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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Operations on elements of 2 sets. Similar to nested forEach loops.

Does this make sense? $$ \{ p \in P_n | \{g \in \Bbb G, q \in P_n: (q * g) \in P_n \}\}$$ Intention: all $p \in P_n$ that are products of $q \in P_n$ and $g \in \Bbb G$ If not what would be the right ...
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Formal definition of Real Numbers

I was recently getting started with mathematic formal notation by defining sets of numbers. In Wikipedia, there are some of them definited as: $\mathbb{Z} = \{\ldots -3,-2,-1,0,1,2,3\ldots\}$ $\mathbb{...
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Showing a relation between two specific elements of two different sets, simultaneously as showing their set-membership

I found myself using the following notation, wondering if it was conventional or clear: $$X \ni x = g \in G$$ Meaning that $x$, from the set $X$, is equal to $g$, from the set $G$. More generally, I'm ...
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What do $\frac{\partial}{\partial z}$, $\frac{\partial}{\partial x}$, $\frac{\partial}{\partial y}$ mean?

Maybe a dumb question but I'm relatively new to this topic so I'd still ask this: I saw it here: https://en.wikipedia.org/wiki/Wirtinger_derivatives In ordinary derivatives, I know $\frac{d}{dx}$ is ...
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Notation for an approximately defined interval

Is there a concise notation to express the following "inequality"? $$a\lesssim x\lesssim b$$ For proper inequalities like $a< x\leq b$ we write $x\in(a, b]$, or sometimes also $x\in]a, b]$...
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Explanation of summation notation when inequality and equality are in the index

I recently came acrossed the following notation: $\sum_{\substack{ j\leq j'=1}}^{2^i-1} a_{j,j'}$ Frankly, I don't know how to interpret the index portion correctly. Say that i=2 (as is the case I ...
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Notation for infinite-dimensional vector output of a function $f: X \rightarrow \mathbb{R}$ on an interval

I have a function $f: X \rightarrow \mathbb{R}$. I want to create a vector of the "outputs" of the function, in the following sense: if $X$ was discrete, e.g., $X = \{x_1, x_2, \dots, x_K\}$,...
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Determinate $d^{k}f_{c}$ for function $f(x,y)$ with k and c given

In my Calculus 3 class I have been given following problem: Determinate: $d^{k}f_{c}$ for function $f(x,y)$ where $c = (c1,c2)$ and $k = 2$. My problem is that I have no clue what does $d^{k}f_{c}$ ...
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Parenthesis needed? in $(S\subset T)\vee(S=T)\iff S\subseteq T$ [closed]

In the statement $$(S\subset T)\vee(S=T)\iff S\subseteq T$$ are the parenthesis formally incorrect correct but deemed best removed correct and optional correct and deemed required for readability ...
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How to express a sequence of matrix multiplications $\prod_{i=1}^{n}M_i$ but starting from $n$

Having square matrices $M_j\in\mathbb{R}^{m\times m}$, with $j\in\{1\,..\,n\}$, what notation should be used to express the following sequence of multiplications?: \begin{align} M_n\to M_{n-1}M_n\to ...
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A question about the meaning of the notation $(\frac{\partial}{\partial \cosh r})^2 F(r,x).$ [closed]

I am studying a book and I found the expression $$(\frac{\partial}{\partial \cosh r})^2 F(r,x).$$ Any help about the meaning of the notation that uses the author? About the exponent, I am sure that he ...
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I canot find what does $\langle,\rangle$ mean in a cooperative game theory book

Does anyone know what here on the page 83 it holds that $ \langle x,y\rangle\geq \langle x,y'\rangle$ in taking the minimum ? I.e. I cannot find the notation $\langle,\rangle$ in the book.
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Notation of math expressions

Let $h \in \mathcal{H}$ be a classifier from a hypothesis class $\mathcal{H}$ trained to infer $\mathcal{y}$ from $\mathcal{x}$, $\mathcal{h} : \mathcal{X} \rightarrow [0,1]^{|\mathcal{Y}|}$. We use $\...
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Shorthand set builder notation?

I was reading some papers and they had the following notation: $$ \big\{k\big\}_{k=0}^n $$ I assume this implies that $$ \big\{k\big\}_{k=0}^n = \{k \in \mathbb{WHAT} : 0 \leq k \leq n\} $$ What is $k$...
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For $f : \mathbb{R} \to \mathbb{R}$, what is $\sup\limits_{x \in A} f(x)$ if $A=\emptyset$? [duplicate]

Let $f : \mathbb{R} \to \mathbb{R}$ and $A$ be an empty set. What is $\sup\limits_{x \in A} f(x)$ equal to? I reckon it should just be undefined or is there an alternative convention?
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Matrix entry notation and plane coordinate notation

Let $A$ be an $m \times n$ matrix, and $a_{ij}$ a matrix entry. Usually the index $i$ indicates the row and $j$ the column of the matrix. So changing the index $i$ makes you move in the $vertical$ ...
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Notation: function returning the element of a partition containing $x$

Suppose $Y=\{y_1,\ldots,y_m\}$ partitions the set $X=\{x_1,\ldots,x_n\}$. I would like to define a function $y: X \to Y$ which returns $y \in Y$ if and only if $x \in y$. Is there a way to write this ...
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What is the difference between the principal square root of $x$ and $x^{1/2}$?

Today I learned that the square root symbol ($\sqrt{}$) represents only the "principal" square root of a number. What about exponents?—if I write $x^{1/2}$, would that encompass the negative ...
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Meaning of Big O notation with Index $[O_{\epsilon}(n^{\epsilon})]$

I found this notation in a book and can't figure what it means. I haven't found any other examples. This is $$O_{\epsilon}(n^{\epsilon})$$ One can see it in context at page $186$ of book. Anyone know ...
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Notation for probability density function in Bayesian context

The Bayes theorem is often quoted as, $$P(\theta|X) = \frac{P(X|\theta)P(\theta)}{P(X)}.$$ In my use case, I'm dealing with Gaussian continuous variables. So, by $P(X|\theta)$ I'm referring to the sum ...
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Understanding random variables as functions

First of all, I have read What is a function and I have understood it basically and it is clear to me that in order to caluclate statistics "things" have to be transformed or mapped to ...
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What is the notation of length of vector?

I am looking for a symbol to express the length of a vector or table, For example we have a vector V, len(V)==5 ( for example). How can I represent it by a symbol ?
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Confusing $\min$, $\max$ notations for $k$-Means

If I have a graph like the one below: and I want to describe at what $k$ does the function stop decreasing the most, it would be $k=2$. But how would I put it notation-wise? Should I do this:$$\...
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Notation for smallest value greater than a number in a sorted set

In a finite set, is there a concept/notation for the smallest value larger than a particular element? For example, I have a sorted set as $ A = \{a_1, a_2, ..., a_k \} $ where $ a_2 > a_1, a_3 >...
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Notation: set X whose elements are smaller than each elements of set Y

Does there exist some notation to indicate that all elements of a set X are smaller than all elements of a set Y?
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Summation: Adding all scores with a min and max (score) of 10 and multiply by (amount of Issues) Y - Notation Formatting Advice

I am new this portal and haven't done any maths for a very long time. For an academic project I am working on, I attempting to create a equation (not even sure that's the right language to use) where ...
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Could someone explain this notation and line inthe transcendence proof of $\pi$?

In the following linked proof for the transcendence of pi https://fermatslibrary.com/s/the-transcendence-of-pi, the author uses the notation $\theta_1(x)=(e^{\alpha_1}+1)(e^{\alpha_2}+1)(e^{\alpha_3}+...
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What does $d^n\textbf{x}$ mean in this context?

I found the following on Wikipedia. Integration over more general domains is possible. The integral of a function $f$, with respect to volume, over an $n$-dimensional region $D$ of $\mathbb{R}^{n}$ ...
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Correct notation to illustrate the product of a weighted path

I am struggling to represent a situation using the correct mathematical notation. I have a graph that consists of a series of nodes (1-4) and edges which represent movement pathways between nodes. The ...
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why is the notation for sums and integrals so different?

An integral over some set $A$ is typically written $$\int_A f(x) \,\mathrm{d}x$$ whereas a sum over a set $A$ is written $$ \sum_{i\in A} f(i). $$ To me, these "feel" different. Why not ...
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Notation for adding elements to ordered set

Say I have an ordered set $E_1 = (e_1, e_2, ..., e_j)$ composed of some elements. I'd like to add a few more elements $E_2 = (e_k, e_{k+1}, ..., e_\ell)$ to this ordered set so as to obtain set $E_3 = ...
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What do you call it when, in the limit, two functions are proportional by a factor of 1?

Two functions $f(x)$ and $g(x)$ have the property that $$\lim_{x\to\infty}\frac{f(x)}{g(x)}=1$$. What is this referred to as commonly? All I know of is Big-$\Theta$ notation, which says $$f(x)\in\...
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Notation $(S,*)$ or $(\mathbb S,*)$ for semigroup leading to group $(\mathbb G,*)$

In a number-theoretic context, is it best to use the notation $(S,*)$ or $(\mathbb S,*)$ for a generic semigroup leading to group $(\mathbb G,*)$ by taking equivalence classes under an equivalence ...
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The set $X$ can be seen as index set?

Let $X$ be a set, and the power set of $X$ be $\mathcal P(X).$ For $\mathcal B \subset \color{red}{\mathcal P(X)},$ does \begin{align} &\quad \ \left\{ U\subset X \mid \exists \Lambda : \mathrm{...
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Abbreviating lots of consecutive indexed summations

I have lots of equations of the following form $$\sum_{r_0}\sum_{r_1}\sum_{r_2}\cdots\sum_{r_N} x_{r_0} x_{r_1} x_{r_2}\cdots x_{r_N} $$ I can use the following notation for the product of $x$s $$x_{...
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The Derivative Operator vs $f'(x)$

For the sake of example, suppose $f(x) = \sin(x) $. Then, $$f'(x) = \frac{d}{dx}[f(x)] $$ But $$f'(0) = 1 \neq \frac{d}{dx}[f(0)] = 0. $$ Why is this the case? Usually, when you have two equivalent ...
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What is the best way to write a function having many arguments?

I'd like to ask you guys what is the best way to write a function having a sequence of arguments? If a function has $x$ and $y$ as its arguments then we write, $$ f\left(x,y\right) $$ What if we have ...
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Notation of there exists

I have the following matrix, which represents a state of a state machine: $$ \mathbf{F} = \begin{pmatrix} f_\mathrm{1} & f_\mathrm{2} & f_\mathrm{3} \end{pmatrix} $$ I now want to show in my ...
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Meaning of this symbol: "inverted product"

I am reading algebraic geometry from notes of a senior and I am struck on this terminalogy. Question: In the definition of Closed subvarities of $\mathbb{P}^n$, the author write $\mathbb{P}^n= \...
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How do I denote a set where the order is important?

I am trying to formulate a special kind of the VRP with a vertex set $N=\{1,...,n\}$, but in this problem there exist a duplicate vertex for each $i \in N$ lets call this set $\tilde{N}$. I want to ...
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How to denote the limit of a function with a vector input?

Suppose that I have a function $y(v)$ such that $v \in \mathbb{R}$ and $\lim_{v \to 0} y(v) = \infty$. How would I use the limit notation if $v$ is instead a vector $V = (v_1,v_2,..,v_N)$. That is, ...
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What is the $T$ parameter of a linear interpolation (lerp) function formally called?

What is the $T$ parameter of a linear interpolation (lerp) function formally called? I only ever see it called $T$. I know it's a ratio or percentage, but does it have a formal name?
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Name and notation of the sum of cardinalities of finite sets

Given a finite set $S = \{s_1, s_2, ..., s_n\}$, where each $s_i$ is a finite set of $|s_i|$ elements, let $$ N = \sum_{s \in S}{|s|} $$ Is there a canonical name and notation for $N$? On a ...
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Mathematical notation on a matrix

Matrix notation question (in reference to "simultaneous interconnection and damping assignment passivity based control"): We select an $n\times n$ matrix called $F_d(x)$. Let $G(x) := F_d(x)+...
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Notation: What is $C(G)$?

$G$ is the order $55$ group described by $G=\langle x,y \mid x^{11}=y^5=1, yxy^{-1}=x^4\rangle$. I am tasked with showing $|G/C(G)|=5$, but I don't know what $C(G)$ is supposed to be. It can't be the ...
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Standard notation for algorithm (instead of function)?

I am writing a math paper and defining various algorithms. A function takes at least one input, and (may) output a result. The notation is $$ \begin{align} f :& \mathbb{R} \to \mathbb{R}\\ &(x)...
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Is Euler's identity really that beautiful or is it notationally contrived to be beautiful?

When I first saw the identity $e^{i\pi} + 1 = 0$, I was awestruck at how $5$ fundamental mathematical constants could combine so elegantly, finding invariance in all their numerical chaos. It was one ...
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notation for replacing values in one matrix with another with condition

I have two matrices of equal size. $'A'$ is a binary image, and '$B$' has values ranging between $0$ and $1$. What's the correct notation which would demonstrate that foreground elements of A (equal ...
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Is this expression correct ? $\sum x_i \xrightarrow{\Delta x_i\ \sim \ \ 0^+} \int x\ dx$

I like math and I always add some mathematical notations, and formulas to my profile picture, which might help the people who are in my contacts to know more about maths. Today I created the following ...
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Integrating w.r.t. to $\mathrm d\mu(x)$ vs $\mu(\mathrm dx)$

When integrating a function $f$ with respect to a measure $\mu$, I often see two different notations $$ \int_A f(x) \mathrm d\mu(x) $$ and $$ \int_A f(x) \mu(\mathrm dx) $$ How are these two different?...
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