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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1answer
20 views

Complexity class notation

In my CS courses we often use Big-O notation to denote the complexity of a certain calculation. However, we often also write stuff like: $$mO(1) = O(m),$$ or: $$O(m) + O(n) = O(m+n) = ...
6
votes
1answer
84 views

Why do so few people use the $\equiv$ symbol for identities?

I've always been taught to write, for example, $$\sin^2(\theta)+\cos^2\theta \color{red}{\equiv} 1,$$ rather than $$\sin^2(\theta)+\cos^2\theta \color{red}{=} 1,$$ and $$x(x+2)\color{red}{\equiv} ...
2
votes
0answers
49 views

Is $P(x,y)$ different from $P((x,y))$?

The symbols $P(x,y)$ and $P(z)$ indicate generally the concept of "predicate", but if $(a,b)$ is (ordered) 2-tuple, and in $P(z)$ I have $z=(a,b)$, Is $P(a,b)$ different from $P((a,b))$? .. Thanks in ...
0
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0answers
21 views

Column or row of a matrix?

The question is so simple, but I cannot find the answer. Is $M_i$ (usually) the $i^{\text{th}}$ column of matrix $M$? Or the $i^{\text{th}}$ row? Since $M_{ij}$ is the $j^{\text{th}}$ element of the ...
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4answers
39 views

Why do you add +- to only one side when you remove square root from both sides?

As the title says, why when you take a square root of both sides of the equation do you add $\pm$ only to the side which is a number, as opposed to an unknown? For example: $$x^2 = 9 \implies x = ...
6
votes
4answers
105 views

Meaning of $\mathbb{R}^0$, $\mathbb{R}^{1/3}$, and $\mathbb{R}^{-2}$.

In mathematics, we take $\mathbb{R}^n$, where $n$ is a fixed positive integer, to represent the Cartesian product $$ \overbrace{\mathbb{R} \times \mathbb{R} \times \cdots \times \mathbb{R}}^{n \ ...
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0answers
64 views

What's the meaning of $544/119 = 4 + 68/119$?

The Art Of Computer Programming Third Edition, page 4 While explaining Euclid's Greatest Common Divisor algorithm for the numbers $544$ and $119$, Knuth calculates the remainder of $544/119$ ...
3
votes
1answer
71 views

Polynomials vs polynomial functions

On my algebra course, sometimes we write, say $$f \in R[X], f= X^2 + X + 1$$ And sometimes we treat polynomials as functions, so $$ f(x) = x^2 + x + 1$$ What is the difference between these two ...
2
votes
3answers
73 views

What does mod mean in mathematics?

I have sometimes seen notations like $a\equiv b\pmod c$. How do we define the notation? Have I understood correctly that $c$ must be an element of some ring or does the notation work in magmas in ...
0
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0answers
100 views

What does a dot in a circle mean?

I'm looking at some formulas involving matrices (in the context of machine learning, but I'm not sure it's relevant) and I came across $\odot$. What could this mean? The context is $M \odot N$, where ...
2
votes
1answer
24 views

Ordered tuples of proper classes

From time to time I encounter notation like this: A triple $\langle \mathbf{No}, \mathrm{<}, b \rangle$ is a surreal number system if and only if ... The confusing part is that a proper class ...
1
vote
1answer
54 views

What does the symbol $\mathbb{R}(\zeta)$ denote?

I know hat $\mathbb{R}[\zeta]$ denotes the ring of polynomials in $\zeta$ with real coefficients. I came across the symbol $\mathbb{R}(\zeta)$. Which ring is this?
1
vote
1answer
38 views

$\mathbb{R}_*$ Notation

What does the notation $\mathbb{R}_*$ denote? I am seeing it used for showing domain of matrices, $M\in \mathbb{R}_*^{a \times b}$, which is different from $N \in \mathbb{R}^{a \times b}$. But I do ...
2
votes
2answers
51 views

How to write a counter in proper notation?

I have made a system where I count every hour that meets a certain condition. I would now like to express this in a simple formula but sadly can not come up with a solution. This is what I need. ...
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2answers
83 views

Are subscripts 0 or 1 based?

Being a programmer by nature I can never remember when looking at a math formula if the subscript that indicates which value in a series starts at 0 or 1. I know computer arrays start at 0 so that is ...
0
votes
1answer
12 views

How to state each individual solution of an expression is evenly divided by n as opposed to their sum being divided by n

Given that for k2n (n=4 and k takes all integer values from 1 to n) If we want to state the sum of all solutions is evenly divisible by n we write: n ∑ k2n = 0 (mod n) k=0 ...
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1answer
27 views

Quick question about contravariant and covariant tensors

I have seen many different notations to denote contravariant/covariant and mixed tensors. For example, I think the notation $\omega^{v}_{\,\,\,\mu}$ stands for a mixed tensor, where one index ...
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0answers
24 views

Notation of the Weierstrass Sigma Function

This is with regards to the Weierstrass $\sigma$-function belonging to a pair of periods $(\omega$,$\omega')$ where $\frac{\omega}{\omega'}\notin \mathbb{R}$. The notation of this function in my ...
0
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1answer
36 views

A notational confusion on gradient

Given a parametrized function $f_{w}: \Bbb R ^{m} \to \Bbb R ^{k}, w \in \Bbb R^d$, I see in a book the following notation $\bigtriangledown ^ {w} f_{w}(.)$ denote its gradient w.r.t. $w$. What is the ...
3
votes
3answers
75 views

Notation for choosing the k smallest elements from a set of integer

Is there any specific notation for picking $k$ elements from a set which are the smallest? Ex: {$1,3,5,7,9,11$} with $k = 3 \Rightarrow$ We want $1,3,5$
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vote
2answers
82 views

What is the difference between a sequence of functions $(f_n)$ and a sequence of functions $f_n(x)$?

What is the difference between a sequence of functions $(f_n)$ and a sequence of functions $f_n(x)$? I am reading my textbook on analysis, and it seems to use 'sequence of functions' to describe both ...
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0answers
34 views

Integration notation

could someone explain the following notation: $u(x,t):=\int^t_0 v(x,t:\tau)d \tau$ It's come up but I don't understand how to interpret the semicoloned tau
0
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1answer
27 views

How to notate a set of point meeting a given condition?

How do you formulate the set of the n closest neighbours within the radius r of point P with proper mathematical syntax?
2
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1answer
48 views

Meaning of dot symbol “·” in this context?

In this context, what does the dot symbol "·" mean? Multiplication? I looked it up on Wikipedia but couldn't find it.
0
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1answer
69 views

Notation for dimension of vector space

Is it an unusual notation to write $|V|$ for the dimension of a vector space $V$? Is it ok to use it if you blur the distinction between the grid for the finite element method and its associated ...
2
votes
1answer
31 views

What is the meaning of $<$ in a preorder?

Let $(P,\le)$ be a preorder, i.e. $P$ is a set and $\le$ is a relation on it that is reflexive and transitive. In this context for myself I can find two interpretations for the symbol $<$ 1) ...
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0answers
45 views

Given a line bundle $M$ and ascheme valued point $a$, what is $M(a)$?

I am reading Olsson's book http://math.berkeley.edu/~molsson/mono020807.pdf . If $M$ is a line bundle on an abelian group algebraic space, he considers scheme valued points $a \in A$ and $M(a)$. I ...
1
vote
1answer
30 views

The origin labelled on a graph: $0$ or O?

When one draws a graph, say in the x,y plane, we label the origin with a circular/elliptical symbol. Now is this a $0$ (zero), or is it O (for Origin), or simply just a circle/ellipse? Can it be ...
0
votes
3answers
65 views

Set question - $ ℤ^+ = ℕ$ [duplicate]

I am not sure whether the following statement is true: $ ℤ^+ = ℕ$ if not, why? Thank you in advance! I appreciate your help!
0
votes
5answers
47 views

Kronecker delta versus identity matrix

How should $\delta_k^j$ be regarded? Is it a scalar that takes on variable values? A 3x3 identity matrix (in 3 dimensions)? The wikipedia article on raising and lowering indices with the metric ...
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2answers
55 views

Integrating a term again and again

So if you have $f''(x) = 24x$ you know you want to integrate it, because it would look much better integrated, so now we have $f'(x) = 12x^2$, but it could still look better, so we integrate it to ...
1
vote
2answers
48 views

What does d f(t,x) = 0 mean?

A differential equation that can be written in the form $d\phi(t, x) = 0$ for some continuous and differentiable function $\phi(t, x)$ is called exact. What does $d\phi(t, x) = 0$ mean?
5
votes
2answers
35 views

What does the colon in the statement $T:t\to−t$ mean?

I don't understand what the colon $T:t\to−t$ means. I understand that it does something, is it a variable or just a symbol that has no significance?
1
vote
1answer
33 views

matrix inverse in tensor notation

Suppose there is a matrix $A$ that transforms vectors, $$ Y = A x $$ Now express this in some other coordinate system, with $x = B z, \,\, y = B w$, so \begin{align*} & Bw = A B z \\ ...
0
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3answers
67 views

What does a tilde underneath an inequality mean?

I've recently come across an expression of the form $$\Large x \lesssim y$$ What does this expression mean? Thank you.
2
votes
4answers
122 views

What does $\exp(f)$ mean?

In several posts around this site, I have encountered the expression $\exp(x)$ where $x$ is an arbitrary expression. What does this notation mean?
4
votes
4answers
478 views

A doubt in Bergman's notes

On pg. 8 of these notes, Bergman says that a group $G$ contains an inverse operation $i:G\to G$, along with $\mu:G\times G\to G$ and a "neutral element" $e$. Hence, a group should be referred to as ...
1
vote
1answer
18 views

Notation for Model-Relation of formulae with free variables

Lets assume we have a formula $\mathsf{path}(x,y)$ with free variables $x,y$, and $\mathsf{acyclic}$ with no free variables on the signature $\tau = \{E\}$ (i.e. Graphs). Informally, what the formula ...
1
vote
1answer
36 views

Is the following use of indices correct?

Is the following use of indices correct? A vector $\langle x_i, x_{i+1},...,x_{i+k}\rangle$ is given. For every such vector a function is defined through $$\mu=\frac{\sum_{j=0}^k ...
0
votes
0answers
52 views

Notation in Linear Algebra

What does $(A\mid b)$ denote in Linear Algebra? Specifically in the context of the following question: "If $(A\mid b)$ is in reduced row echelon form, prove that A is also in reduced row echelon ...
2
votes
1answer
31 views

Notation in Actuarial text

I am not very knowledgeable in Actuarial Sciences, but I was looking at a text, and found a notation, which I cannot find any references to. What does: $_{2|2}q_{1} = 0.288$ mean? I don't think it ...
1
vote
1answer
28 views

Set Theory Elementhood Notation

From How to Prove it: Given $A=\{n^2|n \in N\}$ where $N$ is the set of all natural numbers. I want to express A in terms of elementhood test notation. Velleman says $A=\{x| \exists n \in N ...
0
votes
3answers
57 views

What is this and why is it so important $(x,y) \to (y,x)$?

When doing a $y=x$ reflection the notation is $(x,y) \to (y,x)$. My teacher told us to find out what it is, what it is called, and why it is important? Please help ?
7
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2answers
629 views

History of notation: “!”

Does anyone know where the factorial "!" symbol came from? I can't decide if it is my favorite or least favorite notation in mathematics...
3
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1answer
37 views

Is there standard notation to handle “chains of functions”?

Let $f(x)=g $ $g(y)=z $ Is there standard notation to express z in terms of f(x)? Something like (f(x))(y)?
0
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1answer
50 views

notation used in algebraic topology [closed]

i have some confusion in notations used in my algebraic topology class. $\approx$ homeomorphic $\simeq$ homotopy $\cong$ isomorphic Please correct me for the above.
1
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1answer
28 views

Representing a for loop with modulus in formal notation

I have the following section of code I am writing for research. Basically I need to formally represent the mathematical notation behind a set that follows: ...
0
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1answer
31 views

What is the meaning of the semicolon in $h(x;\theta)$?

The context is machine learning, and the full expression is $h(x;\theta) = \operatorname{sign}(\theta_1 x_1 + \cdots + \theta_d x_d)$. $x$ is a feature vector and $\theta$ parameterizes a set of ...
0
votes
1answer
18 views

Notation for the mean within a set of divided values multiplied by another value

This is my very first crack at notation and I'm very unsure about the results: With two sets with values: QTY: [4, 5, 5, m...], DOSE: [20, 10, 10, n...], and the number 30. The formula is: ((30 / ...
0
votes
2answers
39 views

Problem with symbology

I know this may be a pretty basic question, but what is the difference between $\approx , =,\cong, \text{and}\sim $ ? I had problem while changing schools and now I am confused.