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
21 views

How to notate the restriction of an inverse of a function?

Let $f$ be a function, which is defined on $E\subset\mathbb{R}^n$, and which is mapping to an extended real numbers. That is, $$f:E\longrightarrow\overline{\mathbb{R}}$$ Then, for a subset $A$ of $E$ ...
5
votes
1answer
85 views

Unusual integral notation

When I was learning analysis, I often wondered why I couldn't seem to find anything like $$\iint f(x) (dx)^2$$ in a standard calculus text, and concluded that it should be meaningless – even though, ...
0
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0answers
25 views

Unfamiliar notation in an AQA Core 4 Mark Scheme $p\frac{dx}{dt}=q$ where $p$ and $q$ are functions

I'm told that after $t$ years there are $x$ fish in a lake where the number of fish is decreasing. The rate of decrease of the number of fish is proportional to the number of fish currently in the ...
-1
votes
1answer
73 views

Symbol for “greater than and possibly equal” [closed]

How would I say, in mathematical notation that $x$ is greater than or possibly equal to something? If I was checking an equality, I might suppose it like $$x \stackrel{?}{=}4$$ Or if I was uncertain ...
3
votes
1answer
32 views

Interval notation: infinity, -infinity in closed interval

I was watching a video stream a little bit ago and noticed on an equation without context that had the interval $\left[{-\infty, \infty}\right]$. This was preculiar to me as I've never seen the ...
1
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1answer
24 views

What does $p_b \propto ρ^n _b $ mean

$p_b \propto ρ^n _b $, in fluid mechanics, where $p_b$ is the pressure inside a bubble and $\rho$ is the density. What does that symbol looking like alpha mean?
4
votes
2answers
176 views

What is the notation for the empty matrix?

My question: What is a notation for an empty 0x0 matrix (i.e. the matrix for the only linear map $f:\{0\}\to\{0\}$)? Is it written $()$? How can I distinguish the 0x0 matrix with for example the 0x3 ...
60
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7answers
5k views

What is “Bra” and “Ket” notation and how does it relate to Hilbert spaces?

This is my first semester of quantum mechanics and higher mathematics and I am completely lost. I have tried to find help at my university, browsed similar questions on this site, looked at my ...
0
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1answer
17 views

double summation notation

In a paper I am studying, the author writes $$\sum_{{i=1}\atop {k=1}}^{N+1} C_i \eta_k$$ How are the two indices to be interpreted? In other words, how would this expression be written using sigma ...
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2answers
41 views

Name of the notation where number is expressed as a sum

I have the following general form of a number: Does this notation have a name? Here is the example of using the form:
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1answer
31 views

Meaning of this expression

I found a relation while studying elliptic curves, I could not understand its' meaning. $E[n]$ is a $n$-torsion subgroup then $E[n]\cong Z/nZ \oplus Z/nZ$, What does this $\oplus$ symbol mean? Thanks ...
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0answers
15 views

Acceptable notation? $\lesssim_{n} n^{-\beta}$ for constants NOT depending on $n$

I am preparing a paper and found it convenient to write things like $$ |\text{Expression of a lot of variables}|\lesssim_{n} n^{-\beta} $$ when an inequality is true up to a multiplicative factor that ...
2
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1answer
22 views

Use of $\arg$ function [closed]

I know that $\underset{x}{\operatorname{argmax}}f(x)$ is defined as the value at which $f(x)$ has its maximum. There is also $\underset{x}{\operatorname{argmin}}$. However, in statistics, I often ...
4
votes
1answer
40 views

What is “non-simple applied first-order functional calculus” (60's set theory)

Azriel Lévy says in his 1960 paper Axiom Schemata of Strong Infinity in Axiomatic Set Theory, that the $\sf{ZF}$ set theory is formalized with a finite number of axioms in "non-simple applied first-...
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0answers
32 views

What notation is usually used to denote the category of $R,S$-bimodules

I came across the notation: $\mathbf{Mod}_{(R,S)}$. But generally, when handwritten, long superscripts/subscripts are becoming clumsy. So I'm wondering whether there's an adopted alternative. Is ...
4
votes
1answer
47 views

U-substitution in disguise (notation) — what is $d(2x+1)$?

I would like to know if there is a name for this notation here, which I've seen many times. $$\int \frac{1}{2x+1} \ dx= \frac{1}{2} \int \frac{1}{2x+1} \ d(2x+1)=\frac{1}{2} \ln|2x+1|+c$$ Thanks
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3answers
54 views

Is there a general operator symbol?

I want to write the expression a + b for +, - and * at once. Can it be done? Is there an operator that represents a general operator?
1
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1answer
40 views

Is there a generic hierarchical difference in measure theory between $\LaTeX$ \mathcal and \mathscr?

I am wondering abut $\mathcal F$ perhaps denoting a $\sigma$-algebra, whereas $\mathscr F$ may be a Borel $\sigma$-algebra, or a set of sigma algebras. Also, if there a standardized, or tacit ...
1
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1answer
37 views

The $I$ in the smallest sigma algebra generated by collection of subsets of $\Omega$

In defining a Borel sigma algebra (and if I understand it right) you can depart from the idea that an arbitrary collection of subsets $\mathcal C$ of the sample space $\Omega$, where $\mathcal C$ will ...
0
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1answer
16 views

Notation: $\sum_i \dfrac{\partial A_i}{\partial x_i} \boldsymbol{e}_i$ using $\nabla$.

I would like to write $\sum_i \dfrac{\partial A_i}{\partial x_i} \boldsymbol{e}_i$ using the $\nabla$ operator if possible, where $\boldsymbol{A}=A_1\boldsymbol{e}_1 + A_2\boldsymbol{e}_2 + A_3\...
1
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0answers
26 views

$ \div $ as divergence operator

I am reading Nonlinear continuum mechanics for finite element analysis by Bonet and Wood, and I encountered an operator $ \div $ I am unfamiliar with. They define it as the divergence of a second ...
0
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1answer
26 views

Meaning of symbol “$y\nearrow x$” in CDF Limit

Could somebody explain the meaning of "$y\nearrow x$"? $F_X$ is right continuous, that is, for any $x$, $\lim_{y \nearrow x} F_X (y) = F_X(x)$.
2
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0answers
26 views

use little $o$ notation.

out from apostol's book. $f(x)=o(g(x))$ if $\frac{f}{g}\rightarrow 0$ when $x\rightarrow a$ and gives some properties. 1)$o(g(x))+o(g(x))=o(g(x))$ 2)$o(cg(x))=o(g(x))$ 3)$f(x).o(g(x))=o(f(x).g(x))...
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1answer
21 views

Limit notation where variable does not approach anything

I was reading an example in my probability textbook that states a limit as $$\lim_{n}{P\left\{X \leq 3 - \frac{1}{n} \right\}}$$ where the RV $X=k$ is defined for $ k \in \mathbb{R}$ What exactly ...
2
votes
3answers
68 views

Expected value of $g(X)$.

If $\mathrm{E}(X) = \sum_{x\in I} x\,\mathrm{P}(X=x)$, how can I deduce that $E(g(X)) = \sum_{x\in ?} g(x)\,\mathrm{P}(X=x)$? I don't see why it isn't $E(g(X)) = \sum_{g(x)\in ?} g(x)\,\mathrm{P}(X=g(...
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1answer
16 views

Is it correct to write $argmin(x, y) \sum_i^n |p_{x_i} - x| + |p_{y_i} - y| = argmin(x) \sum_i^n |p_{x_i} - x| + argmin(y) \sum_i^n |p_{y_i} - y| $?

$argmin(x, y) \sum_i^n |p_{x_i} - x| + |p_{y_i} - y| = argmin(x) \sum_i^n |p_{x_i} - x| + argmin(y) \sum_i^n |p_{y_i} - y| $ Is it a legit way of separating argmins to show independence of $x$ and ...
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votes
2answers
54 views

Using the definition of $f$ is $O(g)$ proof: [closed]

I'm studying for my discrete math class and I don't understand how to prove big O notation. I understand that $f$ is $O(g)$ of another if $f(x) \le c g(x)$ holds. How would I go about proving $\sin ...
0
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1answer
32 views

Regarding taking powers of prime ideals in a ring

My question is simple to ask: given some prime ideal $P$ in a ring $R$, we can talk about $P^2, P^3$ etc. but can we discuss $P^0$? Is there a convention that says $P^0 = R$, or is there something ...
0
votes
1answer
13 views

Notation: square brackets with a unique scalar?

my question is purely about notation. I am reading papers in computer science and I see that people use the following notation $[x]$ to denote $\{1,2,\ldots,x\}$. Is that correct? Or does it mean ...
1
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1answer
27 views

How to interpret scientific notation?

I'm having a problem understanding scientific notation. What is the difference between the following: $$\text{5e2, 5e-2, -5e2, -5e-2}$$
1
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1answer
23 views

The sum of $V=U+W$ of a vectorspace V and subspaces $U$, and $V$

I know what the sum of two subspaces is and how we notate but is it ok to write a minus to denote what I hope should be obvious is meant. So we have $V=U+W+Y$ where $V$ is a v.space and $U,W,Y$ ...
0
votes
0answers
46 views

Is the notation $\mathbb{P}(X \cap Y) = \mathbb{P}(X,Y)$ common?

For two random variables $X,Y$, is the notation $\mathbb{P}(X \cap Y) = \mathbb{P}(X,Y)$ common? In a probability class last year we had always used $\mathbb{P}(X \cap Y)$. This year in a stochastic ...
0
votes
1answer
14 views

Clarification of Direct sum meaning with $\geq 3$ subspaces

Let $V$ be a vectorspace and $U_i$ subspaces of V. In the definition of $\oplus_{i \in I} U_i$ it is said that is does not suffice for $U_i$ to be pairwise disjoint. Instead we must have the stronger ...
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1answer
19 views

Explanation Notation Union Probability

Could somebody explain how to create intuition for the probability/union notation below? I don't know how to read it. And is this a situation where events are disjoint, but dependent?
1
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0answers
55 views

On a Probability notation - $\mathbb{E}[X(.)|\mathcal{F}]_G$

What could mean this notation : $\mathbb{E}[X(.)|\mathcal{F}]_G$ ? where G : $\Omega \rightarrow \mathbb{R}$ is a random variable on a probability space $(\Omega,P, \mathcal{F})$. X could be a ...
0
votes
0answers
19 views

Topology; difference between open subsets of $X$ containing $x$ and open neightborhood od $x$?

I see my lecture notes and some texts alternate between the two. What is the difference in saying that "an open subset of $X$ containing $x \in X$" and an "open neighborhood of $x \in X$"?
1
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4answers
54 views

Mathematical Notation of Sequence of Functions

Let's say I have a finite set of functions $F=\{f_1,f_2,f_3,...,f_n\}$ and I want to show a recursive function that is constructed by an arbitrary sequence of applications of functions in $F$ to input ...
0
votes
1answer
19 views

Are these two notations equivalent?

I've seen both $\forall a,b$ and $\forall a\forall b$ of these being used in various posts and wondered if they are equivalent or if there is a subtle difference between the two.
0
votes
1answer
44 views

In sequent calculus, what's going on with sequents with multiple formulae in the succedent?

The sequent proof systems I learned only allowed one formula on the right hand side of the sequent, and $\phi_1, \ldots, \phi_n \Rightarrow \psi$ (or ... $\vdash \psi$) is understood as saying that $\...
1
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1answer
34 views

How can I write a set of equations in summation form?

I have a system of equations as follows: \begin{align} & A_1^{11} + A_1^{12} + A_1^{13} + \cdots + A_1^{1n}=X \\[8pt] & A_1^{21} + A_1^{22} + A_1^{23} + \cdots+ A_1^{2n}=X \\[8pt] & \...
1
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1answer
14 views

Notation To Define A Mapping From A Set to a Relation Between Two Elements Of That Set

Say I had a set $S=\{s_1,\dots, s_n\}$, and each $s_i$ denotes an outcome. If I wanted to define a function, $f$ which takes two elements of $S$, $\{s_i, s_j\}$, and maps it to a relation, either $s_i\...
0
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1answer
22 views

reference request: $C^k(\overline\Omega)$ as restriction of $C^{k}$ functions on $\Omega$

Let $\Omega\subset\mathbb{R}^d$ be an open set. $C^k(\Omega)$ is defined as the space of functions $f:\Omega\to\mathbb{R}$ such that $\partial^nf$ is continuous for $0\leq|n|\leq k$. There are ...
0
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1answer
48 views

Notation for kernel object

When $f: A \mapsto B$ is a morphism in some category with a zero object and limits, we can use $\ker(f)$ to refer to an equivalence class of morphisms to $A$ which satisfy a particular universal ...
0
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0answers
27 views

How to formalise a procedure involving Cartesian products of sets of vectors and transformation in matrices?

I am asking for an help to formalise with the correct notation the following procedure. Let $n\in \mathbb{N}$. Let $\{0,1\}^{n-1}$ be the set of vectors of dimension $(n-1)\times 1$ with each ...
0
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0answers
47 views

Why isn't the identity/unit matrix upright?

I realize this is more of a typesetting problem then a mathematical one. I've already tried the TeX stack exchange and the question got canned. In ISO 80000-2:2009, variables and running numbers are ...
0
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0answers
20 views

Slow time variable

If I have a time variable $t$ such that $t \in[0,5]$ and I introduce a new variable $\tau$ st $\tau =\epsilon t$, where $\epsilon <<1$. Why is $\tau$ called a 'slow time variable', when $\tau \...
0
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1answer
22 views

Basic set notation in combining different ranges of numbers

What is the proper way to specify a set which contains all even numbers between 1 and 10, and all odd numbers between 11 and 30? Would this work? $$ U = \{n, m\ |\ n \ \text{is even},\ 1 \le n \le ...
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1answer
45 views

Big O notation $O(\epsilon)$ [closed]

What does it mean to say that $\tau=O(\epsilon)$? Where $\epsilon$ is small, meaning that $\epsilon \ll 1$.
1
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1answer
25 views

Write the series using sigma notation: $f(x)= 1 - (x^2)/2! + (x^4)/4! - (x^6)/6! +\cdots$

$$f(x)= 1 - (x^2)/2! + (x^4)/4! - (x^6)/6! + \cdots$$ I don't know how to get the signs to work like negative, then positive. I have tried to make it like the following: $(-1)^{n-1} \frac{x^{2n-1}}{ ...
0
votes
1answer
32 views

Set builder notation: defining the number of elements

I have a set L and I have a subset S which is part of L and contains three elements A, B and C. Finally, each of these elements are subsets that consist of their own elements: $A=\{a_1...a_n\}$ $B=\{...