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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19 views

Examples of sans serif Greek used in published mathematics

Unicode contains several thousand mathematical symbols, including individual code points for different maths alphabets. For example, U+1D434 is "mathematical italic capital A" (𝐴), U+1D63C is "math ...
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1answer
30 views

What is the difference between disjoint union and union?

If $S = A \cup B$, then $S$ is the collection of all points in $A$ and $B$ What about $S = A \sqcup B$?, I think disjoint union is the same as union, only $A, B$ are disjoint. So the notation is a ...
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2answers
19 views

What do we call the set of elements fixed by an involution of the second kind?

If $A$ is an algebra over a field $F$, and $\sigma:A\rightarrow A$ is an involution of the second kind, then it seems natural to talk about the set $S=\{a\in A\mid\sigma(a)=a\}$. I am not finding any ...
7
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1answer
55 views

Never seen this notation before: $\int (y-f(x))^2 Pr(dx,dy) $

I have never seen an integral like this: $$\int (y-f(x))^2 Pr(dx,dy) $$ What is that? More precisely what is $Pr(dx,dy)$? And how is that integral defined? I found it in Elements of Statistical ...
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0answers
18 views

Notation for Line Segment vs. Directed Line Segment

This may be nit-picky, but I noticed inconsistencies in a high school math text I was reading, and I'm curious what the world thinks. For the most part throughout this textbook, notation is used as ...
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0answers
20 views

What is the standard notation for the Banach space of functions that vanish at infinity?

Given a topological space $X$ and a real (or complex) valued function $f$ on $X$, we say that $f$ vanishes at infinity if for any $\varepsilon>0$ there is a compact $K \subseteq X$ such that ...
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1answer
24 views

$L^1(0,T;(L^1_{loc}(\mathbb{R^N}))^N)$ - Confused about notations used for space and time dependent vector fields

I found this notation - $L^1(0,T;(L^1_{loc}(\mathbb{R^N}))^N)$ - in a paper of DiPerna and Lions concerning vector fields space and time dependent, "Ordinary differential equations, transport theory ...
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3answers
40 views

What is the `e` notation regarding to decimal numbers?

Say the following number: $\text{1.4E-46}$. What is the e meaning? I'm not talking about euler number. Thanks in advance.
2
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1answer
35 views

Why are contravariant vectors denoted with a superscript (and not a subscript)?

I wonder whether the choice of denoting contravariant/covariant vectors with a superscript/subscript is arbitrary (and could have been made the other way around), or whether there is a specific reason ...
2
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0answers
30 views

Notation: Vectors in a Matrix

In my lecturer's notes, he made use of the notation $$\begin{pmatrix}|&|&|\\\mathbf u&\mathbf v&\mathbf w\\|&|&|\end{pmatrix}$$ to denote a matrix which is made up of the ...
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2answers
52 views

Quantifier notation: $\forall n \implies \cdot$ versus $\forall n, \cdot$

I'm not sure which of the following two notations is the correct one (or, are both correct?). I've seen both being used by different professors. $\forall \varepsilon > 0\ \exists \bar n \colon ...
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3answers
114 views

What do these symbols mean: $\bigcap$, $\bigcup$, $\bigwedge$, $\bigvee$?

I know that some of these symbols are used in set theory like $A \cup B$, but that's not what I'm talking about. I have seen those symbols used in a way similar to $\Sigma$ summation and $\Pi$ ...
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2answers
27 views

What does the colon stand for in this equation

I'm reading Cohen's translation of the Principia. In chapter 7, on page 166 he states "In sec. 1, following prop. 1, there is a lemma, basically stating that if a : a - b = b : b - c = c : c - d ... ...
2
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2answers
40 views

Using Set Builder Notation on a set that jumps in intervals?

I'm new to the world of Discrete Mathematics. I have been reviewing a little on Set Builder Notation and have looked over the following site thoroughly: ...
2
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2answers
35 views

permutations of elements of vector

I'm trying to find a simple representation of a set I can describe with words, but not mathematically... probably a simple question.. Consider some two-element vector $q=(q_1,q_2)$. How can I ...
3
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0answers
54 views

Limits: “does not exist” vs “cannot be evaluated”

Assuming we have a limit which doesn't exist, i.e. $$\lim_{x\rightarrow x_0}{f(x)} \not{\exists}$$ Is the above wording and notation mathematically equivalent to saying "The limit cannot be ...
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1answer
41 views

What symbol is used for product topology?

Let $((X_k,\tau_k))_{k \in N}$ be topological spaces. The product topology $\tau$ on $X = \prod_{k \in N} X_k$ is the coarsest topology that makes all projections $\pi_k:X \to X_k$ continuous. Is ...
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0answers
14 views

Calculating Covariance from data

So wikipedia says that covariance is: $$ E[(X_i - \mu_i)(X_j - \mu_j)^T] \text{ for all i,j in the covariance matrix} $$ Right now, I have my code set up so that I multiply (dot product) my $i$ ...
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0answers
114 views

What does this probability notation denote: $P( Y=y | X =x)$?

What does this probability notation denote: $P( Y=y | X =x)$? I came across this while looking at my notes for optimal prediction rule. Optimal prediction rule: for each $x ∈ X$ , chose $y ...
2
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0answers
42 views

Is it correct to say $\frac{\text{d}}{\text{d}x}[F_{X}(x)] = f_{X}(x)$ or $\frac{\text{d}}{\text{d}x}[F_{X}] = f_{X}$?

Is it correct to say $\dfrac{\text{d}}{\text{d}x}[F_{X}(x)] = f_{X}(x)$ or $\dfrac{\text{d}}{\text{d}x}[F_{X}] = f_{X}$? This notation question comes up in probability, where $F_{X}$ is a CDF of a ...
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0answers
8 views

Index notation convention for 'all the elements in this dimension'

I am writing a paper that involves several matrix operations. An example an be: $$ \sum_i^N Y_{k, i} \cdot V_k $$ The equivalent matlab notation is ...
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1answer
87 views
2
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1answer
42 views

What do the square bracket signify in $\int [\text{d}\pi]f(\pi)$

I am reading this paper which repeatedly includes integrals such as, $$ P_M(\phi \to \phi') = \int [\text{d}\pi][\text{d}\pi'] P_G(\pi)\delta((\phi, \pi) - (\phi'', \pi'')) $$ Note ...
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2answers
50 views

Can't understand notation of formula

The notation in the image features a straight line after the first fraction with some sort of parameter (?) given below. What does this mean in English exactly?
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0answers
29 views

Math notation (filling in values)

I got the following formula: $$\text{D}=e^{-\pi\frac{|\lambda|}{|\omega|}}$$ When I solve $|\omega|$ I get: $$|\omega|=-\frac{|\lambda|\pi}{\ln\left(\text{D}\right)}$$ Now, how can I ...
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2answers
44 views

How would this combinatorial process be written in conventional mathematical notation?

Now, I am going to define an operation on a list of numbers, which I will treat as a set, for want of a better approach. This may not be ideal, or even conventional, so apologies for lack of clarity ...
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1answer
16 views

Notation for conditional expressions

In Recursive Functions of Symbolic Expressions and Their Computation by Machine, Part I, i.e. "the LISP paper", John McCarthy introduces a notation for conditional expression as follows: $(p_{1} ...
2
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1answer
31 views

General clarification for derivative notation

I am a bit confused on the different notations of derivatives, could you help me clear it up? The following can be interpreted as: the total derivative of f wrt x, or equivalently, the derivative ...
5
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2answers
84 views

Is there a symbol for ‘equal if defined’

Can anybody recommend a symbol for ‘equal if defined’ as an asymmetric concept? In contexts where one might write down notation for an undefined quantity (such as $1/x$ when $x$ might be $0$), ...
0
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2answers
49 views

Let $IJ$ be the set of all sums of elements of the form $ij$?

$(15)$ If $I,J$ are ideals of $R$, let $IJ$ be the set of all sums of elements of the form $ij$, where $i \in I$ and $j \in J$. Prove that $IJ$ is an ideal of $R$. This is a question from Abstract ...
2
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1answer
85 views

What does ≺ represent?

I've been reading through Axiomatic Set Theory by Patrick Suppes and stumbled upon this symbol, ≺, on page 97. The definition reads as follows: We now define in the expected manner the relation ≺ of ...
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2answers
88 views

How to formally write down mapping to a category with math notation?

I'm writing a paper and would like to describe my approach with formulas too. However, I have a problem with writing down the following mapping step (just a tiny step of my algorithm). Imagine, that ...
0
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0answers
23 views

Notation: Rows and Columns of Matrix

A purely notational question: It is possible to denote the rows of a matrix $\mathbf{A}\in\mathbb{R}^{n\times m}$ by $\mathbf{w_1},\ldots,\mathbf{w_n}\in\mathbb{R}^{1\times m}$ and the columns by ...
0
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0answers
11 views

How to denote Hermitian off-diagonal matrix

I have a matrix $$\begin{pmatrix} x+w & f^*+w \\ f+w & x+w\end{pmatrix}$$ which I can write $$x \mathbb{1} + w\mathbb{J} + \begin{pmatrix} 0 & f^* \\ f & 0\end{pmatrix}$$ where ...
0
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2answers
36 views

What does it mean when a letter has both superscript and subscript?

I have a formula for Bond Valuation of a Level Coupon Bond, but I don't understand the notation. It looks like: It's the bottom formula in the image below, starting with PV = What does it mean ...
0
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1answer
20 views

Difference between Gamma and partial Omega to describe a domain boundary

Is there a difference between referring to the boundary of a domain $\Omega$ as $\Gamma$ or $\partial \Omega$ ? Or is this just preference or synonyms of the same thing? From my experience, they seem ...
1
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0answers
44 views

What is the meaning of plus sign above brackets?

I was wondering what the meaning of the notation $\ ( )^+ \ $is? For instance, what would be the value for $\ (x)^+ \ $for different values of x, say: $\ x=-1, x=0, x=1 \ $? Thank you in advance. The ...
0
votes
1answer
21 views

Why $T_{\infty}\sin(x^2)$ could be written as $x^2+o(x^4)$?

I am confused about the little-oh notation in the Taylor series. As we know, $T_{\infty}\sin(x)=x - \frac{x^3}{3!}+\frac{x^5}{5!}+\cdots+(-1)^n\frac{x^{2n+1}}{(2n+1)!}+\cdots$ By substitution, I got ...
0
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1answer
31 views

$P\left(X_1 < X_2 < X_3\right) = P\left(X_1 \le X_2 \le X_3|X_1\ne X_2\ne X_3\right)$?

I have 3 independent random variables, $X_i$, distributed on a continuous uniform distribution between 0 and 1. Does the following hold given the assumptions above? $$ \tag{1} P\left(X_1 < X_2 ...
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0answers
26 views

Mathematics notations

When one says "a random permutation of $[n]$" what does "$[n]$" refer to? Here, $n \in \mathbb{N}\backslash\{0\}$. Also, what does a "$c\sqrt{n}$-tuple $S = (s_1, s_2, ..., s_{c\sqrt{n}})$ of ...
0
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0answers
25 views

finding theta notation with limit rule

I'm stuck at this problem while i using limit rule $$f(n) = 4\sqrt x $$ $$g(n) = (\log x)^2$$ $$\lim_{ n \to \infty} {4\sqrt x \over 5{(\log x)^2}}$$ i know that $f(n) = 4\sqrt x $ is asymtoticly ...
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4answers
2k views

What does this double sided arrow mean?

What is $\longleftrightarrow$ used for in mathematics? I know about $\iff$ being used for "If and only if". Are they the same thing? I was watching a YouTube video that said: $$\sum^{\infty}_{n=1} ...
1
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1answer
24 views

Understanding notation when finding the estimates in a linear regression model

I have been taught to calculate the estimates $\beta_0$ and $\beta_1$ (or $\vec{\beta}$) by using the following formula $$\sum_{i=1}^{n} y_i \vec{X_i}^T = \vec{\beta}^T\sum_{i=1}^{n} X_i ...
0
votes
1answer
25 views

What does this type of “division summation” notation mean?

$$\sum_{p | k} f(p)$$ What does this type of summation actually mean? Can you give me an example(s) with $p=1, 2$?
0
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2answers
93 views

what does ≼ or ≺ mean?

I was reading a paper about well-orderings and this came up: Suppose (E, ≤) and (F, ≼) are isomorphic well-orderings. Then there exists a unique isomorphism for (E, ≤) to (F, ≼). I've been scouring ...
0
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1answer
26 views

Matrix notation as vectors

I am a beginner in linear algebra, and I do not know how to interpret the following matrix notation: $$A = [w_1, w_2, ..., w_m, u_1, u_2, ..., u_n]$$ where $w_1, w_2, ..., w_m, u_1, u_2, ..., u_n$ are ...
0
votes
1answer
42 views

What does this matrix notation mean $(A)^\bot$?

I was doing a multiple choice exercise where I had to determine which one of the options was a basis for the matrix $$A = \begin{pmatrix} 4 & 2 & 0 & -1\\ 1 & 2 & 3 & 0\\ 5 ...
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2answers
49 views

Suppose $f$ is differentiable and $f^\prime(x) = 0$, then $f(x) = o(x)$ at $x\rightarrow 0$?

I know there are two ways to understand $o(x)$, in this case we only discuss in the situation of $\lim_{x\rightarrow 0} \frac {f(x)}{g(x)}=0 \Longleftrightarrow f(x) = o(g)$ So I wonder whether it is ...
2
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2answers
71 views

What does $2^{\mathbb R}$ mean? [closed]

Does it mean $2^{\mathbb R}= P{ (\mathbb R)}$ where $P$ is the power set. What does it mean in terms of functions like $l : 2^{\mathbb R} \rightarrow [0,+ \infty]$?
0
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1answer
18 views

Restricting universal quantifiers with conditions?

I want to express "For every $x\in\mathbb R$ that has (at least) one $p\in\mathbb Z$ and $q\in\mathbb N$, such that $x=\frac pq$, $x\in\mathbb Q$ is true" with logical quantifiers, just like this ...