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

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4
votes
2answers
103 views

Hartshorne Lemma(II 6.1)

Hi I am wondering what Hartshorne means by "an open affine subset on which $f$ is regular". This is part of the first sentence in the proof of lemma 6.1 in chapter two of the book "Algebraic Geometry ...
0
votes
3answers
78 views

Idea of a Random Variable

Two weeks into my class, I am still struggling with the idea of a random variable. I can see why random variables make sense when the outcomes are numbers, like monetary gains and losses. But if ...
1
vote
1answer
76 views

Is this definition of diagonal matrix correct?

I need to know if the following definition: Let $A:=\|a_{i,j}\|_{\substack{i=1,...,m \\ j=1,...,m}}$ be a square matrix. $A$ is diagonal matrix if $$i\neq j \implies a_{ij}=0, \quad\forall i,j \in ...
5
votes
4answers
465 views

Confusion about the usage of points vs. vectors

As far as definitions go, understand the difference between a vector and a point. A vector can be translated and still be the same vector, whereas a point is fixed. But I would like some clarification ...
5
votes
3answers
136 views

What does $\mathbb Z_+$ mean?

I am not so sure whether the meaning of $$\mathbb Z_+$$ is very clear. How many different definitions are there? Does the definition that is used depend on whether the writer is English or German? In ...
17
votes
4answers
1k views

$\sqrt 2$ is even?

Is it mathematically acceptable to use Prove if $n^2$ is even, then $n$ is even. to conclude since 2 is even then $\sqrt 2$ is even? Further more using that result to also conclude that $\sqrt [n]{2}$ ...
0
votes
2answers
66 views

Constraint satisfaction problem - Arc consistency

The Constraint satisfaction problem (CSP) is roughly speaking a formalism that defines a finite set of relations over a domain. The relations are simply defined by enlisting elements in certain ...
-1
votes
1answer
54 views

How these statements are defined by a single word?

How to describe these statements in one word? It takes values from a given set of values It is an algebraic expression where the only operation is multiplication. It is the algebraic sum of ...
3
votes
1answer
786 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 ?
1
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2answers
140 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
votes
2answers
258 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. ...
4
votes
1answer
399 views

“Total” degree of a polynomial?

What is the difference between the "degree" of a polynomial and its "total degree"?
0
votes
3answers
127 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
votes
3answers
228 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
votes
1answer
73 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
votes
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
votes
0answers
80 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 ...
0
votes
1answer
68 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
votes
1answer
81 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 $W$ ...
0
votes
2answers
47 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 $P_P=S^{-1}P$, ...
1
vote
2answers
84 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
votes
1answer
173 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
198 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
189 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 ...
12
votes
3answers
373 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
votes
1answer
258 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
votes
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 ...
0
votes
4answers
176 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
votes
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
votes
3answers
503 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 ...
1
vote
1answer
246 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.
1
vote
1answer
59 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
votes
1answer
145 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
votes
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.
7
votes
4answers
407 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 ...
8
votes
3answers
259 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 \; ...
1
vote
1answer
96 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 ? ...
1
vote
2answers
208 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 $$ m[a,b]=F(b+)-F(a), \quad m(a,b)=F(b)-F(a+) $$ ...
1
vote
1answer
165 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
89 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
200 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
votes
2answers
108 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: ...
1
vote
1answer
639 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
103 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 $\epsilon \in \mathbb{R}_{>0}$ ...
3
votes
2answers
116 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
1k 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
votes
1answer
114 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
votes
1answer
108 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 ...
2
votes
2answers
165 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
votes
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 ...