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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2
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2answers
72 views

why notate coset representatives as $a h_1$ to show well-defined-ness?

In Fraleigh, there is what appears to be a classic theorem on cosets. I'm confused about the proof for the converse. Theorem. Let $H$ be a subgroup of a group $G$. Then left coset multiplication ...
2
votes
1answer
79 views

Integrating factor: renaming variable

I'm having trouble following some math in a manuscript. The author is using an integrating factor to begin to solve a differential equation: $$ dX(t)=[\mu(t)+b(t) X(t)]dt + \sigma(t)dB(t) $$ by ...
1
vote
3answers
542 views

Is it possible to change the step interval on a sum?

Suppose we wanted to create a sum that says, from n=0 to n=16, n will add 4 to itself and add the function like a regular sigma form. Quite similar to: ...
1
vote
2answers
268 views

Notation for vector composed of subset of elements of another vector

Suppose I have a vector $\boldsymbol{x} = (x_1,x_2,\ldots,x_N)$ in $\mathbb{R}^N$. I need to express a function $\boldsymbol{y} : \mathbb{R}^N \mapsto \mathbb{R}^{M(\boldsymbol{x})}$ where $M(\...
1
vote
1answer
52 views

Notation for inductive definitions of sets

Is there a formal notation for inductive definitions of sets? For example, like this: $Closure(U,C,A)$ where $U$ is a set, $C$ is a set of constructors (in a simple case, operations on the set $U$), ...
1
vote
1answer
207 views

Notation for vector v in basis x, dimension y

What is common notation for the value of the $n$th dimension of vector $v$, given by basis $x$. Is it something like $$v_{x}^{y}$$ Thanks!
4
votes
1answer
54 views

Difference between $f(x) \neq 0$ and $f(x) \not\equiv 0$

What is the difference between the statements $$ f(x) \neq 0 $$ and $$ f(x) \not\equiv 0? $$
1
vote
1answer
112 views

Doubts with differential geometry notation in Frankel

This is from Frankel's The Geometry of Physics: Problem 2.3(2) Consider the tangent bundle to a manifold $M$. Show that under a change of coordinates in $M$, $\partial/\partial q$ depends ...
5
votes
2answers
410 views

Notation and the name choice for meet and join (in order theory)

I have two simple questions: From where do the names meet and join come from? I don't see any intuition between those names in context of order theory. From where does the notation come? I have to ...
4
votes
2answers
3k views

What does the $\prod$ symbol mean?

This is one of those cases where I would google if I could, but I don't know what to search for. I've come across this symbol a few times, but I have no clue what it means or what it is called. $$\...
1
vote
0answers
50 views

Notation question in Majda and Bertozzi's “Vorticity and Incompressible Flow”

On pg 2, the fluid velocity in the Navier-Stokes system of equations is noted as: $v(x,t) \equiv (v^1, v^2, \ldots, v^N)^t$, where I am assuming that the velocity vector field is time-dependent. The ...
4
votes
1answer
627 views

Lowercase delta in differential-like equation

Preface: The following question comes from an expression seen in a biophysics paper published in Nature protocols. I'm aware that in pure mathematical notation $\delta$ is never used in the context ...
1
vote
3answers
73 views

Definitions of sets

How would I define the set $\Omega_2(x)\ =\ \{4, 6, 9, 10, 14, 15, 21, 22, 25, 26, 33, 34, 35, 38, 39\dots\}$, where $\Omega_2$ is the set of semi primes (ie - numbers with $2$ not necessarily ...
6
votes
3answers
4k views

The difference between $\Delta x$, $\delta x$ and $dx$

$\Delta x$, $\delta x$ and $dx$ are used when talking about slopes and derivatives. But I don't know what the exact difference is between them.
6
votes
1answer
248 views

A question about co-exponentials

An exponential object $B^{A}$ is defined to be the representing object of the functor $$\mathcal{C}\left(- \times A,B\right): \mathcal{C} \rightarrow Set$$ or equivalently, as the terminal object of $...
1
vote
1answer
130 views

Notation for the union of a system of sets

The notation $\bigcup_{i \in I}A_{i}$ denotes the union of the range of a function, and as such, is used only if we are considering some function (an indexed family). The generalized associative ...
0
votes
1answer
100 views

What does this notation mean? $[p]^B$

The question I'm trying to answer is: Find an ordered basis $B$ of $\mathbb C_3[x]$, such that $$[p]^B= \begin{pmatrix}1\\0\\0\\i\end{pmatrix}$$ for the polynomial $p=2+2x+2x^2+2x^3$. The notation I'...
1
vote
1answer
47 views

Is there a standard shorthand for the set $\{0, 1, \dots,k-1,k+1,\dots,n-1,n\}$?

The question title pretty much says it all. In derivations where there's some fancy footwork being done with indices, I find expressions like $$\sum_{i \neq k} x_i$$ a bit too vague. On the other ...
0
votes
1answer
72 views

Help needed for mathematical formulation

I'm trying to write a simple mathematical formulated which expresses the following: Let F={f1,f2,...,fn} be an ordered set of flights, each f being associated with a begin time begin(f) and an end ...
2
votes
3answers
353 views

Is there an existing mathematical notation for representing a repeated number after a decimal point?

Sometimes you encounter numbers likes 1.3332 in mathematics. Is there a more concise notation for representing a number with the same repeating digit after the decimal point that exists in common ...
2
votes
0answers
48 views

What are default topologies on $R^∞$ and $R^ω$?

To extend the original question Difference between $R^\infty$ and $R^\omega$: What are default topologies on $R^∞$ and $R^ω$? I would think that we take product topology on $R^ω$ and limit topology ...
0
votes
1answer
62 views

Definition of a field

I have it defined here that a field $\mathbb{F}$ consists of $\mathbb{F} = (\mathbb{F},+,0,1,\cdot)$ where $\mathbb{F}$ is a set $0,1 \in \mathbb{F}$, $0 \not = 1$ $+$ is a mapping (addition), $+:\...
1
vote
2answers
167 views

Difference between $R^\infty$ and $R^\omega$

I know $R^\omega$ is the set of functions from $\omega$ to $R$. I would think $R^\infty$ as the limit of $R^n$, but isn't that $R^\omega$? The seem to be used differently, but I can't tell exactly how....
2
votes
1answer
78 views

Partial differentials vs normal differential (notation question/clarification only)

In physics, it seems like the use of $\dfrac{dy}{dx}$ and $\dfrac{\partial y}{\partial x}$ are used somewhat interchangeably. My understanding is that, technically $\dfrac{dy}{dx}$ is only ...
1
vote
1answer
45 views

What is $\mathcal{O}_{X,x}$?

I read from Liu's "Algebraic Geometry and Arithmetic Curves the following definition: A ringed topological space consists of a topological space $X$ endowed with a sheaf of rings $\mathcal{O}_X$ on $...
1
vote
2answers
119 views

Notation for “Nested” Sequences?

Let $X$ and $Y$ be two ordered pairs $X = (A,B)$ and $Y = (A,B)$. Then let $A$ and $B$ be two sequences "nested" in the pairs $A = \langle A_1,A_2,...,A_n \rangle$ and $B = \langle B_1,B_2,...B_n \...
14
votes
5answers
17k views

'mod' or 'remainder' symbol valid in maths?

I am trying to write the Euclidean algorithm in the following way: $A = \lfloor A \div B \rfloor \times B + (\text{remainder of}) \: A \div B $ Now is there any symbol I can use to say "remainder ...
2
votes
1answer
133 views

Notation used for multivariate random variables

Let $(\Omega, \mathcal{F}, P)$ be a probability space and $X_1(\omega), \dots, X_n(\omega)$ be random variables defined on the space. Suppose we are concerned with the joint behavior of the variables. ...
1
vote
1answer
71 views

Can subscripts be used like this?

I have a variable named $P$, and another three variables named $P_c$, $P_d$ and $P_u$. Now, if I define this function: $$f(x) = x_c + x_d + x_u$$ Is it correct to say that: $$f(P) = P_c + P_d + P_u$$ ...
1
vote
1answer
90 views

Definition of functions on metric spaces.

In the post Definition of functions, it is stated in the accepted answer that one way to define a function is to define it as the triple $(f, X, Y)$ where $f \subset X \times Y$. My question is what ...
0
votes
1answer
42 views

Math script/writing for a “count of something”

I was wondering if there was a formal way to write a "count of something". For instance in my problem, I have a population of discrete entities (people) U, I use u with a subscript if I need to refer ...
1
vote
1answer
60 views

Notation in functions spaces

I'm wondering if there is a convention for the following. Let $g:\mathbb{R}^2\to \mathbb{R}$ be given and u in $C^1(\mathbb{R})$. I'm looking for a notation for $g(x,u(x))$ is in the space of ...
1
vote
1answer
83 views

From $\mathsf{O}$ to $\mathsf{I}$ via $\infty$

The following is not true mathematics, but a little imaginary story about mathematical symbols. I wonder if there is - in parts - a true (etymological) story behind it. Once there was a symbol $\...
1
vote
3answers
102 views

Is there a notation for $((n!)!)!$?

I wonder if there is a notation for repeating factorials such as $((3!)!)!$. Without the parentheses, $(3!)!$ could be confused with the double factorial $3!!$. Is there is no such notation known, ...
0
votes
1answer
44 views

Explanation about notation

Can anyone explain to me what $C[x^2,x^3]_{(x^2,x^3)}$ means? It is connected with localizations but it is unclear to me what it means exactly.
3
votes
2answers
89 views

“Scoping” rules in mathematics

I've been having a discussion with my friend about scoping rules in mathematical syntax. By scoping I mean a set of rules that define how variable and function names could be chosen so that at any ...
0
votes
2answers
303 views

Probability notation?

I've tried to gather information about these three topics i'm about to show but i couldn't find. I will display them in a question for better understanding and context of the situation. For a ...
2
votes
3answers
93 views

Find the general solution of the system

Given: \begin{cases} x- z = 1 \\ y + 2z - w = 3 \\ x + 2y + 3z - w = 7 \end{cases} My work: Use Gauss method Leading variables: x,y, and z Free variables: w Express the leading variables x,y,and z ...
2
votes
5answers
2k views

Notation of Hessian?

I looked here, and couldn't find an answer: http://en.wikipedia.org/wiki/Hessian_matrix I am looking for a way to denote the Hessian at a certain value. I have a function $f(x)$ where $x \in \...
4
votes
2answers
456 views

Why do some people place the differential at the beginning of their integral? [duplicate]

I've seen several times on this site people writing integrals like $ \int \! dxf(x) $. This seems confusing to me, especially in an iterated integral or next to a long integrand and it's nonstandard ...
1
vote
1answer
75 views

Notation in a paper

I'm looking at a paper from H Fuchs called Optimal Surface Reconstruction from Planar Contours dated 1977. It can be found here. On page 2 of the PDF (stamped as page 694) the notation $+_k$ is ...
0
votes
1answer
130 views

Notation for derivative of a 2 argument function w.r.t its second argument

For functions of one argument, the "Newton" notation for the derivative of that function is concise and unambiguous. For example, if I want to express $$ \lim_{h\to 0} \frac {f(x^2 + h) - f(x^2)}{h} ...
6
votes
1answer
507 views

What is $\gtrless$

I'm reading Papadimitriou et al's Combinatorial Optimization and came across notation I'd never seen before and don't know what it means. The latex markup for it is ...
0
votes
1answer
167 views

Useful maths symbols - tips and made up symbols for quick writing?

I am looking for some symbols for the most common words in maths problems, such as : such as (":"), given (for some reason I am using $\sqsupset$ ), assume (I use $\downarrow$ ) etc and any common ...
1
vote
3answers
130 views

What does the notation $||u||$ mean?

I know this is basic, but I am just a little unsure of this. What does the notation $||u||$ mean? $u$ is a vector
5
votes
3answers
623 views

What does this combination of symbols mean? $\exists !$

I just want to know what this combination of symbols means: $\exists !$ I know ∃ means 'there exists', but what does it mean when it is paired with a '!'? I have written down 'there exists unique" ...
6
votes
0answers
718 views

Typesetting imaginary unit and bessel functions

I know that this issue has been treated in many places, but I have yet to reach something conclusive, hence I am herein seeking your help. Following the 260.3-1993 - American National Standard ...
2
votes
2answers
645 views

Notation for the last X sorted elements of a set

I have a set of ages, $\mathbb{T}=\{30, 33, 39, 40, 41\}$. I'd like to refer to the last $X$ of these ages by $\mathbb{T}^{(X)}$. So, $\mathbb{T}^{(2)} = \{ 40, 41 \}$. But how can I define this $\...
1
vote
1answer
322 views

Combining set builder and summation notation

What's the best notation for the sum of a subset? Given $S = \{1,2,3,4,5,6,7\}$, let's say I want to find the sum of the squares of elements less than 4. Initially I used the following notation: $$\...
1
vote
0answers
56 views

Other shortcuts similar to $\pm$?

In writing proofs, I'll sometimes end up with two separate but similar results, such as $[a] + [b] = [a + b]$ and $[a] \times [b] = [a \times b]$. Out of curiosity, is there some standard notation to ...