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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9
votes
2answers
282 views

Does this math formula with $1/PI\begin{cases}^\infty_{-\infty}\end{cases}$ mean anything?

I happened upon an image on the Internet and I wondered whether the math on the chalkboard means anything. Using Google's image search, I found three other related images. Piecing together ...
4
votes
1answer
51 views

Notation for the set of holomorphic differential forms

I used to write $\Omega^p(M)$ for the set of $p$-forms on $M$ and $\Omega^{p,q}(M)$ for the set of $(p,q)$-forms on a complex manifold $M$. Now some authors use $\Omega^p(M)$ to denote the set ...
1
vote
2answers
39 views

Notation for a unit vector

If I have a vector $\vec{v}$, is there a standard concise notation for the unit vector in the same direction of $\vec{v}$ that is $\frac{\vec{v}}{|\vec{v}|}$?
2
votes
0answers
51 views

Is there a name or symbol for the matrix division resulting in a scalar?

I am not talking about the inverse matrix, $A^{-1}$ which gives $A\times A^{-1}=I$, but rather the operation $\frac{1}{n}tr(\space\cdot \times A^{-1})$, which gives 1 when applied to a $n\times n$ ...
0
votes
2answers
67 views

Matrix notation in handwriting

I understand that typically matrices are printed in bold to distinguish them from other mathematical entities with the same symbols. However I find it difficult to actually handwrite in bold. With ...
1
vote
1answer
33 views

What does the $2$ in $\|Xw-y\|_2$ mean?

I was going through this link and I found the minimization expression. It says $\|Xw-y\|_2^2$. What is the significance of $2$ in the subscript? What does it mean? I understand that the double ...
1
vote
2answers
35 views

Game Theory: Can someone explain the notation used in the definition of “best response”

I am reading a paper which states that that the best response correspondence of a player is mapping: $B_i(s_{-i}): S_{-i} \Rightarrow S_i$ such that $B_i(s_{-i}) \in arg\ max_{s_i \in S_i} ...
2
votes
2answers
60 views

What do the symbols $\mathbb{Z}$ and $\mathbb{Z}_n$ mean on this discrete math problem? [closed]

Currently I have come across a problem set which I cannot decipher or begin to ask or search because I do not know what kind of notation or problems these are. Please circle the best description: ...
3
votes
1answer
68 views

Correspondence between Ext group and extensions (from Weibel's book)

I am trying to understand the proof of Theorem 3.4.3 from Weibel's book Introduction to homological algebra. The statement is the following. Let $R$ be a ring. Given $R$-modules $A$ and $B$, an ...
1
vote
1answer
31 views

Can somebody explain the notation $f \in C^4$

To give some context the full question is: Suppose $f \in C^4$ in a interval containing the root $\alpha$ and that Newton’s method gives a sequence of iterates $\{x_k\}$, $k = 0, 1, 2, \dots$ which ...
1
vote
1answer
29 views

Set builder notation for matching element pairs

I have a set of pairs, $S = \{ \langle a,b \rangle_1, \langle a,b \rangle_2, ..., \langle a,b \rangle_n \} $ where $a$ is not unique amongst the pairs. If I want to express the extraction of all the ...
4
votes
2answers
89 views

How I can express in a pure symbolic way common reasoning? Examples inside.

I have a broad question here, I know, but I will go define it clearly through examples. I want to know how express reasoning in a pure symbolic way, with no words, this is possible? Example: I was ...
0
votes
1answer
21 views

fourier series notation question

Find the fourier series for the given function $$f(x)=-x \quad \text{for } -L\le x < L, f(x+2L)=f(x)$$ this is a question from my book, and im just wondering about one thing and that is what does ...
3
votes
2answers
47 views

What does the multiply sign mean in set

A question about set notation. What does the multiply sign here mean? $$\omega = \times_{i\in N}T_{i}$$
2
votes
2answers
74 views

What does $\mathbb{R} \setminus S$ mean?

What does $\mathbb{R}\setminus S$ mean? I am not getting it what it actually means. I have found it manywhere in real-analysis like in the definition of boundary points of a set. Can anyone tell me ...
1
vote
4answers
102 views

How to write the set of all permutations on a set $n=\{1, 2, \ldots, n\}$

Let $n ∈ N$. Let $S_n$ denote the set of permutations on $\{1, . . . , n\}$. For any $σ ∈ S_n$, define $sign(σ) := (−1)^N$ , where $σ$ can be written as the product of $N$ transpositions. Now, let ...
4
votes
2answers
477 views

What is meant by the delta equivalent sign?

What is the meaning of the delta equivalent ($\overset{\Delta}{=}$) sign? I met this in a communication theory text. It said, signaling rate: $r\overset{\Delta}{=} 1/D$ symbols/s or also called ...
0
votes
0answers
20 views

Meaning of complement of vertex and edge sets

I am reading about maximum bipartite matching and everywhere I look there is thing: $$ U':=U\setminus M $$ Sometimes it's written like this: $$ U':=U\setminus\cup M $$ where $U$ is a set of vertices ...
3
votes
1answer
44 views

What does “$\mathbb{1}$” mean in this document?

I understand everything in this document on the first page except the following: \begin{align} \overline{\int_a^b}f&=\inf\left\{\int_a^b\psi:\psi\geqslant ...
1
vote
1answer
20 views

What is the correct way of writing a vector element?

When using the convention making a label bold to indicate a vector, should you still use the bold if you are only referring to a single element of the vector? for example, which of these methods of ...
0
votes
1answer
34 views

Notation about factors

What is the name (if there is one) of the "full factorization representation" of a number, in which also the powers of the factors are (recursively) decomposed until all the numbers used in the ...
0
votes
1answer
61 views

Reverse Polish Notation Quadratic formula

The quadratic formula is $$\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$ I tried converting this to RPN; I am new to doing this, and I have thus: b-ac*4*-b2^+±a2*/. Am I ...
0
votes
0answers
29 views

What do the ellipses mean in $x^TW_{\dots ij}$

I cam across this notation: $x^TW_{\dots ij}$. I don't understand the notation. Why are there 3 dots? Source: http://jmlr.csail.mit.edu/proceedings/papers/v28/goodfellow13.pdf, top of second page, ...
0
votes
1answer
21 views

Notation for piecewise equations with vertical lines

I feel silly for asking such a simple conceptual question, but it's something that's been bugging me for a while. Is there any notation that makes it possible to define a piecewise equation that ...
1
vote
1answer
42 views

What is the meaning of expressions of the type $f(\cdot)$ (function (dot))?

Simple question, fully expressed in the Title line. Is the dot within the parenthesis intended to mean, "any possible function"?
3
votes
1answer
64 views

Why are power sets called power sets?

Why are power sets called power sets? What is so powerful about them?
0
votes
3answers
48 views

Some questions about notation in “$[T]_\alpha^\beta$”

I just have a few questions about the general meaning of the notation "$[T]_\alpha^\beta$". I would really appreciate if someone would dumb it WAY down to the most basic level (no assumptions, no ...
0
votes
1answer
25 views

S-indexed Family of A's vs. a Family of A's indexed by S

I looked at here, and here, and here, to understand the terminology used regarding indexed family of sets (but I still left with some questions): Take S as the set $S=\{1,2,3\}$ we define: 1) K ...
0
votes
2answers
36 views

Proof $\sum_{i=1}^{n}x ^i=O(n)$

I just started working with Big-O notation. Proof: for which $x$ is $\sum_{i=1}^{n}x ^i=O(n)$ true. First case: $x>1$ $\sum_{i=1}^{n}x ^i=s_n=x\frac{x^{n+1}-1}{x-1}$ and $\lim_{n->\infty} ...
3
votes
0answers
35 views

Notation for Christoffel symbols used by Gödel in “An example of a new type of cosmological solution of Einstein field equations of gravitation”

I have difficult to understand the meaning of the notation used by Gödel in the article cited in the title of this post. You can find it here: http://www.lygeros.org/10552b.pdf In the second page ...
0
votes
0answers
25 views

What does $]-\pi, \pi[$ mean [duplicate]

I sometimes see a notation like $]-\pi, \pi[$, but I don't know how it is different from $[-\pi, \pi]$. Could you tell me what it means?
1
vote
0answers
26 views

Congruence relations and congruence modulo m

What are some congruence relations appart from congruence modulo $m$? Also, is the congruence modulo m called that way because the sets of all the integers $0,1, ..., m-1$ form an isomorphic group ...
0
votes
0answers
35 views

Random Variables: Expected Values and Values computed from a data

Given a random variable $X : \Omega \rightarrow \mathbb{R}$, we have $E[X]$, which is called expected value of the random variable. I have one random variable $X$, ...
0
votes
0answers
16 views

Generalized superfactorial notation.

I would like to know if there's a general shorthand notation for denoting the following product: $$\mathcal{P}=a!\times (a-k)!\times (a-2k)!\times\cdots\times (a-nk)!$$ where $a$ and $k$ both ...
3
votes
1answer
86 views

$\leq$ V.S. $\leqslant$ [duplicate]

Is there a substantial difference between $\leq$ and $\leqslant$? My textbook uses both, but I could not tell why the authors selected one or the other. I asked my teacher, and she said that there was ...
2
votes
1answer
45 views

Do I write $f\in C^{-n}$ for an integrable function?

I have seen in a variety of texts that an $n$-differentiable function $f$ is written \begin{align} f\in C^{n}\Longleftrightarrow f^{\left(n\right)}\in C,\tag{1} \end{align} such as in Widder's ...
0
votes
0answers
16 views

Name for maximal number of pre-images under a function

Let $E$ and $F$ be sets and $f:E\to F$ be any function. I want a name for the quantity $$\displaystyle\sup_{y\in F}\operatorname{Card}(f^{-1}(y)).$$ I am currently calling it the "maximal ...
1
vote
2answers
47 views

Equality symbols in modular arithmetic

E.g., can I write $(a^{p})^{2p} \equiv a^{2p}=a^pa^p\equiv aa\equiv a^2\pmod{\! p}$? I often see equality symbols inbetween mod equivalences. The equality signs point out the equality is not ...
1
vote
1answer
46 views

Ambiguous? Notation for Cross-Correlation

Is it true that $f(t)\star g(-t)$ is not the same as $f(t)\star h(t)$ if $h(t)=g(-t)$? ($\star$ means cross-correlation). The example I was thinking of was if $g(t)=t$. Then $(f(t)\star ...
0
votes
2answers
25 views

How is the cartesian product $E^F$ defined?

For arbitrary sets $E, F$, how is $E^F$ defined? It seems to be the set of all maps from $F$ to $E$, i.e. $E^F := \{\phi : \phi\colon F \rightarrow E\}$? Is that right?
1
vote
1answer
29 views

How to simplify this expression using tensor notaion?

$\nabla^2 (\phi A)-A \nabla^2 \phi -2(\nabla \phi \cdot\nabla)A$ Where $A,\phi$ are any sufficiently smooth vector and scalar fields respectively.
1
vote
1answer
46 views

What does this symbol mean here in this context?

The colon ("$:$" sign) - I am seeing this a lot in Chapter 8 of PDE Evans, like this (page 497 of the 2nd edition): THEOREM 6 (Pressure as Lagrange multiplier). There exists a scalar function $p ...
1
vote
0answers
18 views

set nested element symbol

Given a nested set $S = \{\{e_{11},e_{12},\dots,e_{1n}\},\{e_{21},e_{22},\dots,e_{2n}\},\dots,\{e_{k1},e_{k2},\dots,e_{kn}\}\}$, I want to specify a set consisting of specific subelements, for example ...
3
votes
0answers
45 views

Writing a Series of Equalities and Inequalities across Several Lines

What is the convention for writing a series of successive equalities and inequalities across multiple lines? Let me explain. Let $E_k$ denote an expression; for example, $E_0$ could be a sum or an ...
1
vote
1answer
17 views

Notation: express the minimum absolute value of a scalar function.

I have a pretty simple function that I want to express, but I'm having trouble coming up with a pleasing notation to express it. The function is: $h(y) = \arg \min_x |f(x, y)|$ $g(y) = f(h(y), y)$ ...
0
votes
1answer
13 views

Notation for estimator in statistics

I'm trying to understand the estimator notation $\hat\theta$ and what is correct use. If $\hat\theta = \hat\mu$, is $\hat\mu(X) = \overline X$? Is $\hat p(X,n) = \dfrac{X}{n}$? If not, what is ...
0
votes
0answers
22 views

Notation from Bowen's Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms

On pdf page 39 (page 33 of the text) under Proposition 2.8, the following notation was used $$U(\underline{V})=U(V_{i_0})\cdots U(V_{i_{m-1}})$$ $$U(\Gamma_m)=\{U(\underline{V}):V\in \Gamma_m\}$$ I ...
6
votes
4answers
96 views

$\binom{n}{r}$ versus $^n\mathrm{C}_r$ : which notation is more used?

I know that the notation $\binom{n}{r}$ is more standard to use since we have a $\LaTeX$ command for it while there is no such thing for $^n\mathrm{C}_r$. Now, I'm wondering which notation do people ...
0
votes
0answers
17 views

Alternative interval bracket notation?

I am reading a computer science paper and have come across some notation that I am unfamiliar with. it says to consider functions from $\mathbb{R}^N$ to $]-\infty,\infty]$ is this the same as ...
3
votes
2answers
265 views

What does $×$ mean in this context

I have two definitions, from real analysis - Metric Space: Given a set $X$, a function $d:X×X→\mathbb{R}$is a metric on $X$ if for all $x, y∈X \dots $ Function: Let $A,B$ be two sets. A function ...