is a rule that assigns to every element $a$ in $A$ one and only one value in $B$. Well, I ...
-4
votes
1answer
75 views

Equal Categories

Let $\infty$ be the "category" of all categories, where the objects are categories and the morphisms are functors. I am trying to motivate the definition of equal categories by doing it as follows. ...
11
votes
7answers
368 views

Why associativity $h \circ (g \circ f) = (h \circ g) \circ f$ is required in composition?

An introduction into category theory says that A category is a quadruple $A = (O, \mathrm{hom}, \mathrm{id}, \circ)$ consisting of blah-blah and is subject to the following conditions: (a) ...
0
votes
1answer
36 views

Definition of the vector cross product

As far as I understand the cross product between two vectors $\mathbf{a},\mathbf{b}\in\mathbb{R}^{3}$ is defined as a vector $\mathbf{c}=\mathbf{a}\times\mathbf{b}$ that is orthogonal to the plane ...
0
votes
1answer
115 views

Why are some branches of mathematics called 'theory' and others not?

We say: graph theory , group theory, number theory , set theory, what is definition of theory? We also say abstract algebra, real analysis, but why we do not say abstract algebra theory or real ...
1
vote
2answers
82 views

Wrong pushforward of vector field definition on wikipedia

Wikipedia claims that for $$\mathrm d \varphi_x:T_xM\to T_{\varphi(x)}N\,$$ we have for $X \in T_pM$ and $f \in C^{\infty}(N,\mathbb{R})$ $$\mathrm d\varphi_x(X)(f) = X(f \circ \varphi)$$ whereas I ...
2
votes
2answers
87 views

Is defining an “$S$-scheme” this way just a stylistic choice?

I have a potentially pretty silly question, involving the definition of an "$S$-scheme". For a fixed scheme $S$, the definition of the category of $S$-schemes is a category with: objects: scheme ...
1
vote
0answers
109 views

Is there a special name for functors from a category C to a subcategory of C?

Is there a special name for functors from a category $C$ to a subcategory $S$ of $C$ where $S \ne C$? I'm using $\ne$ pretty informally above. I'd be happy with anything that reasonably captures the ...
1
vote
3answers
105 views

What is the definition of axiom (mathematically speaking)

I'm wondering about the exactly definition of axiom (mathematically speaking). The definition of this term seems a little blur in my mind. For example, the definition of point in the Euclidean ...
-1
votes
0answers
23 views

Mathematical formal expression of find “subfunction” in function [closed]

Imagine if I have a function $s(t)$ and $r(t)$. $s(t)$ may contain $r(t)$ one or more times as $s(t)$ is a quasi-period function. What is the correct expression if I want to say the $s(t)$ contains ...
1
vote
1answer
37 views

Weibel definition 1.4.1. understanding the indexes on splitting maps

The book says: Definition 1.4.1. A complex $C$ is a called split if there are maps $s_n : C_{n+1} \to C_{n+1}$ such that $d = dsd$. The maps $s_n$ are called splitting maps. If in addition $C$ ...
0
votes
1answer
61 views

About odd functions and improper integrals e.g. $\int^{\infty}_{-\infty}\sin x \; dx$

Does $\displaystyle \int^{\infty}_{-\infty}\sin x \; dx$ converge? Since $\sin x$ is an odd function, and we know that in definite integrals $\displaystyle \int^{a}_{-a}\sin x \; dx=0$ then does ...
0
votes
1answer
29 views

Question about negative value using the ratio convergence test for integrals

Find for what $p$, $\displaystyle \int ^{\infty}_0 x^p \arctan x dx$ converges. By parts, it's equal to: $\displaystyle \lim_{b\to \infty}\frac 1 {p+1}x^{p+1}\arctan x |^b_0- \int ^b _0\frac ...
1
vote
1answer
50 views

Metric Space Definition

From my book, the definition given is: Given a set $X$, a function $d: X \times X \to \mathbb{R}$ is a metric on $X$ if for all $x,y \in X \dots$ Then a metric space is a set $X$ together with a ...
1
vote
2answers
32 views

Is the co-domain needed if we have the range? [duplicate]

Why do we need the co-domain if we have the range? I know what both mean. Isn't it just better to use the range instead of the co-domain when defining a function? This question brought up to me when ...
2
votes
1answer
50 views

Uniqueness of a number $area(A)$

I have the following definition of area: Let $A$ be a bounded set from $\mathbb{R}^2$. We say that $A$ has area if there exist two sequences $(E_n)_{n\in \mathbb{N}}, (F_n)_{n\in \mathbb{N}}$ of ...
3
votes
3answers
121 views

(Is it a set?) Set of all months having more than 28 days.

Set is a well defined collection of distinct objects. Is the following is a set? Set of all months having more than 28 days. I'm confused here. Because on one hand I think that it is well ...
1
vote
2answers
26 views

Is there a relation eigenvectors and unitary operator.

I am trying the understand the spectral theorem as given in wikipedia link: https://en.wikipedia.org/wiki/Spectral_theorem I understand that eigenvectors are vectors and unitary operator is a ...
1
vote
1answer
37 views

Dually Dedekind Set and Weakly Dedekind set

$A$ is dually Dedekind infinite (dD-infinite), if there is a surjective non-injective map from $A$ onto $A$. $A$ is weakly Dedekind infinite (wD-infinite), if there is a surjective map from $A$ onto ...
1
vote
1answer
58 views

dual Dedekind-infinity may not imply Dedekind-infinite without AC

It is written in wikipedia: https://en.wikipedia.org/wiki/Dedekind-infinite_set It is not provable (in ZF without the AC) that dual Dedekind-infinity implies that A is Dedekind-infinite. (For ...