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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Symbol for the cardinality of the continuum

The usual symbol for the cardinality of the continuum (i.e. the real numbers) is Fraktur $\mathfrak{c}$. However, I recall some sources also using $\aleph$ (with no subscript). This usage is not ...
3
votes
1answer
276 views

Inconsistent naming of elliptic integrals

This may be a question whose answer is lost in the mists of time, but why is the elliptical integral of the first kind denoted as $F(\pi/2,m)=K(m)$ when that of the second kind has $E(\pi/2,m)=E(m)$? ...
3
votes
4answers
391 views

A short way to say f(f(f(f(x)))) [duplicate]

Is there a short way to say $f(f(f(f(x))))$? I know you can use recursion: $g(x,y)=\begin{cases} f(g(x,y-1)) & \text{if } y > 0, \ \newline x & \text{if } y = 0. \end{cases}$
2
votes
1answer
569 views

What's the meaning of $x \choose y$ in the Catalan number formula?

What's the meaning of $\choose$ in this formula: $$C_n=\frac1{n+1}\binom{2n}{n}=\frac{(2n)!}{n!(n+1)!}\qquad\mathrm{for}\;n\geq 0$$ Is it a division?
1
vote
3answers
252 views

If $V$ is a vector space, that what is $V^\mathbb{N}$?

Notation question (I believe the notes that I'm reading uses pretty common notation): Let $V $ be a vector space over a field $K$. What is $V^\mathbb{N}$? Is it a vector space of infinite dimensions? ...
5
votes
4answers
628 views

Long division notation (census of nations)

The Wikipedia article on long division explains the different notations. I still use the European notation I learned in elementary school in Colombia. I had difficulty adapting to the US/UK notation ...
11
votes
6answers
1k views

In written mathematics, is $f(x)$a function or a number?

I often see notation/wording like "let $f(x)$ be a continuous function" or "let $f(x) \in C^0(\mathbb{R})$". I would say that $\sin$ and $x \mapsto \sin(x)$ are functions, while $\sin(x)$ is a real ...
0
votes
2answers
147 views

Conjecture about the set of Sphenic numbers

Sum of a set of sphenic numbers can't be equal to the sum of any other set of sphenic numbers. By that I meant, Say S is the set of sphenic numbers. Let S$_1$ $\subset$ S. Then there is no such ...
3
votes
2answers
947 views

Definition of $C_0$

This is probably a silly question, but a couple of people that I have talked to have had different responses. Does $C_0$ denote the set of continuous functions with compact support or the set of ...
3
votes
2answers
2k views

Proper notation for distinct sets

Consider I have two sets that neither one is a subset of the other (for example, the set of prime numbers and the set of odd integers). Is there a specific notation that combines the meanings of both ...
6
votes
2answers
2k views

Precedence of set union, intersect, and difference?

Online, I have read contradicting opinions on whether intersect should take precedence over union (by analogy to logical ...
15
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3answers
70k views

What does E mean in 9.0122222900391E-5?

I am not a mathematician(IANAM), however I wish I could be. My question: I often find this at the bottom of pages. Page generated in 0.00013899803161621 Sometimes, I come across Page ...
12
votes
4answers
1k views

Notation question: Integrating against a measure

Suppose $\mu$ is a measure. Is there any difference in meaning between the notation $\int f(x)d\mu(x)$ and the notation $\int f(x) \mu(dx)$? Many thanks.
7
votes
2answers
60k views

what does ∇ (upside down triangle) symbol mean in this problem

Given $f(x) = \frac{1}{2}x^TAx + b^Tx + \alpha $ where A is an nxn symmetric matrix, b is an n-dimensional vector, and alpha a scalar. Show that $\bigtriangledown _{x}f(x) = Ax + b$ and $H = ...
7
votes
4answers
638 views

Can Leibniz Notation Be Treated As a Quotient?

Why is saying $\frac{dy}{dx}\frac{dy}{dx}=y\frac{d^2y}{(dx)^2 }$ not valid? Does Leibniz notation (and thinking of it as an infinitesimal quotient) not work for higher-order derivatives?
7
votes
1answer
315 views

Why do lambda and pi go together?

In measure theory, we have "lambda systems" and "pi systems". Pearl's message passing algorithm has "lambda messages" and "pi messages". Is there a reason that lambda and pi go together?
14
votes
4answers
554 views

Solving (quadratic) equations of iterated functions, such as $f(f(x))=f(x)+x$

In this thread, the question was to find a $f: \mathbb{R} \to \mathbb{R}$ such that $$f(f(x)) = f(x) + x$$ (which was revealed in the comments to be solved by $f(x) = \varphi x$ where $\varphi$ is ...
1
vote
1answer
451 views

Einstein notation - difference between vectors and scalars

From Wikipedia: First, we can use Einstein notation in linear algebra to distinguish easily between vectors and covectors: upper indices are used to label components (coordinates) of ...
6
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4answers
4k views

Why are x and y such common variables in today's equations? How did their use originate?

I can understand how the Greek alphabet came to be prominent in mathematics as the Greeks had a huge influence in the math of today. Certain letters came to have certain implications about their ...
1
vote
1answer
241 views

Formal notation for number of rows/columns in a matrix

Is there a generally accepted formal notation for denoting the number of columns a matrix has (e.g to use in pseudocode/algorithm environment in LaTeX)? Something I could use in the description of an ...
1
vote
2answers
193 views

Notation of homeomorphism from B(H) to B(K), corresponding to unitary transformation of Hilbert spaces

Let U be a unitary transformation from Hilbert space H to Hilbert space K. How do you call a *-homomorphism f from B(H) to B(K), defined by $f(a) = U a U^{-1}$? I'm interested both in a symbol, ...
1
vote
1answer
202 views

Syntax for the convergence of random variables

Can anyone explain the syntax from the following Wikipedia article on the convergence of random variables? In particular: What is the squiggly line? What are $d$, $D$, $L$, $L_X$? Why does it say ...
6
votes
5answers
3k views

Does set $\mathbb{R}^+$ include zero?

I've been trying to find answer to this question for some time but in every document I've found so far it's taken for granted that reader know what $\mathbf ℝ^+$ is.
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votes
8answers
4k views

How did the notation “ln” for “log base e” become so pervasive?

Wikipedia sez: The natural logarithm of $x$ is often written "$\ln(x)$", instead of $\log_e(x)$ especially in disciplines where it isn't written "$\log(x)$". However, some mathematicians ...
15
votes
5answers
2k views

Usage of dx in Integrals

All the integrals I'm familiar with have the form: $\int f(x)\mathrm{d}x$. And I understand these as the sum of infinite tiny rectangles with an area of: $f(x_i)\cdot\mathrm{d}x$. Is it valid ...
9
votes
8answers
2k views

Why are derivatives specified as d/dx?

Is the purpose of the derivative notation d/dx strictly for symbolic manipulation purposes? I remember being confused when I first saw the notation for derivatives - it looks vaguely like there's ...
4
votes
2answers
213 views

$n$ out of $ m$ theorems (some imply the rest)

Is there symbolism (or even a name) for groups of statements in which any fixed-number of them imply all the rest? For example, in linear algebra, a basis is sometimes defined as a set of ...
19
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4answers
4k views

What is the meaning of the double turnstile symbol ($\models$)?

What's the meaning of the double turnstile symbol in logic or mathematical notation? : $\models$
3
votes
1answer
579 views

Why do we use the commutator bracket for Lie algebra's

We define Lie algebras abstractly as algebras whose multiplication satisfies anti-commutativity and Jacobi's Identity. A particular instance of this is an associative algebra equipped with the ...