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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3
votes
3answers
243 views

Math 'equal to?' symbol

My teacher used the following symbol: $\boxed{\overset{\wedge}{=}}$ We had to write down a vector equation, and he said my direction vector $\begin{pmatrix}6\\2\\2\end{pmatrix}$ could be simplified ...
1
vote
3answers
58 views

What is the correct notation for flipping $a$ and $b$ values in a complex?

I'm currently doing some experiments on fractals and in one of my equation I need to flip the real and imaginary components of a complex number, such as : $$ z = a + bi $$ Becomes : $$ z = b + ai ...
0
votes
1answer
72 views

What are sets and classes in maths, and how are they related to $O()$ and $o()$ notation?

Are there many definitions of sets and classes in mathematics, as given in Formal definion of the notations used in measuring time complexity? And in particular, why the notation given in Fedja's ...
0
votes
1answer
41 views

Big-Oh notation proofs [on hold]

a) $f(n) \quad \Omega (g(n))$ b) $f(n) \quad \Theta (g(n))$ c) $f(n) \quad \Theta (g(n))$ d) $f(n) \quad \Theta (g(n))$ Am not sure why I got lots of $\Theta (g(n))$ , am I correct in the ...
0
votes
0answers
34 views

Modular forms are arithmetic objects

What does arithmetic object exactly means? In an article, I found the following statement: modular forms are arithmetic objects. What this should means? Bests.
0
votes
1answer
45 views

What is difference between ≈ and ~?

I'm reading a quantum mechanics book, and it has the following equation: $$ \Delta x \approx \frac{\lambda}{\sin\alpha} \sim \frac{h}{mc\sin\alpha} $$ What is the difference between $\approx$ and ...
0
votes
1answer
35 views

What is the correct notation for showing something is a union of many sets?

If I wanted to show this would it look something like Whatchamacallets = (Things + Stuff + Pandas)
1
vote
0answers
26 views

Tilde Notation in Additive Combinatorics

A, B are some finite subsets of a abelian group. |A + B| ~ |A| The problem is I couldn't precisely understand what does ~ mean in that case. For example, it could be found in notes of Tao's ...
8
votes
4answers
1k views

Meaning of $\geqslant$, $\leqslant$, $\eqslantgtr$, $\eqslantless$

What do slanted inequality signs mean? Specifically, these are $\geqslant$, $\leqslant$, $\eqslantgtr$, $\eqslantless$. Is there any place I can look this up? I've searched Wikipedia and the web and ...
1
vote
2answers
28 views

Rationale for expressing as a direct sum and a direct product

In "Ireland and Rosen" page 35, it says if $R_1, R_2, ..., R_n$ are rings, then $R_1 \oplus R_2 \oplus \dots \oplus R_n = S$ is the direct sum of the $R_i$. Later in a proposition it says if $S = ...
-2
votes
0answers
55 views

$O$, $\Omega$, or $\Theta$ functions of f(n) and g(n)? [on hold]

I am new to this topic, but I managed to try some problems. I am not sure of my answers, so I want to be checked here before I hand in; a) $f(n) \quad \Omega (g(n))$ b) $f(n) \quad \Theta (g(n))$ ...
1
vote
2answers
23 views

Notation for an integer being square-free

An integer $n \neq 0$ is square-free if $n$ is divisible by no prime square. Can you figure out any notation for simplifying this long description? It may be guessed that $p^{2} \nmid n$ for any ...
0
votes
0answers
23 views

What is the notation for pull-back and push-forward of an exponential map?

So there is a nice notation for a one-parameter group of transformations $\Phi_t$ corresponding to its infinitesimal generator $\boldsymbol X$: $$\Phi_t = \exp \left(t \boldsymbol X \right)$$ But ...
6
votes
3answers
100 views

Confused about the $\pm$ sign?

I have multiple questions about the $\pm$ sign, since it seems to confuse me in general... Question 1: Say I have $15=\pm(a+x)$, Can I use the distributive property so it becomes $15=\pm a \pm x$? ...
1
vote
1answer
45 views

$f\in C(\mathbb{R})$. What does it mean?

$f\in C(\mathbb{R})$. What does it mean? My guess is "Differentiable on $\mathbb{R}$" but I'm not sure.. Thanks.
0
votes
0answers
13 views

Compounding unary operators

I am working with the symmetric group $S_5$. I have 3 unary operators defined: $R$, $T$, and $O$, and I'm writing about their composition. Suppose I want to denote the compound operation of "$T$, ...
0
votes
0answers
28 views

Question on notation - real analysis

"There exists an open interval J containing a such that f(x) > 0 and g(x) > 0 for x in J \ {a}." What does the notation J \ {a} mean? Thanks
0
votes
1answer
20 views

Notation of list expansion to a tuple

I have a set $S$ that I want to expand to a $|S|$-tuple. How is the notation for that? Currently I have something like that: $$ T = (f(x) : x \in S) $$ An example: $$ S = \{A,B,C\}\\ T = (f(A), f(B), ...
5
votes
3answers
230 views

Why do statisticians like “$n-1$” instead of “$n$”?

Does anyone have an intuitive explanation (no formulas, just words! :D) about the "$n-1$" instead of "$n$" in the unbiased variance estimator $$S_n^2 = \dfrac{\sum\limits_{i = 1}^n ...
0
votes
1answer
16 views

The meaning of notation $\|x - x^*\|$

I was just wondering what $\|x - x^*\|$ in the following equation means: $$B(\epsilon) = \{x : \|x - x^*\|<\epsilon\} $$ Thanks.
0
votes
1answer
21 views

Expressing a solution in interval notation

I am faced with this problem. I am told to express the answer in interval notation. |3x| > 12 I solve like usual, by doing this: ...
2
votes
1answer
16 views

Understanding this characterization of Schwartz's space

While reading up on tempered distributions I came across this definition of Schwartz's space: $S(\mathbb{R}^n) = \{ f \in \mathbb{C}^\infty : \underset{x \in \mathbb{R}^n}{sup} \: \underset{\lvert ...
1
vote
0answers
33 views

formal notations

I have a problem of expressions with proper notations. If you can help me to give a formal and exact expression, it will be helpful. The specific description is Given a set of non-negative ...
21
votes
2answers
309 views
+100

What is the Coxeter diagram for?

I understand that Coxeter diagrams are supposed to communicate something about the structure of symmetry groups of polyhedra, but I am baffled about what that something is, or why the Coxeter ...
0
votes
1answer
65 views

Is $C(C(\mathbb R))$ notation for the set of continuous functions mapping $C(\mathbb R)$ to itself?

Given that in general functional analysis we have $C(\mathbb{R})$ being the set of all continuous functions, $f: \mathbb{R} \to \mathbb{R}$. However, could I use $C(C(\mathbb{R}))$ notationally to be ...
1
vote
1answer
62 views

What is $1 / \mathbf{Set}$ if $1$ is a one-element set and $\mathbf{Set}$ a category?

What does $1 / \mathbf{Set}$ denote? A pointed set is a set $X$ equipped with an element (a basepoint) $x \in X$. Let $\mathbf{Set_*}$ be the category of pointed sets and basepoint-preserving ...
1
vote
1answer
22 views

Understanding the notation of a response to a question about extending partial orderings.

I found a question and two answers that both are very complete but I cannot understand the meaning behind some of the symbols that are used. They are different from what I am familiar with. The ...
5
votes
0answers
35 views

Symbols to represent each distinct symmetry of polyhedra

Is there a pictorial or symbolic way to represent each distinct symmetry of a polyhedron?
4
votes
4answers
135 views

Notation for the set of all arguments corresponding to local minima.

The notation $$\mathop{\mathrm{arg\, min}}_{x \in X} f(x)$$ is sometimes used for the set of all $x \in X$ corresponding to global minima of the function $x \in X \mapsto f(x).$ Is there notation for ...
3
votes
3answers
55 views

Index notation for inverse matrices

I have a question: There is an standard way to write the inverse of a matrix in index notation?. The reason is that I don't want to write $(A^{-1})_{ij}$ or $(A^{-1})_i^j$ or $(A^{-1})^{ij}$ using ...
-4
votes
1answer
97 views

What does the “@” symbol stand for, when used between two numbers? [closed]

In this equation, I am confused on what the "@" symbol stands for. In other words, what is the verbal description of this equation? $$s@t = 1$$ Context I encountered this question on the GMAT prep ...
-4
votes
2answers
49 views

How to set braces in $\log_2(-x^2)$? [closed]

How should one use parentheses in $ \log_2(-x^2)$? Two alternatives: $\log_2(-(x^2))$ for example: $\log_2(-(2^2)) = \log_2(-4) \implies \text{error}$ $\log_2((-x)^2)$ for example: $\log_2((-2)^2) ...
2
votes
1answer
117 views

Meaning of notation $\operatorname{ord}_Q(g)$ in “Algebraic Curves” by Fulton

I didn't understand this notation in the chapter 7 page 93 of Fulton's algebraic curves book: What the author means by $\text{ord}_Q(g)$? Maybe he would like to say $\text{ord}_Q(G) := ...
3
votes
1answer
54 views

What does $F = 2^W$ mean?

I'm reading the book Reasoning about uncertainty and having some problems with the notation. $F = 2^W$ where $W$ is a set and $F$ an algebra. What this mean?
1
vote
1answer
75 views

Why $1\frac{1}{2}\ne \frac{1}{2}$?

Why mathematicians have chosen notation such that in algebra $1\frac{1}{2}=\frac{3}{2}$ but $x\frac{y}{z}=\frac{xy}{z}$, instead of $x\frac{y}{z}=\frac{xz+y}{z}$?
0
votes
1answer
76 views

What does this symbol $\sum\bigoplus M_i$ mean?

Let $R$ be a ring and $\{M_i\}$ a family of $R$-modules, then what does this symbol $\sum\bigoplus M_i$ mean? This symbol appeared in the following paper, Eben Matlis, Injective modules over ...
2
votes
2answers
660 views

Bar symbol over a matrix

So I am reading a paper (not online) and I come across a definition: $$\mathbb E=R\bar R$$ Where R is a complex matrix. I am thinking that it means complex conjugate, but I honestly have never seen ...
9
votes
2answers
514 views

Why do we need the absolute value signs when integrating the square of a function?

Why do we need the absolute value signs in the definition of square-integrable function? As seen below: $$ \int_{-\infty}^{\infty} \lvert f(x) \rvert^2 dx < \infty $$
0
votes
1answer
43 views

How to write the union of sets

This is just a question about notation(and I can not write it pretty well in Latex either). Is $X=(0,+\infty)\subset\Bbb{R}$ and $Y=\Bbb{R}$. Then $X\times Y= (0,+\infty)\times \Bbb{R} =$ ? ...
29
votes
10answers
2k views

What could be better than base 10?

Most people use base 10; it's obviously the common notation in the modern world. However, if we could change what became the common notation, would there be a better choice? I'm aware that it very ...
1
vote
0answers
25 views

Rearding notation of (Relatively)Projective/ (Relatively)Injective in Group cohomology

I am reading Group cohomology from Serre's Local Fields. I got confused with the notation he used... We know that : $A$ is Projective module if $Hom_R(A, \_)$ is exact $A$ is Injective module if ...
10
votes
4answers
365 views

Notation problem in integration: write $dx$ or ${\mathrm{d}}x$?

I have a question. When I write the integral of a generic function $f(x)$, do I have to write $$\int f(x) \color{red}dx$$ or $$\int f(x) \color{red}{\mathrm{d}}x \quad ?$$ Why? Thank you!
1
vote
1answer
31 views

Asymptotics of logarithm: $\frac{1}{n}\ln(a+o(1)) = \frac{1}{n}\ln(a)+o(\frac{1}{n})$

I am having problems with the use of the little oh notation my professor is adopting in the solutions to some exercises. As an example I do not understand why $$ \frac{1}{n}\ln(a+o(1)) = ...
1
vote
1answer
2k views

Boolean algebra operation precedence?

In my discrete mathematics class we wrote down the truth table for some Boolean functions and in that table they go in the following order: ¬, ∧, ∨, →, ~, ⊕, |, ↓ So, I assumed that this is the ...
1
vote
4answers
41 views

Notation for two-vertex graph with m edges

Is there standard notation for the graph on two vertices with $m$ edges between them?
2
votes
0answers
34 views

Specifying types of variables in pure mathematics and applied mathematics

In pure mathematics, we can write such as "an integer $a$ ..." to specify that $a$ is a given integer or $a$ runs through the ring of integers. But in contexts where mathematics is applied ...
11
votes
3answers
706 views

History of notation: “!”

Does anyone know where the factorial "!" symbol came from? I can't decide if it is my favorite or least favorite notation in mathematics...
3
votes
1answer
52 views

Formally correct way to define asymptotic notations

I found an algorithm book which tries to define asymptotic notations as sets and then used notations like $n=O(n^2)$. Is there a mathematically correct way to define asymptotic notations like $O(n), ...
1
vote
1answer
27 views

Explanation of notation $f(t)\in L_{\infty}$ in a control theory textbook

In a control theory textbook I saw the following notation : $$f(t)\in L_{\infty}$$ Since I am not familiar with this kind of notation could someone explain What does it mean?
2
votes
1answer
34 views

Two integers with the same prime factors notation

Let $m,n\in \mathbb{Z}$, what is the notation usually used to say that $m,n$ have the same prime factors, i.e. $m=p_1^{m_1}p_2^{m_2}\cdots p_2^{m_r}$, $n=p_1^{n_1}p_2^{n_2}\cdots p_r^{n_r}$ for some ...