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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5
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2answers
343 views

What is the proper notation for a general number of nested summations?

A sum over one index: $\sum_i f(i)$ A sum over two indices: $\sum_i \sum_j f(i,j)$ A sum over many indices: $\sum_{k_1} \sum_{k_2} \underbrace{\dots}_n \sum_{k_n} f(\mathbf k)$?
0
votes
2answers
49 views

What is the difference between $[H, g]$ and $[h, g]$?

I am working on this problem, where $[H, g]$ is the commutator group: Let $H$ be a subgroup of $G$, show that $[H, g] = [H, \langle g \rangle]$. Before solving it, I need to understand the ...
0
votes
0answers
54 views

Name the maps in a commutative diagram

When writing a formal paper sometimes one needs to construct complicated commutative diagrams, such as My question is, should one always give names to all maps in the diagram (perhaps except those ...
1
vote
1answer
41 views

What does “$C^{\infty}$” convergence mean?

I'm studying first notions about several complex variables. As a consequence of the (generalized form) of the Cauchy esteem for holomorphic functions, the book says that in the space $\mathcal ...
1
vote
1answer
37 views

Can one use the following notation in integrals?

I read from theoretical physics lecture notes the following: http://theory.physics.helsinki.fi/~fymmi/Luennot4_1-9.pdf $$\Gamma(p)\Gamma(q)=4\int_0^\infty dr r^{2p+2q+1}e^{-r^2}\int_0^{\pi/2}d\varphi ...
0
votes
1answer
37 views

what is the name of this operation: $x^T\otimes B$

So the question is simple: how the following operation is commonly called? $x^T\otimes B$, each element of matrix B is multiplied by the array $x^T$, so the result is a matrix. I'm not even sure if I ...
3
votes
1answer
58 views

A quickie about set theory notation

I'm reading the first chapters of my discrete mathematics textbook and I couldn't help but wonder (perhaps I haven't seen enough examples) -- is it more appropriate to write that $a$ is an integer and ...
1
vote
1answer
54 views

What is the operator priority in set theory?

Say I have three arbitrary sets $A,B,C$. Which statement is true ? $A \times B \cup C = (A \times B) \cup C $ $\quad $ or $\quad$ $A \times B \cup C = A \times (B \cup C) $ And the same ...
1
vote
0answers
45 views

Mathmatical notation for a term of a polynomial

If I have a polynomial $f(x) = ax^n + bx^{n-1} + cx^{n-2} \ldots zx^0$, is there any mathematical notation for one term, such as the $x^3$ term. For example, if I have a polynomial of $f(x) = x^6 + ...
0
votes
1answer
46 views

Summation notation with ambiguous subscripts

I'm reading a paper which has the following description; Say we have a time series of correlated sequential observations of the random variable $X$ denoted $\{x_n\}_{n=1}^N$ from a stationary, time ...
1
vote
1answer
39 views

Need help with notation — finite set of random primes

I need help with notation for a finite set of random primes. Edit I've inserted my take on the format from the answer. Does it work? My attempt:$$\{X\in\binom{\mathbf P_{3,100}}{20}\},$$ ...
0
votes
0answers
27 views

is it okay in a journal to express the vector-scalar division like this

Assume you want to show that a vector is divided by a scalar and then the norm is taken, i.e. $\|\frac{x}{c}\|$ where x is the vector, and c is the scalar. So is it okay to show it like this? I ...
1
vote
0answers
37 views

Notating the components of the $\hat{\beta}$ matrix when $\hat{Y}$ is multidimensional

This is a question on The Elements of Statistical Learning. We have from the linear model $$\hat{Y} = X^{T}\hat{\beta}$$ where $\hat{Y}^{T} = \begin{bmatrix} \hat{Y}_1 & \hat{Y}_2 & \cdots ...
1
vote
3answers
195 views

What Does $y=A\exp(6x)$ mean?

So my professor used this and I don't really know what this equation means. $A$ is a positive constant, different $A$'s give different curves and these curves form a family $\mathcal{F}$. Given a ...
0
votes
1answer
51 views

Why is $(f(x))'$ shortened $f'(x)$

Why is $(f(x))'$ shortened $f'(x)$? This makes the chain rule look awkward, as $(f(g(x)))'\neq f'(g(x))$, but rather $f'(g(x))\!\times\! g(x)$, and makes it difficult to remember. It's also an ...
4
votes
1answer
141 views

What does a standalone $dx$ mean?

Some literature uses $dx$, in the context of differential equations, in a confusing way without defining what it really stands for: $Mdx + Ndy = 0$ Does it mean one of the following or something ...
0
votes
1answer
52 views

$\bigcup_{i \in I} \mathcal{P} (A_i)$

This is Velleman 3.7, Problem 4 Below is the problem, verbatim. Suppose $ \{ A_i \mid i \in I\}$ is a family of sets. Prove that if $\mathcal{P}(\bigcup_{i \in I} A_i) \subseteq \bigcup_{i \in I} ...
0
votes
1answer
77 views

What is the name of this graph operation? (creating $k$ connected copies)

I'm looking for the name of this natural graph operation, which is kinda similar to Cartesian product, but not quite, as the copies of the graphs are not fully connected. Instead, it creates a $k$ ...
39
votes
6answers
3k views

What do mathematicians mean by “equipped”

I am a mathematical illiterate so I do not know what people mean when they say equipped. For example, I say that Hilbert space is a vector space equipped with a inner product. What does that ...
0
votes
3answers
56 views

What does the notation $\min_x$ mean?

I have a problem in which I need to find $\min_x(f(x))$. What does this notation mean?
1
vote
2answers
591 views

What is cos²(x)?

This looks odd to me. I need a definition. Is it just the square of $\cos(x) $ ? Like $\ \cos^2(x) = \cos(x) \cdot \cos(x) $ ? Then why don't you write it like that: $\cos(x)^2 $ ?
1
vote
1answer
84 views

Confusion about integration notation

This is probably a silly question but I've never seen this notation: For a > 0, compute $$\int\int_{x/y \leq a} 2e^{-(2x+y)} dx dy$$ What is $x/y \leq a$ there for? This is from my statistics ...
1
vote
2answers
27 views

Question about alternate subset notation

I am reading a new text, and I have come across the notation '$\Subset$' as well as '$\subset$'. Am I correct in assuming that '$\Subset$' is an alternative method of specifying an improper subset ...
1
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0answers
55 views

How to avoid fractions when writing expressions?

When I'm writing expressions in an articles the fractions when not segregated in a single equation usually are a problem. One example is $\frac{x}{\log y}$ that can be written as $x\log y^{-1}$. ...
1
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0answers
66 views

The maximum notation as regards the absolute value?

We know that $\max(\textbf{A})$ gives the maximum element of the array $\textbf{A}$. What is the notation, or a short formula, if we seek the element having the largest absolute value? e.g., ...
11
votes
2answers
1k views

Why is there this strange contradiction between the language of logic and that of set theory?

In standard probability theory events are represented by sets consisting of elementary events. Consider two events for which (as sets) $A \subset B$. If an elementary event $x \in A$ takes places then ...
7
votes
5answers
147 views

Does the interval notation $[a,b]$ imply that $a<b$?

If showing an interval $[a,b]$, are $a$ and $b$ implied to be such that $a,b\in\mathbb{R}$ and $a\le b$, by simply writing that, or must they be specified as such?
7
votes
4answers
435 views

What is $1 + 999999…$ (an infinite string of $9$s)?

I'm doing a programming assignment in Haskell, and it involves adding "infinite" lists. At the bottom of the assignment, our lecturer has written ... "in some sense an infinite string of $9$s is the ...
1
vote
1answer
64 views

The Sum Of Sets Or Intervals

I came across the sum of set, ie $$ A + B = \{\mathbf{a}+\mathbf{b}\,|\,\mathbf{a}\in A,\ \mathbf{b}\in B\}. $$ What does that mean? I can understand unions etc of set, but not the sum. For ...
1
vote
0answers
74 views

What does this || symbol mean?

I am trying to understand this paper, but in this paper the following || symbol is frequentlly used. I wonder to know what does it mean, what are these two ...
1
vote
1answer
33 views

Specific treatment for the first and last element of sequence in a function?

Let $A = \langle a_1,\dots,a_n \rangle$ be a sequence. I have a function that given any element $a_k$ it will return the values of $a_{k-1}+a_k+a_{k+1}$ with the exception of the first and last ...
0
votes
0answers
38 views

Interpretation of an integral of a function $f$

When we think of a Riemann integral, it is usually defined as $\lim_{\Delta x_{k}\rightarrow 0}\sum_{k = 1}^{n}f(x_{k}^{*})\Delta x_{k} = \int_{a}^{b}f(x)~dx$. This means that $f(x)$ should be ...
1
vote
1answer
31 views

Sigma notation abstract limits

What is meant by the following notations? $\sum\limits_{k=0}^{p+1}k^3$, $\sum\limits_{k=0}^{p-2}k^3$ I need to use this notation to prove a statement is true for $\sum\limits_{k=0}^{p+1}k^3$ when ...
1
vote
2answers
198 views

What does this symbol mean?

This is from Discrete Mathematics and its Applications What is the symbol used in 9c, 9d, 9f, 10c, 10f, 10g? I looked through the chapter section and the closest symbol I saw to this is the subset, ...
1
vote
2answers
116 views

Is there convenient notation for Viète's formula?

Is there a convenient way to write Viète's formula $\displaystyle \frac2\pi= \frac{\sqrt2}2\cdot \frac{\sqrt{2+\sqrt2}}2\cdot \frac{\sqrt{2+\sqrt{2+\sqrt2}}}2\cdots$ using sigma and/or pi notation a) ...
1
vote
2answers
158 views

What does a prime (apostrophe) mean before a predicate?

I found this statement in a paper by John McCarthy: $$ \forall x.ostrich\ x \supset species\ x ={}^\prime{}ostrich $$ I can't figure out what the prime indicates.
12
votes
5answers
1k views

Why does notation for functions seem to be abused and ambiguous?

I really need to clear up a few things about function notation; I can't seem to grasp how to interpret it. As of right now, I know that a function is roughly a mapping between a set $X$ and a set $Y$, ...
1
vote
4answers
70 views

Is $S$ a monoid, or is $(S,*)$ a monoid?

If I have a set $S$ with operation $*$ as a monoid. Would I say I have a monoid $S$ with the binary operation $*$ or would I say I have a monoid $(S,*)$ where the binary operation $*$ does ...
1
vote
1answer
39 views

How to denote the result of application of a function on items from other multiset?

Let $A$ be a set, i.e. $A=\{2,3\}$. Then it is common to denote by $\{f(a)|a\in A\}=\{f(2),f(3)\}$ the result of application of function $f$ on items from $A$. But this work just for sets. How is ...
1
vote
1answer
66 views

What does $d\zeta_1\wedge\cdots\wedge d\zeta_n$ mean in the context of Cauchy formula (on polydiscs)?

A Polydisc of center $z^o=(z_1^o,\dots,z_n^o)\in\Bbb C^n$ and multiradius $r=(r_1,\dots,r_n)\in(\Bbb R^+)^n$ is defined as $$ P_{z^o,r}:=\prod_{j=1}^n\Delta_{z_j^o,r_j} $$ where ...
1
vote
1answer
32 views

Notation of functions

For example, we have a definition of a cryptosystem: We define a cryptosystem as tuple $\langle M, K, C\rangle$ where $M$ is a set of texts that are not ciphered, $K$ set of keys, $C$ set of texts ...
1
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0answers
32 views

Which mathematical symbol should I use to decribe the relationship between two vectors?

There are two vectors, $\mathbf{A}=(a_1,a_2,\cdots,a_n)$ and $\mathbf{B}=(b_1,b_2,\cdots,b_n)$. If $a_i=0$, then $b_i=0$. If $a_i\neq 0$, then $b_i\geq a_i$. Which mathematical symbol should I use to ...
0
votes
4answers
138 views

Can someone tell me what this sign means?

Here it is ∝ Have never seen before and google search hasn't given any results.
1
vote
1answer
81 views

what mean $\overline{\text{span}(e_1,…,e_n)}$?

In functional analysis, what mean $$\overline{\text{span}(e_1,...,e_n)} \ \ ?$$ is it the closure of $\text{span}(e_1,...,e_n)$ ? How does it work ? Even if it's that, I don't understand how ...
3
votes
1answer
143 views

Is there a multiple function composition operator?

Is there a commonly-accepted operator which defines multiple function composition? I have not been able to find one on any of the related Wikipedia pages. In one of my proofs, I've been finding it ...
0
votes
1answer
43 views

Currying syntax clarification - how to work through an example of currying?

I understand currying from a computer science background, so I'm happy explaining currying with a before and after example in specific languages, eg, in Java ...
1
vote
0answers
25 views

Is there standard notation for partial and total functional derivatives?

Consider a functional of multiple functions $A\left[y,z\right]=\int f\left(y\left(x\right),z\left(x\right)\right)dx$. I would like to distinguish between partial and total functional derivatives, ...
2
votes
2answers
158 views

'Smaller than infinity' notation

I've been coming across some papers (written in the 1960s - 1970s) that use the following peculiar statement: Let use denote by $H$ the space of all grid-functions $w_r$ for which: $$ ...
3
votes
0answers
157 views

Row-normalized and column-normalized matrix notation

I'm searching for the mathematical, algebraic notations of a row-normalized and column-normalized matrix. For example, let us consider the following matrix A: $$ A = \begin{pmatrix} 2 & 7 \\ 4 ...
1
vote
0answers
31 views

How to correctly index a sequence

I have a problem that might seem quite trivial, but I'm having a hard time getting my head around it: I have to construct a vector of the form $$\overline{\pi}=[\pi(0),\{\pi(i,j)\}]$$ where ...