Definitions are the core of mathematical precision; they come to answer "what is X" in mathematics. Into this category fit questions regarding equivalence of definitions; clarifications regarding complicated definitions; as well questions with purposed definitions for mathematical notions with ...
4
votes
1answer
44 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
159 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 ...
1
vote
2answers
33 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
41 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
36 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
69 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 ...
4
votes
3answers
113 views
Proving if $f(x)$ is differentiable at $x = x_0$ then $f(x)$ 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 $f(x)$ be continuous at $x=x_0$ is $\lim\limits_{x\to x_0} f(x)=f(x_0)$.
(1) If $f(x)$ ...
3
votes
1answer
29 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
41 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 ...
0
votes
2answers
65 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
42 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
52 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
77 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 ...
8
votes
2answers
97 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
28 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
42 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
127 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
63 views
Is there a theory of extensible definitions?
We can define $+$ as a function $\mathbb{N}^2 \rightarrow \mathbb{N}$, and then prove the theorem:
Thm. The range of $+$ is $\mathbb{N}$.
If we later wish to extend $+$ to a function $\mathbb{Z}^2 ...
17
votes
11answers
2k 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
45 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
47 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 ...
10
votes
3answers
254 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
32 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
113 views
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 if $f(x)$ is not a constant, then a small $\epsilon$ will give me a small $\delta$?
5
votes
2answers
84 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
47 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 ...
3
votes
2answers
42 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
30 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
41 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
215 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
0answers
24 views
Definition: “A contour respects causality”
When doing a contour integral, what does "the contour respects causality" mean?
0
votes
1answer
102 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
62 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
43 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
38 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 \\
...
3
votes
3answers
102 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 ...
2
votes
1answer
72 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
32 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 ...
2
votes
1answer
26 views
The definition of $p$ capacity of a set $A\subset\mathbb{R}^n$
I am having a bit of difficult understanding the definition of the $p$-capacity of a set $A\subset\mathbb{R}^n$ and I was wondering if anyone would be able to clarify whether I have the right idea or ...
1
vote
0answers
45 views
How can I show using the $\epsilon-\delta$ definition of limit that $u(x,y_0)\to l$ as $x\to x_0?$
Let $u(x,y)\to l$ as $(x,y)\to z_0=(x_0,y_0).$ How can I show using the $\epsilon-\delta$ definition of limit that $u(x,y_0)\to l$ as $x\to x_0?$
Choose $\epsilon>0.$ Then ...
1
vote
1answer
92 views
Definition of correspondence
A one-to-one correspondence is an alternative name for a bijection between two sets, but to what does the term 'correspondence' alone refer? As far as I can see, it seems to be another term for ...
1
vote
1answer
50 views
Defining Test-Objects
In various categories one encounters what are referred to as 'test-objects'. For example, singleton set serves as test-object (in testing the equality of functions) in the category of sets. In the ...
1
vote
1answer
41 views
geometrically finite hyperbolic surface of infinite volume
I am starting to read some papers involving analysis on hyperbolic manifolds. In these the notion of a
"geometrically finite hyperbolic surface of infinite volume"
is mentioned frequently and I am ...
5
votes
1answer
83 views
Defining $\Bbb{Q}$ without the axiom of infinity
(TL;DR version: I want a meaningful definition of $\Bbb{Q}$ without $\sf{Inf}$.)
In the "conventional" construction of the rationals, we define $\Bbb{Q}$ as follows:
$\omega$ is the first limit ...
2
votes
2answers
39 views
Name of the order with irreflexivity, antisymmetry and transitivity?
I have an order otherwise poset aka partial order but it is irreflexive so relationships such as 1R1 and 2R2 are impossible. What is the name of this order?
1
vote
2answers
37 views
Equivalence relation proofs: general or specific?
I'm confused about whether a specific example must exist to prove an aspect of an equivalence relation.
For example: if a set, $A$, only contains one element, $A = \{1\}$, and a relation, $R$, on ...
0
votes
2answers
48 views
Definition of logarithm in complex domain
My first question is:
What is the proper definition of logarithmic function $f(z)=\ln{z}$.
where $z\in \mathbb{C}$.
quoting Wikipedia.
a complex logarithm function is an "inverse" of the ...
0
votes
1answer
28 views
Difference in reference of L space in Fubini Tonelli
I think my understanding of how $L^+$ and $L^1$ spaces are defined (I'm using Folland) is a little hazy. For example, in the Fubini-Tonelli theorem:
$\textbf{For the Tonelli part:}$ We start with if ...
1
vote
3answers
84 views
Cantor Set and Fractals
I have read that the Cantor set is considered a fractal. I am referring to the Cantor set in which the middle third of a real line is removed recursively. I see that this is recursively defined, but ...
3
votes
1answer
40 views
What does it mean for the determnant of a matrix to be independent of the vector space
Explain what it means for the determinant of the matrix, representing an operator $F$, to be independent of the basis of the vector space. Prove this property of the determinant.
I'm not exactly ...
