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
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1answer
144 views

What group does $\mathbb{G}_m$ denote?

What group does $\mathbb{G}_m$ denote? I saw it used here.
1
vote
3answers
64 views

Set notation - problem

I have problem with a notation used in an information theory course. Let $N = \lbrace 1,\dots, n \rbrace$ for $n\in\mathbb{N}$. What does $2^{N}$ mean/denote?
20
votes
3answers
14k views

When log is written without a base, is the equation normally referring to log base 10 or natural log?

For example, this question presents the equation $$\omega(n) < \frac{\log n}{\log \log n} + 1.4573 \frac{\log n}{(\log \log n)^{2}},$$ but I'm not entirely sure if this is referring to log base $...
0
votes
0answers
111 views

Correct formal notation for sets containing sets

Informally, I have the set $A$ that contains sets of $B$ - for every element in $A$ there is a set $B$. If $n$ is the number of elements then $A_n \le B_n$. $B$ is not a subset of $A$ and $A$ is not ...
2
votes
1answer
103 views

How can we simplify $(a\pm b)(c\pm d)$?

Is there any way to simplify the expression: $$(a\pm b)(c\pm d)\quad ?$$ At first, I tried just distributing the values, and through FOIL I achieved: $$ac \pm ad \pm bc \pm bd$$ This works when the ...
2
votes
1answer
129 views

Standard notation for the the collection of all *minimal* elements of the set of all upper bounds?

Let $(P,\leq)$ denote a poset and suppose $X \subseteq P$. Then the minimum element of the set of all upper bounds of $X$ can be denoted $\operatorname{sup} X$, or $\bigvee X$. Is there a similar ...
2
votes
1answer
3k views

What are the components of a function?

Can someone tell me what is meant by The action of f on input x is written out in component form is $f(x) = (f_1 (x), \dots, f_m(x) )$ What is the component of a function? So if $m = 2$, and say $f(x)...
0
votes
1answer
109 views

Express this mathematically

Sorry it has been a while. I have three variables, $X, Y, Z$. Every $X$ has a different number of $Y$ and every $Y$ has a different number of $Z$ so how do I express the total number of $Z$? For ...
0
votes
2answers
69 views

How does the different use of | cause a difference between these equations?

I have the equations $$sMAPE = n^{-1}\sum_{i=1}^n\left|\frac{y_i - f_i}{(y_i+f_i)/2}\right|$$ and $$sMAPE = n^{-1}\sum_{i=1}^n \frac{|y_i-f_i|}{(|y_i|+|f_i|)/2}$$ but I don't understand what $|$ means....
1
vote
3answers
377 views

If $\Delta$ is the Laplace operator, what are $| \Delta |$ and $|\Delta +1|$

Assuming $\Delta : H^2(\mathbb{R}^n) \to L^2(\mathbb{R}^n)$ be the Laplace operator, then: What is the exact definition of $| \Delta |$? What is $|\Delta +1| $ also? This answer to another ...
1
vote
1answer
26 views

Question about interpreting asymptotic results

Suppose I have the following asymptotic: $$\{n \in \mathbb{Z} \cap [1, X] : f(n) \text{ is good}\} \sim cX$$ as $X \rightarrow \infty$ for some absolute constant $c$ where "good" means some desired ...
3
votes
1answer
540 views

When using sigma notation, why is “1” put as value?

First of all, I never really learned exactly what Sigma notation is, and although I understand the basics, there are some things that are confusing me. I'm trying to convert the following algorithm to ...
1
vote
2answers
299 views

what does the inverse membership symbol means?

I know that the symbol $$ \in $$ stands for membership, but what does the symbol $$ \ni $$ stand for? Because I know that in the set membership, one symbol stands for subset and the other ones for ...
1
vote
1answer
52 views

Notation issue regarding differential equations

I am given the following problem: Find a basis of solutions for the equation: $u^{iv} + 2u'' + 3u = 0$ The notation is an exact duplicaticate of what our professor used in his notes. Does anybody ...
0
votes
1answer
85 views

How to formally write a property of a specific coloring of a graph.

I have a doubt about how to write a statement in rigorous logical notation. I have a graph whose nodes are partitioned in some subsets $V_1,...,V_k$ and I have a function $Z:V\rightarrow \mathbb N$ ...
1
vote
3answers
159 views

Is ∞ considered defined?

$\infty$ (Infinity) is not a number, but infinity is considered to be defined, right? There are expressions in mathematics such as: $\frac x0,0^0, \frac\infty\infty,$ which are not defined because ...
1
vote
2answers
55 views

Is it legal to write $A_\mu = (a, b, c, d)$?

In special relativity (theoretical physics), one uses a lot of four-vectors. With regular vectors, I would say the following is okay: $$ \vec A = \begin{pmatrix} a \\ b \\ c \end{pmatrix} \qquad A_1 = ...
4
votes
3answers
257 views

Difference between $\mathop{\text{div}} \vec x$ and $\vec \nabla \cdot \vec x$?

I often see the following notations for the divergence in theoretical Physics: $$ \mathop{\text{div}} \vec x \qquad \left\langle \vec \nabla, \vec x \right\rangle \qquad \vec \nabla \cdot \vec x $$ ...
9
votes
5answers
8k views

Negating A Mathematical Statement

Regard this statement $ x \ge 0$. According to my teacher, by negating this statement, it will become $ x < 0$. Why is this so; why does the $\ge$ morph into $<$, and not into $\le$?
2
votes
3answers
427 views

Matrix notation differences

So I'm in an elementary matrix and linear algebra class and during lecture my professor uses this notation for a matrix: $$\left(\begin{matrix}a & b\\c&d\end{matrix}\right)$$ but in my ...
5
votes
1answer
660 views

About Math notation: the set of the $n^{th}$ natural numbers

It's true that I can write the set of the $n^{th}$ natural number as $[ n] $? For example, $[10]= \{1,2,3,4,5,6,7,8,9,10 \}$. And in which math contex this is use?
4
votes
0answers
48 views

Symbol for function composition [duplicate]

Possible Duplicate: History of $f \circ g$ Choice of symbols can be an indicator of intellectual allegiance. Consider how, back in the day (and before LaTeX regularised things so much!), the ...
1
vote
1answer
5k views

What does upside down “v” ($\wedge$) mean in this equation?

I have a simple question, but it is hard to google it. I have this equation here: $$y(t, x) = \sum_{i=1}^{d}(|x_i| \wedge t)^{2} $$ Here $x$ is a size $d$ signal and $t$ is just a scalar. I am not ...
2
votes
1answer
170 views

Product rule smart notation

Imagine we have a product of functions $f_1\cdots f_m$. We know a rule to compute the derivative. On the other hand, we also have a rule or formula to compute the $n$-th derivative of $fg$ but my ...
1
vote
4answers
10k views

Can someone identify this math font?

I'm doing my master's thesis and trying to find a good font for math symbols and formulas... I would like to use the following font style: Can anyone recognize this? How to get X the way it is in ...
3
votes
1answer
471 views

Integration Antiderivative vertical bar [duplicate]

Possible Duplicate: What is the name of the vertical bar? When taking a definite integral, the first step is finding the anti-derivative. Once you have gone through all the steps to complete ...
3
votes
2answers
5k views

What does $C[0,1]$ mean?

In the context of real analysis, I have found this question: For each $$f \in C[0,1] $$ there is a series of even polynomials , which converge uniformly on $[0,1]$ to f. What is $C[0,1]$ ? Is it ...
1
vote
2answers
189 views

Understanding the notation $N/(N_{r}\bigcup N_{v})$ in graph theory

Currently I'm dealing with a graph problem but I don't understand one specific notation. What does the following mean: $$N/(N_{r}\bigcup N_{v})$$ $N$, $N_{r}$, $N_{v}$ are sets of nodes. $N$ is the ...
0
votes
1answer
63 views

Given a function space with a norm , what is the meaning of writing $||.||$ when the used norm is $||.||_\infty$

Example 1 Given $$C_{0}(\mathbb{R}^{n})=\{f\in C(\mathbb{R^n} \ | \ \ \exists R \ge 0 \ \text{such that } f(x)=0 \ \text{for} \ ||x||\ge R \}$$ and $$||f(x)||_{\infty} = \max_{x\in R^n}|f(x)| $$ ...
3
votes
1answer
59 views

When does a solution to $a^x\equiv b\pmod m$ exist, and how is the smallest solution denoted?

Given fixed integers $a,b,m$ such that $\gcd(a,m)=1$, how do I know if there exists an integer $x$ such that $a^x\equiv b\text{ mod } m$, also if a solution does exist, what is the typical notation ...
2
votes
1answer
80 views

Precedence of $\times$ and $\cup$.

In topology, there is a very strong need of describing subsets of some product $X\times Y$ by means of unions and products. For example, this is a very convenient way of describing subsets of the ...
1
vote
1answer
59 views

Question on Notation

In Loring Tu's text An Introduction to Manifolds, Exercise $2.4$ asks us to show that $D_1\circ D_2$ need not be a derivation while $D_1\circ D_2-D_2\circ D_1$ is always a derivation. My question is ...
2
votes
1answer
77 views

Set notation for infinite subsets.

In set notation, how can one express an infinite set of subsets where each subset has exactly two elements $\{an-1, an+1\}$ where $a$ is a constant and $n\ge1$ and the $n$ value for each subset is one ...
0
votes
2answers
314 views

Can an uncountable union be written as a limit?

Say there is a nested, uncountable sequence of subsets $\{A_i\}_{i \in I}$ of a given set $A$. Can the union of those subsets $\bigcup_{i \in I} {A_i}$ be written as the limit as $i \rightarrow \infty$...
3
votes
2answers
349 views

Generalization of a product measure

Let $(X,\mathfrak B(X))$ and $(Y,\mathfrak B(Y))$ be measurable spaces and further let $\mu$ be a measure on $\mathfrak B(X)$ and let $K$ be a kernel, i.e. for any $x\in X$ we have $K_x$ is a measure ...
1
vote
3answers
846 views

What does $\inf\int$ mean?

What does notation $$\inf\int f(x) \,\mathrm{d}x$$ stand for? I noticed it in a question on this site. Any keywords, links or stories about this or similar notations will be appreciated :) Sorry ...
1
vote
4answers
682 views

Notation using “such that”

Is it correct to say "$a_t=\alpha:\alpha\in A_{\sigma}$" with the following meaning: $a_t$ is equal to $\alpha$ and $\alpha$ belongs to set $A_\sigma$.
3
votes
5answers
1k views

How to represent each natural number?

Assume we get the set of natural numbers $\mathbb{N}$ from any model of the Peano axioms. We're given the symbols: $0,1,2,3,4,5,6,7,8,9$, or rather, we're given $0$ and we choose to use the symbols $...
10
votes
4answers
458 views

From a mathematician's point of view, what is the purpose of '$dx$' in $\int f(x)\ dx$?

I've done a bit of searching and found a fairly well written explanation, but at the end, the author noted that this explanation seems to work fine for a physicist's purposes - but a mathematician ...
0
votes
1answer
257 views

I don't understand this 'semi colon' notion in regards to PDE solutions

In solving first order PDE's with solution $u(x,y)$, when constructing a graph of $u$ as a union of initial curves $C_s$ emanating from the initial curve $\Gamma$. My lecture notes say, for each $s \...
2
votes
1answer
84 views

Sigma with Or Underneath

This is the notation found in Spivak's Calculus: Chapter 23: Infinite Series Theorem 9 What does $\displaystyle \sum_{i \text{ or }j > L} |a_i||b_j| $ mean?
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vote
3answers
65 views

Conventional name for cartesian product of n spaces

I'd like to represent mathematically tuples having $n$ elements (where $n$ is not a constant) as vectorial space. $$\mathbb{F} \times \underbrace{\mathbb{F} \times \mathbb{F} \cdots \mathbb{F}}_{\...
17
votes
5answers
1k views

Notation Question: What does $\vdash$ mean in logic?

In a "math structures" class at the community college I'm attending (uses the book Discrete Math by Epp, and is basically a discrete math "light" edition), we've been covering some basic logic. I've ...
12
votes
2answers
2k views

Is there a collection of alternative mathematical notation? (Semi-soft Question)

I'm interested in alternative systems of notation for mathematics. I've often heard how mathematical notation is illogical, inconsistent, filled with grandfather clauses that serve no purpose, and ...
6
votes
1answer
83 views

what does $\alpha(N)$ mean in this article?

I'm trying to understand this article: http://imgur.com/a/HfJoY but I'm unsure what $\alpha(N)$ means in this context? Is it the algebraic multiplicity, that's pretty much the only $\alpha$ I have ...
2
votes
3answers
132 views

Is $f = x^2$ or only $f(x) = x^2$ correct?

I currently study special relativity and some authors write stuff like: $$ r^\mu = \left(ct, \vec x\right) $$ This is awful since $\mathsf r$ is a vector, and $r^\mu$ ist just a single component of ...
5
votes
0answers
690 views

Is there a name or definition for this popular notation?

I'm sorry if this is a silly question. I've done quite a bit of searching and have not found any definition or name for this symbol/usage, despite immense popularity and convenience. The sources I've ...
1
vote
1answer
217 views

“Discovering” the indefinite integral's notation

I'm currently reading Keisler's Elementary Calculus -- An Infinitesimal Approach, which develops the main results usually thought in undergrad calculus using Robinson's hyperreal numbers (instead of ...
3
votes
1answer
127 views

Strange Notation: $f(x,y) = \ln (x + \text {s} \space \overline {x^2 + y^2} $)

I am currently working on my first calculus assignment of the quarter, and immediately ran into a strange notation which neither my teacher discussed nor is it mentioned in any previous parts of the ...
2
votes
1answer
1k views

Notation — What does “Gauss” brackets mean

In a paper I'm trying to understand, from a time series $x(1),x(2),\ldots,x(n)$ a new set of time series is created: $$x^m_k=x(m),x(m+k),x(m+2k),...,x\left(m+\left[ \frac{n-m}{k}\right]k\right) \:\:\:...