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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2
votes
2answers
116 views

useful notation for pullback

Let $f:A\to C\leftarrow B:g$ be morphisms in a category. There exists in literature a useful notation for the morphisms $\bar f:A\times_C B\to B$ and $\bar g:A\times_C B\to A$ in terms of $f$ and $g$? ...
0
votes
0answers
139 views

Time series notation

I'm developing formal software requirements specifications for processing time series data and thus need to mathematically describe time series and operations on time series. Is there establish ...
3
votes
2answers
84 views

Conventions on definitional if(f)

When defining a term it seems common to use 'if' when the stronger 'iff' is also true. For instance: Definition 1: A set $A$ is open in $(X,d)$ if $\forall x \in A$, $\exists \epsilon \gt 0$ such ...
1
vote
1answer
198 views

Notation for Multiple summation

Is there an alternate way to represent the multiple summation given below? $\sum_{i_k=k}^{n} \sum_{i_{k-1}=k-1}^{i_k} \dots \sum_{i_2=2}^{i_3} \sum_{i_1=1}^{i_2}$ It guess it is wrong to write it as a ...
0
votes
1answer
75 views

correct expansion of a sum using multiple indexes

I have looked for a similar posting but haven't found anything... but then I am also a bit unsure of how to search because I've never posted a math question before. In my introductory finite element ...
2
votes
3answers
298 views

What 's the differece between $\cot(x)$ and $\arctan(x)$? [duplicate]

I know that $\displaystyle \cot(x)=\frac{1}{\tan(x)}$ and $\space \displaystyle \arctan(x)=\tan(x)^{-1}=\frac{1}{\tan(x)}$ What is the difference between these two function? Is $\cot(x)$ the ...
4
votes
2answers
253 views

Mathematical symbol for “has”

Just out of curiosity, I was wondering if there was a symbol for "has" so intead of saying $x \in A$, we could say something like "$A$ has $x$", they both mean the same thing but I was just wondering ...
5
votes
1answer
117 views

Why do we write second derivatives like $\frac{d^2x}{dt^2}$ [duplicate]

Why do we write the second derivative of $x$ with respect to $t$ as $\frac{d^2x}{dt^2}$? It's never been explained to me, and I've never found a particularly good explanation. What's up with this ...
2
votes
1answer
111 views

Getting rid of the set builder notation in the expression $\{ f(x) \mid P(x) \} = \{ g(x)\mid Q(x) \}$

The set-builer notation is used to have $$\{ x \mid P(x) \} = \{ x \mid Q(x) \}$$ denote $$\forall x\ \big(P(x) \Leftrightarrow Q(x) \big).$$ And some people write $$\{ x \in U \mid P(x) ...
4
votes
1answer
107 views

Nomenclature and notation for some aspects of weighted directed graph.

I'm having some problem with nomenclature some structures and quantities related to weighted directed graph. Suppose that $A \in \mathbb{R}_+^{N \times N}$ is the weighted adjacency matrix of a ...
2
votes
1answer
45 views

A question about the validity of a notation

I am writing a paper and using such a notation. Do you think that it is mathematically a reasonable notation? $$ \hat{{\cal{P}}}_{i}=\{\hat{Q}: ...
4
votes
4answers
299 views

Confusion about the usage of points vs. vectors

As far as definitions go, understand the difference between a vector and a point. A vector can be translated and still be the same vector, whereas a point is fixed. But I would like some clarification ...
2
votes
1answer
155 views

Notation for set of all closed sets

Is there a common notation for the set of all closed sets of a topological space? I have been using $(X,\tau)$ to denote a topological space with $\tau$ being the topology, set of all open sets. I ...
2
votes
1answer
105 views

What does the notation $[V]^2$ mean (in graph theory)?

In graph theory, a graph is a pair $G=(E,V)$ of sets satisfying $E\subseteq[V]^2$. But what is $[V]^2$? I suppose that it is the same as $V\times V=V^2$, but I do not know where the square brackets ...
1
vote
1answer
61 views

Is this notation for Stokes' theorem?

I'm trying to figure out what $\iint_R \nabla\times\vec{F}\cdot d\textbf{S}$ means. I have a feeling that it has something to do with the classical Stokes' theorem. The Stokes' theorem that I have ...
2
votes
1answer
455 views

Is there a symbol for matrix multiplication operator?

Title says it all. Is there any specific operator symbol for matrix multiplication? Not just write down side by side but symbols like cross ($\times$).
0
votes
2answers
116 views

Horizontal bar notation for isomorphisms or bijections

I have seen in many books, particularly on category theory, the use of an horizontal bar to indicate some sort of equivalence, but I have not seen a proper definition in any context. For example: $X ...
3
votes
3answers
568 views

What is the notation for the set of all $m\times n$ matrices?

Given that $\mathbb{R}^n$ is the notation used for n-dimensional vectors, is there an accepted equivalent notation for matrices?
0
votes
1answer
68 views

Postfix notation in MAPLE

I usually use MATHEMATICA as a computer algebra system and there I heavily employ the postfix notation (achieved by "//" at the end of some command), e.g. ...
2
votes
0answers
35 views

Is it possible to call such properties “cocycle conditions”?

Let $\mathcal O = \{\alpha,\beta,\ldots\}$ be a family of objects of general but similar type (numbers, functions,e.t.c.). Let $\{ X_\alpha \mid \alpha \in \mathcal O\}$ be a family of linear spaces ...
0
votes
2answers
322 views

What does apostrophe as a suffix denote?

I was just curious as to what "$'$" denotes; i.e. $x' = y$, as in $x'(t) = x(t)$ which has the solution $x(t) = c_1\;e^t$. I've found out that it has something to do with differential equations, but ...
4
votes
2answers
218 views

$^t$, $^\dagger$, $^*$, $^H$, $^⊤$, and $^T$ : Which is which, and what do each mean?

I think this question's answer(s) will be of profound use to the future generation of human beings who happen to stumble upon the website math.stackexchange.com. What are the differences between ...
4
votes
5answers
119 views

The derivative of $e^x$

We all know that the derivative of $e^x$ equals $e^x$. I found this notation on Wikipedia: $$\dfrac{d}{dx} e^x = e^x$$ Why isn't this expression $$\dfrac{dy}{dx} e^x = e^x$$ Is it because ...
4
votes
4answers
304 views

Calculus Leibniz' notation

I'm currently doing integration by parts, and I'm finding that the notation is what makes it tough for me. So I looked it up and found that: $$\int u(x)v'(x)dx= u(x)v(x) - \int u'(x)v(x)dx$$ But the ...
1
vote
1answer
31 views

Proper Notation for Predicate Variable Not In Universe?

Let's say I have the simple statement Q(x): x + 1 > 2x. For a universe of all integers, I can easily compute any truth value. If my universe of discourse changes to, say, x | x < 1, what is the ...
1
vote
1answer
60 views

Clarification on 2 'E's for expected value in a conditional probability

The text I am reading defines the Expected Prediction Error as the squared difference between the actual Y value and the predicted Y value (f(X) in the text). Then it conditions on X. The trouble I'm ...
5
votes
4answers
155 views

What is the use of, and intuition behind, writing $\frac{d^2}{dx^2}$ for the second derivative?

Is it possible to take a second derivative without taking the first derivative before? Why do we multiply the $d$ and $dx$ operators? Like, does $\dfrac{d^2}{dx^2}$ really mean $\dfrac{d}{dx} \cdot ...
3
votes
1answer
109 views

A question about Fraenkel's 1922 paper

In Fraenkel's 1922 paper, what does the angle bracket mean that he uses on page 254 on line 3 in the expression $\phi(x) = \{\{\langle x \rangle, \langle o \rangle \}, \mathfrak U x+ \{\langle ...
0
votes
1answer
70 views

Tree graph intuition + mathematical notation

Let $T$ be a tree with at least two vertices. Consider a path of maximal length in $T$. Prove that the start and end point of this path have degree 1 by contradiction : If we let the start and ...
5
votes
1answer
103 views

Is the variant direct image mathematically significant?

Preimages have the property that for an arbitrary function $f : X \rightarrow Y$ and all $B \subseteq Y$ it holds that $$f^{-1}(B^c)=[f^{-1}(B)]^c.$$ However, the analogous statement for direct ...
2
votes
3answers
159 views

What does 'i-th' mean?

I have seen a problem set for the tower of hanoi algorithm that states: Each integer in the second line is in the range 1 to K where the i-th integer denotes the peg to which disc of radius i is ...
1
vote
0answers
73 views

Simplified matrix notation

There is a mathematical notation to define an array that you can write using standard keyboard characters on one line? for example:   1 3 8 7 10 17 22 6   5 10 23 8 11 98 7 12
0
votes
1answer
253 views

Symbol for Mutual Inclusive events

is there any symbol for mutual inclusive(opposite of Mutually exclusive) events in probability. I meant to say is for OR we have U symbol in Set Theory. Likewise ...
1
vote
0answers
216 views

Notation in Financial Math

I am very close to showing part b of this question and think that the reason my solution doesn't match up is because I don't understand a piece of notation.The notation I don't understand is ...
3
votes
2answers
112 views

Show that a local ring is equicharacteristic iff it contains a subfield

A local ring $(A,\mathfrak m)$ is equicharacteristic if $\operatorname{char} A=\operatorname{char} \kappa (m)$. Need hints to solve the following question: A local ring is equicharacteristic ...
1
vote
2answers
93 views

Replacing the value of a function with the value of the limit - is this a standard construction?

Consider a partial function $f : X \rightarrow Y$ where $X$ and $Y$ are topological spaces and $Y$ is Hausdorff. Note that, although the source of $f$ is $X$, the actual domain of $f$ is a (not ...
2
votes
2answers
48 views

How can partial derivatives feature in the definition of a function?

I have a map $f(t,g,h)$ where $f:[0,1]\times C^1 \times C^1 \to \mathbb{R}.$ I want to define $$F(t,g,h) = \frac{d}{dt}f(t,g,h)$$ where $g$ and $h$ have no $t$-dependence in them. So $g(x) = t^2x$ ...
0
votes
0answers
96 views

Notation question: sampling a set randomly or using a function

What is the correct and elegant notation for sampling a set? For example, I have a set, $D$, and would like to draw randomly $n$ elements into a new set, $S$: $S = \{j \in D \,|\, \text{Random}() ...
8
votes
0answers
3k views

Symbol for Approximately Equal To [closed]

There seems to be confusion/incoherence around the ‘Approximately Equal To’ symbol in Unicode, LaTeX, on Wikipedia, and elsewhere. Let me summarise: $$≈ \tag{double tilde}$$ Learnt in school (UK) ...
3
votes
0answers
306 views

Notation for the pushforward measure

Given a measure space $(X,\Sigma,\mu)$, a measurable space $(Y,\Xi)$ and a measurable map $f\in\Sigma/\Xi$, the pushforward measure $\nu:=\mu\circ f^{-1}$ is given by $$ \nu[A] = \mu[f^{-1}(A)] $$ ...
3
votes
7answers
602 views

Symbol for “if and only if”: $\implies$ or $\iff$?

I was wondering about the iff sign in maths. I've never learned about it in school & see it a lot online. Usually the sign looks like this: $\implies$, but in math.stackexchange I always see this: ...
9
votes
3answers
342 views

Understanding the differential $dx$ when doing $u$-substitution

I just finished taking my first year of calculus in college and I passed with an A. I don't think, however, that I ever really understood the entire $\frac{dy}{dx}$ notation (so I just focused on ...
1
vote
4answers
211 views

What is rigorous notation for functions?

I have seen many ways to denote a function: $f(x)=x^2, y=x^2, f: x\mapsto x^2$ and so on. What is exact notation for functions? Please include lethal doses of rigor, set theory, and of course ...
1
vote
3answers
74 views

Double Integrals. Simple question, don't understand their wording.

Make a sketch of the region over which $$\int_0^{\pi/2} dx \int_0^{\sin(x)} dy$$ Would this be the same as $$\int_0^{\pi/2} 1 dx \int_0^{\sin(x)} 1 dy$$ Which simply evaluates to ...
0
votes
1answer
158 views

Letters for complex numbers

Suppose that I am writing a proof or some other piece of mathematical writing, and wish to introduce $n$ distinct complex numbers, for some positive integer $n$. What are the complex numbers called? ...
1
vote
0answers
169 views

How to denote a random variable and the set of possible values

I'm curious what the most common way is to denote a random variable and the set of possible values that it can take on. I doesn't seem correct to say, for instance: $r.v.\ X \in \{\ldots\}$, because ...
3
votes
0answers
39 views

notation for invariation

Let $\Lambda = \{T \in \operatorname{Her}_2(\mathcal{O}) ; T \ge 0\})$ and $\mathcal{O}$ the maximal order of some quadratic imaginary number field. I write $T[U] := U^* \cdot T \cdot U$ where $U$ is ...
3
votes
1answer
103 views

writing $M : \Gamma_{n,0} \backslash \Gamma_n$

Let $\Gamma_n = \operatorname{Sp}_n(\Bbb Z)$ and $\Gamma_{n,0} \subset \Gamma_n$ be a subgroup. We write $$ M = \begin{pmatrix}A & B\\ C & D \end{pmatrix} \in \operatorname{Sp}_n(\mathbb{Z})$$ ...
2
votes
2answers
104 views

How do you define definition symbol :=?

How is "$:=$" defined formally and why? "$\iff$", "$=$", ...?
2
votes
1answer
406 views

Integral sign with circle (AND arrow on the circle) through it

I know from multivariable calculus that the integral sign with circle in its middle means integrating along a closed path. So when I encountered in complex analysis the above integral sign but with ...