Questions tagged [notation]
For questions about mathematical notation, i.e. the symbols used to represent mathematical objects and operations.
295 questions
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4
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Writing an equation with cases when case description is long
This question seeks advice on what is common or at least acceptable in typing a certain inequality in a research paper.
I have an inequality that looks like this:
\begin{equation*}
f_{a,b,c}(x)\leq
\...
- 868
13
votes
2
answers
716
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Earliest use of $\lesssim$ for inequality up to constants
When I was a graduate student, I was highly influenced by the work of Bourgain, particularly his 1991 paper
Bourgain, J., Besicovitch type maximal operators and applications to Fourier analysis, Geom. ...
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1
vote
1
answer
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Notation in 5.6.3: General Sobolev Inequalities in Evans PDE book
I am currently reading Evans's PDE book. I reached section 5.6.3, General Sobolev Inequalities, which goes as follows:
THEOREM 6: (General Sobolev Inequalities).
Let $U$ be a bounded open subsetof $\...
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3
votes
0
answers
227
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Unknown notation involving prime ideals of number fields
I'm reading the paper
R. R. Laxton, “On a problem of M. Ward,” Fibonacci Quart., 12 pp. 41–44 (1974),
which can be downloaded for free from the Fibonacci Quarterly website: https://www.fq.math.ca/12-...
- 143
7
votes
2
answers
415
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Conflicting notation for periods of elliptic functions
It seems that there is absolutely no general agreement on the notation
for the periods $\omega_1$ and $\omega_2$ of Weierstrass elliptic functions.
Even if Serge Lang's book on elliptic functions is ...
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8
votes
1
answer
376
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Old notation: subsets (of the plane) denoted with capital E?
So I’m reading an old paper of Kinoshita, “A solution of a problem of R. Sikorski” (1953). The paper is very short and concerns a certain 0-dimensional nested triple of subsets of the plane. There are ...
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15
votes
1
answer
1k
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Unknown classical notation
I was looking through Théotiste Lefevre's Guide pratique du compositeur et de l'imprimeur typographes, vol. 1 (1880), when I came across a symbol I didn't recognize. Does anyone know what this ...
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2
votes
1
answer
369
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Best notation for tensor product with associativity
I want to express the product of a three-dimensional array by two one-dimensional vectors over some ring $R$:
$$r = A \cdot b \otimes c$$
where
$A \in R^{\ell \times m \times n}$
$b \in R^n$
$c \in R^...
- 147
0
votes
0
answers
120
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Name for linear combinations of Chebyshev polynomials
The term "Chebyshev polynomial" is ambiguous in the sense that it can refer to one of the basis functions $T_n(x)$ when interpreted as linear combinations of monomials, or, to a linear ...
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1
vote
1
answer
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Origin of the pattern of prime symbols in writing short exact sequences
In many sources that discuss short exact sequences, there is a curious notational convention to write them with the following pattern of prime symbols:
$$0\to M'\to M\to M''\to 0 \, .$$
That is, the ...
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0
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228
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Is there a good or commonly accepted short notation for the set of differentiable, but not necessarily continuously differentiable maps?
Every once in a while I find myself in need of some short notation for the set of differentiable, but not continuously differentiable maps, say, $X \to Y$. Always having to specify "...
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0
votes
1
answer
136
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What is the standard notation for bilinear, biquadratic, etc... spaces?
A typical notation for the polynomials of degree $k$ is $P_k$. The space $P_k$ is considered well-suited for interpolation on simplices, although that is hard to put into practice in full generality.
...
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5
votes
0
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Why $f^\lambda$ in the hook-length formula?
This is my first question on this site so I apologize if it’s not adequate for it.
I just learned the hook-length formula for the number $f^\lambda$ of Standard Young Tableaux of shape $\lambda$:
$$f^\...
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6
votes
1
answer
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Does $\mathsf{SVC}^\ast$ exist?
$\mathsf{SVC}(S)$ is the assertion that for all sets $X$ there is an ordinal $\eta$ and a surjection $f\colon\eta\times S\to X$. I would like to denote by $\mathsf{SVC}^\ast(S)$ the same assertion but ...
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4
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1
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Notation for weak derivatives
I remember that, as a student, I felt a bit uncomfortable because I had to use the same notation (say $f'$, $D^\alpha f$, $\frac{\partial f}{\partial x^j}$, $\nabla \cdot f$ etc...) for classical and ...
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5
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0
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Adjunction symbol
What are the reasons for the adjunction symbol $F\dashv G$ for a pair of functors $F:C\to D$ and $G:D\to C$? There is no explanation or motivation in the article of Kan where adjunctions are ...
0
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0
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178
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Notation $\le_{a,b,n,\ldots}$ in Analysis
In modern Analysis, especially Functional Analysis, one proves, or one uses inequalities of the form
$$F(X)\le_{a,\ldots,n}G(X).$$
The meaning of the subscripts in the inequality sign means that there ...
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-2
votes
1
answer
157
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Is this single-variable function being called with two variables (and if so, how do I handle that), or am I misreading the notation? [closed]
While working on a project, I came across a paper that includes this sum on page 15 as the definition of a support function $r$ for a surface with tetrahedral symmetry:
$$
r(ξ, \bar{ξ}) = \frac{1}{\#𝒢...
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3
votes
1
answer
495
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Unpacking the plethystic substitution $h_n[n\mathbf{z}]$ in a paper by Aval, Bergeron and Garsia
I'm not familiar with the formalism of plethysm, so I need some help in unpacking the plethystic substitution $h_n[n\mathbf{z}]$ found in eqns. 5.6 and 5.9 of "Combinatorics of labelled ...
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3
votes
1
answer
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Cartesian product of $(k-2)\text{-times } [\text{Interval}_1] \times [\text{Interval}_2] \times [\text{Interval}_2]$
This is a soft question, hoping that it is still appropriate for this forum.
I need to describe twice the following region of $\mathbb{R}^k$ (i.e., we are in a $k$-dimensional Euclidean space, where $...
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0
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0
answers
160
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Definition of term functions, in universal algebra
According to the definitions in Sankappanavar's universal algebra :
Assume $p$ is a term, then $p(x_1,x_2,...,x_n)$ indicates that the variables occurring in $p$ are among $x_1,...,x_n$. But there is ...
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2
votes
0
answers
311
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Notation for spectral sequences [closed]
Every single spectral sequence I have seen in my life was denoted by $E$. Even when there is more than one spectral sequence, people tend to use the same letter with some workaround (e.g. a fourth ...
- 2,202
0
votes
0
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143
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Merging two composable walks in a graph
Let $G$ be a graph (i.e., an undirected graph in which we allow for loops and parallel edges). Denote by $V$ the vertex set, by $E$ the edge set, and by $\psi$ the incidence function of $G$, and let $\...
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votes
8
answers
5k
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Examples of bad notation and its consequences [closed]
An example of bad mathematical notation that comes in my mind and has caused complications throughout history is the notation for imaginary numbers. The original notation used to represent imaginary ...
1
vote
0
answers
104
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Minus sign inside derivative operator, notation problem
Hello fellow mathematicians. Can anybody help me understand what the minus (-) sign in this derivative means? Its the usual d/dy but with a minus added d-/dy. I can't find references, the book cited ...
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2
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0
answers
211
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What do you call $x$ such that $\textrm{dim} f^{-1}(f(x))>0$?
Let $f:V\to W$ be a morphism between varieties, with $\dim \overline{f(V)} = \dim V$. What do you call the closed proper subvariety $S$ of $V$ consisting of points $x$ such that $\textrm{dim} f^{-1}(f(...
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1
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Notations for open and closed sets
I am wondering why a standard notation for open sets is $G$ and that for closed sets is $F$. I mean, $F$ precedes $G$ in the alphabet, whereas open sets are usually introduced before closed ones.
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3
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How does one write the "gothic" letters ($\mathfrak{g}$) in handwriting?
Most mathematical notation is designed with handwriting in mind in the first place, and typography must then try to follow, not always very successfully. However there is a particular type of notation ...
3
votes
0
answers
242
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What is the meaning of big-O of a random variable?
I encountered this problem in a book "Pattern Recognition and Machine Learning" by Christopher M. Bishop. I excerpt it below:
screenshot of the book
In the excerpt, the big-O notation $O(\xi^...
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1
answer
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Name for extension of the symplectic group
Let $S_g$ denote an ortientable surface of genus $g$. Let $\operatorname{Diff}(S_g)$ denote the group of diffeomorphism (that need not fix the orientation). Is there a name for the image of $\...
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2
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1
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What is the meaning of $\alpha^{+L}$ for $\alpha$ an infinite countable ordinal?
Condition (a) of lemma 3.4 in the paper “Countable ranks at the first and second projective levels” [M. Carl, P. Schlicht, P. Welch] is
$\alpha^{+L} = \omega_1,$
where $\alpha$ denotes any infinite ...
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0
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Notation for function that is constant with respect to a parameter
I am wondering if there is a common notation for a function that does not depend on a particular parameter. I am wondering about notation both for applying the function ($f(x, y)$) as well as defining ...
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1
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1
answer
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"Variable and fixed" in categories
We often find in Grothendieck terminology the words variable and fixed (or absolute).
For example in SGA 4 studies variable topological spaces, groups, and categories as examples of morphisms of topos....
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2
votes
1
answer
572
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First use of corner quotes for Gödel numbers
Who first used the corner quotes, $\ulcorner$ and $\urcorner$, for the notion of Gödel number? They can also be written as\Godelnum with Sam Buss's macro.
They were ...
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2
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0
answers
482
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Degree of a morphism between affine varieties
(Context: rewriting a joint paper with a coauthor.)
We are defining the degree of a morphism $f:A^m\to A^{n}$ to be $\max_{1\leq i\leq n} \deg(f_i)$, for $f_1,f_2,\dotsc,f_{n}$ the polynomials ...
- 22k
1
vote
1
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Notation for infinite cartesian products
This is a soft question, feel free to delete it if deemed inappropriate for the site. What is the best notation for the cartesian product of an infinite number of copies of the same set $E$? Maybe one ...
- 9,152
1
vote
0
answers
72
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Notation for dominating (or uniformly bounded) function
While I developing a new statistical estimator, I wondered is there any good notation for dominating (or uniformly bounded) function.
A situation like this. For some true function $f:\mathbb{R} \to \...
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0
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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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2
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0
answers
286
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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, ...
- 1,077
-3
votes
1
answer
291
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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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13
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1
answer
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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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2
votes
1
answer
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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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vote
1
answer
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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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1
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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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vote
0
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126
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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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2
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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 ...
5
votes
2
answers
525
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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 ...
5
votes
0
answers
769
views
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 ...
- 147
2
votes
0
answers
133
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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
vote
1
answer
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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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