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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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 = ...
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2answers
25 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 ...
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0answers
27 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 ...
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3answers
105 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 $15=\pm a \pm x$? ...
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0answers
35 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 ...
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1answer
47 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.
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0answers
14 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 "$T$, ...
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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
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1answer
34 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: ...
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1answer
17 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.
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1answer
17 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 ...
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0answers
37 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 ...
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1answer
22 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), ...
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1answer
80 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 ...
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1answer
66 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 ...
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1answer
25 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 ...
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0answers
40 views

Symbols to represent each distinct symmetry of polyhedra

Is there a pictorial or symbolic way to represent each distinct symmetry of a polyhedron?
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1answer
56 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?
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1answer
81 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 $x\frac{y}{z}=\frac{xz+y}{z}$?
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1answer
86 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 ...
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1answer
45 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 $X\times Y= (0,+\infty)\times \Bbb{R} =$ ? ...
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2answers
530 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: $$ \int_{-\infty}^{\infty} \lvert f(x) \rvert^2 dx < \infty $$
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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 ...
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1answer
35 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)) = ...
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3answers
85 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
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0answers
35 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
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1answer
55 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), ...
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4answers
376 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
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1answer
37 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 ...
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4answers
48 views

Notation for two-vertex graph with m edges

Is there standard notation for the graph on two vertices with $m$ edges between them?
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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 ...
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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
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0answers
26 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
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1answer
41 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 $F[x_1, \ldots, x_n]$. ...
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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
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0answers
21 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.
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0answers
16 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?!
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1answer
90 views

What does $\frac12(D_{2p}\times D_{2p})$ mean in group theory?

Reading a thesis, I have come across the (unexplained) notation $$\frac{1}{2}(D_{2p}\times D_{2p})\cong (p\times p):2,$$ where $D_{2p}$ is a dihedral group. What does this "$\frac12$" notation mean? ...
1
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0answers
37 views

Statistical symbols: should greek letters be used for population or for a sample?

When finding mean and the stansard deviation, do you use the Greek symbols for a population, or a sample. When do you use "s" and xbar, for a population, or a sample? (I am taking AP Statistics)
2
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1answer
85 views

Why we use $\mathbb{R}^{m \times n}$ notation instead of $\mathbb{R}^{n \times m}$?

I just realised, that I use all the time the notation $\mathbb{R}^{n \times m}$, and all books and papers use $\mathbb{R}^{m \times n}$. $\mathbb{R}^{n \times m}$ is more sympathetic for me, because I ...
0
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1answer
43 views

Integral notation

I have encountered the following integral: $\int_{x-d}^{x+d}f(y)dy$ I am trying to figure out what is the role of $d$ in this integral. Is the $d$ at the beginning of the integral the same as the ...
3
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0answers
12 views

Equivalent to proportionality sign for additive constants

Short question Is there an equivalent to the proportionality sign $\propto$ for additive constants? The proportionality relation $y\propto x$ implies that $y=kx$ for some constant $k$. Is there a ...
6
votes
2answers
416 views

Meaning of a set in the exponent

Let $ D = 2^\mathbb{N} $, i.e., D is the set of all sets of natural numbers. What's the meaning of this definition? Intuitively, I would suggest that $ D = \{1,2,4,...\} $ but the explanation ...
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0answers
41 views

notation for minimum and maximum?

I'm trying to figure out the correct notation for this situation for use in Machine Learning. I have various ratings (for texts): ...
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1answer
48 views

Standard Notation For The Set of All the Morphisms Of A Category

Let $\mathscr C$ be a category. Let $\text{Ob}(\mathscr C)$ be the set of all the objects of $\mathscr C$. Is there a standard notation for $\bigcup_{A,B\in\text{Ob}(\mathscr C)}\text{Mor}(A,B)$? ...
6
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5answers
506 views

Functional Notation.

I have some doubts regarding function notation: First If I present a function I write:$f(x)$ If I write it's inverse:$f^{-1}(x)$ So why doesn't$f(f(x))=f^2(x)$ Second If $\frac{df(x)}{dx}=f'(x)$ ...
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1answer
41 views

Just a translation issue.

I'm italian and my professor of spectral theory wrote the list to the arguments to be studied in italian. The problem is that all the literature is in english and often the translation are a bit ...
5
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2answers
78 views

What is this notation for a function? I've never seen it written like this before.

What does this mean? $$ f=\{ (x,y): y= x+2 \}$$ I don't understand what "$(x,y):$" means in regard to the problem. Also why is the $y$ inside of the $f(x)$ function. Isn't it supposed to be outside? ...
0
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1answer
98 views

Does “arbitrarily small” mean very close to zero or very negative?

In mathematical writing, does “arbitrarily small” mean very close to zero (like $0.000001$) or very negative (like $-1000000$)? Are there better phrases to distinguish these two cases?
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1answer
31 views

What notation to use for a sequence of integers that end with digit 5?

I need to solve a low high school home work and I ask a question about the most correct notation. The problem is to build a set of circles with $r$ and $d$ such that $d=5, 15, 25, 35,...d_{+_1}$ and ...