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.

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3
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1answer
4k views

what does z subscript something mean

Decide a positive integer $N \in\mathbb Z$. Generate a uniformly distributed random positive integer sequence: $$v_1, v_2, \ldots,v_n\in\mathbb Z_N$$ My question is, what does $\mathbb Z_N$ really ...
2
votes
1answer
109 views

Any name for an isosceles triangle sides

Is there an English translation for Finnish words kanta and kylki? Namely, if $ABC$ is an isosceles triangle with $AB=AC$ then $BC$ is kanta in Finnish and $AB$, $BC$ are both kylki.
0
votes
1answer
63 views

What is being maximised in the channel capacity formula?

The channel capacity formula is given as such: $$C=\max_{p(x)}I(X,Y)$$ Does this mean that it is the maximum probability multiplied by the mutual information, or is something else being maximised ...
1
vote
3answers
79 views

Difference between $\land$ and braces

I was wondering what are the difference between the $\land$ and $\begin{cases} \\ \\ \end{cases}$ symbol. As I know, they both mean "and". So far, I've noticed the $\land $ on statements (not sure ...
2
votes
2answers
700 views

Probability notation

Hey guys, I was just wondering why in my textbook(A First Course in Probability, 8th edition) and basically everywhere I've looked at when we have some random variable(assume for the sake of the ...
1
vote
1answer
31 views

How to name a matrix with restricted input values?

How should I refer to a matrix with a restricted domain of possible values that can be stored inside?
0
votes
2answers
513 views

Elementary Set Theory - Relations

I'm not exactly sure what to search for this problem I'm having, as I don't know the keywords, so I figured the best action would be to ask a question. I have this question: ...
1
vote
2answers
115 views

What does $\mathbb{\bar C}$ denote in complex analysis?

What does $\bar A$ denote when $ A \subseteq \mathbb{C}$? I've seen it used in some places as the algebraic closure, other places as $\bar A = A$ \ $ \partial A $ and other places again as $\bar A = ...
0
votes
2answers
35 views

Notation for number of value changes in a sequence

Let $A=\{a_{1}, a_{2}, a_{3}, a_{4}, ...,a_{n}\}$ be a finite sequence , where $a \in \mathbb{N}$. I would like to know the notation for something similar to a change rate. If I programmed, what I ...
4
votes
2answers
155 views

Summation and Product Bounds

If I have a sum or product whose upper index is less than its start index, how is this interpreted? For example: $$\sum_{k=2}^0a_k,\qquad \prod_{k=3}^1b_k$$ I want to say that they are equal to the ...
0
votes
1answer
145 views

Should brackets be placed around an exponentiated factorial?

For example, one can derive an approximation of $\pi$ from Stirling's approximation with one additional term as $$ \lim_{n \to \infty} \frac{72n(n!)^2}{n^{2n} e^{-2n} (12n+1)^2} $$ but is it correct ...
2
votes
1answer
55 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
28 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
51 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 ...
2
votes
3answers
565 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
75 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
335 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 ...
4
votes
1answer
2k 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 := ...
4
votes
1answer
81 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
356 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 + ...
3
votes
2answers
69 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 ...
2
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0answers
309 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
111 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 ...
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0answers
173 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 ...
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vote
2answers
201 views

What is $\ln 0$?

What is $\ln 0$ ? Is it $-\infty$ or indeterminate?
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0answers
42 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
76 views

Notation in numerical methods

What are the differences in using $h$ and $\triangle t$ to represent a time step?
1
vote
1answer
76 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 ...
1
vote
1answer
66 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
57 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
125 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 ...
1
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0answers
53 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
126 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
73 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
360 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
81 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
37 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
148 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!
4
votes
2answers
1k 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 ...
3
votes
3answers
4k 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 ...
3
votes
1answer
123 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 ...
5
votes
2answers
1k 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
40 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
56 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
163 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
58 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
1k 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 - ...
2
votes
1answer
657 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 $\sqrt{-1}$, ...
2
votes
1answer
47 views

Problem with a summation suffix.

Please can someone tell me why we drop the summation of i here? $S = \sum_{i,j}(y_{ij}-\mu -\alpha_i)^2$ $\frac{dS}{d\alpha_i} = \sum_{j}(y_{ij}-\mu -\alpha_i)$ It's part of a question which is ...
3
votes
2answers
149 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 ...