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
40 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
18 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
30 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
56 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
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0answers
25 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
16 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
32 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
55 views

Why are power sets called power sets?

Why are power sets called power sets? What is so powerful about them?
0
votes
3answers
40 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
24 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
34 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
28 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
24 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
28 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
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0answers
13 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
78 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
43 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
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0answers
13 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
32 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
45 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
24 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
24 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
44 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
14 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
40 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
15 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
12 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
15 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 $$U(\bar{V})=U(V_{i_0})\cdots U(V_{i_{m-1}})$$ $$U(\Gamma_m)=\{U(\bar{V}):V\in \Gamma_m\}$$ I double ...
6
votes
4answers
93 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
13 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
40 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 ...
1
vote
2answers
20 views

Notation for a vector with constant equal components of arbitrary dimension

Is there a standard notation to specify vectors of the form $(c,c,\ldots,c)$, where $c$ is some constant?, e.g. suppose I have the vector $(c,c,c,c,c)$ which has 5 components. Is there a standard ...
0
votes
0answers
33 views

What's the “real” definition of variational derivative

This seems a notational question. Given a functional $S[f]=\int L(f,f';x)\ dx$, I want to derive $\delta S[f]$. There are quite a lot of literature interchanging integrate and variation. That is, ...
1
vote
2answers
66 views

What does $(n|p)=1$ mean?

My number theory book mentions the following condition: $(n|p)=1$, where $p$ is prime. What does $(n|p)=1$ mean? I used to think $n|p$ implies that $n$ divides $p$.
2
votes
2answers
31 views

Notation for replacing a matrix column with a vector

Let $A$ be an $n\times n$ matrix. Let $v$ be an $n\times 1$ matrix. Is there a notation to signify replacing the $j$-th column of $A$ with $v$? If not, what is the accepted way to denote this?
1
vote
1answer
17 views

What does the notation $\{ \pm 1 \}^X$ in relation to functions and hypothesis classes means in the context of PAC learning over half spaces?

I was reading the following paper (on PAC learning over half-spaces) and encountered the following notation for a hypothesis class (on page 4): $$\mathcal{H} \subset \{ \pm 1 \}^X$$ However, it was ...
0
votes
1answer
13 views

Notation in Meade's paper “ODE models for the parachute problem”

This refers to ODE models for the parachute problem by Douglas B. Meade. On page four the author says: When $k$ is piecewise constant, the snatch force has a jump discontinuity at the instant of ...
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0answers
42 views

Index notation for group multiplication and commutators

Are there any serious errors in the following discussion of notation? I will be grateful for any corrections, suggestions and pointers towards the relevant literature. The index notation for inner ...
1
vote
1answer
19 views

What is the notation for a sequence of elements (non number elements)?

I am new to math and am exploring how to formally represent a sequence of events. I want to be able to say "an event sequence $E_s$ is a sequence of events $\langle e_1, e_2,\ldots, e_n\rangle$". Just ...
0
votes
0answers
35 views

Symbol for the area of an Ellipse

Is there any commonly used symbol for the area of an ellipse? I'm thinking of $A(E)$ or $\mathcal{A}(E)$? Thanks!
0
votes
0answers
23 views

Function Mapping Definition in Relation

So I have two relations: i) Ordinary relation ii) Disjunctive relation. Definition of Ordinary relation: $\Sigma$ be a finite set of attribute names, where for any attribute name $A\in \Sigma $, ...
0
votes
1answer
11 views

Notation for the sum of product indexing over all possible string of ternary digits of length $k$.

Let $f_0, f_1$ and $f_2$ be three functions. What is a shorthand notation to write $$ \underbrace{f_0 f_0 \cdots f_0}_{\text{k times}} + \underbrace{f_0 f_0 \cdots f_0f_1}_{\text{k times}} + \cdots ...
0
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0answers
91 views

Notation on De Jong and Starr's paper

I reading De Jong and Starr's paper "Divisor classes and the virtual canonical bundle for genus 0 maps". I am quite confused about what would be the correct definition of functor ...
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1answer
61 views

How to do If and Else in math?

Hello all I would like to learn how to show if/else logic in math notation so here is the problem I'm faced with. then ...
6
votes
4answers
94 views

What does this symbol $\otimes$ mean? [duplicate]

I came across the symbol $\otimes$ as below and I would like to know what this symbol $\otimes$ means: $\text{.... the projection operator P is given by: }$ $$P = I_nd - \nabla G^T(\nabla G ...
0
votes
1answer
20 views

using a formula to make a statement with the use of greater than, less than symbol

I am trying to make statement using the greater than less, less than symbol. I am unsure if the use of $\lt\gt$ symbols have to facing the same direction in an equation. Could you say: $x \lt x + y ...
0
votes
1answer
53 views

what does the notation $\|x\|$ where x is a vector mean?

I have come across something that requires minimizing $\|x\|$ where $x$ is a vector? What does the notation $\|x\|$ mean? Is it the two norm or something else?
1
vote
1answer
26 views

Right notation to recurse over a sequence or list

I have a function $f(x, a)$ which is invoked over all the elements of a sequence feeding the result to the next call, with $x$ being the next element in the list and $a$ the accumulated result. What ...
2
votes
3answers
102 views

Is $\mathbb{N} = \mathbb{Z}^+$?

Because $\{1,2,3,4,\ldots\}$ contains all natural numbers, which are also all positive integers.