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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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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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1answer
32 views

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
42 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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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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16 views

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
33 views

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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114 views

What is difference between $\{\{\}\}$ and $\{\varnothing\}$?

I need to know what is difference between $\{\{\}\}$ and $\{\varnothing\}$
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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
20 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
41 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
23 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
67 views

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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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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0answers
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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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0answers
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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
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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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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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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
57 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
44 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
23 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
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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
40 views

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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$d_1:ax+2y+4=0$; $d_2:3x-4y+12=0$; $|d_1d_2|=?$

$d_1:ax+2y+4=0$ $d_2:3x-4y+12=0$ $d_1$ // $d_2$ $|d_1d_2|=?$ This is a slightly modified version of a university entrance examination problem set up by the experts at this specific ...
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28 views

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
49 views

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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22 views

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 ...
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4 views

Is my interpretation of the expression on the expectation is a correct one?

Consider the following expression $$E_{x\sim p_{data}(x)}[ f(x) ] $$ I am understanding it as 1) if X is a collection of a discrete random variable $(X_1, X_2,......, X_n)$, and x is generated from ...
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Meaning of mathematical notation for summation with two variables.

Is the notation $$ \sum_{x, y} $$ mathematically equivalent to: $$ \sum_{x}\sum_{y}$$
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1answer
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An acceptable way to express the maximum of a union of two finite sets

Suppose that $A= \{a_1, \ldots, a_n\}$ and $B= \{b_1, \ldots, b_n\}$ are two sets of real numbers. In such a situation, I usually find myself writing $\max\{a_i, b_i\mid i=1, \ldots, n\}$ as ...
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2answers
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Summation Notations - (Discrete Math) I'm having trouble

I've been working on a single sigma notation problem for about 40 minutes now, and I'm having trouble conceptualizing the solution to this problem. I look at examples, and it seems that I begin to ...
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95 views

Why are |vertical lines| used to mark matrix determinants?

This notation is sometimes used to denote the determinant: $$ \begin{vmatrix}a & b \\ c & d\end{vmatrix} = ad-bc$$ Why? Where did this notation come from? Was there any relationship between ...
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1answer
26 views

What is the $\oplus$ sign mean in $R^{\oplus I}$?

I read a defenition of a "finite generated module": An $R$-module $M$ is said to be finite generated if there exists a surjective homomorphism $R^{\oplus I}\to M$ for some finite set $I$. For $I=\{1,....
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Prove by induction on $n$ that $(1)(2)+(2)(3)+…+n(n+1)={1\over 3}n(n+1)(n+2)$

So after testing myself with this question, I was unable to solve it. I was able to prove the base case $n=1$, but I was pretty lost on the induction step. I took a look at the solution and here it is:...
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1answer
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Prove that $(1+x)^n>1+nx$ for all integers $n\geq2$, where $x>-1$ and $x\neq0$.

I would like to get feedback on whether this proof is valid or not and would like to know if my usage of set-builder notation is correct. I get the feeling that the discrete mathematics class I'm in ...
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1answer
70 views

notation in abstract algebra: $\mathbb{Q}(\sqrt{2})$

What does $\mathbb{Q}(\sqrt{2})$ mean? Is it $\{ a+b\sqrt{2} \mid a, b \in \mathbb{Q} \}$ ?
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What does the summation mean in the general Leibniz rule for 3 functions?

I stumbled upon this https://en.wikipedia.org/wiki/General_Leibniz_rule on the nth derivative of products of functions. I am wondering if anyone is knowledgeable enough to state the nth derivative for ...