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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1answer
101 views

question about Graph Theory notation

I'm just starting to learn graph theory. I have two questions about notation: 1). For a graph $G$ we denote the vertex set $V$ and the edge set $E$ by $G=(V,E)$. So we have a graph $G=$ ...
2
votes
2answers
120 views

Ambiguous notation for squared matrix

What does $A^{2}$ mean for square $A$? Is it $AA$ or $AA^{T}$? Sometimes, the result may differ. Or there is no uniform approach?
8
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2answers
544 views

Notation for intervals

I have frequently encountered both $\langle a,b \rangle$ and $[a,b]$ as notation for closed intervals. I have mostly encountered $(a,b)$ for open intervals, but I have also seen $]a,b[$. I recall ...
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2answers
82 views

How to describe a n-tuple of sequences

When I write computer programs, I often use something called a multidimensional array. I think the concept would be equivalent to an n-tuple of finite sequences. Suppose I have the following four ...
2
votes
1answer
55 views

What does $K^{1/p}$ for a field $K$ mean?

In the proof of the finite generation of the invariant ring of a finite group acting on $k[x_1,\dots,x_n]$, at one time there is a symbol I don't understand. The situation is as follows. $k$ is a ...
5
votes
1answer
638 views

Difference between “for any” and “for all”?

In a textbook (on economics, not "pure" mathematics), one definition requires that some condition holds for any $x,\ x' \in X$, and right afterwards another one requires that some other condition ...
4
votes
2answers
159 views

How do I understand the meaning of the phrase “up to~” in mathematics?

I am reading a book that explains elementary number theory: Number Theory: A Lively Introduction with Proofs, Applications, and Stories by James Pommersheim, Tim Marks and Erica Flapan. The authors ...
0
votes
1answer
40 views

Notation to describe the adding of a constant to all terms of a sequence

I've been struggling to get down the proper mathematical notation for sequences. Suppose I have the following sequence: $$A = (4, 3, 7, 3, 1)$$ How do I describe the addition of a constant to all ...
3
votes
1answer
66 views

How to combine these 2 notations to be as simple as possible?

Consider the following question: Find the antiderivative of $f(x) = 6x \, (x^2+1)^5$. I have been using 2 notations and I would like to combine them. Notice that $[x^2 + 1]' = 2x$, so $6x \, dx ...
5
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2answers
292 views

Modern approaches to mathematical notation

I'm interested if there is literature on projects which try to improve formal notation, especially for doing mathematics on an advanced level. For example, I'm thinking along the lines of diagrammatic ...
1
vote
0answers
222 views

What is meant by an integral with $\Omega$ as its range: $\int_{\Omega}$

What does it mean when $\Omega$ is range of integral? $$\iint G \frac{\partial U}{\partial n} - U(J k G) ds =\int_{\Omega}{G\left(\frac{\partial U}{\partial n} -j k U\right)R^2 dw} $$ I do not ...
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2answers
34 views

Limiting the scope of an operator

With the product rule, one has to limit the scope of operators (without using $f'$ or $\partial f/\partial x$): $$ \frac\partial{\partial x} u(x) v(x) = \left(\frac\partial{\partial x} u(x)\right) ...
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2answers
56 views

Tensor field notation

Let $L$ be a finite dimensional vector space. A tensor of type $(p,q)$ on $L$ is an element of the tensor product $L^{\otimes q}\otimes (L^{*})^{\otimes p}$. How to interpret the following formulation ...
0
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2answers
66 views

Shorthand Notation for Bicubic Graphs

I remember that I once read/heard that graphs may be given a shorthand notation when you take the upper diagonal of the adjacence matrix, put the resulting lines next to each other and convert ...
6
votes
2answers
202 views

The notations change as we grow up

In school life we were taught that $<$ and $>$ are strict inequalities while $\ge$ and $\le$ aren't. We were also taught that $\subset$ was strict containment but. $\subseteq$ wasn't. My ...
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1answer
89 views

backwards membership notation set theory

This is more of a notational/historical question. I had a course last quarter where the professor would write things like $A \in x$ for $A$ a subset of some bigger space $X$ and $x$ an element of $X$. ...
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3answers
114 views

Notation of “defined for all complex numbers except the negative integers and zero”

The Euler and Weierstrass forms of the gamma function are : ...
2
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1answer
71 views

Is the notation $[x,\to[$ common?

I recently started reading Topology and Groupoids by Ronald Brown and this notation came up. The notations is $$[x,\to[ \; =\{z \mid x \leq z\}$$ and a similar notation for other type of intervals. I ...
0
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1answer
77 views

What is the semantic of square brackets after the set denoting coefficients of polynomial?

I have the following excerpt: Unless stated otherwise, we assume all polynomials take integer coefficients, i.e. a polynomial $f \in \mathbb{Z}[{\bf y}, x]$ is of the form $$f(y, x) = a_m · x^{d_m} ...
3
votes
1answer
80 views

Simplifing formulas using tensor notation

Im trying to symplify formulas like: $$\operatorname{div}(\operatorname{rot}\vec{F}),\qquad \operatorname{rot}(\operatorname{rot}\vec{F}) $$ or something more strange like: ...
10
votes
3answers
336 views

Meaning of $\int\mathop{}\!\mathrm{d}^4x$

What the following formula mean? $$\int\mathop{}\!\mathrm{d}^4x$$ I know that this $\int f(x)\mathop{}\!\mathrm{d}x$ is the integral of the function $f$ over the $x$ variable, but the following ...
0
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1answer
93 views

Concise notation for the successor of a cyclic index

Very frequently we index "cyclic objects" using the integers. For instance, we might say that the vertices of a polygon are $x_1, x_2, \dotsc, x_n$, where the "next vertex" after $x_n$ is $x_1$. This ...
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0answers
27 views

Equivalence of Notation for Momentum Continuum

This wikipedia page for Magnetohydrodynamics lists the conservation of momentum in a continuum as: \begin{equation} \rho \left( \frac{\partial}{\partial t} + \vec v \cdot \nabla \right) \vec v = \vec ...
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1answer
51 views

Help identify this operator

Except the whole operator is more compact. It is used between two fractions like.... x / y mystery operator x /y . I don't know how to write it on my computer. ...
-1
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1answer
90 views

Quick question: local-to-global spectral sequence [closed]

What does "$\implies$" mean in the following: $E_2^{p,q} = H^p(\mathcal Ext^q(F,G)) \implies Ext^{p+q}(F,G)$ Could you explain the meaning of the whole thing? (Let's say $F$ and $G$ are sheaves on ...
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0answers
51 views

Canonical way of denoting the set of all (totally) ordered subsets

Given a finite set $S$, we usually denote the set of all subsets of $S$ by $\mathcal{P}(S)$, i.e. the power set of $S$. I need to denote the set of all totally ordered subsets of $S$, let us call it ...
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3answers
1k views

Learning to read complex math formulas

could anybody point me to a book or article where I could learn how to read formulas like this one: I have no idea what that means.
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2answers
91 views

Meaning of $\bar{i}:=i+n\mathbb{Z}$ in Modular Arithmetic

I am starting to learn graph theory and ran into the following definition: The set $\mathbb{Z}/n\mathbb{Z}$ of integers modulo $n$ is denoted by $\mathbb{Z}_n$; its elements are written as ...
2
votes
1answer
52 views

Notation for function being differentiable at a certain point

This question describes a notation for a function $f(x)$ being (continuously) differentiable on some domain $A$. Often, I see the requirement that some function $f(x)$ be differentiable only (or ...
3
votes
1answer
871 views

What does the notation min max mean?

Min clearly means minimum and max maximum, but I am confused about a question that says "With $x, y, z$ being positive numbers, let $xyz=1$, use the AM-GM inequality to show that min max $[x+y,$ ...
2
votes
1answer
63 views

What does an expression $[x^n](1-x)^{-1}(1-x^2)^{-1}(1-x^3)^{-1}(1-x^4)^{-1}…$ mean?

I came across the function that describes number of partitions of $n$ (I mean partitions like $5=4+1=3+2=3+1+1$ and so on. There was defined a Cartesian product: ...
1
vote
1answer
231 views

How to write two for loops in math notation?

I have a vector of numbers that looks like this: [1, 2, 3, 4, 5] For every number the vector, I would like to multiply each number by every other number and find ...
0
votes
1answer
67 views

Joining finite sequences [duplicate]

How do I describe the joining of two finite sequences in mathematical notation? For example, suppose the following: $$ A=(a_i)_{i=1,2}=(4,2)\\ B=(b_i)_{i=1,2}=(9,5)\\ C=(c_i)_{i=1,...,4}=(4,2,9,5) $$ ...
2
votes
1answer
68 views

prime notation clarification

When I first learned calculus, I was taught that $'$ for derivatives was only a valid notation when used with function notation: $f'(x)$ or $g'(x)$, or when used with the coordinate variable $y$, as ...
0
votes
2answers
133 views

I don't understand this notation… - Series with ln

I found this notation in my book $$ \sum\limits_{i=1}^n \ln^n3 $$ and I don't know how to interpret it. Is it $$ \sum\limits_{i=1}^n \ln((1^n)\cdot3)\;? $$ And btw, how to check if this series ...
0
votes
1answer
83 views

question about epsilon, delta limit definition

Sometimes, when describing the closeness of $x$ to $a$ as being less than $\delta$, it's stated as $|x-a|<\delta$ and sometimes it's stated as $0<|x-a|<\delta$. What is the " $0<$ " part ...
4
votes
1answer
93 views

$\delta$ Notation in linear algebra

In this equation below, what is $\delta_{l,q}$ denoting? Is $\delta$ a standard notation, or anything to do with all one's or the basis matrix etc? $$A_{ij}=\delta_{l,q}\left(\sum_{h=1}^n B_{l,h} + ...
2
votes
2answers
64 views

Default metrics for $c_0$ and $l^{\infty}$

In my book there is a question like Let $\{a^{(k)}\}$ be a convergent sequence of points in $l^1$. Prove that $\{a^{(k)}\}$ converges in $l^{\infty}$. Now I don't see it mentioned anywhere what ...
1
vote
1answer
47 views

What does $\|$ mean in this definition of the total derivative of a function?

What does $\|$ mean in this definition of total derivative? picture:
2
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0answers
44 views

Notation for n-array function with domains of different types.

I'm wondering what notation I should use to express a function R that maps elements from different sets to either 0 or 1. Is the following a reasonable usage? $$ R: S_1 \times S_2 \times ... \times ...
2
votes
2answers
118 views

useful notation for pullback

Let $f:A\to C\leftarrow B:g$ be morphisms in a category. There exists in literature a useful notation for the morphisms $\bar f:A\times_C B\to B$ and $\bar g:A\times_C B\to A$ in terms of $f$ and $g$? ...
0
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0answers
140 views

Time series notation

I'm developing formal software requirements specifications for processing time series data and thus need to mathematically describe time series and operations on time series. Is there establish ...
3
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2answers
84 views

Conventions on definitional if(f)

When defining a term it seems common to use 'if' when the stronger 'iff' is also true. For instance: Definition 1: A set $A$ is open in $(X,d)$ if $\forall x \in A$, $\exists \epsilon \gt 0$ such ...
1
vote
1answer
205 views

Notation for Multiple summation

Is there an alternate way to represent the multiple summation given below? $\sum_{i_k=k}^{n} \sum_{i_{k-1}=k-1}^{i_k} \dots \sum_{i_2=2}^{i_3} \sum_{i_1=1}^{i_2}$ It guess it is wrong to write it as a ...
0
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1answer
77 views

correct expansion of a sum using multiple indexes

I have looked for a similar posting but haven't found anything... but then I am also a bit unsure of how to search because I've never posted a math question before. In my introductory finite element ...
2
votes
3answers
309 views

What 's the differece between $\cot(x)$ and $\arctan(x)$? [duplicate]

I know that $\displaystyle \cot(x)=\frac{1}{\tan(x)}$ and $\space \displaystyle \arctan(x)=\tan(x)^{-1}=\frac{1}{\tan(x)}$ What is the difference between these two function? Is $\cot(x)$ the ...
4
votes
2answers
264 views

Mathematical symbol for “has”

Just out of curiosity, I was wondering if there was a symbol for "has" so intead of saying $x \in A$, we could say something like "$A$ has $x$", they both mean the same thing but I was just wondering ...
5
votes
1answer
117 views

Why do we write second derivatives like $\frac{d^2x}{dt^2}$ [duplicate]

Why do we write the second derivative of $x$ with respect to $t$ as $\frac{d^2x}{dt^2}$? It's never been explained to me, and I've never found a particularly good explanation. What's up with this ...
2
votes
1answer
111 views

Getting rid of the set builder notation in the expression $\{ f(x) \mid P(x) \} = \{ g(x)\mid Q(x) \}$

The set-builer notation is used to have $$\{ x \mid P(x) \} = \{ x \mid Q(x) \}$$ denote $$\forall x\ \big(P(x) \Leftrightarrow Q(x) \big).$$ And some people write $$\{ x \in U \mid P(x) ...
4
votes
1answer
108 views

Nomenclature and notation for some aspects of weighted directed graph.

I'm having some problem with nomenclature some structures and quantities related to weighted directed graph. Suppose that $A \in \mathbb{R}_+^{N \times N}$ is the weighted adjacency matrix of a ...