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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0
votes
2answers
47 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
850 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 ...
2
votes
2answers
39 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 ...
1
vote
1answer
31 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
92 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
54 views

How do I 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. (The blue writing is what I have so far) I am just beginning to learn the whole concept of set ...
0
votes
0answers
56 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
175 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 $\cdot$) ...
21
votes
12answers
4k 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 ...
0
votes
3answers
44 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 ...
1
vote
3answers
25 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
56 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
15 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
28 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
1answer
36 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
1answer
124 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 ...
11
votes
6answers
820 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 ...
0
votes
1answer
25 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 ...
-3
votes
1answer
47 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 ...
2
votes
2answers
53 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
134 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 ...
2
votes
1answer
23 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
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 $X$ ...
3
votes
2answers
86 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 ...
0
votes
2answers
65 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. $$\Large\inf\limits_{\epsilon>0}\qquad \sup\limits_{\epsilon>0}$$
1
vote
4answers
88 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
35 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
41 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 ...
1
vote
1answer
20 views

Is this true: $\frac{f(x)}{1-c-o(1)}= \frac{f(x)}{1-c}(1-o(1))$

Let $f$ be a function, for example $f(x)=log(1+x)$ and let $c$ be some constant $>0$ (for simplicity, we may assume that it is different from 1). Is this true: $$\frac{f(x)}{1-c-o(1)}= ...
3
votes
1answer
32 views

Name for a nonlinear version of bilinear form

A map $b:X \times Y \to \mathbb{R}$ is called a bilinear form if $b$ is linear in both arguments. Is there a name for a form $b$ which is linear in only one argument and may be nonlinear in the ...
0
votes
2answers
61 views

A Series might be a number or a sequence – Is there a better notation?!

Take the expression $\sum_{k=1}^\infty a_k$. Sometimes this expressions refers to the sequence of partial sums $\left(\sum_{k=1}^n a_k\right)_{n\in\mathbb N}$ and sometimes to the limit of this ...
2
votes
1answer
42 views

Succinctly writing some code in vector/mathematical notation - how?

I’m trying to learn how to correctly represent some code I have in vector notation. Apologies if it’s a bit convoluted, keep in mind I’m trying to learn how to better communicate it (!) The code ...
1
vote
1answer
66 views

What groups are that? What does : mean?

What are the groups 2^6 : 3 . S_6 or 2^4 : A_8 ? Are they some subgroups of S_6 or A_8? I believe that 2 . A_n is the double cover of A_n, and "multiplying" with a number gives a covering group. But ...
0
votes
1answer
21 views

Representing a combinatorial sum with an equation

I am trying to represent a situation with an equation that is fairly conceptually simple, but I am not sure what is the proper way to represent it as a formal mathematical equation. I have a set of n ...
1
vote
1answer
31 views

E Scientific Exponential Notation

Gday, I have a question regarding scientific notation. Today I learnt that $a\operatorname{\mathbf{E}}b$ is the same as $a\cdot10^b$ and since myself and examiners (I'm in year 12) like neat working ...
2
votes
0answers
41 views

Proper names for different representations of the same formula

I would like to know what to call formulas that are all on one line and what to call the same formulas that are on multiple lines. One line example: P ÷ TVD ÷ 0.052 Multiple line example: ...
0
votes
0answers
31 views

In a chain of equalities/inequalities, is each line referring to the previous one?

In a chain of equalities/inequalities, or equalities/containments: $$\begin{align*} A & = B\\ & \subsetneq C\\ & = D \end{align*}$$ to which element is the last part refering to? In ...
2
votes
2answers
54 views

Notation - “' sign” in summation

I am studying logistic regression - and i saw the following symbol: c' (in sum symbol of the denominator). What does this mean? I thought it might be all other elements of the vector except the one ...
12
votes
5answers
272 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 ...
0
votes
0answers
21 views

Notation for “set of leaves of a tree” when leaves are “repeated”

Having this tree, I need to specify the number of leaves (6) and the set of leaves. Question: is this the correct notation to specify the set of leaves when there are some repeated or is there ...
-2
votes
3answers
63 views

Logic notation question

If I have some statements $P$ and $Q$ and I have the following logic formula: "$P$ or $Q$", does that mean only one of them is true or both of them can be true?
4
votes
3answers
178 views

Typesetting the mathematical expression “const.”

How do I write in the right way, that something is constant? I think, usually one does write = const., which should be an abbreviation for the Latin ...
1
vote
1answer
47 views

what is the handwritten notation for a random vector?

Not sure if this is a maths question but didn't know where to turn. I'm learning probability theory on my own using a textbook. It uses capital letters with subscripted number to denote random ...
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 ...
2
votes
3answers
97 views

Is there any difference between 'all real numbers' and '$(-\infty, \infty)$'

I've just thought about this. All the textbooks I've been looking at for pre-calc, the domains are always written as 'all real numbers', whereas my calculus textbooks would rather write them as ...
1
vote
2answers
336 views

Linear Algebra Done Right Example 1.7

In the book Linear Algebra Done Right I came across this example for the sum of vector spaces. Did he say the second $W + U$ is still equal to 1.7 because it doesn't matter whether you say $(x+y, ...
4
votes
1answer
69 views

What is the set $\mathbb{R}^{\nvDash}$ defined as?

The set $\mathbb{R}^{\nvDash}$ is used in my multivariable-calculus assignment, but I do not know what the superscript '$\nvDash$' means on the set of real numbers. The context in which it used is ...
7
votes
1answer
408 views

Integral notation for degree homomorphism on algebraic cycles

In Fulton's Intersection Theory, he develops the notation $\int_X$ for the degree homomorphism from $A^*(X)$ to $\mathbb{Z}$, and I was wondering if there was a reason for the notation. Is this in any ...
0
votes
1answer
21 views

Show that the conditional probability $P(\cdot | C)$ satisfies the axioms of probability, and is thus, indeed, a probability function.

The Statement of the Problem: For a fixed event $C$ such that $P(C) \gt 0$, show that the conditional probability $P(\cdot | C)$ satisfies the axioms of probability, and is thus, indeed, a ...