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.

learn more… | top users | synonyms

0
votes
1answer
32 views

Meaning of such that.

The use of this term confuses me, I've seen this like $A=\{(x,y) :x,y\in\Bbb R\ \text{and } P(x,y) \}$ and $B=\{(x,y)\in \Bbb R^2:P(x,y)\}$ for some predicate $P$. Is there any difference between ...
0
votes
2answers
41 views

Does anyone know when I would use this symbol ($\supseteqq$) and meaning?

Does anyone know what this symbol means? Where would one use it? Someone recently asked me but I do not know what it means. I have seen it with just one line underneath to denote subset. With an ...
0
votes
1answer
41 views

Some doubts about right ideals of a ring

I would like to know whether the following paragraph regarding right ideals and modules is correct. Any comment or help is welcome: A right ideal of $R$ is just a submodule of the right $R$-module ...
20
votes
12answers
3k views

What do sine, tan, cos actually mean?

I know that $\sin\theta=\frac{y}{r}$ and $\cos\theta=\frac{x}{r}$. My question is: is $\sin$ a function of $\theta$, as in $\sin (\theta$)? If yes, why is there no $\theta$ on the right hand side of ...
2
votes
2answers
99 views

Can someone show me why mathematicians use $d\mu$ instead of $dx$ for Lebesgue Integral over $u(x)$

I am an engineer and I learned my Lebesgue integral from an engineering text which dumbed down a lot of stuff, most prominently all Lebesgue integrals were introduced as $\int_\Omega u(x) dx$ instead ...
1
vote
2answers
21 views

How does the product of sets of complex numbers give a character?

I'm working through this "Introduction to Banach Algebras" and just after proposition 8.2 they say: If $A$ is a commutative Banach algebra, $a\in A$ and $\phi\in M(A)$, then $\phi(a)\in sp(a)$. ...
3
votes
2answers
40 views

Notation for union / intersection (in the same way $\pm$ stands for plus / minus) - is this a good idea?

Note: $F$ is a class of sets. I was solving a problem in Apostol's Calculus Volume 1. It is to show that $$B-\bigcup_{A\in F} A=\bigcap_{A\in F}(B-A)\qquad\text{ and }\qquad B-\bigcap_{A\in F} ...
13
votes
8answers
3k views

How would you count a base > 36 system?

When I am counting in a base greater than ten, I can use the letters of the alphabet. What do I use when I run out of those? What comes after z?: 0, 1, 2, 3... 9, a, b, c, d... x, y, z, (?) And ...
0
votes
1answer
17 views

Notation for polynomials and equating coefficients

I am reading a paper that defines $P_k(s|t)$ as a polynomial of degree $k$ in $s$ given $t$. Does this mean that each term is of the form $f_{k}(t)s^{k}$? (What does "given $t$" mean?) The paper ...
0
votes
0answers
36 views

Improper use of “for all”?

I am applying a method from Fernandes (2009), Classification trees for species identification of fish-school echotraces (ICES Journal of Marine Science, 66: 1073–1080), and I have a suspicion about a ...
0
votes
0answers
36 views

Formulation: Smooth vs. finitely often differentiable

I treat the cases $f\in C^\infty$ and $f\in C^k$ in different sections of my thesis. While I am happy with the title smooth functions for the first section, I am not so sure if finitely often ...
0
votes
2answers
58 views

How to denote an even number in mathematics? [on hold]

I need a sign for an even number (not $a\cdot 2$) in my formula. I tried to google it, but I saw only $2a$. Please tell me if there is a special sign?
1
vote
1answer
51 views

Inserting parentheses to produce different values

Without grouping symbols, the expression $$\verb/2 ⋅ 3 ^ 3 + 4/$$ has a value of $58$. Insert grouping symbols in the expression $\verb/2 ⋅ 3 ^ 3 + 4/$ to produce the indicated values. ...
0
votes
2answers
42 views

What's the meaning of the $R(f(x),g(x))$ in $\int R(f(x),g(x))?$

Usually when I'm reading about integration, there is a notation for integrals on some forms, for example: $$\int R(\sin(x),\cos(x)) \;dx$$ Obviously I've deduced that this represents functions that ...
9
votes
3answers
797 views

Must all Lebesgue integrable functions really be invertible?

I am studying Lebesgue integration after a course on Riemann integration, and the definition of measurable function is given as follows: $f:{\mathbb R}\rightarrow {\mathbb R}$ is measurable if the ...
12
votes
5answers
267 views

Is $1 : 7 = 1 / 8$ or is it $1/7$?

In a certain (non-mathematical) Stack Exchange, when I wrote $n : m = n / m$ where $n$ and $m$ are positive integers, one of the moderators said "No! $n : m$ is usually the notation for "$n$ parts in ...
2
votes
2answers
35 views

Is there accepted notation for the pushforward measure that doesn't mention $\mathbf{P}$?

Let $(\Omega,\mathcal{F},\mathbf{P})$ denote a probability space, $(S,\mathcal{M})$ denote a measurable space, and $X : (\Omega,\mathcal{F},\mathbf{P}) \rightarrow (S,\mathcal{M})$ denote a measurable ...
0
votes
0answers
18 views

Notation sumation confusion

I am reading paper about additive schwarz preconditioner, where following notation is used in order to obtain matrix C $$C_i = \sum_k (I^k B^k (P^k u_i)R^k)$$ . My question is, what's dimension of ...
13
votes
1answer
352 views

Looking for an approach to mathematical notation wherein the universe is divided into disjoint worlds.

Is there a rigorous approach to mathematical notation wherein the "universe" is divided into disjoint "worlds," and the meaning of notation is world-dependent? This would solve a few pesky problems. ...
1
vote
1answer
16 views

How to calculate a posterior probability with a given Gaussian Mixture Model?

I'm building a GMM-based classifier in speech processing and I'm using GMM as a probabilistic scoring mechanism (therefore I don't intrinsically care about the underlying mixture components). For ...
1
vote
1answer
66 views

Should I use set notation or list notation when writing out a basis of vectors?

I think in Sheldon Axler's Linear Algebra Done Right, he makes a comment about why the technically correct way is to write vectors in lists, such as $(v_1, ... v_n)$, while many books use set ...
2
votes
1answer
22 views

Determine the domain and range of the following relations using set builder notation.

I have been given the following relations to find the domain and range of using builder notation. I am just beginning to learn the whole concept of set builder notation, and I am running into a ...
0
votes
1answer
30 views

What is usually understood as DOMAIN and CODOMAIN of a Relation

Suppose I have a relation declaration as $R \subseteq A \times B$, such that $A=\{1,2,3,4\}$ and $B=\{10,20,30,40\}$. And suppose that the definition of this relation is $R=\{(1,20),(3,40)\}$ We ...
0
votes
3answers
43 views

What does P|a means?

In the proof for the existence of unlimited prime numbers, i saw the following let n be the number of prime numbers as P1,P2,P3,.......Pn let a = P1P2P3....Pn+1 a > Pn and a is not a prime number a ...
0
votes
0answers
55 views

What does $D^n$ refer to?

I'm not sure what object $D^n$ is in the following exercise: "Write down an explicit homeomorphism between $D^n/S^{n-1}$ and $S^n$." Thanks!
0
votes
3answers
159 views

Why do mathematicians use $\oplus$ instead of $+$?

What is the historical reason for using $\oplus$ instead of $+$ to denote operations that are generally thought of as addition? Similarly, why is $\otimes$ used instead of $\times$ (or just $\cdot$) ...
0
votes
1answer
117 views

Four mathematical notations for fraction $1/999$ and how to show/present they are equal

I need some help for notation. I need to present fractions in four different format and I'd like to get it right. I just take $1/999$ for example, but of course it could be any fraction with positive ...
6
votes
1answer
219 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
3answers
24 views

Set numbering representation

I want to represent proper element of sets. For example, $$ A=\{1,2,3,4\} $$ $$A(2)=2$$ But I wonder that this expression is right. Because I know that the set has not order. How can I do this??
4
votes
2answers
53 views

Notation of an infinite union

Is there any difference between: $$ \bigcup_{n =1}^\infty a_{n} \\ \bigcup_{n \in \mathbb{N}} a_{n} $$ From my understanding they both define an infinite union. Is this correct?
0
votes
1answer
14 views

Operator for comparing an n-tuple

Suppose you have to compare the following two finite ordered list of elements (tuples): $(\psi_{i}, R_{i}, A_{i}, \eta_{i})$ and $(\psi_{i}^{*}, R_{i}, A_{i}, \eta_{i})$ and for instance it turns out ...
1
vote
1answer
9 views

Name for map associated with simplicial complex

Given a simplicial complex $\Delta$, implied by the construction process there are associated maps sending euclidean standard simplices into the simplicial complex $\imath: \Delta^n \to \Delta$. What ...
1
vote
1answer
23 views

Interpretation - exponential loss formula

i stumbled upon the following formula while studying the concept of exponential loss. I try to understand this formula. I read it as follows: The function that minimizes the expected exponential loss ...
0
votes
0answers
61 views

When to use $\Delta$ or $\delta$ in formulas? [closed]

I use this symbol to denote the interval of created packets (variable, not constant) in the field of computer science. In formulas, which symbols should I use, $\Delta$ or $\delta$? Or other symbols? ...
10
votes
6answers
790 views

Notation of the differential operator

I see the differential operator both with upright and italic d in different books/articles. So I'm curious about $$ \int x^2 \, dx \quad \text{vs.} \quad \int x^2\, \mathrm{d}x,$$ and ...
-3
votes
1answer
45 views

What does it mean by a function $ f(x)=\exp(O(|x|^2)) $ for $|x|$ large?

Given $f(x)$ is continuous in $(-\infty,\infty)$ and $ f(x)=\exp(O(|x|^2)) $ for $|x|$ large. Now I have an I expression like ...
0
votes
1answer
24 views

Matrix,Linear algebra,polynomial,finite field,notation

In the book by Arora and Barak,Computational Complexity,on page 168,1st paragraph, there is a notation which I do not understand. They write For every $n \times n$ matrix $A$,and $i\in [n]$,we define ...
0
votes
2answers
63 views

Meaning of symbols like $\inf\limits_{\epsilon>0}$

I am very confused at the precise definition of the following symbols. A reference or explanation would be great. $$\Large\inf\limits_{\epsilon>0}\qquad \sup\limits_{\epsilon>0}$$
2
votes
2answers
47 views

What is the symbol to denote that two triangles are similar?

Does there exist a unique symbol to denote that two triangles are similar to each other without resorting to using the phrase "is similar"?
3
votes
4answers
113 views

How to notate all integers $\gt 1$ except products of $2, 3 , 5$?

What is a notation for all whole numbers greater than $6$ which are not a product of $2, 3 , 5$? $7$ would the first, then $11, 13, \ldots$ also $7\times 7$ or $11\times 11$ would be included. As a ...
0
votes
0answers
44 views

What does “Ad” mean in a proof? [closed]

My prof tend to use the word Ad. when writing proofs. Can someone please explain what "Ad." mean in a proof?
3
votes
2answers
82 views

Notation for Tautologies

I've been stuck for a while in this question and so far I don't understand the flaw of my reasoning please if you guys could help me out. See, this is my context. From the definition of argument we ...
2
votes
1answer
22 views

Notation - matrix calculatoin

i am self-studying linear algebra, and came across the following statement about singular value decomposition (X = $USV^T$). I am somewhat confused on how to interpret the (|) and (--) symbols. Should ...
0
votes
0answers
14 views

Using subscripted symbols in functions [migrated]

Using subscripts would significantly improve the structure of my code. Until now I used the following two lines to implement this. ...
0
votes
1answer
24 views

Notation for matrix that is partially unknown.

I have a matrix with some elements known and some unknown. I am using the notation $A(X)$ where $X$ are the unknown elements (not sure if relevant but I will be solving for the unknown part $X$ ...
1
vote
4answers
85 views

Difference between $f(x(t))$ and $f(t,x)$

Why is there a difference between the two differential equations: $\overset{.}{x}(t)=f(x(t))$ and $\overset{.}{x}(t)=f(t,x)$ ?
0
votes
0answers
34 views

Succinct notation for specifying that eigenvalues must have negative real part?

Is there a succinct way to denote that all eigenvalues of a matrix $A$ have negative real parts? If the eigenvalues were real, I could simply write this as $$-1 < A < +1$$ since we have the ...
2
votes
0answers
39 views

Riemannian Geometry notational tricks or alternatives

I am interested in learning tricks that people have developed to speed up / clean up calculations in Riemannian Geometry. I am hopeful about this question because there is often a lot of symmetry in ...
3
votes
1answer
44 views

What's Being Returned Here?

I'm working my way through this paper, and I'm having a bit of trouble understanding what it's telling me to do. Here's the specific excerpt that's tripping me up: A (finite) one-shot game is a tuple ...
6
votes
2answers
447 views

Index notation for tensors: is the spacing important?

While reading physics textbooks I always come across notation like: $$J_{\alpha}^{\quad\beta},\ \Gamma_{\alpha \beta}^{\quad \gamma}, K^\alpha_{\quad \beta}.$$ Notice the spacing in indices. I can't ...