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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26 views

Why do we use r to represent vector-valued functions?

Many standard calculus texts use r as the default function name when defining vector-valued functions, e.g., $\textbf{r}(t)=\langle x(t),y(t),z(t)\rangle$. For scalar-valued functions, we default to ...
0
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2answers
37 views

How to express open interval in roster notation?

For example, an open interval such as $(a, b)$ means $a$ and $b$ are not included. If I have $[a, b)$ I know $a$ is included but $b$ is not. I need to express this in roster notation, which is a list ...
3
votes
1answer
43 views

Why do we write $df/dx$ instead of $df/dx(x)$?

I was just thinking about how, i.e., if $f\colon\mathbb R\to\mathbb R$ is defined by $f(x) = x^2$, then it's customary to write $$ \frac{df}{dx} = 2x. $$ But since the derivative is itself a function ...
0
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2answers
22 views

null empty set has 2 subsets?

The question in the book was: How many subsets does $\{\emptyset\}$ have? a) 0, b) 1, c) 2, d) 3. The answer was c. How can an empty set have 2 subsets?
2
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1answer
79 views

Question about notion $d\mu = fdv$ in Real Analysis of Folland

I'm reading the book Real Analysis of Folland, chapter 3 about signed measure, and there's some notion that confused me. In this book, he defines that $dv = fd\mu$ if $v(E) = \int_E{fd\mu}$, and ...
3
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0answers
15 views

Characterize in terms of fibre

I am not familiar with the notion "characterize" in the following context. Does this mean to redefine or?.... Any help would be appreciated. Thank you. For a function $f:X\to Y$, and y an element of ...
0
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0answers
12 views

Tensors, indices and matrix notation - is there a common convention?

For a tensor named T with two indices, there are four possibilities: $T_{ij}$ , $T_i^{\ j}$, $T^i{\ _j}$ and $T^{ij}$. Is there a common convention as to how these tensors would be represented as ...
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0answers
11 views

Notation for the ith row and column of a matrix

When noting the $i^{th}$ scalar of a vector $\mathbf{x}$ one usually does it as $x_i$, since it is a scalar When doing this for matrices that are being denoted in bold, let's say $\mathbf{A}$, how ...
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1answer
30 views

Is there a notation for “Bounded Kleene star”?

I understand that Kleene star is defined as: $$V^*=\bigcup_{i = 0 }^\infty V_i = V_0 \cup V_1 \cup V_2 \cup V_3 \cup \ldots.$$ (given $V$ is a formal language, $V_0 = \{\varepsilon\}$, and $V_k$ is ...
0
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2answers
52 views

Is there a mathematical symbol for “and”? [duplicate]

I have a statement as such: $\mathbb{Z_+} \triangle E = \{ x \in \mathbb{Z_+} : x \space \% \space 2 \neq 0 \space and \space x \in \mathbb{Z} : x < 0 \space and \space x \space \% \space 2 = 0 ...
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1answer
44 views

what is the difference between $f(x;y)$ and $f(x|y)$? [on hold]

I was wondering whether there is a rigorous difference between $f(x;y)$ and $f(x|y)$, or if both mean the same.
2
votes
2answers
28 views

Integration of a scalar function with respect to a vector

I have a scalar function that takes $n$ arguments, $f(x_1, x_2,x_n) = f(\mathbf{x})$, and two vectors also with $n$ elements, $\mathbf{z} = (z_1, z_2\cdots,, z_n)$, and $\Delta\mathbf{z} = (\Delta ...
0
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0answers
12 views

How to deduce this fact from the existence of factorized regular conditional probabilities and disintegration of probability measures?

In the lecture we had a theorem about the disintegration of probability measures in the following formulation: Theorem: Given two standard Borel spaces $(S_i,\mathscr S_i),i=1,2$ let $(S,\mathscr ...
0
votes
1answer
22 views

Identity element of word addition

I realize this is rather an arbitrary question, but it's important to me, that I understand it and get it right, and I'm not finding the answer anywhere else. I'm working through "A Book of Abstract ...
0
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1answer
38 views

What do double parallel lines on vectors mean? [closed]

What do the lines mean in the notation $\|u\|$ where $u$ is a vector?
3
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2answers
83 views

Does “$\dots$” imply countability?

I am given an arbitrary set $S$. If I say the following: "Suppose that the elements of $S$ are labeled $x_1,x_2,x_3,\dots,$" am I notationally implying that the number of elements in ...
0
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1answer
37 views

For two functions $f,g$ from $X$ to $\mathbb R$ how should I interpret $f \wedge g$?

For two functions $f,g$ from $X$ to $\mathbb R$ how should I interpret $f \wedge g$? I came to such a term while reading the axioms of fuzzy topology.
2
votes
2answers
39 views

Index notation interpretation for matrices

I want to understand the how to interpret the matrices which are represented by index notation. Here is my matrix $𝜎_{𝑖𝑗}+𝜎_{𝑖𝑘}𝑤_{𝑘𝑗}−𝑤_{𝑖𝑘} 𝜎_{𝑘𝑗}$ All the matrices in the equation ...
2
votes
2answers
70 views

Notation: $\varphi$ and $\phi$

Is it bad style to use $\phi$ and $\varphi$ in the same paper (for different things, of course)? I'd like to use $\phi$ for a function and $\varphi$ for a particular function value.
0
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1answer
18 views

Notation: Codomain of a probability density function

I need some help with the correct notation for the codomain of a probability density function. Consider the following problem. Let $$ F : V \to (0,1), \, x \mapsto \int\limits_{\inf V}^{x} f(t) \, ...
1
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0answers
32 views

Abuse of notation ? $(A\mid M_{n\times p})$ to denote a set of matrices…

Let $A\in M_{n\times m}$. Would it be considered an abuse of notation to write $$\left(A\mid M_{n\times p}\right)\subseteq M_{n\times (m+p)},\tag{1}$$ where $\mid$ denotes matrix augmentation ? By ...
0
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0answers
31 views

Notation for does not converge?

Is it okay to write that $x_n \not \rightarrow x$ to indicate that the sequence $(x_n)$ does not converge to $x$? Or should this notation be discouraged?
0
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1answer
12 views

Definition of a function whose codomain is set of probability measure over cartesian product with dependency between sets in the product

I am thinking about the following function: $$ p : A \to \Delta \big( F(x, y(t) ) \times T \big) ,$$ where $t \in T$ denotes continuous time, and $\Delta (X)$ denotes the set of all probability ...
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0answers
14 views

Notation for compactly supported functions

I teach a course in real analysis and applications to partial differential equations in which I spend some weeks talking about Sobolev spaces. I have always used the symbol $C_0^\infty(\Omega)$ to ...
0
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0answers
21 views

$\mapsto$ functional notation and probability distributions

I have a question concerning mathematical writing. If we have a function $f: X \to Y$, we can write it \begin{align} f: \ & X \to Y \\ & x \mapsto f(x), \end{align} where $f(x)$ can ...
0
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1answer
36 views

Is this “truncating” matrix function well known?

I'm working with a kind of "truncating" matrix function $\tau_r:M_{n\times n}\to M_{n\times r}$, where $r\leq n$, defined by $\tau_r(A)=B$, where $b_{ij}=a_{ij}$ for $j\leq r$. Is this a well known ...
0
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0answers
11 views

Notation of multivariate functions and parameters

I have some trouble having a standard notation for multivariate functions and parameters. I'm dealing with the derivation of the Master equation. Some authors write the probability density function ...
1
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2answers
37 views

Notation Question $n$ $ < <$ $m$

What does the following notation mean: $n< < m$ , where $n$ and $m$ are numbers?
1
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1answer
77 views

What does $\propto$ mean in the following equation?

I have seen this question but I still have problem with the meaning of this symbol. From this book: The time-dependent angle may be defined from the components of the wave vector in order to ...
0
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0answers
23 views

If $C^H$ is the conjugate transpose of $C$ then what is meant by $C^{-H}$?

If $C^H$ is the conjugate transpose of $C$, i.e., $C^H=\overline{C^T}$ then what is meant by $C^{-H}$?. Assume that $C$ is a square matrix. I can't find a definition for this anywhere?. Can anybody ...
1
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0answers
14 views

What is the mapping of Z-transform?

Recall that given a series $x(k)$, the Z-transform $\mathcal{Z}$ is defined as: $$\mathcal Z(x(k)) = \sum_{k =0}^{\infty} x(k) z^{-k}$$ where $x(k)$ satisfies $|x(k)| \leq M\rho^k$, $M, \rho$ real ...
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 ...
2
votes
1answer
48 views

A Type of Union I do not understand

I am not able to read this following union. Could someone please help me to understand it? $\bigcup\limits_ {k \geq n}${${x \in X: |f_k(x)-f(x)|\geq q} $ }
1
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3answers
66 views

Acronyms and words as variables and in mathematical notation

I am unsure if this question is warranted. Often mathematical symbols and objects are represented by a single character, e.g. variables are most often single characters like $x$, and often to ...
1
vote
1answer
69 views

Use of exclamation point

I'm quite puzzled by the use of an exclamation point in this paper. The authors introduce the following linear constraints to a quadratic program: $ \sum_k a^{(l)}_k b_j (\mathbf{x}_k) = r_j^{(l)} $ ...
5
votes
3answers
5k views

Notations involving squiggly lines over horizontal lines

Is there a symbol for "homeomorphic to"? I looked on Wikipedia, but it doesn't seem to mention one? Also, for isomorphism, is the symbol a squiggly line over an equals sign? What is the symbol with a ...
0
votes
4answers
390 views

Can exact numbers be written in scientific notation

I've learned in scientific notation you have last number as probable. Meaning it could be anything... so in $3$ significant digit number such as follows $3.34$ has '$4$' which could be anything. ...
23
votes
7answers
29k views

How does one denote the set of all positive real numbers?

What is the "standard" way to denote all positive (or non-negative) real numbers? I'd think $$ \mathbb R^+ $$ but I believe that that is usually used to denote "all real numbers including infinity". ...
0
votes
0answers
32 views

Notation/definition problem for commutative binary operation

I'm trying to describe/define the commutative binary operation on a three-element set which when the operands are the same, gives the same element and when they are different gives the element which ...
0
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1answer
31 views

What is the “T” set?

I am reading an engineering paper and it references a "T" set in the same way that one would reference the set of complex numbers with $\mathbb{C}$, or the set of real numbers with $\mathbb{R}$". What ...
2
votes
2answers
105 views

What does $e$ mean in this expression?

I've seen this formula $$1RM = \frac{100 \cdot w}{48.8 + 53.8 \cdot e^{-0.075 \cdot r}}$$ but I don't know what does the $e$ means. The $w$ stands for weight. The $r$ for repetitions but I think ...
12
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4answers
1k views

Why not write $\sqrt{3}2$?

Is it just for aesthetic purposes, or is there a deeper reason why we write $2\sqrt{3}$ and not $\sqrt{3}2$?
2
votes
2answers
43 views

Notation of expectation and random variables

I'm trying to understand the notation used at p18 of The Elements of Statistical Learning. I suspect errors in notation. What do the authors mean and, if any notational errors, what would be the ...
0
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0answers
25 views

Mathematical notation for first and second maximum

I have a vector $f_x = f_{x_1}, f_{x_2},\cdots, f_{x_n} $ having the frequencies for bin $x = x_1,x_2,\cdots,x_n$. Now I want to address two bins having highest frequencies. I address the highest ...
14
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1answer
6k views

What's more common? Re / Im or Fraktur-R / Fraktur-I for real / imaginary part?

Title says it all. What's more common? Is there one to prefere (maybe due to some norm)? This: $\operatorname{\mathfrak{R}} z, \operatorname{\mathfrak{I}} z$ or that: $\operatorname{Re}z, ...
0
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1answer
23 views

Meaning of $^sB$, s an element, B a subgroup

Let $G = SL_2(\mathbb{F}_q)$, $B$ the subgroup of all upper triangular matrices, $s = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}$. What does $^sB$ mean? I read it from page 4 of C. ...
0
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1answer
15 views

Notation in Kronecker-Weber theorem

Sorry for the dumb question but I don't understand a notation. I'm reading the notes of Culler about the Kronecker Weber theorem (see here) and at page 3 we have a finite extension of number field ...
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0answers
24 views

Function notation of y [closed]

could you expain what the following function means in layman terms: y(x ; W) Cheers
2
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2answers
89 views

Set theory formula

I picked up a copy of Jech's Set Theory at my school library and I'm reading through it and taking notes. Right at the beginning, though, he mentions something called a 'formula'. Here's the quote: ...
3
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0answers
42 views

Standard notation for indices in group theory?

I've seen three notations for indices in group theory, namely $(G:H)$, $[G:H]$ and $|G:H|$. Is there any of these notations that is standard?