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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4
votes
3answers
69 views

Difference between $\sqrt [q]{x^p}$ and $x^{p/q}$?

Is there difference between $\sqrt [q]{x^p}$ and $x^{p/q}$ ? There seems to be no difference when just dealing with positive reals, but what about when $x$ is negative (or even complex)? p.s. ...
0
votes
1answer
84 views

What does $(\mathbb C\backslash\{0\})\times\mathbb R$ mean?

What does $$(\mathbb C\backslash\{0\})\times\mathbb R$$ mean, and how does it look like? Thanks.
2
votes
2answers
44 views

Typesetting of per-degree

What is the best (most accepted) way to typeset something like $5$ per (angular, not temperature) degree (in $\LaTeX$)? I came up with the following options (using ...
0
votes
2answers
110 views

Is “zed” a valid name for $\emptyset$?

I've always known that $\emptyset$ is called an empty set or null, until recently, when I heard someone calling it zed. I looked it everywhere but couldn't find this naming. Is "zed" a valid name ...
1
vote
2answers
34 views

Set of all $a\in\mathbb Z$ that are coprime to $b\in\mathbb Z$

$\newcommand\Z{\mathbb{Z}}$I'm looking for a standard or reasonable notation for $$ \{a\in\Z : a\perp b\} = \{a\in\Z : \gcd(a,b)=1\} $$ My problem is that: It is not $\Z/b\Z^*$, since for me, ...
2
votes
1answer
21 views

What is the name for the property that a subset of a set follows the same rules as the set?

I have a set that follows a certain property and I want to say that the subsets of this set also follows the property. What is this called? I know that closure under an operation means that performing ...
1
vote
1answer
30 views

Clarifying the PDE notation C^1([0,T], X).

In studying nonlinear hyperbolic PDE, I've come across the following spaces: $C([0,T],H^s(\mathbb{R}^n))$. $C^1([0,T], H^s(\mathbb{R}^n))$. $L^p([0,T],H^s(\mathbb{R}^n))$. I presume that $(1)$ ...
1
vote
2answers
57 views

What is $\Bbb R^{\times}$?

I'm doing some sheets for my Abstract Algebra class and I can't seem to remember the group defined as $\mathbb{R}^{\times}$. It's obviously some variation of $\mathbb{R}$ but I'm away from college on ...
6
votes
1answer
55 views

Infinite Continued Fraction Notation

I can't find anywhere via googling; is there some sort of $\sum$ like notation for infinite continued fractions? In other words, for a sum we do this: $$ 1+x+x^2+x^3+... = \sum_{n=0}^\infty x^n $$ ...
1
vote
2answers
171 views

A Theorem About Compactness and

My first exposure to any sort of topology is from Spivak's Calculus on Manifolds. I think I understand compactness conceptually, I'm just finding the rigor a little bit elusive. My first question ...
0
votes
1answer
21 views

Formal way to add a set to an existing collection

If we have an existing collection of sets $\{G_\alpha\}$ (possibly uncountable) and a $X$ that we would like to add to the collection, what is the formal way to do this? Note that I'm not looking for ...
1
vote
1answer
25 views

Summation notation for time series

I need to add together the results of a function for four consecutive years prior to the start of a project. The function is calculated for individual years. It is (10 × (A + B + C) ÷ ...
3
votes
0answers
100 views

Why do Mathematicians use $u$ and $v$ as variables?

I'm sure this has happened to you as well: you are reading some hand-written work, the variables used are $u$ and $v$, and at some point the handwriting becomes unclear and you cannot distinguish the ...
1
vote
1answer
29 views

Mathematical notation - cummulative summation

The Mertens function can be calculated in Mathematica by: Accumulate[Table[MoebiusMu[k], {k, 1, n}]] and is written as $$M(n)=\sum_{k=1}^n\mu(k)$$ Could someone ...
1
vote
2answers
42 views

Could someone explain this notation?(set theory)

The expression is : $$\bigcap_{s \in S}L_s$$ Where $s$ means student, $S$ is the set of all students and $L(x, y)$ means "$x$ likes $y$". So where I'm reading it from ("How to Prove It A Structured ...
0
votes
0answers
21 views

What does $\Gamma_\infty$ denote in modular forms

I am catching up on some notes for an online modular forms course. The slides have just suddenly started using the notation $\Gamma_\infty$, without even defining $\Gamma_k$ (except for $k=1$), never ...
0
votes
2answers
32 views

Notation question about defining a set

If I'm given: $\{ n^2 + n + 1 \mid n \in \mathbb N\} \subseteq \{ 2n + 1 \mid n \in \mathbb N\}$ Does this mean: "The set $A$ where each element is made by putting a natural number in that formula ...
1
vote
1answer
29 views

Split long relation over two line using boolean operator

Normally, when you have a long equation, you can split it on two lines. Suppose that $a$ and $b$ are very long expression. Then, for example: $$ x = a - b$$ can be rewritten as $$ x = a + $$ ...
1
vote
1answer
30 views

Shorthand method for expressing the limit of something

Solving limits takes a lot of steps sometimes, but I feel bad leaving out the limit each time I do something and rewrite "=". Is there a shorthand method for writing the limit? $$\lim_{x \to p}f(x) = ...
1
vote
1answer
64 views

Simplification of a nested sum

I have a nested sum like so: $$\underbrace{\sum_{k_1=k_0}^{k^*} \ ... \sum_{k_n=k_{n-1}}^{k^*}}_{\text{n times}} 1\quad\ \text{with}\ \ n, k_0, k^* \in \mathbb{N},\ k^*\geq k_0$$ Is there a general, ...
0
votes
1answer
40 views

What does the notation $O(x^n)$ mean?

I am reading a book about Padé Approximations, and I am trying to understand the following line: We denote the $[L/M]$ Padé approximant to $A(x)$ by $A(x) - P_L(x)/Q_M(x) = O(x^{L+M+1})$ where ...
2
votes
2answers
27 views

Qualification of a Universal Quantification

Let us say I have a predicate, $P(n)$, and I want to say that it holds for every integer greater than $2$ (an example would be $P(n) = 2n>2+n$). Let us furthermore say that the UOD (universe of ...
0
votes
1answer
30 views

Notation for nested sigmas (summations)

Is there any standard notation, other than an ellipsis, for a chain of nested sigma summations? For instance, I have: $$ \sum_{b_0=0}^{L} \sum_{b_1=0}^{L-b_0} \sum_{b_2=0}^{L-b_0-b_1} \cdots ...
1
vote
1answer
66 views

What does $\nabla ^\perp$ mean?

As in the title, I am reading a paper and they use the notation $\nabla ^\perp$ without explaining what it means. Is this standard notation for something? I have not seen it before! Thanks in ...
2
votes
1answer
31 views

$\varepsilon$-neighborhood set notation as $V_{\varepsilon}(a)$ and proof question

Two questions regarding the set $V_{\varepsilon}(a)$. It is defined in the book I am currently using as $$V_{\varepsilon}(a)=\{x\in\mathbb{R}:|x-a|<\varepsilon\}$$ 1). Why the $V$? I suppose ...
0
votes
2answers
45 views

Notation of the square root in textbooks

About a year ago, I started to notice that a lot of textbooks of recent date seem to favor the notation $x^\frac{1}{2}$ over $\sqrt{x}$, while I agree that notating the square root as an exponentation ...
3
votes
1answer
34 views

Conditioning versus subscripting for a Markov process

I tried to see if this question had been asked anywhere, but it's a bit hard to set up the search terms properly. I want to confirm my a priori notion that the notations $P_{X_\tau}[G]$ and ...
0
votes
3answers
44 views

Give the truth table of a single binary connective which is adequate.

This might be a silly question, but I am confused. I know there is a theorem saying the only single binary connectives which are adequate are NOR or NAND, so I could use either of them. And then the ...
1
vote
2answers
58 views

Sigma hierarchy of logical formulae

In some papers on mathematical logic I've found references to hierarchy like $\Sigma_1^0$-sentence and so on. What does it mean? What is $\Sigma_n^m$, what is $n$ and $m$ here?
-1
votes
1answer
34 views

What is P(Y) here?

A multivalued map, f: X -> Y, from a set X to a set Y, is a map f: X -> P(Y). Multivalued maps will be also called multimaps. I don't understand what a multimap is in category theory and I think the P ...
4
votes
1answer
116 views

The meaning of $\rightsquigarrow$ in math?

In order to write a scientific paper, I would like to use the symbol \rightsquigarrow or \leadsto which look as: $\rightsquigarrow$ or $\leadsto$ I'm wondering about the usual meaning and use ...
0
votes
1answer
20 views

How to write a compact list of indices for subsets from finite combinations

I am using the binomial coefficient to calculate the number of 2-outcomes from $n$ items i.e. $\binom{n}{2}$ combinations. I am looking for a way to write the list of combinations compactly in terms ...
2
votes
3answers
73 views

Less than or equal sign

If I know for two numbers a and b that $${a < b }$$ Then is it correct to say that $$ a \leq b $$ I know that the second statement is true as long as the first one is. It seems OK as it is ...
0
votes
1answer
23 views

How to describe a set of coordinates of variable length?

I need to describe a set of coordinates with up to 8 dimensions. A problem is asking me to describe an event from a experiment involving sampling. The catch is that the experiment doesn't end until a ...
2
votes
2answers
44 views

Is there a commonly accepted notation for k-subsets?

The question says it all. I have once seen the following notation used for $k$-subsets of a set $S$ but I failed to verify that it is commonly used and I was also unable to find any evidence for a ...
0
votes
0answers
21 views

Expressing three recursive forms into one using parameters?

I have the following recursive function that takes three forms and I want to express it in one form: Initial: $f(x) = m * f(x-1)$, $f(0) = value.$ Forms: 1 - $f(t) = m * f(t-1)$. where t is at ...
1
vote
1answer
52 views

About a notation in Rudin's “Real and Complex Analysis”

On page 14. Suppose $\{f_n\}$ is a sequence of extended-real functions on a set $X$. Then $\sup\limits_nf_n$ and $\limsup\limits_n f_n$ are the functions defined on $X$ by $$ ...
0
votes
3answers
61 views

Function Notation question that needs an answer

$f(x)= f(x+1)+3$ and $f(2)= 5$, determine the value of $f(8)$. I don't understand how $f(x)$ can equal $f(x+1)+3$
0
votes
0answers
19 views

Mean square convergence notation

Convergence in probability is often stated as lim as n->inf. P(|Xn-X|>e)=0 I am confused because Xn is defined as a sequence of numbers and X as an constant. Isn't the right definition that the ...
2
votes
2answers
48 views

Set builder notation

I'm not sure of the correct notation if someone could please help. Say you wish to have a set comprised of the union of two sets, such as $$h=\Big\{ \begin{bmatrix}x & y\\y & x\end{bmatrix}: ...
1
vote
3answers
34 views

Notation of this set in a set?

I am currently struggeling with the following notation: For $\epsilon \in (0,1)$ and $p \in (0,\infty)$, consider the following subset of $L ^p$: $M(p,\epsilon)=\{f \in L^p:m \{x:|f(x)| \ge \epsilon ...
0
votes
1answer
32 views

Constructing sets involving predicates. Let $P(x),Q(x)$ be predicates over a set $X$?

Let $X$ be a set and $P(x),Q(x)$ be predicates over $X$. Consider the sets $$Y = \{y\in X\mid P(y)\}$$ $$Z = \{z\in X\mid Q(z)\}$$ Complete the following sentences with quantified propositional ...
0
votes
2answers
85 views

Can logic and mathematics be used together?

I have been wondering if logical symbols ($\to$, $\sim$, etc.) can be used with traditional mathematical notation ($+$, $/$, etc.) in the same equation. For example, would the following equation be ...
2
votes
1answer
38 views

How to express MATLAB indexing in math notation?

I am currently writing a paper concerning discrete arrays. Given the following MATALB statement ...
3
votes
3answers
285 views

What does “$\cong$” sign represent?

I came across this sign when reading some papers. I looked up Wikipedia. It says "The symbol "$\cong$" is often used to indicate isomorphic algebraic structures or congruent geometric figures." So ...
1
vote
1answer
48 views

The Mysterious Discrete Math Operator

I am working on some discrete mathematics and came across this strange operator on two sets. $R \circ S$ I have only seen this circle operator with function compositions, so is this "Set ...
3
votes
0answers
71 views

What symbol expresses “less than approximately”?

Suppose, I want to state that $a$ is less than $b$. However, I do not know $b$ exactly, but only that it is approximately $c$. With other words I want to state that $a$ is lesser than some value which ...
1
vote
3answers
79 views

How does ZFC describe addition?

Surprisingly, the Wikipedia article on addition doesn't contain the answer. I looked elsewhere online for it, but didn't find it. Intuitively, the cardinal of the union of two sets seemed ...
0
votes
1answer
33 views

Is there a way to treat arrays as sets?

I was doing some F.O.L. problems and I noticed that quite a lot of them could be easily solved if I could just treat a given array as a set. Example: $B(1..n)$ is a permutation of $A(1..n)$ The ...
0
votes
1answer
15 views

Meaning of Clo(A), Int(A), Rint(A)

I've just stumbled upon this notation in a text about optimalisation with no explanation as to what they mean (suggesting they are widely used and well known?). $A$ is a set (a convex set in this ...