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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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
0
votes
3answers
48 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
vote
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
vote
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
votes
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
vote
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
votes
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
votes
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))...
0
votes
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(...
0
votes
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 ...
-2
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
votes
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
vote
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
vote
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 ...
1
vote
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
vote
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
vote
4answers
53 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
vote
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
vote
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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 ...
-2
votes
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
vote
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
31 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=\{...
1
vote
0answers
29 views

Question regarding terminlogy and wording of the derivative

When doing calculus, we typically say that we "take the derivative of a function f(x)." However, rigorously, f(x) is not a function but rather the value of the function f evaluated at x. Thus, in ...
0
votes
0answers
14 views

Notations of conditions

Assume I have the Mean Square Error: $\mathcal{L}(X, Y) = \frac{1}{\lvert X\rvert}\sum_{i=0}^{\lvert X\rvert}\left(f(\mathbf{x}_{i}) - \mathbf{y}_{i}\right)^2 \space \space ,\space \space \lvert X\...
2
votes
1answer
69 views

Why don't we use “dx” as the limiting variable when teaching the definition of the derivative with respect to x?

For a function $f(x)$ that is differentiable at $x$, the derivative of $f$ with respect to $x$ at the point $x$ is usually given as $$\frac{df(x)}{dx} = \lim_{h\to0} \frac{f(x+h)-f(x)}{h}.$$ I think ...
0
votes
0answers
28 views

representing a periodic circular shift for a vector formally

I'm writing an algorithm in which the operation of circular shift to a given vector $x=[1 \ 0 \ 0\ ... 0]$ is needed on periodic basis i.e. every $\Delta t$ a circular shift will occur. How this ...
1
vote
1answer
60 views

Can I ask about copyright here? [closed]

I don't know it is okay if I post about copyright here. If there is any problem, please leave a comment. I will delete this post. Thank you. I am a graduate student studying engineering and ...
0
votes
1answer
33 views

Levi civita and kronecker delta properties?

I'm trying to grasp Levi-civita and Kronecker del notation to use when evaluating geophysical tensors, but I came across a few problems in the book I'm reading that have me stumped. 1) $\delta_{i\,j}...
1
vote
0answers
50 views

Falling Factorial Notation

If $(x)_{n}$ refers to $$x(x-1)\cdots(x-n+1)$$ then what does $(xy;x)_{n}$ refer to? Is it $$xy(xy-1)\ldots(xy-n+1)?$$ Thanks. The notation in question is used on page two of this paper.
0
votes
0answers
10 views

Understanding falling factorial notation with semicolons.

I have a question about the following notation: $$\prod_{j=1}^{k/d}(z^dt^{dj}q^d;q^d)_{\infty}$$ I'm confused as to what I do with $q$ and the other variables especially when the semicolon is ...
3
votes
1answer
167 views

What exactly is a p'-group?

In the context of finite group theory I understand that $p$-group is a group whose order is a power of $p$ ($p$ a prime number) but I am unclear on the exact definition of a $p'$-group. This ...
4
votes
2answers
72 views

Difference in use between $d$, $\partial$, $\operatorname d$, $\varDelta$ and $D$ for derivatives.

While reading different sources on implicit differentiation (and thereafter differentiation in general), I came across many different "d's" being used for (or similar to) the familiar $$\frac{dy}{dx}$$...
0
votes
1answer
19 views

Quick way to denote the different solutions of an equation.

Let's say that we conclude from $u^2+u-2=0$ that (over $\mathbb R$) $$u=- \frac{1}{2}\pm\sqrt{\frac{1}{4}+2}$$ What are the notational possibilities for this situation? Should I write $$u^2+u-2 =0$$ ...
0
votes
1answer
22 views

FUNCTIONS : Theoretical Doubt

I am currently learning calculus of one variables , and i have come across a symbol $$f(x,y).$$ Can anybody explain the meaning of this ? Thanks!
1
vote
3answers
46 views

Question regarding Sum Notation in the least squares formula [closed]

I'm attempting to figure out the difference between Σx^2 and (Σx)^2 in this least squares regression formula http://i.imgur.com/HwxnM28.jpg. Any ideas? I figure there must be a difference.
0
votes
2answers
125 views

Notation in commutative algebra

I am doing some exercises on commutative algebra and came along the following expressions, which were not elaborated on. Is someone familiar with them? The first is for $p$ a prime number $\mathbb{Z}...
0
votes
2answers
78 views

What is $\operatorname{syt}$?

I came across the following definition of the set on this web page But what is $\operatorname{syt}$?