Question on the meaning, history, and usage of mathematical symbols and notation. Please remember to mention where (book, paper, webpage, etc.) you encountered any mathematical notation you are asking about.
2
votes
1answer
41 views
Interpretation: existence of $2$ elements in a set. Simple question.
I'm ask to decide if there exists $2$ elements $a$,$b$ in a set such that $a+b =8$.
It seems to me that many understand that $a$ and $b$ need to be two different elements. How should I understand ...
1
vote
1answer
17 views
Asymptotic Approximation and Sign Convention
When I write the asymptotic approximation of a function, does the sign convention matter? i.e. suppose I have (though the formula might not make sense) $$f_n(x)=x^2+\dots-O(n),$$
If my function is ...
0
votes
3answers
38 views
Reference a vector component
If I have the following cartesian vector:
$$
\vec{v_1} = \pmatrix{1 \\ 2}
$$
How would I reference the second vector component of: $$ \vec{v_1} $$
Is it okay to do it this way: $$ \vec{{v_1}_y} = 2 ...
0
votes
3answers
33 views
Notation: permutation and its inverse
Consider the sequence $S = (A, B, C, D, E)$ and the permutation $\pi = (4, 1, 3, 5, 2)$:
Which of the following is true?
$$ \pi(S) = (B, E, C, A, D) \quad and \quad \pi^{-1}(S) = (D, A, C, E, B) ...
0
votes
2answers
31 views
Inner product convention for $\ell^p$?
So I'm reading through some analysis problems and one is discussing $\ell^p$ (the space of $p$-summable sequences $x: \mathbb Z^+ \to \mathbb C$ such that $\sum_{n \in \mathbb Z^+}|x_n|^p < ...
1
vote
1answer
83 views
What does the notation $C(\mathbb{R})$ mean?
I thought that $C(\mathbb{R})$ was the set of all $$\mathbb{R}\to\mathbb{R}$$
functions that are continuous, but I may of seen a case that the function
was $$\mathbb{R}\to\mathbb{C}$$
Was the use of ...
2
votes
1answer
41 views
What is the difference between a probability distribution on events and random variables?
For the purpose of simplicity, assume everything below is only in the discrete domain.
A $\text{probability space}$ is usually defined as a triple $(\Omega , 2^\Omega , P)$ where
$\Omega := ...
0
votes
0answers
19 views
conditional matrix form
I am deriving a matrix form for part of an equation which demands a conditional form but I have trouble in making it so that to be acceptable in scientific communities. Let's assume that we have ...
4
votes
1answer
46 views
Is the Knuth arrowup notation defined for non-natural exponents?
I recently found out about Knuth's arrowup notation. Wikipedia, among other websites, only shows a definition for $a \uparrow^n b$ where $n \in \Bbb{N}_0, a \in \Bbb{R}, b \in \Bbb{N}$ as following:
...
2
votes
2answers
32 views
Einstein Summation with multiple terms
I know the basics of Einstein Summation but i've got an equation here that is a little more complex than the easy examples i'm only finding on this subject:
$C = (p-nT) \partial_\gamma u_\gamma + ...
2
votes
2answers
55 views
Help with unknown notation
In Ahlfors' Complex Analysis, page 19 it says (in relation with the Riemann sphere):
"writing $z=x+iy$, we can verify that: $$x:y:-1=x_1:x_2:x_3-1, $$ and this means that the points ...
1
vote
0answers
60 views
French notational differences
I wish to read some French probability / measure theory papers. I do not wish to be caught out with different notations. For example if "compact" in French has a weaker meaning than in English ...
0
votes
1answer
22 views
congruence modulo and equality
why in cryptography most of the equalities written in the form of
$$a:=b$$
why not we write $a=b$
why in congruence modulo $a \equiv c \pmod b$ that bracket is put. Is it refers the priority.
can ...
1
vote
0answers
31 views
Is there a common notation for the labelled degree of a vertex?
Let $G$ be an undirected graph with labelled edges. The labelled degree of a vertex $v \in V(G)$ is the number of edges incident to $v$ with distinct labels.
The definition of the labelled degree ...
1
vote
2answers
101 views
1
vote
0answers
24 views
Notation for Restriction of Permutation
Suppose $\sigma$ and $\tau$ are permutations such that $\sigma(x)\not=x\implies \sigma(x)=\tau(x)$. Intuitively, I would like to think of $\sigma$ as a restriction (or projection) of $\tau$ onto a ...
1
vote
1answer
34 views
Notation in numerical methods
What are the differences in using $h$ and $\triangle t$ to represent a time step?
1
vote
1answer
35 views
Can we write $\sqrt[w]{z}=z^\frac{1}{w}$ when both $w$ and $z$ are complex numbers? [duplicate]
Let $w$ and $z$ be complex numbers defined in terms of real numbers $a$, $b$, $c$ and $d$ as follows:
$$ w = a+bi \\ z = c+di $$
Can we analogically write
$$ \sqrt[w]{z} = z^\frac{1}{w} \qquad ...
0
votes
1answer
17 views
Mixing asymptotic notations
I have a function $f(x) = g(x) - h(x)$ and I know that $g(x)=\Omega(\hat g(x))$ and $h(x)=O(\hat h(x))$. Is it well-defined to express this in asymptotic notation, as $f(x) = \Omega(\hat g(x))-O(\hat ...
1
vote
1answer
39 views
Limit of a seqence $\{f_n \}_{n\in \mathbb N}$ of functions?
I don't really know how mathematicians talk about this concept.
I try to explain better what I mean with limit of a sequence of functions:
Given a countable set of functions $\{f_n \}_{n\in \mathbb ...
2
votes
0answers
49 views
“Product” bundle notation.
Let $\newcommand{\Spin}{\operatorname{Spin}}M$ and $M'$ be two manifolds, equipped with a principal $\Spin_n$ and $\Spin_{n'}$ bundle called $P$ and $P'$, respectively.
Then there is an induced ...
0
votes
0answers
15 views
Order of operations
Looking for a source that lists every possible proper order of operations.
The reason for this is that I made a mistake with fractional exponents, I don't want to make this mistake again.
3
votes
1answer
56 views
What is Baire's zero-dimensional metric space?
I'm not familar with metrizable spaces. I met a notation: Baire's zero-dimensional metric space. Could somebody explain it for me? Thanks ahead:)
2
votes
0answers
35 views
What does the notation $\mathbf{Lt}$ signify in a limit?
The author of a book I'm reading defines the Dirac delta-function as
$$
\delta(\omega_0-\omega) = \frac{2}{\pi}\mathbf{Lt}_{t\rightarrow \infty} \frac{\sin^2\left(\frac 1 2 ...
1
vote
3answers
28 views
Help with notation for tuples
How would I represent a set of tuples $(a_1,a_2,a_3,...)$, where each element $a_i$ is a positive integer in the interval $1\leq a_i \leq A_i$
The only thing I can think of is defining the set of ...
3
votes
1answer
40 views
Understanding fourier notation $F(\partial_x)$
Can somebody please help me understand some of the notion in the equations below, taken from a published paper on image de-blurring.
I have an energy $E(H)$ defined over an image $H$, a point-spread ...
0
votes
1answer
24 views
How do differentiate between function arguments and multiplication?
Say I have the following:
$$ H\left(\frac x{x_o}\right) $$
How do I see that it is the $H$ function at $x/x_0$ and not the quantifty $H_n$ premultiplied by $x/x_0$?
In Mathematica, I would have ...
2
votes
1answer
51 views
What does $\Omega^\bullet(M)$ mean?
What does $\Omega^\bullet(M)$ mean?
I know that $\Omega^k(M)$ is the set of all differential k-forms.
Thanks in advance!
2
votes
1answer
51 views
what does it mean for a matrix to be greater than another?
I am reading these notes on viscosity solutions, here is a theorem:
Let us assume $u\in C^2$ is a classical solution of
$F(x,u,Du,D^2u)=0$, $x\in \Omega$
then $u$ is a viscosity solution whenever ...
0
votes
1answer
52 views
Is there any standard notation for specifying dimension of a matrix after the matrix symbol?
I want to explicitly specify dimension of matrices in some expressions, something like
$$\boldsymbol{A}_{m \times n} \boldsymbol{B}_{n \times m} = \boldsymbol{C}_{m \times m} \, .$$
Is there any ...
0
votes
1answer
52 views
Notation Clarification of Koch Curve
I am having trouble making sense of the notation used to describe the Koch Curve in the book Getting Aquanted with Fractals. The link will take you to a preview of the book which describes the ...
2
votes
1answer
81 views
What is the standard notation for a set of equivalence classes?
What is the standard notation for a set of equivalence relations? Specifically, I have a pair of objects, call them $x$ and $y$ and I denote the ordered pair as $\left(x,y\right)$. I have a set of ...
0
votes
1answer
18 views
Single variable true or false statements
If I have a true or false statement S, depending on a varible x, is there some standard function or opperation in formal logic, that takes the statement S and the variable x, and outputs $1$ if ...
3
votes
1answer
22 views
Notating each element of a vector which already has a subscript
If I had a vector $\mathbf{x}$, I would denote element $i$ as $x_i$.
However, if my vector already has a subscript, for example $\mathbf{x}_j$ or $\mathbf{x}_{10}$, how should I show element $i$?
I ...
3
votes
2answers
82 views
Meaning of $\log$
If you write $\log{x}$ rather than ${\log_a{x}}$ for some base $a$, does it have a particular meaning? Sometimes I see people leave off the base by mistake when posting questions and it seems from the ...
2
votes
1answer
50 views
Help with Notation
Let $S_{1},S_{2},S_{3},....$ be a sequence of mathematical statements, each of which is dependent on the same variable, such that, at any given moment exactly one and only one of the statements can be ...
3
votes
2answers
102 views
Less than all positive numbers, greater than all negative numbers, and not zero; what is $\ast$?
A part two, you could say, of my previous question.
I was watching a Numberphile video about the favorite number of some mathematicians, and at one point, the creator of ViHart said the following -
...
1
vote
1answer
206 views
$j^2 = 1$, but $j \neq \pm 1$; what is $j$?
I was watching a Numberphile video about the favorite number of some mathematicians, and at one point, the creator of MinutePhysics said the following -
Similar to the way that $i$ is $\sqrt{-1}$, ...
3
votes
2answers
72 views
Question about the radical of the Jacobson radical.
I am confused about the notation $\operatorname{rad}^2 A$. It can be considered as $\operatorname{rad}(\operatorname{rad}(A))$ or as $(\operatorname{rad}(A))^2$. Are ...
0
votes
3answers
42 views
Summation and brackets
Is it possible to write $\sum_{i=1}^k y_i\log p_i +(n_i-y_i)\log(1-p_i)+\log \binom{n_i}{y_i}$ as one mathematician said it is correct but another said that one should write $\sum_{i=1}^k\left ( ...
0
votes
0answers
52 views
What do angle brackets ($\langle\rangle$ ) mean in mathematics/statistics (autocorrelations)?
Okay, so the logarithmic return on a stock is given by:
$$r_τ (t) = \ln P(t+τ) - \ln P(t),$$ where τ is the interval of time.
I have no problem calculating that. My question comes to the following ...
1
vote
1answer
40 views
Iwaniec Kowalski Notation
On page 532 of the book analytic number theory by Iwaniec and Kowalski, the following notation is used:
$C^{~\infty}$ and $\tau(n,\chi)$.
Could anyone tell me what these represent? (the former is ...
1
vote
1answer
42 views
What's this matrix called?
In an inner product space, $v_1,\dotsc,v_n$ are linear independent iff the matrix $A_{ij} := \langle x_i | x_j \rangle$ is invertible. What's the name of this matrix??
2
votes
4answers
55 views
Meaning on the sign << [duplicate]
What does the sign '<<' mean, for example angle X << 1?
This was in a brilliant.org problem. I won't post the full problem since that would make me a cheat but the first line of the ...
2
votes
1answer
70 views
What's wrong with this proof of unique factorization?
I've never seen this proof of the unique factorization theorem (aka the fundamental theorem of arithmetic). This doesn't mean much, since my reading in number theory is scant.
I like the proof, ...
0
votes
0answers
24 views
What are $\sigma$ and $\tau$ in $n = 2^{\sigma(n)}\,\tau(n)$ called?
Every non-zero integer $n$ can be factored uniquely as $2^s t$, where $s$ is a non-negative integer, and $t$ is an odd integer.
In other words, there exist functions $\sigma:\mathbb{Z}\backslash ...
2
votes
2answers
27 views
Nesting of different Asymptotic operators
Is it possible to nest big-oh notation with omega-notation? I came across this here, while doing calculations on an exercise:
$$
f(x) \in O(\Omega(\log x))
$$
I'm really unsure on how to properly ...
4
votes
2answers
126 views
In set theory, what does the symbol $\mathfrak b$ mean?
In set theory, what does the symbol $\mathfrak b$ mean? Could somebody tell me something basic about $\mathfrak b$? In particulat, I want to know the relation between $\mathfrak b$
and $\mathfrak c$.
...
0
votes
2answers
85 views
What does $a^b$ mean?
Notation $a^b$ seems to be ubiquitous in mathematics and I think that most of us take it for granted. But, at least to me, it seems that it means totally different things depending on the context.
...
3
votes
1answer
39 views
Notation: What is the scope of a sum?
I would interpret $\sum_{i=1}^2 x_i + y$ as $x_1 + x_2 + y$, but I would interpret $\sum_{i=1}^2 x_i + y_i$ as $x_1 + y_1 + x_2 + y_2$. I realize this is a little inconsistent. Should the latter be ...
