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

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2
votes
2answers
27 views

What is the definition of “prime ideal decomposition”?

I'm reading about Sunada's theorem in the book Geometry and Spectra of Compact Riemann Surfaces (Peter Buser) and I encountered this paragraph: If R is an algebraic number field and if $p \in ...
2
votes
0answers
16 views

$\omega$-limit set of a point $x \in X$

I would like to verify whether the following definition of the $\omega$-limit set of a point $x \in X$ is correct: $$\omega(x) = \{ y \in M : \exists \text{ sequence }\{t_j\}, \text{ where } t_j ...
1
vote
1answer
11 views

What are those weighed graphs called?

Let $G = (V, E)$ be a directed graph, and define the weight function $f : V \sqcup E \to \mathbb{R}^+$ as follows: sum of weights of vertices is 1, if a vertex has edges coming out of it, their ...
4
votes
1answer
36 views

What definition of Reed-Solomon code is correct?

In the discuss with @Evinda we have the contradiction with the definition of Reed-Solomon Codes (not generalized case) over the finite field $\mathbb{F}_q$. We have two papers and ...
0
votes
1answer
25 views

How to lift an action along a covering

Took from Lawson-Michelsohn "Spin Geometry", pages 80-81. Let $E\to X$ be an oriented $n$-dimensional vector bundle over a manifold $X$, and let $P_{SO}E$ be the associated $SO(n)$-bundle. Assume ...
0
votes
1answer
68 views

A kind of planar figures

Studying issues related to the planar shapes I've found some attribute, useful for my investigations: Any segment with origin in mass center and end point on figure's boundary is contained within ...
1
vote
1answer
17 views

Is this definition of a Euclidean frame well-defined?

Going through my lecture notes on geometry I find a definition of a Euclidean frame which doesn't seem to have been formed correctly (most likely written down wrong). So I've taken it upon myself to ...
6
votes
4answers
467 views

What are some physical, geometric, or otherwise useful interpretations of divergent series?

I don't understand what ideas such as Abel, Cesàro summation or other types of sum 'regularization' help us describe. What is the practical application to discussing the 'sum' of sequences that are ...
0
votes
3answers
102 views

What is the usual definition of “ordinal number”?

My definition for the ordinal number is "von-Neumann ordinal". I thought this is the only definition for the ordinal, but i found some other definitions in wikipedia. What is the usual definition of ...
2
votes
1answer
2k views

What is the definition of a geometric progression?

If the first term in our geometric progression (GP) is $k$, and the common ratio is 0, then our sequence is $\{k, 0, 0, 0, 0,\ldots\}$. Is there anything wrong with this statement? So, is $\{0, 0, ...
2
votes
1answer
384 views

Question about Definition of Boundary in Stokes' Theorem

I was wondering if what my teacher said was correct and complete in that in Stokes' Theorem the "boundary curve" of a surface can be defined as the mapping along the boundary of the two-dimensional ...
1
vote
2answers
151 views

Is the empty family of sets pairwise disjoint?

„A family of sets is pairwise disjoint or mutually disjoint if every two different sets in the family are disjoint.“ – from Wikipedia article "Disjoint sets" What about the empty family of sets? Is ...
1
vote
2answers
78 views

Definition of algebraic structure

Is there a definition of algebraic structure? Wikipedia says: a set (called carrier set or underlying set) with one or more finitary operations defined on it. In particular, what is the ...
3
votes
1answer
1k views

What is unitary space

In https://www.encyclopediaofmath.org/index.php/Unitary_space, unitary space seems to be Hilbert space. But in http://www.answers.com/topic/unitary-space, "finite dimensional" is required. My question ...
-5
votes
0answers
15 views

definition of a limit in terms of sequences and epsilon delta [on hold]

link to the question i) for every sequence Xn in the real number system with limit negative infinity, we have limit f(Xn)=infinity n->infinity ii) for any M>0 and N<0 there exists a a> N such ...
3
votes
2answers
5k views

What does $×$ mean in this context

I have two definitions, from real analysis - Metric Space: Given a set $X$, a function $d:X×X→\mathbb{R}$is a metric on $X$ if for all $x, y∈X \dots $ Function: Let $A,B$ be two sets. A function ...
-1
votes
2answers
74 views

Is this definition of Mersenne Primes correct? [on hold]

According to my understanding, the definition of Mersenne Prime is the following: A Mersenne Prime is a prime number that is obtained by using the formula $2^n-1$, where $n\in\mathbb{N}_+$
1
vote
1answer
100 views

The maximum notation as regards the absolute value?

We know that $\max(\textbf{A})$ gives the maximum element of the array $\textbf{A}$. What is the notation, or a short formula, if we seek the element having the largest absolute value? e.g., ...
3
votes
2answers
48 views

Properties of sin(x) and cos(x) from definition as solution to differential equation y''=-y

I recently came across the interesting definition of the sine function as the unique solution to the Initial Value Problem $$y'' = -y$$ $$y(0) = 0, y'(0) = 1$$ (My first question would be why this ...
0
votes
0answers
20 views

Hatcher's notation and definition for the boundary of a singular chain complex

I have a hard time following Hatcher in general, but especially his notation in this case with the bar (page 108 in my pdf copy): For the boundary of a singular $n$-simplex, $\sigma$,he writes: ...
2
votes
0answers
20 views

Question Janelidze and Tholen's 'Beyond Barr Exactness: Effective Descent Morphisms'

I have some questions about Janelidze and Tholen's paper Beyond Barr Exactness: Effective Descent Morphisms. First of all, they remark at the end of the introduction: Throughout this chapter ...
1
vote
3answers
63 views

$0^0$ is undefined, but sometimes defined as $1$?

When typing a calculation to Wolfram Alpha it always takes $0^0$ as undefined, but for some calculations I need to preform, I specifically need it to be equal $1$. Is there a way to make it ...
1
vote
2answers
25 views

Extension of Up/Down/Right/Left in n dimensions

I'm wondering if there are extensions of the right/left/up/down concepts in n dimensions, i.e. "if $x_0$ points ahead of me, I'm looking LEFT/RIGHT along $x_1$, UP/DOWN along $x_2$, XXX/YYY along ...
0
votes
1answer
25 views

Definition of Polish Topology

Let $\xi$ such that $1 \leq \xi < \omega_1$ and $f$ be of Baire class $\xi$. In this paper (Section $5$), the author defined $$T_{f,\xi}=\left\{ \tau^{\prime} : \tau \subset \tau^{\prime} \text{ ...
22
votes
12answers
8k views

How to represent the floor function using mathematical notation?

I'm curious as to how the floor function can be defined using mathematical notation. What I mean by this, is, instead of a word-based explanation (i.e. "The closest integer that is not greater than ...
7
votes
3answers
213 views

In the definition of a functor, why is it necessary that $F(id_{A})=id_{F(A)}$?

A functor $F$ is defined to be a mapping from category $\mathcal{C}$ to $\mathcal{D}$ such that: (1) $F(f\circ_{\mathcal{C}} g)=F(f)\circ_{\mathcal{D}} F(g)$ (say, for a covariant functor). (2) ...
0
votes
1answer
33 views

How to calculate $f'(0)$ from $f(x)=x|x|$ directly from the definition of derivatives as a limit?

How to calculate $f'(0)$ from $f(x)=x|x|$ directly from the definition of derivatives as a limit? So $\lim\limits_{h\to 0}\frac{f(x+h)-f(x)}{h}=\lim\limits_{h\to ...
35
votes
20answers
2k views

Problem with basic definition of a tangent line.

I have just started studying calculus for the first time, and here I see something called a tangent. They say, a tangent is a line that cuts a curve at exactly one point. But there are a lot of lines ...
0
votes
2answers
47 views

How to formally define $\lim\limits_{x\to\infty}f(x)=\infty$?

How to formally define $\lim\limits_{x\to\infty}f(x)=\infty$ I suppose it's $\forall K\in\Bbb{R},\ \exists N\in\Bbb{R},\ x\gt N\implies f(x)\gt K$. Could someone correct it?
1
vote
2answers
31 views

Definition of interior

An interior point is defined as the following in the Euclidean space. If $S$ is a subset of a Euclidean space, then $x$ is an interior point of $S$ if there exists an open ball centered at $x$ ...
1
vote
2answers
59 views

Is there a symbol for always less than (or just always?)

For e.g, the quotient of $\frac{1}{n}$, $q$, where $n \gt 1$, $q$ will always be less than $1$. $$\frac qn\le n$$ etc. I can't really write $\frac {q}{n} < n$, because whilst true, it doesn't ...
1
vote
1answer
14 views

What does it mean for a boundary to be analytic in the context of a PDE?

I am reading a paper where they assume the boundary of a domain is "Analytic". They never define it. Is this a standard definition, and, if so, what is it?
0
votes
0answers
21 views

Commutative rings and PI algebras

Any commutative ring $R$ with unity is a PI ring (polynomial identity ring). When could one take $R$ as a PI algebra? Essentially, what is the relation between an arbitrary commutative ring and a ...
1
vote
0answers
20 views

Is a finite dimensional vector also a covector?

I am reading something where it seems they have defined a covector/linear functional as a map $F: \mathbb{R}^n \to \mathbb{R}$ where $F$ is a vector For example, $F(x) = \begin{bmatrix} 1 & 2 ...
3
votes
4answers
621 views

Why does an isomorphism need to be a homomorphism?

In many books I read I found isomorphism defined as a 'bijective homomorphism'. I do not understand why is it that existence of inverse or order preserving requires the property of a homomorphism, ...
3
votes
2answers
572 views

Definition: Mathematical way to define the “left” and the “right”

How do we define the left and the right, (such as left/right handness) from a mathematical way of definition? How can we explain this definition to some inhabitant of a distant stellar system who has ...
0
votes
0answers
30 views

Definition of an Integral Domain in the second edition of Herstein's Topics in Algebra

I think a definition in Herstein's Topics in Algebra needs to be modified and I am asking this question to make sure I am not missing something. Herstein defines an integral domain as a commutative ...
0
votes
1answer
26 views

An equivalent definition of connectedness in General Toplogy

I'm studying General Topology by the book of James Munkres and there he defined connectedness this way: "Let X be a topological space. A separation of X is a pair U, V of disjoint nonempty open ...
8
votes
5answers
355 views

The derivative of $e^x$ using the definition of derivative as a limit and the definition $e^x = \lim_{n\to\infty}(1+x/n)^n$, without L'Hôpital's rule

Let's define $$ e^x := \lim_{n\to\infty}\left(1+\frac{x} {n}\right)^n, \forall x\in\Bbb R $$ and $$ \frac{d} {dx} f(x) := \lim_{\Delta x\to0} \frac{f(x+\Delta x) - f(x)} {\Delta x} $$ Prove that ...
1
vote
2answers
32 views

the definition of random variable

If we supposed that X is a random variable, is X - X a random variable? Could the outcome of an event is only 1? Cause X-X has only one outcome, and the possibility of it is 1. How about X + X?
1
vote
1answer
37 views

What is the catch when introducing measure theory using $\sigma$-ring instead of $\sigma$-algebra?

I am currently using Matthew A Pons book Real Anaysis for Undergrad for introduction of measure theory In my opinion this book is unbelievably clear for almost all the sections EXCEPT the section ...
0
votes
1answer
27 views

Condition on symplectic form: $(d\alpha)^n \neq 0$?

I started to read about contact and symplectic forms and I came across this answer here. It seems to state that the definition of symplectic form is that $d\alpha$ is non-degenerate if and only if ...
0
votes
1answer
38 views

What is the definition of binary sequence?

Can I write an infinite binary sequence like so: ...0111001001, ...10010 because I saw some people write infinite binary set from left to right like so: 1011000... , 101111... But I was not sure if ...
0
votes
1answer
24 views

Is a function, in part, defined by it's domain?

I have read the definition of a function from two sources.Both sources state that a function defines a relationship between the input and the output. However, the first source states that it is the ...
0
votes
1answer
27 views

Combinatorial Geometry explanation

I do not understand what is going on in $(4)$: for every flat $E \in \mathcal F$, $E \ne X$, the flats that cover $E$ in $\mathcal F$ partition the remaining parts. What is meant by "the flats ...
0
votes
1answer
28 views

Definition of matrix transformation

I really understand the definition of linear transformation, but I'm not sure about the definition of matrix transformation. Could it be that a matrix transformation is defined as a linear ...
1
vote
2answers
126 views

A Pure Maths Approach to Thinking About Vectors

Introduction Generally most students are introduced to the concept of Vector as something that has both a "magnitude and direction" and Scalars as something that only has a "magnitude and no ...
1
vote
1answer
25 views

Explanation of defintion of ${[n] \choose k}$

I am reading a definition in a paper, but am not sure of how to interpret the following definition: If $K \in {[n] \choose k}$, then let $\operatorname{Path}(K)$ denote the set $$\{S: S \text{ is ...
1
vote
5answers
62 views

Does excluding or including zero from the definitions of “positive” and “negative” make any consequential difference in mathematics?

I was absolutely certain that zero was both positive and negative. And zero was neither strictly positive nor strictly negative. But today I made a few Google searches, and they all say the same ...
0
votes
0answers
12 views

Terminology for a Complex Semicircle

A while back I read about a special type of number system termed (if I remember correctly) as "degrees of sign". The idea was that numbers sat on a series of 0 to 180 degree rays. Positive ways the 0 ...