# Questions tagged [notation]

For questions about mathematical notation, i.e. the symbols used to represent mathematical objects and operations.

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### Notation for multidimensional sorting function

Given a matrix $X\in\mathbb{R}^{n \times m}$ I would like to denote a row ordering function $s:\mathbb{R}^{n \times m}\rightarrow \mathbb{R}^{n \times m}$, which sorts rows in ascending order by the ...

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### Are there standard short notations for ascending and descending cyclic permutations?

In a paper I am currently writing I use cyclic permutations of the form
$$
(k,k+1,\dots,\ell)
$$
and
$$
(\ell,\ell-1,\dots,k)
$$
of consecutive elements quite a lot (I added the commas to avoid ...

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### Why is $H$ the standard notation for mean curvature?

I am curious about the origin of the notation $H$ to denote the mean curvature of a surface in $\mathbb{R}^{3}$.
I suppose that the symbol $K$, which is commonly used to denote the Gaussian curvature, ...

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### What is the basis for the quantifier notation? [closed]

The symbols $\forall, \exists$ are the ones officially used to denote universal and existential quantifiers respectively. I understand that the choice of $\exists$ was made by Peano, while of $\forall$...

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### Who introduced the notation for $\beth$ numbers and when?

Georg Cantor, when developing the basics of set theory, noted that there are two ways to increase cardinality: power sets and successors (or, in modern terms, the Hartogs operation).1
Eventually the ...

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### What does the subscript 'x' of a matrix mean? [closed]

The 3x6 matrix G is as follows,
$\text{G} = [\text{V}_\times| I_{3\times3}]$
$\text{V}$ is a skew matrix of a vector with 3 elements about a 3D point. The dimension of $\text{V}$ is 3x3.
$I$ is the ...

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1
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### The meaning of $L_p^l(\Omega)$ in a paper of Bogovskii on Sobolev spaces

On the first page of the old paper Solution of the first boundary value problem for an equation of continuity of an incompressible medium of Bogovskii, the notations $W_p^l(\Omega)$ and $L_p^l(\Omega)$...

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### Why aren‘t op and co switched?

When reading through Loregian and Riehl - Categorical notions of fibration, on p. 3 there is a remark that confuses me about notation. Given a $2$-category $\mathcal C$ one usually defines $\mathcal C^...

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### Common notation for function over infinitely many variables? [closed]

For a document about reinforcement learning, I want to write the joint probability density over the entire trajectory of states and actions like $p(s_0, a_0, s_1, a_1, s_2, \dotsc)$. However, this ...

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### Is it improper to define matrices as being $n \times m$ rather than $m \times n$? [closed]

For whatever reason, I have always defined matrices as being $n \times m$, and that is how I have been defining matrices throughout my dissertation. Recently however, I have noticed that nearly every ...

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### About the maximum number of leaves adjacent to a vertex in a tree

Let $T$ be a finite tree graph with the set of vertices $V(T)$. For an arbitrary vertex $ v \in V(T)$, I define $l(v)$ to be the number of leaves connected to $v$.
In my study, I need to define the ...

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### What does $\omega^*$ mean? [closed]

I've recently found in some short article (source below) the symbol $\omega^*$ (generally, starred ordinal number), but without explanation what that symbol means. From the context I understood that ...

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### Name for the theory of words with equal length, prefix, successors

I've worked with this theory for a while, but I've never been quite sure what to call it:
$$(\Sigma^*, =_{el}, \preceq, (S_a)_{a \in \Sigma})$$
Where
$\Sigma^*$ is the set of finite words on finite ...

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1
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### Pronunciation: the Erdős–Rado partition notation

The Erdős–Rado notation $a \rightarrow (b)^c_d$ is common in partition calculus / combinatorial set theory, as well as its negation $a \not\rightarrow (b)^c_d$. In that field, is there a standard way ...

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### Reference request: Different definitions of Big O notation

This question might sound strange, but I would like to settle this problem once and for all.
For as long as I can remember, I was introduced to the Big O notation by this definition:
Def. 1: Let $f, g$...

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### Terminology and notation for generated subgroups

I would like to think about formation of the smallest subgroup (or monoid, or whatever) $H$ of $G$ containing two given subgroups $A$ and $B$ as an operation on subgroups, and I wonder if there is a ...

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### Higher order Leibniz rule and ordered multiindex notation

Although I think this is probably known, I am making here a short exposition on the multiindex notations I am using to make this question self-contained. I note that there is at least two different ...

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### What does square bracket superscript star mean in basic group theory typically?

I'm reading some paper where they haven't really defined their notation very well (or I've missed something). You can see the image below.
What does the square bracket and star mean precisely? The ...

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### Notation for H is isomorphic to a subgraph of G

Is there a notation for the statement $H$ is isomorphic to a subgraph of $G$? I was thinking of using $H<G$, but I'd like to use standard notation if possible.

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### Can NBG be interpreted in this system that use new notation for class-abstractions?

We introduce a new symbol $\lambda$ to denote class-abstractions, and we add the following rule:
if $\phi$ is a formula that use $``\mu"$, and in which the symbol $\sf y$ doesn't occur; then: $\lambda ...

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### What is meant by this notation of the real forms of $E_6$?

There are five real forms of the exceptional Lie group, $E_6$. Four of them are notated as in the following:
The split form as EI or $E_{6(6)}$
The quasi-split form as EII or $E_{6(2)}$
EIII or $E_{...

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### Proof of Theorem 9.2 of the book Cubic Forms by Yu. I. Manin (end of page 37)

I warn that I first posted this question in Mathematics Stack Exchange but it got no attention at all. I think that it fits better there by its explanatory nature but maybe the book being reference is ...

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### Finitely-generated conjugation action on a subgroup that is not normal... what is that?

If $H \lhd G$, then $G$ acts on $H$ by conjugation. I need to talk about this action but in a situation where $H$ is not (necessarily) normal. When $H \leq G$, there is a "partial action" of ...

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### How to denote a partial derivative?

This question is related to Was Jacobi the first to notice the ambiguity in the partial derivatives notation? And did anyone object to his fix? and Suggestions for good notation .
When there are two ...

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### Good notation for finite partial functions from $\omega$ to 2

I'm working in computability theory and need to use partial functions with finite domain from $\omega$ to 2 as approximations in my current paper. Normally this is simply done using $2^{< \omega}$ ...

2
votes

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answer

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### Temporal generalization of graphs: density vs $n$ and $m$?

In short: we generalize graphs to the temporal case, but fail to fully preserve the usual relation between density, number of vertices, and number of edges; how to make better?
Context.
We propose a ...

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229
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### Confusing notation for sets of unordered vs ordered pairs

Given two finite sets $X$ and $Y$, one may consider the ordered pairs $(x,y)$ with $x\in X$ and $y \in Y$. Then, $(x,y) \not= (y,x)$, and $(x,x)$ exists if $x\in X$ and $x\in Y$.
One may also consider ...

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### Ideals with certain properties

I recently isolated the following definition, which I believe it should have appeared somewhere.
Let $\kappa$ be a cardinal, and let $X$ be a set with $\kappa^+\leq |X|$.
Definition: An ideal
$\...

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### Typesetting of symbols and "operators" denoting sets [closed]

Question:
what are the conventions for typesetting sets of certain objects, especially the vertices and edges of a graph or those adjacent to an edge or vertex.
For vectors and matrices there is the ...

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### Explanation of a formula to calculate the zenith distance of sun and moon [closed]

I am studying tidal accelerations and referring to a well known paper by I M Longman :
Formulas for computing.." J Geophys Research 64 (12) Dec 1959.
At Eq 12 he writes a term "1336.rev"...

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### Origin of the symbol for the tensor product

I have recently realised that the Paleo-Hebrew (and Phoenician) graph for the Hebrew letter ט (Teth) is $\otimes$. This made me wonder if there is any relation between the choice of the symbol and the ...

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### Notation for the regular and the adjoint representation of a finite group, in particular the symmetric group

The (left) regular representation of a finite group $G$ is the action on itself by left multiplication, $g\cdot h = gh$.
The adjoint representation of a finite group $G$ is the action on itself by ...

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### Why are fundamental weights denoted by omega?

In my field (and many others, I believe) the absolutely standard notation for the fundamental weights of a root system is lowercase omega: $\omega$. Recently I was taken aback to receive a copyedited ...

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### Question about the notation $N_{\chi}(\alpha, T)$, the number of zeroes of the $L(s, \chi)$ in a rectangle

I am confused with what seems to be a standard notation in analytic number theory and I'd appreciate any clarification. I am interested in the zero density estimates, for example link.springer.com/...

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### Name and properties of $\mathrm{lcm}(\{1,\,\cdots,\,n\})$ [closed]

one of the most prominent functions of the first $n$ natural numbers is the factorial $n!$ that denotes their product.
Today however I wondered whether the least common multiple $\mathrm{lcm}(n):=\...

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### What is the p-adic Plancherel measure?

What I know as the Plancherel measure for a group is a measure on the spectrum of $G$, aka the set of irreducible representations - at least for finite groups, this makes perfect sense.
Now, this ...

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### Who introduced the heart ($\mathcal{C}^\heartsuit$) notation in the context of $t$-structures on triangulated categories?

In the context of $t$-structures
([Wikipedia],
[nLab],
[Notes I],
[Notes II],
[HA, Definition 1.2.1.11)],
[BBD, Définition 1.3.1]),
one often writes $\mathcal{C}^\heartsuit$ for the heart of a ...

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### Reference request: Dictionary of the Leibniz notation

Is there any published, somewhat comprehensive, list of (almost?) all the many ways in which the Leibniz notation ($dx,$ $P(dx),$ $d\mu(x),$ $du\wedge dv,$ etc., etc.) gets used in the various areas ...

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### A question about chaining Vinogradov notation

This is not a research question, but I hope it is still legitimate to ask for this platform. Suppose $A(x)$, $B(x)$, $C(x)$, $D(x)$ are positive-valued functions of $x$, and
$A(x) \ll B(x)$ and $ C(x) ...

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### Terminology: Almost stable states

I have a question about fixed points which are almost stable.
I have an increasing transition function $f:[0,1]\rightarrow[0,1]$ where $f(0)>0$ and $f(1)<1$ but I don't necessarily have ...

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### What are the benefits of writing vector inner products as $\langle u, v\rangle$ as opposed to $u^T v$?

In a lot of computational math, operations research, such as algorithm design for optimization problems and the like, authors like to use $$\langle \cdot, \cdot \rangle$$ as opposed to $$(\cdot)^T (\...

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### Notation question: bigraded direct sum of graded objects

In some work I'm doing I have two graded modules $M$ and $N$ (graded on $\mathbb Z$, say) and need to take, not the usual direct sum, but the bigraded sum consisting of all $M_p \oplus N_q$ (so graded ...

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### Notation for induced subgraphs

For a graph $G=(V,E)$, is there a standard notation for the induced subgraph on $V \setminus \{v,w\}$ where $v,w$ are the endpoints of some edge $e$? I know $G[V \setminus \{v,w\}]$ is an option, but ...

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### Marcinkiewicz-Mihlin-Hormander Fourier multiplier theorem

I'm trying to understand the hypothesis of the Marcinkiewicz-Mihlin-Hörmander multiplier theorem. See for instance Theorem A in this paper of Elias Stein.
Theorem A: Assume that $m: (0, \infty)\to \...

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1
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### Unknown notation in "Boolean function complexity" by Stasys Jukna [closed]

I am currently reading Boolean Function Complexity - Advances and Frontiers by Stasys Jukna and on page 7 of the latest edition there is a paragraph titled Boolean functions as set systems with the ...

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### Why did mathematical notation stay so hard to read? [closed]

One of the first things you learn in a programming 101 course is to write readable code, and to name your variables properly. This notion has seemingly never translated into mathematics. Everywhere ...

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### Are measures better thought of as densities than differentials?

The standard notation for integrating with respect to a measure $\mu$ is:
$$\int f(x)\,d\mu(x).$$
But I've wondered if it could be better written as:
$$\int f(x)\mu(x)\,dx$$
where $\mu(x)$ is now ...

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### Examples of improved notation that impacted research?

The intention of this question is to find practical examples of improved mathematical notation that enabled actual progress in someone's research work.
I am aware that there is a related post ...

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### Need help in understanding meaning of a notation and theorem used in research paper due to a reference being in German Language

I thought of utilizing this lockdown period to study research papers in number theory by myself.
I began reading the research paper By T Estermann ->" On Goldbach Problem : Proof that Almost all ...

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### Why are orthogonal matrices so often denoted $Q$?

I apologize for the stupid question in the title. Of course, we can baptize a particular given matrix as we want but, for example, the QR-decomposition has a fixed meaning.
My humble guess is that ...