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

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3
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1answer
680 views

Projective Normality

What is the significance of studying projective normality of a variety ? How does it relate to non-singularity, rationality of a variety ?
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2answers
105 views

Definition of notation $\mathbb Z_n$

What does the notation $\mathbb Z_n$ mean, where n is also an integer. I have only seen n being a positive integer up to now. some examples are $\mathbb Z_2$ or $\mathbb Z_3$ This is the context: How ...
2
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2answers
215 views

Definition of $C^k$ boundary

Can someone give me a resonable definition of $C^k$ boundary, e.g., to define and after give a brief explain about the definition. I need this 'cause I'm not understanding what the Evan's book said. ...
3
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1answer
315 views

“Total” degree of a polynomial?

What is the difference between the "degree" of a polynomial and its "total degree"?
0
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3answers
122 views

Understanding the definition of chain

I'm reading Pedersen's 'Analysis Now' at the moment and I'm quite confused about the definition of chain given there. I'm used to the definition that for a partially ordered set $X$ a chain $C$ is a ...
3
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3answers
201 views

What's the difference between $(\mathbb Z_n,+)$ and $(\mathbb Z_n,*)$?

I noticed that $(\mathbb Z_n,+)$ and $(\mathbb Z_n,*)$ are not the same thing. For example 2 is not invertible in $(\mathbb Z_6,*)$. However, since $\bar2+\bar4=\bar0$, thus it is invertible in ...
0
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1answer
72 views

Why Integral domains haven't an unified definition? [duplicate]

We can define integral domains as: rings without zero divisors commutative rings without zero divisors commutative rings with identity and without zero divisors I don't know why integral domains ...
2
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0answers
107 views

Is there any compelling reason why $0$ *shouldn't* be a natural number? [duplicate]

This seems to be (from what I've heard from various math people of various statures) a heated debate. One of my previous professors proclaimed very strongly that $0$ is not a natural number. Another ...
3
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0answers
76 views

Definition of small $o$

In one of my homework assignments (in intro to applied mathematics) there is the following definition: Given two functions $f(\epsilon),g(\epsilon)$ that are defined in $D=(0,\epsilon)$ we say ...
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1answer
58 views

Two definitions of totally bounded uniform spaces

Wikipedia gives this definition of totally bounded uniform space: a subset $S$ of a uniform space $X$ is totally bounded if and only if, given any entourage $E$ in $X$, there exists a finite cover of ...
3
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1answer
78 views

maximal antichain

I don't understand the definition of Jech (set theory) for "maximal antichain". Let $B$ a boolean algebra and $A$ a subalgebra of $B$. $W\subseteq A^+$ is a maximal antichain if $\sum W=1$ and $W$ ...
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2answers
45 views

Definition of localization of rings

I'm trying to understand this definition of Hungerford's book: The definition is simple, I think I understood what the author means, but... What is $P_P$? because we will have $P_P=S^{-1}P$, ...
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2answers
75 views

Definition of diffeomorphism functions

I see the definition of Diffeomorphism in Wikipedia homepage, but I don't understand whether "the differential of f (Dfx : Rn → Rn) should be bijective at each point x in U" or "f itself" should be ...
2
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1answer
140 views

Tetration and its inverse to various exponents

I've recently seen in my studies tetration, or the next operation in the addition, multiplication, exponentation... series. I've also heard much discussion about how to extend this operation to ...
2
votes
1answer
183 views

What is the exact definition of polynomial functions?

I'm trying to understand the difference between polynomial functions and the evaluation homomorphisms. I noticed that I don't know what's the exact definition of a polynomial function, although I've ...
2
votes
2answers
174 views

Normal subgroup if conjugate subgroup is subset

I find this explanation in Isaacs' Algebra: Lemma. Let $H\subseteq G$ be a subgroup. Then $H$ is a normal subgroup if $H^g\subseteq H$ for all $g\in G$. The reader should be warned that this ...
10
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3answers
354 views

How to write $\pi$ as a set in ZF?

I know that from ZF we can construct some sets in a beautiful form obtaining the desired properties that we expect to have these sets. In ZF all is a set (including numbers, elements, functions, ...
0
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1answer
240 views

Mapping on a set with respect to function composition

In Isaacs' Algebra, I found the following exercise Let $G$ be a group of mappings on a set $X$ with respect to function composition. Find an example where $G$ is not a subset of ...
2
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1answer
62 views

If for every $x_n$ such that $x_n \rightarrow x$, there exists a $x_{n_k}$ such that $Tx_{n_k} = Tx$, is $T$ continuous?

Let $X$ and $Y$ be Banach spaces and $T$ be the (possibly nonlinear) map $T\colon X \rightarrow Y$. $T$ is continuous if for every $x_n \in X$ such that $x_n \rightarrow x$, then $Tx_n \rightarrow ...
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4answers
171 views

Does Euler totient function gives exactly one value(answer) or LEAST calculated value(answer is NOT below this value)?

I was studying RSA when came across Euler totient function. The definition states that- it gives the number of positive values less than $n$ which are relatively prime to $n$. I thought I had it, ...
2
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2answers
63 views

Is there even a point in defining the notion of a 'metric' (as opposed to a metric space), etc.?

When we define a new mathematical structure, we generally double up on definitions. We define structures (think: metric spaces, partially ordered sets, etc.) and also the ingredients that they're ...
8
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3answers
463 views

What does area represent?

Since any two Euclidean shapes have an infinite number of points inside of them, shapes with different area have the same infinite number of points in them (and any object has the same number of ...
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1answer
215 views

Anonymous graphs and graph embeddedness

What are anonymous graphs, what is graph embeddedness, and how do they relate to each other? Very confused - I could not find short answer. Thanks.
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1answer
57 views

question about Morse theory in Hilbert space

This is sade to be the Morse theory in Hilbert space ,and i want to know the definition (or where i can find it ) of : The qth singular relative homology groupe The qth critical group Please; ...
0
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1answer
140 views

Calculate the determinant of a multilinear operator

How to calculate the determinant of a multilinear operator? Is it something different from the determinant of the linear operator? Thanks.
2
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1answer
1k views

What is the mathematical definition of index set?

I find some descriptions http://en.wikipedia.org/wiki/Index_set and http://mathworld.wolfram.com/IndexSet.html . But can't find any definition.
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4answers
358 views

What is the relationship between “recursive” or “recursively enumerable” sets and the concept of recursion?

I understand that "recursive" sets are those that can be completely decided by an algorithm, while "recursively enumerable" sets can be listed by an algorithm (but not necessarily decided). I am ...
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3answers
253 views

Why is the universal quantifier $\forall x \in A : P(x)$ defined as $\forall x (x \in A \implies P(x))$ using an implication?

And the same goes for the existential quantifier: $\exists x \in A : P(x) \; \Leftrightarrow \; \exists x (x \in A \wedge P(x))$. Why couldn’t it be: $\exists x \in A : P(x) \; \Leftrightarrow \; ...
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1answer
88 views

Ordinary differential equations with double resonance

I want to know what is the definition of "resonance, double resonance" in ordinary differential equations with double resonance for exemple this : what it means the probleme is resonant in infity ? ...
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2answers
197 views

Definition of Lebesgue-Stieltjes measure on $\mathbb R$

Let $F:\mathbb R\to\mathbb R$ be a non-decreasing, left-continuous function. Let $a,b\in\mathbb R$, then define the Lebesgue-Stieltjes measure $$ m[a,b]=F(b+)-F(a), \quad m(a,b)=F(b)-F(a+) $$ ...
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1answer
155 views

Ordered pairs in sets

So I am asked to explain why $(x,y,z)=(x,(y,z))$ is a reasonable definition of an ordered triple. However, I am in a limbo at the moment let alone with trying to explain an ordered pair,(x,y), ...
4
votes
1answer
87 views

Confused on definition of strong induction

I found the following statement in Munkres' Topology: Theorem 4.2 (Strong induction principle). Let $A$ be a set of positive integers. Suppose that for each positive integer $n$, the statement ...
2
votes
3answers
196 views

The negative square root of $-1$ as the value of $i$

I have a small point to be clarified.We all know $ i^2 = -1 $ when we define complex numbers, and we usually take the positive square root of $-1$ as the value of "$i$" , i.e, $i = (-1)^{1/2} $. I ...
2
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2answers
106 views

Help to understand the ring of polynomials terminology in $n$ indeterminates

In the Hungerford's book, page 150, the author defines a ring of polynomials in "n" indeterminates in the following manner: After the author defines the operations in this ring with a theorem: ...
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1answer
584 views

Why is a CDF right-continuous at “a” in [a,b), when property Pr(a<X≤b) doesn't even require point “a” to exist, and “b” could carry baggage?

c.f. wikipedia:Cumulative distribution function properties "Every cumulative distribution function F is (not necessarily strictly) monotone non-decreasing (see monotone increasing) and ...
2
votes
4answers
99 views

Definition of open set/metric space

On Proof Wiki, the definition of an open set is stated as Let $(M,d)$ be a metric space and let $U\subset M$, then $U$ is open iff for all $y\in U$, there exists $\epsilon \in \mathbb{R}_{>0}$ ...
2
votes
2answers
109 views

Why the terms “unit” and “irreducible”?

I'm trying to understand why in a ring we choose the names unit to an invertible element and irreducible element in this definition Maybe historical reasons? For example, I suppose the second ...
6
votes
3answers
982 views

If $f$ is differentiable at $x = x_0$ then $f$ is continuous at $x = x_0$.

Claim: if $f$ is differentiable at $x = x_0$ then $f$ is continuous at $x = x_0$. Please, see if I made some mistake in the proof below. I mention some theorems in the proof: The condition to ...
3
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1answer
96 views

Compatible PDEs

If we have an overdetermined system of pdes what does one have to check to be sure that they are compatible? Suppose each pde is derived from a different Hamiltonian and we have that the Poisson ...
2
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1answer
101 views

Is there a rigorous definition of a Young tableau?

In all combinatorics and algebra texts that I have seen so far, the notion of a "Young tableau" is defined in a somewhat informal fashion. The most common approach is stating that a Young tableaux is ...
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2answers
139 views

Definition of metastability

I was reading Terence Tao's blog post on analogies between soft and hard analysis when I saw that the soft analysis statement "$x_n$ is convergent" corresponds to the hard analysis statement "$x_n$ is ...
3
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0answers
51 views

Problem of understanding

Please , can someone help me to understand this part of text please What it means "... for some $k\in \mathbb{N} \cup \lbrace 0 \rbrace$ and uniformly fore a.e $t\in [0,2\pi]$" and please how to ...
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1answer
85 views

Definition: Gauss Sum - Where is the error?

In my algebraic number theory lecture we defined Gauss sums as follows. However, I am quite unsure whether this definition is correct. My intuition says "there is a mistake somewhere". I tried ...
3
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1answer
154 views

(Non) equivalence of regular cardinal definitions

The usual definition of a regular cardinal is "$\kappa$ is regular if $cf(\kappa) = \kappa$", which, assuming the axiom of choice, is equivalent to this definition: "$\kappa$ is regular iff it cannot ...
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2answers
163 views

Why do we use open intervals in most proofs and definitions?

In my class we usually use intervals and balls in many proofs and definitions, but we almost never use closed intervals (for example, in Stokes Theorem, etc). On the other hand, many books use closed ...
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1answer
124 views

Good source for Triebel-Lizorkin spaces?

I'm trying to look into different types of function spaces. At the moment, at least for function spaces involving integration, I only have $L^p$ and $W^{k,p}$. The next function spaces I thought I'd ...
2
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1answer
191 views

On the definition of divisors in Riemann Surfaces

The sum notation for a Divisor $D$ in a Riemann Surface $X$ (as in Miranda's "Algebraic Curves and Riemann Surfaces") is $$ D=\sum_{p\in X} D(p)\cdot p $$ That is, $D$ assumes the value $D(p)$ at $p$. ...
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2answers
2k views

Understanding big O notation

I'm not a mathematician by any stretch and I'm trying to translate some maths terms into simple maths terms. Please don't laugh, I do consider this complicated! The equations in question are ...
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2answers
91 views

Is there a theory of extensible definitions?

We can define $+$ as a function $\mathbb{N}^2 \rightarrow \mathbb{N}$, and then prove: Theorem 1. The range of $+$ is $\mathbb{N}$. If we later wish to extend $+$ to a function $\mathbb{Z}^2 ...
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12answers
5k 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 ...