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

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1answer
60 views

Limit Point of a Set

Definition. A point $x$ is a limit point of a set A if every $\varepsilon$-neighborhood of $x$ intersects the set A in some point other than $x$. I understand the definition in that $x$ is our limit ...
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1answer
117 views

Is the value of the sum $1-1+1-1+1-1+\cdots$ does not exist? [duplicate]

Let $a_k:=(-1)^k$ where $k\in\mathbb{N}$. $\mathbb{N}$ is the set of all non-negative integer. And we define the partial sum $S_n:=\sum \limits_{k=0}^{n}a_k$. Notice that the sequence $\{S_k\}$ ...
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2answers
47 views

Help with understanding the definition of operation

I'm having trouble understanding this excerpt from Wikipedia, which defines an operation: Mainly, I don't understand what is meant by $V \subset X_1 \times...\times X_k$. Why does an operation ...
2
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1answer
100 views

Definition of an $n$-tuple agreeing with the Kuratowski's definition of an ordered pair

Is there a nice and elegant definition of an $n$-tuple ($n$ is a nonnegative integer) in ZFC, which will at the same time agree with the Kuratowski's definition of an ordered pair, i.e. $\left ( a,b ...
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1answer
66 views

Solvable Group, which Quotients need to be Abelian?

In Wikipedia it says a group $G$is solvable if it has a subnormal series $\{e\}=G_1\lhd G_2,\dots \lhd G_n=G$ where $G_i$ is a normal subgroup of $G_{i+1}$ and all the factor groups are abelian. My ...
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2answers
85 views

How do i define 'complex rational function'?

http://en.wikipedia.org/wiki/Rational_function I don't get the definition in wikipedia. It would be great to define "complex rational function" with the domain $\overline{\mathbb{C}}$, namely ...
3
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1answer
38 views

which is the usual definition of Argument?

Let $z\in\mathbb{C}\setminus\{0\}$ Then, there exists a unique $\theta \in [0,2\pi)$ and $\phi\in(-\pi,\pi]$ such that $z=|z|e^{i\theta}=|z|e^{i\phi}$. Between $\phi$ and $\theta$, which is the ...
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2answers
39 views

Does the definition of “mean”/“average” require the result to be in the domain set?

If I have a function that calculates the mean value of a set of elements that is an arbitrary subset of some set $X$, does the mean, by definition, have to also be in $X$? (In other words, if the mean ...
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1answer
49 views

Fiding a derivative

I need to find the derivative of $\sqrt{x^2+3x}$ using the definition of derivative. e.g. $\frac{f(x)-f(a)}{x-a}$ as x->a. Normally I get these but the $x^2$ is messing me up. I am at $$\lim ...
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2answers
209 views

Total/Gaussian curvature is intrinsic, yet mean curvature is extrinsic, why?

What characteristics define the total/mean curvature to be intrinsic/extrinsic accordingly? What is different geometrically about these curvatures that cause them to be defined as this?
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1answer
29 views

Question about measure on set that is not in $\sigma$-algebra

I think I have problem with badly written book, or I just can't understand statement. Let $(X,\mathcal{A},\mu)$ be any measure space and $\mu^*$ be outer and $\mu_*$ inner measure inner measure ...
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2answers
452 views

Geometrically reduced variety

What is a geometrically reduced variety (or geometrically reduced algebraic set if you will, a variety has not been assumed to be irreducible in this definition)? I tried looking up on the internet as ...
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1answer
154 views

Why does an odd number plus one, not necessarily entail it being even?

Why does an odd number plus one, not necessarily entail it being even? For example, $\sqrt{5} + 1$ is not even.
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1answer
109 views

Links between Minkowski metric, Hamming distance and Levenshtein distance

I know that Hamming distance is a particular case of Minkowski metric (with the specific definition of the subtraction). Also it seems that Hamming distance is a particular case of a Levenshtein ...
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2answers
171 views

I Need Help Understanding the Formal Definition of A Limit

I had been taught the formal definition of a limit with quantifiers. For me, it is very hard to follow and I understand very little of it. I was told that: $$\text{If} \ \lim\limits_{x\to a}f(x)=L, \ ...
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2answers
464 views

Construction of the Hyperreal numbers

Several times I have seen questions/answers here about using the correct definition of derivatives. There are also questions about whether or not $1/0$ is defined. Sometimes there is a discussion ...
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1answer
65 views

Positive definite kernel vs. positive definite function

What is the difference between positive definite kernels and positive definite functions? As I understand it, a positive definite kernel is a positive definite function if it is translation ...
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1answer
105 views

Is there a rigorous definition of the term “coordinate system”?

You hear the term coordinate system thrown around a lot, and we all know the usual examples (polar coordinates in $\mathbb{R}^2$, spherical coordinates in $\mathbb{R}^3$, etc.), but in truth I have no ...
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1answer
119 views

Is this a correct definition of the natural numbers in ZF?

Set $s$ is a natural number if $s$ is transitive and for every $x$, $y$ and $z$ $y\in{s}\rightarrow(y$ is transitive$)$, and if $x\in{P}s\wedge(x$ is transitive$)\wedge{z}\in{P}x\wedge(z$ is ...
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0answers
110 views

How is this the definition of equi-integrable?

Let $Q=(0,T)\times\Omega.$ I am completely lost with this: No definition of equi-integrability I have seen looks anything like this. Can someone help me please? Presumably it is a fact that a ...
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95 views

Defining “Penon Infinitesimals”.

In this lecture (which is accompanied by these slides), right near the end (so page 9 in the pdf of the slides; I don't think you have to watch the lecture), P. Johnstone refers to the "Penon ...
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1answer
321 views

What is the most accurate definition of the hyperboloid model of hyperbolic geometry?

For simplicity, let's focus on the two-dimensional case (the hyperbolic plane in 3-space). I have seen the hyperboloid model defined as variations on the following i) The positive sheet of a ...
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2answers
513 views

Closed immersion definition

Hartshorne defines a closed immersion as a morphism $f:Y\longrightarrow X$ of schemes such that a) $f$ induces a homeomorphism of $sp(Y)$ onto a closed subset of $sp(X)$, and furthermore b) the ...
2
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1answer
677 views

About the epsilon definition of a convergent sequence. Is this definition equivalent?

I read that it is appreciated to include the context and motivation of a question. I may have overdone this a little bit in this question. To summarize, my question is: Are the two blockquoted ...
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2answers
38 views

Let $x\in \mathbb{R}$ and $z\in\mathbb{C}$. And define equality ($x=z$) iff $x=(0,x)$. Is this equality well defined?

Let $x\in \mathbb{R}$ and $z\in\mathbb{C}$. And define equality ($x=z$) iff $x=(0,x)$. Is this equality well defined ? Okay. It is easily shown that something goes wrong if you define equality in ...
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4answers
605 views

Definition of General Associativity for binary operations

Let's say we are talking about addition defined in the real numbers. Then, by induction we define $\sum_{i=0}^{0}a_i=a_0$ and $\sum_{i=0}^{n}a_i=\sum_{i=0}^{n-1}a_i+a_n$ for $n> 1.\:$ Now, how do ...
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1answer
68 views

About definition of series associated to sequence!

let $f:\Bbb{N}\to \Bbb{R}$ a sequence, $n \in \Bbb{N}$ and $S \in \Bbb{R}$, $S$ is $n$th-partial sum of $f$ if $$S=\sum_{i=0}^nf(i)$$ let $g:\Bbb{N} \to \Bbb{R}$ and $h:\Bbb{N} \to \Bbb{R}$ two ...
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3answers
93 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
155 views

What does “weakly compact” mean when applied to subsets $X \subset Y$?

Let $X$ be a subset of a Banach space $Y$. Please can you give me a definition of what "$X$ is weakly compact" means? I want one which is in terms of sequences and boundedness, as opposed to one with ...
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1answer
117 views

Multiplication of positive fractional numbers.

I am reading this answer. We know that multiplication of two positive rational number $\dfrac{a}{b}$ and $\dfrac{r}{s}$ respectivly is defined as follows: $\dfrac{a}{b}\times\dfrac{r}{s} = ...
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3answers
343 views

Equality of positive rational numbers.

I am reading the second article Rational Numbers of the book "A Treatise on Advanced Calculus" by Philip Franklin. I have mainly 3 questions regarding this article. I am writing all these $3$ ...
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3answers
295 views

An example of an ordered, uncountable set in $\Bbb R$?

Is there an example of an ordered, uncountable set in $\Bbb R$? My Calculus professor, who likes to keep things simple, defined a sequence in $\Bbb R$ as an "ordered and infinite list of real ...
2
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0answers
1k views

Proper functions

I'm a bit confused with the definition of proper function, i.e.: "$f: X \to Y$ is proper if the inverse image of every compact set in $Y$ is compact in $X$." Can anyone provide a more intuitive ...
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2answers
131 views

How to proof that $\lim_{h \to 0}\frac{e^h-1}{h} = 1$ using the definition $e = \lim_{n \to \infty}(1+\frac{1}{n})^n$?

In other words, how I can prove that these two definitions of $e$ is equal? I saw these two definitions while trying to find proofs for $\frac{d}{dx}e^x$ and $\frac{d}{dx}\ln x$; some use the former ...
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2answers
397 views

Integral of a differential 1-form along a curve (clarification on the definition)

Let's denote with $(e_1,\dots,e_d)$ the usual basis of $\Bbb R^d$, and with $({e_1}^*,\dots,{e_d}^*)$ the dual basis of its dual space $\Bbb {(R^d)}^*$. Let $U$ be an open subset of $\Bbb R^d$ and ...
2
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1answer
75 views

Embedded and Non-Parametric Surface definition

What does it mean for a minimal surface to be embedded? For example the Scherk surfaces? How would I define what 'an embedded surface' is? And also what does it mean for a surface to be ...
16
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1answer
686 views

Definition of the nth derivative? [First post]

If the definition of the derivative is $$ f^\prime(x) = \lim_{\Delta x \to 0} \dfrac{f(x+\Delta x) - f(x)}{\Delta x} $$ Would it make sense that the nth derivative would be (I know that the 'n' in ...
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1answer
60 views

About definition of recursive sequence

Can I define the recursive sequence in the following? Let $f: \Bbb{N} \to \Bbb{R}$ a sequence and $k \in \Bbb{N}$ and $\forall x \in \Bbb{N}(f_x:=f(x))$ , $f$ is recursive sequence if ...
2
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1answer
1k views

Definition of “the surface measure”?

Let $\mu_n$ be the $n$-dimensional Lebesgue measure. Let $||\cdot||$ be a norm on $\mathbb{R}^n$. Define $S^{n-1}=\{x\in\mathbb{R}:||x||=1\}$. I have proven that $\forall A\in\mathscr{B}_{S^{n-1}}, ...
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0answers
51 views

Indecomposable groups vs. indecomposable objects

An object $X$ in a category $\cal C$ with an initial object is called indecomposable if $X$ is not the initial object and $X$ is not isomorphic to a coproduct of two noninitial objects. A group $G$ ...
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4answers
447 views

$\hom(V,W)$ is canonic isomorph to $\hom(W^*, V^*)$

Introduction My Semester just started and we have a new Professor for Linear Algebra II (replacing our former Professor). Apparently we are behind our schedule and thus we only had a brief ...
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2answers
52 views

Point as an element of an affine space vs point as an element of a topological space?

I am searching for the "most natural" definition of a (geometrical/space) point as an element of "something" in mathematics (I am trying to design a small computational geometry library on strong ...
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0answers
58 views

definition of a fiber bundle

I came across the definition of a fiber bundle in May's "A Concise Course in Algebraic Topology" (http://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf, Chapter 7, section 4). There were two ...
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1answer
360 views

Tangent space to a surface at boundary points

Let $M$ be a $2$-dimensional compact oriented surface in $\mathbb R^3$ with boundary $\partial M$. For any $p\in M \setminus \partial M$ tangent vectors are defined as speed vectors of smooth curves ...
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1answer
55 views

Definition of “point” and “vector” in $\Bbb{R}^n$, and a model for $\Bbb{A}^n(\Bbb{R})$..

Can I use the following definitions? let $(a_1,a_2,..,a_{n+1}) \in \Bbb{R}^{n+1}$, $(a_1,a_2,..,a_{n+1})$ is a point of $\Bbb{R}^n$ if $$a_1=1 \wedge \forall i \in \{2,...,n+1\} (a_i \in \Bbb{R})$$ ...
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4answers
242 views

Have any definitions in mathematics been redefined

Based on certain intuitions and motivations we make certain definitions and then proceed to use these concepts in further developing our intuition. For example, we have an intuition that a line has ...
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2answers
404 views

What is the zero subscheme of a section

Let $X$ be a scheme, $\mathcal F$ a locally free sheaf of rank $r$ and $s \in \Gamma(X, \mathcal F)$ a global section of $\mathcal F$. Question: What is the zero subscheme of $s$? I can't believe ...
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3answers
64 views

I'm confused about the definition of poset.

The definition of poset : $\qquad$A set with a partial ordering. Partial ordering is a binary relation $\preceq$ over a set ($P$). If I understood the definition of relation correctly, then ...
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1answer
32 views

What is the name for the property that a subset of a set follows the same rules as the set?

I have a set that follows a certain property and I want to say that the subsets of this set also follows the property. What is this called? I know that closure under an operation means that performing ...
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2answers
113 views

About definition of “direct sum of $p$-vector subspaces”

In the books 1 and 2, in Somme directe d'une famille de sous-espaces vectoriels, I am reading the following: 1) let $E,F$ two vector subspaces of $V$, $E+F$ is direct sum, $E+F \doteq E\oplus F$, if ...