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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15
votes
4answers
933 views

Donald Knuth's summation notation confuses me.

I do not understand a lot of cases of Knuth's summation notation in Concrete Mathematics. In general, I cannot seem to get a grasp on the commutative law as applied to manipulating sums. The ...
1
vote
1answer
750 views

Greek sigma symbol correct representation?

I wish to display to people that a number is a total. I have decided to use the greek symbol for sum: Σ Do I prefix or suffix numbers with this symbol? Given ...
9
votes
1answer
724 views

Who came up with the arrow notation $x \rightarrow y$?

I read that the arrow notation $x \rightarrow y$ was invented in the 20th century. Who introduced it? Each map needs both an explicit domain and an explicit codomain (not just a domain, as in ...
3
votes
1answer
149 views

Matrices of Trace $0$

The set of all $n$-square matrices with trace $0$ is a subspace of the set of all $n$-square matrices. Is there a standard notation and/or name for this subspace?
0
votes
1answer
6k views

What does the right arrow mean in the example provided.

My second question here. Does it show that I know very little about mathematics? :) I'm doing Project Euler Question 14 and would like to know what the right arrow → means in: ...
2
votes
1answer
197 views

Can the output of a mathematical function be another mathematical function?

Apologies if this is actually mathematical gibberish. I'm very familiar with mathematical functions at a simple level. A function relates a set of inputs to a set of outputs. What I'm trying to ...
2
votes
1answer
2k views

How to write the set of all natural numbers, that can be divided by two?

I would like to write as a mathematical expression such sentence: x is the set of all natural numbers that can be divided by 2 without the remainder. $\{x \in \mathbb N: What \ here? \} $ Thanks in ...
6
votes
1answer
592 views

What do these arrows mean? (Froda's Thm)

I was reading "A Course in Probability Theory" by Kai Lai Chung, and in the book he was discussing discontinuity of monotonic functions, and after doing some searching online to learn more about the ...
2
votes
1answer
331 views

Notationally, what is the difference between $\Pr(X = x)$ and $P(X = x)$? When should I use each?

I'm talking specifically about probability theory. I was reading some stuff about probabilistic graphical models, and they kept switching the notation in this book, but I couldn't discern the ...
17
votes
7answers
3k views

What does $dx$ mean?

$dx$ appears in differential equations, such us derivatives and integrals. For example, a function $f(x)$ its first derivative is $\dfrac{d}{dx}f(x)$ and its integral $\displaystyle\int f(x)dx$. But ...
4
votes
1answer
231 views

Is there a name for this $k$-fold vector product?

Let $V$ be a set of vectors of length $n$. Define a $k$-fold product on $V$, $$ \Upsilon(\{v_1,\ldots,v_k\}):=\sum_{j=1}^n\prod_{i=1}^k v_{ij}, $$ where $v_i\in V$ and $v_{ij}$ is the $j^\text{th}$ ...
2
votes
1answer
6k views

how can i calculate the number with“e”? [duplicate]

Possible Duplicate: What does E mean in 9.0122222900391E-5? if here e is Euler then it is 2.718... but it couldn't be ,the number being smaller every rows ...
3
votes
1answer
330 views

Disproving a big O equation

As a homework assignment I am trying to prove/disprove the next statement: Let $f(x)=O_a(g(x))$, then $\forall A,B\in\mathbb{R}\rightarrow A\cdot f(x)=O_a(B \cdot g(x))$ Which I think is wrong ...
1
vote
2answers
110 views

Is there a language-agnostic way to formularize “if”-conditions?

How can you formularize "if"-conditions in a language-agnostic way? I'm talking about formulas like $$ x=\begin{cases} 2,&\text{ if } n \le 2 \lor m=3\\ 3,&\text{ if } n \gt 2 \land m=n ...
1
vote
1answer
115 views

What does the degree symbol in $K_n \subset K_{n+1}^{\circ}$ mean?

What does the degree symbol in "Let $(K_n)$ be a sequence of compact subsets such that $K_n \subset K_{n+1}^{\circ}$ mean?
1
vote
1answer
332 views

Notation (sets, partitions)

I have a set of vertices $V$ such each vertex $v\in V$ is labelled with a color $col(v)$. I want to consider the class of partitions of $V$ that have the following property in common: all vertices ...
2
votes
3answers
1k views

Using setminus notation with set elements

The "correct" way to write a set without a specific element is as follows: $S \setminus \{s\}$ But in some contexts this is cumbersome to write/type or read, and it detracts from the flow of the ...
3
votes
1answer
135 views

What is this function on ordinals called?

Define a function $f:\mathbf{ON}\to\mathbf{ON}$ as follows: for each ordinal number $\alpha$ let $$f(\alpha)=\operatorname{ord(}\lbrace \beta<\alpha|\text{ }\beta\textrm{ is a limit ...
1
vote
1answer
117 views

Notation for the set of all isomorphisms between two modules.

Let $R$ be an arbitrary ring. Given two $R$-modules $A$ and $B$, we may denote the set of all $R$-homomorphisms from $A$ to $B$ by $\operatorname{Hom}_{R}(A,B)$. If in addition we know that $A$ and ...
4
votes
1answer
146 views

Notation in Sacks' 'Higher Recursion Theory'

I'm having trouble with the notation in Sacks' Higher Recursion Theory. I've asked specific questions from page 4. Instead of reading my question in detail and trying to understand my confusion (which ...
2
votes
1answer
74 views

Definition of a Set in terms of Other Sets

I have a set $S(k)$ well defined only if $k>0$, and another set $X$ well defined only if a condition $cond$ is satisfied. I want to define $S(k)$ as the union of $S(k-1)$ and $X$ provided that ...
2
votes
1answer
80 views

Notation confusion in “On Some Results of Atkin and Lehner” article

I'm currently reading an article on Automorphic Forms, and I'm a bit confused about some of the notation used. The article is "On Some Results of Atkin and Lehner" by William Casselman, published in ...
1
vote
1answer
188 views

What is the nature of the definition symbol?

A question about definitions and also notation, illustrated with a trivial example: Let $a,b,c\in\mathbb{N}$ and $$a:=2,$$ and $$b:=1$$ I postulate the the following formula holds ...
2
votes
1answer
458 views

Markov Process: Have you seen this notation and do you know what it means?

Ok, I've already posted this a minute ago, but my text deleted itself while I was editing it :-( So next try: Can you help me to understand the notation my professor uses to describe Markov ...
5
votes
2answers
247 views

Analytic versus Analytical Sets

Browsing MathOverflow I came across a question about analytical sets. Through the discussion following a comment made by our very own Asaf, I learned that bold face $\mathbf{\Sigma^1_1}$ and light ...
2
votes
4answers
334 views

What is the meaning of $\exp(\,\cdot\,)$?

What is the meaning of the notation $\exp(\text{expression})$ ? I think that it's something of the form $a^\text{expression}$ but does it mean that the base $a=e$ or can it be any base?
5
votes
1answer
218 views

Notation in Atiyah - Macdonald

I am now going through some problems in Atiyah - Macdonald Chapter 3. In problems 21 and 23 of chapter 3, they use the notation $A_f$ to mean something I don't know. I have not seen this before. ...
0
votes
1answer
362 views

Notation for extracting the index of an element from a set

OK I edited the question. Sorry for the wrong terms. What is the correct notation such that a specific function maps an element of a specific sequence/list/n-tuple to its index? I have researched ...
1
vote
2answers
742 views

What does this symbol mean? (looks like a lower-right corner — subject: manifolds theory)

Below I have posted an excerpt of Lee's Book "Introduction to Smooth Manifolds" (page 371). I don't know what the symbol means that looks like a lower-right corner, and I cannot find it via the index, ...
2
votes
3answers
747 views

notation of distribution

I have a question: Does $$N(0, x)$$ mean a normal distribution with mean $0$ and variance $x$? Or standard deviation $x$? The notation seems ambiguous sometimes.
1
vote
1answer
427 views

Permutation Group Notation

I have a question about notation: For a permutation $\sigma\in S_n$, define the following quantities: $\text{ret}(\sigma)$ as the number of indexes $1\leq i\leq n$ such that $\sigma(i)\lt ...
1
vote
2answers
247 views

summation notation for general sets

I'm working through an academic game theory paper and stumbled upon this summation notation in a proof and I'm not quite sure what it means: $$\sum\limits_{j \in M \backslash\ \{i\}}$$ There is a ...
5
votes
5answers
11k views

Notation for “is divisible by”

Is there a standard way of writing $a$ is divisible by $b$ in mathematical notation? From what I've search it seems that writing $a \equiv 0 \pmod b$ is one way? But also you can write $b \mid a$ as ...
5
votes
3answers
204 views

Dealing with many entities that need a symbol

What does one do when one needs a lot of symbols and one has exhausted the useful symbols of the latin and greek alphabets? (I say useful symbols because letters like iota (ι) and upsilon (υ) seem too ...
2
votes
3answers
62 views

Clarification regarding a derivative symbol

I came across the following expression: $$\frac{\partial^{i_1+\cdots+i_m}P(x_1,\ldots,x_m)}{\partial x_1^{i_1}\cdot\cdot\cdot \partial x_m^{i_m}}$$ for $P(x_1,\ldots,x_m)$ a polynomial in $m$ ...
2
votes
2answers
557 views

Set operators vs. Logical operators while discussing probability theory.

Let $S = \{1, 2, \dots, 10\}$ Let $A$ be the event that a number selected from $S$ is even. Let $B$ be the event that a number selected from $S$ is a multiple of $3$. If any number is equally likely ...
2
votes
1answer
150 views

Is the notation for the definition of a set stricly formalized?

In the 800 pages set theory book by Jech, he uncommented starts using $$Y=\{ u\in X : \phi(u)\}$$ as equivalent to $$Y=\{ u:u\in X \wedge\phi(u)\}$$ on the first few pages. The fact that, in the ...
2
votes
1answer
231 views

What does $ \langle Y_{lm} | Y _{\lambda\mu} \rangle = \delta_{l\lambda} \delta _{m\mu} $ mean?

In Rotation Matrices for Real Spherical Harmonics. Direct Determination by Recursion, I can almost completely understand the recurrence relations described, but for one part. The $Y^l_m$ function is ...
1
vote
1answer
107 views

Notation question (shorthand for dot products?)

I am reading a paper and trying to understand a calculation and all of a sudden I bump into the following term: $D^2_y p(t,y(t,x))(\partial_t y(t,x),y^\epsilon(t,x))$ where $p$ is a scalar field ...
0
votes
1answer
157 views

indicator function?

Does anyone know what $ 1_\omega v $ means where $v \in L^2((0,T) \times \Omega )$ and $\omega \subset \subset \Omega$? It should be an indicator function of $(t,x)$, but not sure how to interpret ...
0
votes
1answer
54 views

How to specify that if a denominator should not be 0

How can i specify in the otherwise condition of the following equation \begin{cases}\ U & \text{if $A_{U_{Max}} = A_{U_{Min}} =$ Null},\\ \ \frac{A_{U_{Max}} - A_{U_{Min}}}{U\sqrt{2}} & ...
1
vote
2answers
2k views

What does the syntax $P(X=n)$ mean?

I'm reading a text book about probability and this syntax $P(X=n)$ keeps resurfacing. I've googled around, but I keep getting results on binomial distributions. What does $P(X=n)$ mean? Does it mean ...
6
votes
4answers
1k views

History of $f \circ g$

$f \circ g$ is usually interpreted as $f(g(x))$ although, as Google shows, $g(f(x))$ is used frequently too. My question: Does anybody know who was the first mathematician to use this symbol and what ...
1
vote
1answer
120 views

Correct precedence of division operators

Say i have the followingv operation - $6/3/6$, i get different answers depending on which division i perform first. $6/3/6 = 2/6 = .33333...$ $6/3/6 = 6/.5 = 12$ So which answer is correct?
2
votes
1answer
823 views

Inserting if else in a mathematical expression is it possible

Hope you guys can help me out. I have a mathematical expression which I constructed its: $$ A_S = \frac{A_{U_{Max}} - A_{U_{Min}}}{U\sqrt{2}} $$ Is there a way in which I could mathematically specify ...
4
votes
2answers
611 views

Necessity of Brackets for Integration

Suppose I want to integrate $f(x)+g(x)$. Can this be written as $\int f(x)+g(x)\, dx$ or are brackets necessary, i.e. $\int \left(f(x)+g(x)\right) \,dx$?
1
vote
1answer
582 views

Square bracket notation in equation

I'm attempting to understand a paper[1] on contracting geometry for computer graphics. I have an undergraduate Physics degrees to give some indication of my experience, though that was 6 years ago ...
0
votes
2answers
301 views

Mathematical set notation

How would I write mathematically (using set notation): Let $F$ be the set defined by the distinct values of the function $f(x)$ for $a \le x \le b$ ($a$, $b$, $x$ and $f(x)$ are all natural ...
-1
votes
1answer
72 views

$T(n) = n^{O(1)}$ iff exists $k > 0$ such that $T(n) = O(n^k)$

I must use O notation to show that: $T(n) = n^{O(1)}$ iff exists $k > 0$ such that $T(n) = O(n^k)$ But, I don't understand what mean: $n^{O(1)}$
1
vote
1answer
619 views

How to denote a function mapping of two parameters to an output?

I am wondering how to denote something like this: I have a function mapping two sets to a set of vectors and am currently denoting it as so: $p: [F, \Sigma] \to S $ I am wondering if this is ...