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Questions tagged [notation]

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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What does $\mathbb{R}_0^+$ mean?

What does $\mathbb{R}_0^+$ mean? I know that the superscript indicates all real numbers $\gt0$ but what does the subscript indicate in set notation? Thanks! EDIT: The context is $f:\mathbb R\to\...
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2answers
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Weird Notation in Differential Geometry (Variational Vector Field)

I can't seem to understand what my professor wants to say with his notation. So I hope that some of you can help me. We're looking at some variational curve $\gamma_t : I \to \mathbb{R}^2, \ t \in (-\...
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1answer
20 views

How do you render the union of a simple closed curve and its interior?

In this question there is confusion about the answer because it appears (to me, anyway) that the origin was supposed to be rendered as not on or inside a simple closed curve; but the question is ...
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1answer
36 views

Some confusion about tensor notation

I'm learning about tensor notation but I've become a little confused by an example in my notes as it seemingly contradicts what I've read elsewhere. What I know: Superscripts denote column vectors ,...
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2answers
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A contradiction arising due to set-builder form.

I have noticed a kind of contradiction arising due to set-builder form involving universal set$(U)$ while I was performing operations on set theory. Here it is - Let us consider an arbitrary non-...
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28 views

What are the correct mathematical expressions for these headers?

Table Z is the function which adds a and b (just a simple example). Column D shows the increase from the last value in absolute value and Column E in percentage. I would use Delta(Star)Z and Delta%Z ...
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2answers
65 views

What is the difference between a definition and an equivalence class?

In what way is 'the definition of $x$ is $y$' ($x:=y$) not the same as '$x$ is equivalent to $y$' ($x=y$)? I can find no justification for making the distinction aside from 'it feels right'. It seems ...
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34 views

What does the forward slash mean set notation mean?

I have an equation that looks like this: $L(\hat{x}_i, y_i, S) =\begin{cases} \hat{x}_i - y_i & \text{if $i \in S1$} \\ \infty & \text{if $i \in S / \{S2,S3\}$} \end{...
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Choice of symbols: $O_p(G)$, $O^p(G)$, and $O_\infty(G)$

For a finite group $G$ and a prime number $p$, several normal subgroups are defined as follows: $O_p(G)$ = the largest normal $p$-subgroup of $G$ ($p$-core) $O^p(G)$ = the smallest normal subgroup $N$...
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1answer
36 views

Rewriting expressions using ∑?

How would you write this expression using $\sum$ notation? $(x + 1)(x − 3)(x + 5)(x − 7)· · ·(x + 101)$ Thanks!
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1answer
25 views

Cycle notation for cyclic orders

Is there a convenient cycle notation for cyclic orders (https://en.wikipedia.org/wiki/Cyclic_order)? For example: Definition. A set of four elements $a, b, c, d$ of a cyclically ordered set is a 4-...
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About the notation <S,R> for ordered sets or <G,+> for groups. Is this notation absolutely rigorous or is it simply a convenient shortcut?

One can often read things like: let (S,R) be an ordered set, that is the set S ordered by the relation R; or, let (G,+) be a group, that is a set G together with an operation verifying such and such ...
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1answer
43 views

Is there a common notation for a matrix with one entry equal to one and zero otherwise?

Kind of like a unit vector, but it's a matrix. For example, if the notation is $A_{ij}$, then this matrix has all zero elements, except $a_{ij} = 1$. Is there a common notation/term for this kind of ...
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2answers
40 views

Symbol for the number of elements in a set

I would like to express below concisely and mathmatically TP is a set of real numbers $$C = \{ x \in TP : x > threshold \}$$ c_count = len(C) Basically, in ...
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Notation to truncate a Laurent series

Suppose that, as $x$ vanishes, the function $f(x)$ is approximated by the Laurent series $$f(x) \approx \frac 1 x - i\pi - \frac x 4 + \frac{i\pi x^2}{12} + \cdots$$ Question 1: can I write the ...
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2answers
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What does the notations $X \times_{Z} Y$ mean?

Right now I am learning how to create commutative diagrams in Latex, and since I am also studying category theory right now, I have come across the notation before in category theory texts like ...
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3answers
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Using 0/1 instead of T/F in propositional logic. Is there any interest in doing so? ( either at the language level or at the metalogical level)

Is there any interest in using 0/1 instead of T/F in propositional logic? Does it allow things the T/F notation doesn't? Does it make easier or simplyfy in any way the exposition of logical theory?...
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143 views

What is difference between $\{\{\}\}$ and $\{\varnothing\}$? [on hold]

I need to know what is difference between $\{\{\}\}$ and $\{\varnothing\}$
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1answer
27 views

On vector notation

I've read that, when expressing a quantum mechanical operator into a particular representation, the more-formally correct notation is $\rightarrow$ rather than $=$, e.g. the momentum operator ...
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1answer
21 views

asymptotic proportionality

Is there a well established notation for asymptotic proportionality by a non-zero and non-unitary constant (for null contant we have little-oh, for unitary constant of asymptotic proportionality we ...
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1answer
42 views

Notation for reversed rows and/or columns of a matrix?

In this answer I am using a transformation of matrices, and I would like to know if there is a notation for this. Given a matrix $A$, let $B$ be the same as $A$ with rows in reverse order, $C$ with ...
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1answer
24 views

is there a convention of the usage of square brackets and round brackets?

I am learning the justification of Sample variance $${\displaystyle {\begin{aligned} \operatorname {E} [\sigma _{Y}^{2}]&=\operatorname {E} \left[{\frac {1}{n}}\sum _{i=1}^{n}\left(Y_{i}-{\frac {...
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2answers
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Congruence, Equal, and Equivalence

I know this is very basic problem about math. But sometimes confusing. What is the difference among Equal Sign $\left(\,=\,\right)$ Congruence Sign (we saw this on number theory) $\left(\,\equiv\,\...
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101 views

Beach-ball like differential operators of a two-dimensional function

I'm looking for references, known names of, and other useful pointers and insight about (pairs of) differential operators that are "beach-ball like" because they sample a 2-dimensional function in ...
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28 views

Mathematical notation nth smallest element of a set

I want to put the following in formula-form: "$ T_i$ should be the sum of the $ b $ smallest durations out of all durations of the predecessors of $i$ (which is the set $P_i$)" Does a mathematical ...
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Need help with a symbol, seems like inverted $\in$.

Need help with a symbol that appears in logic notations, as it appears (for the first time, in the book) on page 2 of the book : Elements of Real Analysis, by Denlinger. The logical statement ...
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1answer
20 views

What is the meaning of the $0<x, y<\infty$ bound in this joint PDF?

I'm given a joint PDF, \begin{equation} f_{X,Y}(x,y) = \begin{cases} e^{-(\frac{x}{y}+y)}y^{-1} & \text{ , } 0<x,y<\infty\\ 0 & \text{ , otherwise} \end{cases}. \end{equation} From ...
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meaning of double headed arrow

I am studying properties of discrete time Fourier transform and I encountered a notation shown highlighted in attached photo What is meant by this notation? Is it meaning equality? screenshot of ...
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28 views

Is this matrix notation correct?

I have a bunch of matrices that I denote as follows: $$\mathbf{A_j}=(a_{c,k}^j)\in[0,1]^{d\times q}$$ The reason I don't use a tensor of order $3$ is because $d$ will vary for each $j$, which is why ...
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A new notation for operations and some questions

let $x^{/1/}=x+x$ (addition) $x^{/2/}=x.x$ (multiplication) $x^{/3/}=x^x$ (exponentiation) $x^{/4/}=^xx$ (tetration) and so on..... I have the following questions: $(1)$ Can we define an ...
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1answer
46 views

Subset notation in Rudin's “Principles of Mathematical Analysis”

I've just began reading Walter Rudin's "Principles of Mathematical Analysis". Early on in the text is introduced subset notation, with which I am familiar. My problem is that Rudin doesn't seem to ...
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Meaning of notation $f(\epsilon) = \mathrm{ord}(1)$ as $\epsilon \to 0$?

What is the meaning of $$ f(\epsilon) = \mathrm{ord}(1) $$ as $\epsilon \to 0$? I have struggled to find the definition of this term online or in any notes I have access to.
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1answer
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Notation for partial function composition

Given functions $f\colon X\to Y$ and $g\colon Y \to Z $, their composition, that is the function $h\colon X\to Z, x \mapsto g(f(x))$ is denoted by $h=g\circ f$. Let's call this the functional form, ...
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1answer
55 views

How to split three numbers with conditions?

Let $a, b, c \in \mathbf{N}_1$ and triples $(a, b, c)$ are formed from them randomly, for example, $(1, 5, 6)$, $(1, 2, 3)$, $(1, 2, 4)$. The order of numbers does not matter in a triplet. We can use ...
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1answer
58 views

How to find $\frac{∂f}{∂y}$ for Functions of the form $f(x,y,z(y))$?

This is the first time I'm using math.stackexchange. Please excuse me and correct me if I'm not doing things in the right format. So my question is this: given a function of the form $f(x,y,z(y))$, ...
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1answer
38 views

Question about notation: The order notation

For example, when we Taylor expand say $e^x$ about $x=0$, we would write $$e^x = 1+x+\frac 12 x^2 + \frac 16 x^3 + \mathcal O(x^4) \qquad \qquad \text{as } x \rightarrow 0$$ with the use of the "Big-...
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1answer
47 views

What is the meaning of sentence E | x?

How to read this math notation? Let closest(j) be the set of all points that are closest to centroid cj. The algorithm proceeds during several passes, during each of which it updates each centroid ...
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What notation is available for denoting points on a path?

I have a closed-path motion on the plane, it looks like A-B-C-D-A on the figure. Each segment is a straight line. The point A ...
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1answer
24 views

Writing an inequality in set notation

I'm having trouble writing a solution in set notation. Iv been asked to write an equation to describe a region from a graph but express the answer in set notation. My solution is y>x^2-4. How do I ...
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Notation to indicate a function is an aggregate of a family of sub-functions?

I have a bunch of component cost functions $C_g(x)$ and an aggregate cost function which is defined as a weighted sum of components $\sum_{g=1}^G \omega_g C_g(x)$. Currently I write $\mathcal{C}(x) = \...
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1answer
20 views

Notation for permutations of function composition

Given a set of functions $F$ of which all are functions in $X \to X$, where certain functions, say $a, b \in F$ can be composed together: $a(b(x)) = (a \circ b)(x)$. What is the notation for the set ...
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7 views

Formal notation for an unordered set used for define weighted median

For $i=1\ldots n$ let's define the couple ($X_i$, $W_i$). I want to define a weigthed median as the highest value of $m$ fulfilling $\sum_{\{i|X(i)\leq m\}} W(i) < \frac{1}{2} \sum_{\{i\}} W(i)$. ...
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1answer
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What symbol means continues irrationally?

Typically we use an ellipsis (...) to show that a number continues on; e.g.: 1/3 = 0.33333... Even in the English language, it is used to indicate that something ...
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2answers
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$d_1:ax+2y+4=0$; $d_2:3x-4y+12=0$; $|d_1d_2|=?$

$d_1:ax+2y+4=0$ $d_2:3x-4y+12=0$ $d_1$ // $d_2$ $|d_1d_2|=?$ This is a slightly modified version of a university entrance examination problem set up by the experts at this specific ...
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Notation for Divisors in Liu's AG & AC

I have a question about a notation in Liu's "Algebraic Geometry and Arithmetic Curves" in the excerpt below (or look up at page 276): What is here means by $(f)_0$ and $(f)_{\infty}$? Does anybody ...
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1answer
7 views

Denoting the Conditional Minimum

This questions concerns notation. I have a $2\times n$ matrix $X$. Is there a smooth way to denote the minimum of the $2$nd row conditional on elements in the first row being equal to some constant $c$...
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Nth-order Laplacian

Mind that I'm coming from a mostly physics background, so this may in fact be common mathematical notation that I simply haven't come across in my own field. I've seen the symbol $\nabla^2$ "applied" ...
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Notation question for push forward substitution in iterated integral

Consider the following iterated integral \begin{gather*} \int_{0}^{1}\int_{0}^{\pi}y\sin(xy)d xd y.\tag{1} \end{gather*} It is standard to evaluate (1) as follows. \begin{align*} \int_{0}^{1}\int_{...
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1answer
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Rationale behind this notation for integrations?

The marginal distribution formula from Wikipedia is: $$ p_X(x) = \int_y p_{X,Y} (x, y) dy $$ It makes sense. But some use the following notation: $$ \mathbb{P}_{X} = \int_Y d\mathbb{P}_{XY} $$ Why do ...
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Notation for scalar coefficients of a vector

I want to know if there is some existing notation for the "magnitude" of a vector when it is complex. For example, suppose I have the following vector: $$\mathbb{v}=(2+i) \ \hat{x}$$ Is there any ...