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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Bach Tensor Definition and Notation

I encountered the following in a paper (http://arxiv.org/pdf/math/0310302v3.pdf) and I'm hoping someone could confirm my interpretation of it. The paper defines the Bach Tensor in local coordinates ...
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2answers
89 views

Set theory formula

I picked up a copy of Jech's Set Theory at my school library and I'm reading through it and taking notes. Right at the beginning, though, he mentions something called a 'formula'. Here's the quote: ...
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2answers
43 views

What is the meaning of the notation of function

More specifically, what is meant by the function $T: \mathbf V \to \mathbf W$? I saw it in the discussion of linear map in Axler's Linear Algebra Done Right but could not understand this notation. ...
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2answers
36 views

If $A$ is a set and $\mathcal B$ is a set of sets, is there some shorthand for $\left\{A\times B:B\in\mathcal B\right\}$?

Let $A$ be a set and $\mathcal B$ be a set of sets. Suppose we want to define $$M:=\left\{A\times B:B\in\mathcal B\right\}\;.$$ Is there some shorthand for $M$ as we've got for $$X\times ...
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1answer
47 views

Is there a mathematical operator to truncate negative values to zero?

Is there a mathematical symbol that truncates a value x to 0 if it is negative, and leaves it untouched otherwise? Something which is logically equivalent to $\max(x, 0)$?
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1answer
33 views

Notation in Munkres' Elements of Algebraic Topology

What is $ R^{N} $ in section 1 of chapter 1 of the book Elements of Algebraic Topology by J.R. Munkres? Is $ N $ some natural number?
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2answers
46 views

What is the difference between $x \bmod y$ and $x \pmod y$?

I'm currently taking calculus I. So I'm new with is all notation, and looking through the Internet I always thought $x \bmod 3$ means the remainder when you divide $x$ by $3$. Am I wrong, and is ...
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1answer
30 views

Symbol to denote length of geometric vector

I have seen both $\left|\vec{u}\right|$ and $\left\|\vec{u}\right\|$ when referring to the Euclidean length of a geometric vector $\vec{u}$. Which notation is preferred. Is it true that the latter ...
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4answers
101 views

What is the function “mod”

Surfing this site, I have often seen many functions and expressions involving $\bmod$ and I have no clue about its meaning. What does that $\bmod$ mean?
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1answer
35 views

Trouble Understanding Notation in Reinforcement Learning Paper

I'm looking at this (warning: this is a download of a pdf) paper and am having trouble parsing the notation on top of page 11, steps 4.1 and 4.2. $\forall i \leq t \in T$, $\forall$ $x_i$, $a_i$ ...
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1answer
31 views

Query description into mathematical notation

I need to formalize two query descriptions into mathematical notation. The source of this description, if anyone is interested in broader context. Query 1: Frequent routes The goal of the query ...
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1answer
37 views

How to evaluate this combination of sums and integrals?

I am reading a book on PDEs, and I am near the beginning where the author is talking about the heat equation and, specifically, solving the non-homogenous equation $u_t={\alpha}^2u_{xx}+f(x,t).$ The ...
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2answers
48 views

What is the correct way to write this matrix equation?

Given an $n \times m$ matrix $X$ and $m \times m$ matrix $A$, I would like to define the vector $y$ as $$y_i = X_{i,*} A (X_{i,*})^T$$ where $X_{i,*}$ is the $i$th row of $X$. Is there a simpler ...
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0answers
28 views

Given a matrix M, is there a name for the matrix MM^T?

One can make a symmetric square matrix out of any m-by-n matrix $M$ by computing the matrix $MM^T$ (or $M^T M$). Is there a name for this operation? I want to call it "symmetrizing" the matrix, but I ...
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0answers
22 views

Vector in vector notation

I'm a bit confused as far as notationally differentiating between row and column vectors goes. Suppose I define a column vector $$\boldsymbol{a} = (a_{1}, a_{2})^{T}$$ and another column vector ...
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2answers
51 views

What does $A^{B}$ mean? [duplicate]

Assume, that A and B are finite sets. What notion $$A^{B}$$ does mean? Have been looking for awhile now.
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1answer
60 views

What are the meanings of the various turnstiles

It is easy to find the meanings of $\vdash$ and $\models$ (see this question and Wikipedia) but what of the (triple?) turnstile $\Vvdash$ and the (vertical double?) turnstile $\Vdash$? Do they have a ...
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1answer
51 views

What do these group theory notations mean: $\overline{3}\otimes\overline{2}$, $\overline{2}\oplus\overline{3}$

Can you explain or give a good reference to explain notations like $$\Large\overline{3}\otimes\overline{2}\qquad\qquad \overline{2}\oplus\overline{3}$$ and combinations of these. Thank you.
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0answers
37 views

Notation in probability theory: conditional on multiple events or joint of event with an conditional one

It might be a quite dumb question and if so, I apologize in advance (I am kind of a newbie in probability theory ). But once in a while it bothers me and I can't find the answer to it. Ok, now the ...
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1answer
22 views

Notation for a projection of a differential form

Let $\omega = a_1 dx_1 + a_2 dx_2 + b_1 dy_1 + b_2 dy_2$. Is there any established notation to denote a mapping that "filters out" the $dy_i$-Terms? To be more precise, I invent my own one. Assume ...
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5answers
66 views

Is there any notation for general $n$-th root $r$ such that $r^n=x$?

As we know that the notation for the $n$-th principal root is $\sqrt[n]{x}$ or $x^{1/n}$. But the principal root is not always the only possible root, e.g. for even $n$ and positive $x$ the principal ...
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1answer
20 views

Intervals of integers modulo n

Do the following related concepts appear anywhere in literature? Denoting an "interval" in the integers modulo $n$ by $[i,j] = \{i, i+1, \dotsc, j\}$. For example, in modulo 6, $[5,3] = ...
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0answers
29 views

Standard notation for the set of children of a node in a rooted tree

In graph theory, given a rooted tree $T$ and a node $a \in V(T)$, is there a standard way to refer to the set of all children of $a$? I have seen $CHILDREN_T(a)$ being used, but this seem quite clumsy ...
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23 views

Proper formulation of one-to-one and onto proofs for group isomorphism

I have to construct an isomorphism for the two groups. I have the isomorphism itself but I'm not sure if my formulation is correct in regard to proving the mapping being 1-1 and onto and I don't want ...
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1answer
48 views

How to express double orthogonal complement?

Let $V$ be a Hilbert space and $U \subseteq V$. Then $U^\perp = \{\mathbf{v} \in V|\forall \mathbf{u} \in U, \langle \mathbf{u}, \mathbf{v} \rangle = 0 \}$. My question is, how do you express ...
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0answers
36 views

What does this matrix notation mean?

What does $|\textbf{M}|$ mean, where $\textbf{M}$ is a matrix? I am under the impression that you can element-wise divide $\textbf{M}$ by $|\textbf{M}|$ to normalize it in some way, kind of like how ...
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1answer
37 views

What is the proper use of Leibniz notation for one-sided derivatives?

The only notation I've seen has been restricted to either Lagrange's prime notation or Euler's $D$. Here are some of the variants: $$f'(a^+):=\lim_{x\to a^+}\frac{f(x)-f(a)}{x-a}$$ ...
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1answer
71 views

What does $\Bbb R/2\pi$ for a set mean?

I simply cannot figure out what this means. I read this on an article about the scalar product of $2\pi$ periodic functions. it says that < f,g > goes from $\Bbb R/2\pi \to \Bbb C$ (complex) Do ...
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1answer
115 views

Can ∂x and ∂y in a derivate be seen as ∂ times x or ∂ times y?

I'm watching some tutorials on machine learning and know just enough calculus to have an intuition on what a derivative is, but that's it. But this question is bugging me so much that now I'm pretty ...
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54 views

Set Theory Notation: What does it mean to “\” one set with another? [duplicate]

What does the "\" operator mean in the above context?
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7answers
154 views

The meaning of the symbol $\infty$ in Spivak's calculus book

Spivak in "Calculus" writes ... symbols of $\infty$ and $- \infty$ are purely suggestive: there is no number $``\infty"$ which satisfies $\infty \geq a$ for all numbers $a$. What is the meaning ...
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1answer
58 views

In the expression $p^2=4q_1$, what does the small $1$ mean?

In the image below there is $p^2 = 4q$ and then a small $1$. What is the name/meaning of this notation? I have never seen it before and can't find what the meaning of it is. Help is appreciated! See ...
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1answer
27 views

Notation question $|X^2|$

I am studying a little bit of set theory, and one of the questions in the book (in Efe A. Ok's real analysis book) asked to show that $\dim(X,\succeq)\leq |X^2|$, where $X$ is a finite set and ...
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1answer
60 views

Is there a short symbol that denotes integration?

I want to illustrate partial integration, see below. With derivatives we can just write $(term)'$. Is there something similar for integration? The best I could come up with is $\int(term)\mathrm{d}x$. ...
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0answers
32 views

What does a * mean after a letter signaling a result?

I have this equation from a biology publication. It's about an ecosystem model. $$\frac{df}{dt}=w(f)f(1-f)-vf$$ $w(f)$ is a function and $vf$ is a multiplication. Then what does $f^*=0$ or ...
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1answer
37 views

Meaning of “$\triangledown$u*ñ=0 on the boundary”

I'm doing homework for my PDE class, I'm coming across this notation and I don't what the ñ means: $\triangledown$u* ñ=0. I have tried to google it, but unfortunately questions like this don't really ...
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1answer
22 views

Symbol representing a vector composing of two vectors

I have a vector including two vector's elements. How do I simply represent a vector with elemental vectors. Formally, I have three vectors $x, a=(a_i), b =(b_i)$ and $x=(a_1, a_2, ...
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1answer
23 views

Question concerning big-Oh and small-Oh notation

What would the notation $a_n = (1+ o(1))b_n$ stand for? (And similarly for $a_n = (1 + O(1))b_n$).
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2answers
55 views

Equal sign or approximation sign?

I want to solve the equation $2\enspace \sin^2\enspace \theta+2\enspace \sin\enspace \theta-1=0$, $0\leq\theta<2\pi$. Using the quadratic formula, I found one of the solution to be ...
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4answers
247 views

How to write $\aleph$ by hand

So far, I've only seen the symbol $\aleph$ in its printed form and am wondering how this symbol could be written by hand on paper or on a board (in mathematical contexts, of course). Whenever I try to ...
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1answer
16 views

About the exact form of a gaussian kernel

Traditionally we define a gaussian function at a point x (assuming mean to be 0) as follows $$g_{\sigma}(x) = \frac{1}{\sqrt{2\pi \sigma^{2}}} \exp\left(\frac{x^{2}}{2\sigma^{2}}\right)$$ In ...
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2answers
65 views

Correct notational use of $:=$

Suppose I use the notation $:=$ to define some symbol the first time it is introduced, e.g. $A:=\Im(f+f^2)$. If later on I make use/remind the reader of the symbol definition, e.g. ...
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3answers
262 views

Arnold Trivium Problem 39

We find in Arnold's Trivium the following problem, numbered 39. (The double integral should have a circle through it, but the command /oiint does not work here.) Calculate the Gauss integral ...
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1answer
36 views

Fourier analysis notation - Sh and Ch

I reading something dealing with Fourier analysis and don't know what "Sh" and "Ch" indicate. Thanks!
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2answers
79 views

Symbol in Linear Algebra

I'm newbie in linear algebra and I do not understand the symbol that is selected with blue color. What does this symbol means? What is the purpose to use this symbol? What context is this symbol ...
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0answers
41 views

Given bifunctor $F$, what is the name of the functor with switched arguments?

Sorry for the unspecific title. Here the actual question: Given categories $\mathcal{A},\mathcal{B}$, let $S$ be the canonical functor $\mathcal{B} \times \mathcal{A} \to \mathcal{A} \times ...
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1answer
32 views

What does the notation $\mathcal{O}_{\mathbb{P}^n}(1)$ mean?

I have tried looking at my sheaves notes but couldn't find anything.
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2answers
294 views

Where can I find the official rule for multi-line expressions?

Consider this simple exercise: $$1+1+11+1+1 = 15\tag{A}$$ But what if it were a very long expression? Let's assume that it is, then $$\begin{equation*} \begin{split} 1+1+\; & \\ ...
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0answers
41 views

“Projective tangent space” to a projective variety

Is there an established notation for the linear subvariety tangent to a projective variety $V$ at a point $x$? I've seen this called the "projective tangent space" in some places. The closest thing ...
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1answer
33 views

What is the remainder of an n-th root called?

I feel like there should be a better word than remainder, but I don't know it. What do you call the thing that's left over when performing an $n$-th root? For example, $\sqrt[3]{29}$ is $3$ with 2 ...