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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5
votes
4answers
815 views

Ambiguity of notation: sin(x)^2

Several people have told me that $\sin(x)^2 = \sin(x^2)$. However, on several computing platforms, such as the TI-84 and Wolfram|Alpha, $\sin(x)^2 = \sin^2(x)$. Can I safely conclude that the notation ...
-5
votes
1answer
33 views

function notations

can you explain how you get this answer? and explain the answer? the distance in feet $d(t)$ a dropped object falls in $t$ seconds is given by the function $d(t)=16t^2$. suppose you drop a ball from ...
3
votes
2answers
64 views

What does$ \|x\|_{\infty}$ mean?

In one of my homework-assignments in analysis, I have stumpled upon $ \|x\|_{\infty}$. I know x is a vector, but what does the infinity-symbol imply? The whole problem is actually this: $\| ...
0
votes
3answers
56 views

How to formulate “The $n$ smallest”

I know how to formulate the set of all $x$ with minimal distance to $y$ with $d(x,y)$ being the distance function: $\{x \mid \arg\min d(x,y)\}$ But how do I formulate the set of the $n$ closest $x$ ...
1
vote
3answers
62 views

Set notation and mappings question

Good evening. I have a question. Suppose I have two sets, $A=\{1,2,3,4\}$ and $B=\{5,6\}$. I want to write the notation for a function that takes each element in $A$ and assigns to it a value in $B$. ...
0
votes
2answers
44 views

Is there a symbol, or abbreviation for coefficient of x

Doing some binomial expansions with algebra where I need to equate different coefficients together but don't know what to write: [Coefficient of $x^3$] = _ $k=+1.5$
2
votes
2answers
28 views

Notation for writing down products of sets of combinations

I am writing a paper in which I come across an expression analagous to; $$ \prod_{k=0}^{n} (x-r_k) $$ I wanted a nice way of writing down how the $r_n$ relate to the coefficients in the resulting ...
3
votes
2answers
75 views

Weird integral symbol : $\mathrel{\int\!\!\!\!\!-}$

What does this integral sign mean ($\int$ with line going through the middle)? $$ \mathrel{\int\!\!\!\!\!\!-} $$ (It had something to do with the Beckenbach-Radó Theorem)
0
votes
1answer
34 views

Notation for a functor between comma categories

Suppose we have two categories $D$ and $S$, as well as two functors $K,L:D\to S$ and a natural transformation $\varphi:K\to L$. Given another category $C$ and a functor $Y:C\to S^D$, is there a nice ...
0
votes
3answers
37 views

Einstein Summation - does the following equality hold: $a_{ij} x_i y_j = a_{ij} y_i x_j$

Does equality hold when $x_i = y_i$ and $x_j=y_j,$ and $ i, j = 1, ..., n $.
0
votes
2answers
31 views

Einstein Summation: How do I show $a_{ij} (x_i + y_j) \not= a_{ij}x_i + a_{ij}y_j $?

Einstein Summation: How do I show $a_{ij} (x_i + y_j) \not= a_{ij}x_i + a_{ij}y_j $?
3
votes
0answers
40 views

Notation $(a,b)$ for $]a,b[$.

Is there any logic or justification for the notation $(a,b)$ to represent $]a,b[$? To me this notation is very ambiguous and confusing because it looks like a couple of numbers and not an interval. ...
2
votes
2answers
93 views

Some notation regarding “::”

I'm reading through some geometry proofs, and I can see something like $AB^2:PM\times EB::BC^2\ :CD\times PQ$ So I understand that $A:B$ is equivalent to $\frac{A}{B}$, but what does the $::$ mean?
2
votes
2answers
29 views

Notation on partial deriviatives

If I need to find $$\frac{\partial ^{2}g}{\partial u \partial v}$$ Then do I want to perform $$ \frac{\partial} { \partial v}\ \big( \frac{\partial g}{\partial u} \big) $$ or $$ \frac{\partial g} ...
0
votes
1answer
23 views

What's the domain of this function?

$$f(x,y,z) = \frac{1}{x-z}$$ I see two possible domains: $$D = \{x,y,z \in \mathbb{R}\mid x \neq z\}$$ or $$D = \{x,z \in \mathbb{R}\mid x \neq z\}$$ Is it equally valid to say the function has ...
2
votes
0answers
17 views

How to explain Borel sets and Stieltjes integral to beginner maths student?

The problem is that I know by the definition what Borel sets and Stieltjes integral are but I'm not good to explain in layman terms what they are. Is there easier answer that "write down the ...
0
votes
1answer
33 views

Notation symbol $x$ for functions

On the Modern Stream Ciphers slide #6, the following expression is used: $$ \{0,1\}^s × R ⟶ \{0,1\}^n$$ What does $×$ mean? I've seen $×$ used in a few other contexts, and I suspect it means ...
1
vote
1answer
15 views

wedge product with and without a second pair of vectors

I am starting to study wedge products, and am stuck on notation. The Bachman book on differential forms says $$ \omega \wedge \nu ( v_1, v_2 ) $$ "gives the area of the parallelogram spanned by ...
1
vote
0answers
44 views

Vector space basis change: is this “index-free” notation correct?

There are already quite a number of questions on basis change in a vector space. Nevertheless, to fully grasp the underlying idea I made up the following notation and I have some doubts on it (note: ...
0
votes
2answers
40 views

What does this mean exactly?

I am having trouble understanding the following notation, which I encountered studying combinatorics: \begin{equation} \sum_{A \subseteq \left[ n \right]} \prod_{a\in A}x_a \end{equation} where ...
0
votes
0answers
34 views

What's the right way to write big-O?

I always write $\mathcal{O}(n)$ (\mathcal{O}(n)). But I frequently see $O(n)$ (O(n)), probably because it's shorter and more ...
0
votes
0answers
16 views

Distributed to symbol for frequency distributions

How would you write down that some random variable $X$ is distributed to a frequency distribution. For the normal distribution e.g. I often see sth. like that: $X\sim \mathcal{N}(\mu,\sigma^2)$. Is ...
0
votes
0answers
31 views

Terms for particular equivalence relation and partition?

Let $T$ be a set of sets. Let $\equiv$ be an equivalence relation on $\bigcup T$ defined by the formula $$a\equiv b \Leftrightarrow \forall X\in T:(a\in X\Leftrightarrow b\in X).$$ Let $S$ be a ...
6
votes
1answer
108 views

What is the mathematical truth behind the Leibniz notation in differentiating twice or more?

So $f: \mathbb{R} \to \mathbb{R}$ is $n>1$ (or more) times differentiable. The notation of the first derivative makes perfect "sense" with regard to what's going on: $$\lim_{h \to 0} ...
1
vote
2answers
43 views

Partial derivative in two dimensions

I am struggling with section 3.3 of the following thesis https://smartech.gatech.edu/xmlui/bitstream/handle/1853/29610/grigo_alexander_200908_phd.pdf. Page 21 is fine, then the problems occur in ...
1
vote
2answers
40 views

What do $\Bbb N^*$ and $\Bbb Z(p^n)$ mean in this paper?

There is a theorem: in this paper: http://journals.cambridge.org/download.php?file=%2FJAZ%2FJAZ78_01%2FS1446788700015548a.pdf&code=2ffd5c5100675caf83c2e95bce65491e But there is no explanation ...
-1
votes
0answers
15 views

Expression : $\|f\|_{X\cap Y} = \| f\|_{X} + \|f\|_Y ?$

I want to know the meaning of the expression of the norm $$ \|f\|_{X\cap Y}, $$ where $X,Y$ are Banach spaces. Does this mean $$ \|f\|_{X\cap Y} = \| f\|_{X} + \|f\|_Y ? $$
0
votes
1answer
31 views

Are several equality operators okay to use in mathematics?

For example, is it okay to say: $x=y=z=1$ if $x,y,z$ all equal $1$?
0
votes
0answers
27 views

What is an open property?

From an academic paper, "the existence of elliptic or hyperbolic 2-periodic orbits is an open property". I have never seen the term "open property" used before, moreover the paper gives no ...
3
votes
2answers
58 views

Is it proper to write $\int \partial x$

For single variable function, you write $\int dx$ But for multivariable function, can you write $\int \partial x$?? I've never seen the latter, can someone explain why?
3
votes
2answers
45 views

Notation for “should be equal to”

Suppose I have some (possibly complicated) expression depending on one or more parameters, and a value which this expression should have for the solution I'm interested in. How do you write that, in ...
1
vote
1answer
27 views

Notation for repeated composition of functions

I have a repeated composition of functions ${T_n}(z) = {\tau _0} \circ {\tau _1} \circ {\tau _2} \circ \cdots \circ {\tau _n}(z)$ By analogy with $\sum\limits_{i = 1}^n {} ,\prod\limits_{i = 1}^n ...
0
votes
1answer
30 views

How are the essential upper and lower limits defined?

What means \begin{equation} \operatorname*{ess\,lim\,inf}_{x\to x^*} F(x) \end{equation} and \begin{equation} \operatorname*{ess\,lim\,sup}_{x\to x^*} F(x)? \end{equation} Sorry I also do not know in ...
0
votes
2answers
34 views

Are 400 and 45 correct for CD and VL in roman,respectively?

can VL and CD be roman numerals for 45 and 400 respectively? By the way,I already tried CCCC for 400.
0
votes
2answers
43 views

What is the notation for the number of elements in a set?

Let's say S = {1, 2, 3}. There are 3 elements in S. How do I express this in notation? I tried using google but I could not find what I was looking for.
1
vote
1answer
26 views

Notation for proof with Tensors

I'm working on proving For a second order tensor $\mathbf{A}$,$\mathbf{u}\cdot\mathbf{A}\cdot\mathbf{u}=0$ for all vectors $\mathbf{u}$ if and only if $\mathbf{A}$ is skew symmetric. Now, I ...
0
votes
0answers
11 views

Notation question about scalar products and bilinear forms

Quick notation question. Is it necessary to distinguish between a scalar product and say a bilinear form $A: V \times V^* \rightarrow \mathbb{R}^n$. Would it be recommended that say you define ...
0
votes
0answers
64 views
+50

Problem with notation in a thesis

I am struggling with section 3.3 of the following thesis https://smartech.gatech.edu/xmlui/bitstream/handle/1853/29610/grigo_alexander_200908_phd.pdf. Page 21 is fine, then the problems occur in ...
-2
votes
1answer
22 views

Is there a notation for incomplete quotient

If $n, m > 0$ are integers and $m \nmid n,$ how to denote the incomplete quotient this division?
1
vote
1answer
32 views

What does this function notation mean?

My text tells me that the general term of a sequence can be looked at like a function: $ f:\mathbb{N}\rightarrow \mathbb{R} $ What does that mean translated into common english?
1
vote
1answer
30 views

Question About Group Theory Notation

I am having trouble understanding what "Universal Cover of $\mathbb{Z} \times \mathbb{Z}$" mean exactly. Thanks
0
votes
1answer
54 views

How to find the domains of functions $f(x) = x-5$, $g(x) = \sqrt{x-5}$, and of their sim?

I've been studying on Study Plan Practice, on MyMathLab for my College Algebra class. We're going over the Algebra of Functions right now and several things don't make much sense. The question is: ...
4
votes
1answer
76 views

Origin and usage of $\therefore$ and $\because$

I've recently read a book which used the sign $\therefore$ (for "therefore"). It was more or less clear from the context what was meant, but I looked it up among the AMS LaTeX symbols just to be ...
0
votes
0answers
20 views

The space of alternating multilinear forms

I was just wondering if there is a standard (or even just usual) notation for the space of alternating $k$-linear forms on an $F$-vector space. I know that this space is naturally isomorphic to the ...
3
votes
2answers
61 views

In composition of two mappings, can the outer mapping access the arguments of the inner mapping?

In composition of two mappings, can the outer mapping access the arguments of the inner mapping? Here is an example to illustrate my question and my thought. E.g. $f: \cup_{n \in \mathbb N} \mathbb ...
3
votes
2answers
49 views

Complex number (square) root notation.

A mathematician told me that the notation $\sqrt{a+bi}$ isn't used, instead we use $w=z^2$ and substitute. Is this correct? If yes, is there any particular reason we don't want imaginary numbers under ...
1
vote
2answers
33 views

How does one interpret functions of topological spaces?

Let $f: X \to Y$ be a map of sets. We are given that $X$ is a topological space. We are to show that there is a topology on $Y$ making $f$ continuous, and moreover, determine if this topology is ...
0
votes
1answer
27 views

Type theoretic existential introduction and proof with subtypes

I'm working through a book[1], on type theory and categorial grammar (for linguistic applications). Sadly, I ran into problems pretty early on. I'd be very grateful if someone could Explain the ...
0
votes
0answers
34 views

Would the growth rate for base 2 and 10 logs be the same?

Since $\log_{2}(x) = \frac{\log_{10}(x)}{log_{10}(2)}$ and $\log_{10}(2)$ is just a constant, would their growth rate be the same?
1
vote
0answers
35 views

What does this operator $\odot$ mean

I read this about the second fundamental form in Wikipedia and I’ve no idea what does $\odot$ mean? Does anybody know? $$II=-dN\cdot dP=\omega^3_1\odot\omega^1+\omega^3_2\odot\omega^2$$