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

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0answers
15 views

Derivative of function of one variable with respect to function of two variables

I'm looking to find the derivative of a function of one variable with respect to a function of two variables: $$ \frac{df(x)}{dg(x,y)} $$ I'm not entirely sure whether this is possible in the first ...
0
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1answer
25 views

Taylor series for exponential function.

The Taylor series for $e^x = \sum_{i=0}^\infty \frac{x^i}{i!}$. Then as $e^0 = 1$, if one evaluates the Taylor series at $x=0$ we find that $e^0 = \sum_{i=0}^\infty \frac{0^i}{i!} = \frac{0^0}{0!} + ...
1
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0answers
33 views

How to describe a polynomial relation on $\mathbb{P}(\bigwedge^k V)$, and if the Zariski topology is canonical

I am working with the space $\mathbb{P}(\bigwedge^k V)$, where $V$ is some $n$ dimensional vector space over some field K. In here I want to define a variety, ie a solution to a set of polynomials. ...
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1answer
19 views

Different versions of Markov's inequality

I have some doubts regarding some different versions of Markov's inequality. From Wikipedia's definition of Markov's inequality we have: If $X$ is a nonnegative random variable and $a>0$, then ...
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1answer
68 views

How does one “join” two graphs in graph theory?

I am asked to find the join of two graphs in graph theory. But I cannot find the exact definition! I know that in lattice theory, we join every vertex of a graph to every vertex of another graph to ...
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1answer
28 views

Is assuming the gluing axiom for any pair of sections equivalent with gluing arbitrary collections of sections?

In Griffiths and Harris the gluing axiom for sheaves is given as For any pair of open set $U,V$ and sections $\sigma\in\mathcal{F}(U)$, $\tau\in\mathcal{F}(V)$ s.t. $\tau|_{U\cap V}=\sigma|_{U\cap ...
3
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0answers
54 views

Can these definitions of the words “problem” and “solution” be formalized, and if so, has this been done? If so, where can I learn more about it?

I had a thought. Define that: Vague Definition 0. A problem consists of: a set $X$ a set $Y$ a function $f : X \rightarrow Y$ a way $\overline{X}$ of representing the elements of $X$ ...
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3answers
33 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 ...
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11answers
3k views

What is exactly the difference between a definition and an axiom?

I am wondering what the difference between a definition and an axiom. Isn't an axiom something what we define to be true? For example, one of the axioms of Peano Arithmetic states that $\forall ...
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2answers
3k views

What is a third proportional?

I searched online, couldn't find anything clear. If I had two numbers, $a,b$, what is their third proportional? Apparently it can be either $c$ such that $a/b=b/c$ or $b/a=a/c$, but obviously these ...
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1answer
34 views

Definition of convergence of a sequence

Can this be a valid definition? For each $\epsilon>0$, there exists $K \in \Bbb N$ such that if $n\ge K$, then $|a_n-a|\le\epsilon$. Should I use "$<\epsilon$" or "$\le \epsilon$"?
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0answers
31 views

What is the structural hierarchy in mathematics?

By structural hierarchy, I mean the mental concept in which things are 'done' in mathematics. At the top, you have mathematics itself, which is a collection of systems, like arithmetic, algebra, ...
0
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1answer
31 views

Compactness and sequences in $\mathbb{R}^{n}$

Why is it that: If $A$ is a compact set and $\left ( a_{n} \right )$ a sequence in $A$, then there is a subsequence $\{a_{n_k}\}$ such that $\lim_{k\to\infty} a_{n_k}=a$ with $a\in A$. I get ...
3
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1answer
26 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 ...
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2answers
30 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 ...
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1answer
12 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 ...
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1answer
39 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 ...
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1answer
28 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 ...
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1answer
70 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
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1answer
19 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 ...
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4answers
471 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 ...
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3answers
103 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 ...
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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, ...
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1answer
396 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 ...
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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 ...
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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 ...
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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 ...
3
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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 ...
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2answers
77 views

Is this definition of Mersenne Primes correct? [closed]

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
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1answer
101 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
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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 ...
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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
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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 ...
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3answers
64 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 ...
0
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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
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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
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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
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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
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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 ...
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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?
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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 $x$ ...
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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 ...
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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?
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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 ...
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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
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4answers
622 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
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2answers
577 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
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0answers
31 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 ...
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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 ...
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5answers
356 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 ...