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

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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
170 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
62 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
415 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
183 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
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
votes
1answer
138 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
942 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.
6
votes
4answers
309 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 ...
7
votes
3answers
227 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
83 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
189 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
147 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
85 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
191 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
104 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
503 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
96 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
102 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
862 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
91 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
93 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 ...
1
vote
2answers
132 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 ...
1
vote
1answer
81 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
votes
1answer
138 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 ...
10
votes
2answers
150 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 ...
1
vote
1answer
98 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
votes
1answer
171 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 $p$. ...
3
votes
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 ...
5
votes
2answers
89 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 ...
19
votes
12answers
4k 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 ...
2
votes
0answers
157 views

Taylor series of a vector field?

Can someone give me the definition of the Taylor Series of a vector field in $\mathbb{C}^2$? Thanks!
1
vote
1answer
183 views

$P \cap N$ is a Sylow $p$-subgroup of $N$, where $N$ is normal in $G$ and $P$ is a Sylow $p$-subgroup of $G$?

In 'A Course in Group Theory' by Humphreys, Proposition 11.14 says that if $G$ is a finite group, $P$ is a Sylow $p$-subgroup of $G$ and $N$ is a normal subgroup of $G$, then $P \cap N$ is a Sylow ...
11
votes
3answers
433 views

It is possible to define our intuitive notion for probability in subsets of $[0,1]$

I've always heard and read the sentence: If you pick a real number $x\in[0,1]$ at random, the probability to obtain a rational number is $0$. What is the meaning for that? Is this the "real" ...
1
vote
1answer
93 views

Two questions on $\limsup$: do nested ones commute and sending $n$ to $-\infty$

As for the second question, this is really just me wondering on definitions. For a function $f:\mathbb{R}\to\mathbb{R}$ define $$\limsup_{x\to+\infty}f(x):=\lim_{x\to+\infty}\sup_{z\geq x}f(x).$$ ...
1
vote
3answers
146 views

Is a function always a monotonically increasing function

For $\lim_{x \rightarrow a}f(x) = L$, if $f(x)$ is not a constant, is $\delta(\epsilon)$ always a monotonically increasing function? Alternatively, how does the definition of a limit guarantee that ...
5
votes
2answers
161 views

Does this qualify as a statement?

Is this a statement? All positive integers with negative squares are prime. What do we need to qualify as such?
1
vote
0answers
88 views

Which cut does the “minimum cut” refer to?

My course notes give the following definitions; could someone please verify that the last definition is non-standard? (I've spent all evening googling, and isn't "minimum cut" a concept related to cut ...
4
votes
2answers
95 views

How do different definitions of ellipse translate to the same thing?

There are 2 definitions of an ellipse that I know. One definition goes: The locus of a point moving in a plane such that the ratio of its distances from a fixed line (directrix) and a fixed ...
1
vote
1answer
35 views

Confusion of one definition in Fourier analysis

The symbol occurs on Page 22 of Bahouri's book Fourier analysis and nonlinear differential equations. As defined there, $$f(D)a:=\mathcal{F}^{-1}\{f\mathcal{F}a\}.$$ The question comes from the ...
2
votes
0answers
178 views

Are edge cuts, vertex cuts, and cut sets all variously called “cuts”?

I've seen "cut" being used to refer to all three, in different places, and sometimes in the same book. Which does "cut" most commonly refer to? p.s. I am aware that "cut" itself can be defined to ...
2
votes
0answers
229 views

Lemma of Whitehead

this is the lemma of Whitehead And i really don't understand the proof How to see that $k$ is well defined (i.e how to write an element from $X\cup_{\varphi_i}e^{\lambda} , i=0,1$ ) and how to ...
0
votes
1answer
112 views

How information works?

I am really confused after reading wikipedia... What I don't get is how can something "bring" information, and in mathematics, how a mathematical object (like a set) can "have" information. For ...
3
votes
6answers
148 views

Motivation for creation of complex exponentiation

I am curious how mathematicians came to develop complex exponentiation. How is the rule for complex exponentiation derived?
1
vote
1answer
75 views

What is a “distinguished automorphism” of a field?

Math people: The title is the question. The reason I am asking is that I am trying to determine exactly what fields can be used for an inner product. I posed that question at ...
0
votes
1answer
138 views

What is a parallel vector space and how do I show it is isomorphic to the solution space?

How can I create an isomorphism between the solution space and a parallel vector space. I'm not sure how to define the vector space and the isomorphism. $$ \begin{bmatrix} -2 & 4 \\ ...
4
votes
3answers
212 views

Definition of $\exp(x)$

I've been thought at school that the definition of $\exp(x)$ is the unique function satisfying $\exp'(x)=\exp(x)$ and $\exp(0)=1$. But I've never found this definition somewhere else. On Wikipedia ...
3
votes
1answer
202 views

The relationship between inner automorphisms, commutativity, normality, and conjugacy.

An inner automorphism of a group $G$ is defined to be a function $f: G \to G$ such that for $x\in G$ $f(x) = a^{-1}xa.$ I have three somewhat broad questions about this: Why is it related to ...
1
vote
1answer
43 views

Are these two statements(theorems) equivalent?

I am given this theorem: Let $H$ be a check matrix for a linear code $C$. Then $C$ has minimum distance $d$ iff. there exists a set of $d$, but no set of $d-1$, linearly dependent columns in ...