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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2
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1answer
35 views

Notation for multiple chained indefinite integrals

The reduction formula for $\int\frac{x^ndx}{Ax + B}$ is given by $$ \int\frac{x^ndx}{Ax+B} = x\int\frac{x^{n-1}dx}{Ax+B} - \int\left(\int\frac{x^{n-1}dx}{Ax+B}\right)dx. $$ Is there any special ...
22
votes
2answers
2k views

Math symbol in German thesis from 1963

I have the following math symbol in a German thesis written in 1963. Is it anything more than just a function name? It is used in the following context and then goes on to state that "If the ...
1
vote
1answer
55 views

Unfamiliar notation. (actuarial science)

I am a math instructor self studying for the actuarial exam and I am trying to understand the following notation that I have encountered today. $$E[X \land d]$$ The explanation in the book told me ...
1
vote
0answers
21 views

Standard notation for Differential Equation Solutions

Quick question: I was solving a differential equation, and wanted to know which of these expressions is in the standard notation for an answer to a differential equation: (a) $y^{3/2} - x^{3/2}=7$ ...
1
vote
2answers
143 views

What is “ terminating reciprocals”

I meet a problem "What kinds of numbers have terminating reciprocals in base 60?", but I don't know what is "terminating reciprocals". Please help me, thanks.
0
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2answers
60 views

Why does the author define these “logical notations” for set logic with “if then” and &?

In Section 1.1 of "Set Theory for Computer Science", the author defines $ \forall x \in X. P(x) $ and $ \exists x \in X.P(x) $ as shorthand for $ \forall x.(x \in X \Rightarrow P(x)) $ and $ \exists ...
3
votes
3answers
327 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
73 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
78 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 ...
2
votes
1answer
159 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
53 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
41 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
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 = ...
1
vote
2answers
26 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
31 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
110 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
0answers
40 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 ...
1
vote
1answer
49 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
15 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
30 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
44 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: ...
0
votes
1answer
18 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
38 views

The meaning of $\mathbb R^n$ in $f:\mathbb R^n \rightarrow \mathbb R$

I'm just making my way in Math and I apologise for the ease of this question. I don't understand what $\mathbb R^n$ in $f(x):\mathbb R^n \rightarrow \mathbb R$ actually means.
2
votes
1answer
19 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
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0answers
38 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 ...
0
votes
1answer
24 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), ...
0
votes
1answer
84 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
67 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
26 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
42 views

Symbols to represent each distinct symmetry of polyhedra

Is there a pictorial or symbolic way to represent each distinct symmetry of a polyhedron?
3
votes
1answer
57 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
85 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
89 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 ...
0
votes
1answer
48 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 $X\times Y= (0,+\infty)\times \Bbb{R} =$ ? ...
9
votes
2answers
538 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: $$ \int_{-\infty}^{\infty} \lvert f(x) \rvert^2 dx < \infty $$
1
vote
0answers
30 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 ...
1
vote
1answer
37 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)) = ...
3
votes
3answers
119 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 ...
2
votes
0answers
36 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 ...
2
votes
1answer
63 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), ...
10
votes
4answers
382 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!
2
votes
1answer
40 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 ...
1
vote
4answers
56 views

Notation for two-vertex graph with m edges

Is there standard notation for the graph on two vertices with $m$ edges between them?
0
votes
3answers
78 views

What is $X^{\omega}$ where $X$ is a set?

I fail to find a duplicate. If it exists, please link me in the comments and I will delete the question. In my recently bought topology book, they use $X^{\omega}$ where $X$ is a set. However, this ...
0
votes
2answers
54 views

How to denote that an equation is true?

If I have a simple equation such as this: $$x+5-1=x+4$$ how can I denote that this equation is true? More specifically, if I refer to that equation as P(x), then is there a mathematical notation for ...
0
votes
0answers
31 views

Notation for parametric family of joint densities

This is copied from my textbook; Assume that the statistical model for the MVR $\textbf{Y}=(Y_{1},Y_{2},\ldots , Y_{n})^{T}$ is given by the parametric family of joint densities: $$\{ ...
0
votes
1answer
48 views

Notation for vector space of polynomials of bounded degree

Is there standard notation for the vector space of polynomials in $n$ variables with coefficients in a field $F$ and with degree at most $D$? Without bounding the degree, it is $F[x_1, \ldots, x_n]$. ...
0
votes
0answers
17 views

Notation for mimimal sum when choosing elements from two sets

I'd be grateful for any pointers on the following I am wondering if there is any standard notation (or neat suggestions) for the following. I have two sets $\{t_1, t_2, \ldots , t_k\}$ and $\{s_1, ...
0
votes
0answers
22 views

Notation of axis

I have a graph where I rescaled the axis by dividing by 10. In the label on the axis, should I put "(x10)" or "(/10)"? I don't know the correct semantics of these labels in graphs.
0
votes
0answers
17 views

Probabilistic Graphical Model Diagram Notation, what does the box mean?

I'm just learning about probabilistic graphical models, I know the circles represent random variables, shaded being observed and unshaded being latent variables. But what does the box mean?!