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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1answer
17 views

Translating between Set Theoretic and Interval notation

Given an open interval (or closed it makes little difference to the question) on the Real Field $(a, b)$, where $a,b$ are real numbers, and an arbitrary predicate $P(x)$, which is true for all ...
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3answers
38 views

What is $\frac{\mathrm{d}f(x,x)}{\mathrm{d}x}$

Related: Is $\frac{\partial}{\partial x} f(x,y(x))$ ambiguous? If $f(x,x) = x^2$, is it correct to say that $\dfrac{\mathrm{d}f(x,x)}{\mathrm{d}x} = 4x$? I know that $\dfrac{\mathrm{d}f(x,y)}{\...
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2answers
48 views

Is $\frac{\partial}{\partial x} f(x,y(x))$ ambiguous?

Given $f(x,y) = xy^2$, then clearly $\dfrac{\partial}{\partial x} f(x,y) = y^2$ However, if we set $y^2 = x$, then we get $f(x,y(x)) = x\times x$ By definition $\displaystyle \dfrac{\partial}{\...
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0answers
13 views

Is there a notation for space of all random variables?

Is there a common notation for the set of all random variables of the form $X:\Omega\to\mathbb K$? I'd like to use it as a shorthand when i need to specify, e.g., that some function actually maps ...
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0answers
18 views

Total derivative using $(x,y)\mapsto f(x,y)$ notation

Related to What is the difference between $\frac{\mathrm{d}}{\mathrm{d}x}$ and $\frac{\partial}{\partial x}$? Is it possible to make sense of $(x,y)\mapsto f(x,y)$ when taking the total derivative ...
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5answers
288 views

What is the difference between $\frac{\mathrm{d}}{\mathrm{d}x}$ and $\frac{\partial}{\partial x}$?

Is there not any difference between $\frac{\mathrm{d}}{\mathrm{d}x}$ and $\frac{\partial}{\partial x}$ as long as your function has one variable? $f(x) = x^3\implies \left\{\begin{align}&\dfrac{\...
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0answers
24 views

Understanding Matrix and Vector Notation

I am trying to understand the Matrix and Vector Notations on page 2 here: (the page is also pasted below, to make it easier to explain the problem). Problem: For equation (2), I think it should be $\...
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1answer
32 views

What is the proper notation for vectors that uses only integers? $\Bbb Z^n$?

How do I denote vectors, similar to $\Bbb R^n$ but with integers instead of the reals? Just $\Bbb Z^n$? Would vectors of exclusively positive integers be $\Bbb Z^{{+n}}$ then? (since $\Bbb Z^+$)
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0answers
87 views

Math notation to show two numbers in a range that added together get the max of the range [closed]

I am completely new to math notations, it's been about 30 years since high school, and I am writing a research paper (completely on my own, not for a degree). I basically want to show that two real ...
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1answer
49 views

The $n=0$ term in the power series $\sum_{n=0}^\infty a_n x^n$

This question is about the definition and notation for the $0^{th}$ term of the power series: $$\sum_{n=0}^{\infty} a_n x^n$$ There are two possible ways to interpret this term: 1) It is just a ...
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0answers
18 views

Understanding the Degree Distribution of Watts-Strogatz Model

In the Watts-Strogatz model, the degree distribution is given as: $P(k) = \sum\limits_{n=0}^{\min\left\{k-\frac{K}{2},\frac{K}{2}\right\}} C^n_{\frac{K}{2}} (1-p)^{n}p^{\frac{K}{2}-n} \frac{\left(\...
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1answer
46 views

Clarification of the topology lemma “Any continuous and open injection of the open disk extends over the circle”

My elementary topology 1 class last semester used the book "Topology: Point-Set and Geometric" by Paul Schick, and covered through the end of chapter 8. I am working through the rest of the book on ...
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1answer
23 views

What does $ K^{\alpha}$ mean?

This is in context of a statement in galois theory: If $F \subseteq K \subseteq L$ and $K$ is splitting over $F$, then $K^{\alpha}=K$ for each $\alpha \in Aut(L)$.
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3answers
144 views

How to suppress the words “if” and “then”?

My math teacher keeps making us write mathematical sentences with "regular" words. I always ask her if it is possible to supress them but she always says "no" or she starts laughing. Take for example ...
2
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1answer
27 views

Symbol to show that implication is one-sided

Sometimes A => B is true, but B => A is not true and this fact is important and not obvious. Is there a short symbol to write it in order not to write "A => B and not B => A"
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1answer
49 views

The best notation for this identity involving pentagonal numbers $\omega(n)$ and the $3x+1$ map

Let the $3x+1$ map $$ f(n) = \begin{cases} 3n+1 & \text {if $n$ is odd} \\ \frac{n}{2} & \text {if $n$ is even} \end{cases} .$$ Now we read the Wikipedia's page for the Collatz ...
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0answers
35 views

What does $F[x]$ mean?

Lemma: $F$ is a field only if $F\left [ x \right ]$ is a Principal Ideal Domain. This is a theorem from Ring; divisibility of integral domain. What does $F\left [ x \right ]$ mean?
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2answers
114 views

What does a norm of a polynomial space mean?

When talking about polynomial vector space, the following example was provided. A polynomial of degree $n$ in two variables is $$p(X)=\sum_{0\leq k+j \leq n} a_{j,k}x_1^jx_2^k$$ where $k+j=n$ and ...
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0answers
35 views

Is there a math notation/ term for “$f(x_n) \to 0$ iff $g(x_n) \to 0$”?

I have two real-valued functions $f,g$ defined over the $N$-dimension Real Euclidean space: $$ f,g: \mathbb{R}^N\to\mathbb{R}. $$ They satisfy this property: $$ \forall x_n \in \mathbb{R}^N: f(x_n)\to ...
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2answers
54 views

Is there a name for the logical scenario where A does not necessarily imply B, but B implies A?

A real life example of this is the 'Active' status on Facebook Messenger. (For those interested see this article here, and some Quora answers here for details.) When you are actively using Facebook ...
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2answers
24 views

Describing sets from running indices

For a paper I have a set of particles which I usually reference by $ p $. The paper is physics related, so I haven't made any real formal definition of the set, and always just relate to them as "the ...
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0answers
28 views

Notation: should Markov chains steps be noted by uppercase or lowercase letters?

I'm reading the chapter about perfect sampling of the "Monte Carlo Statistical Methods" by Robert and Casella, 2004. I've got an issue about notation, when they talk about random mappings, they say $$...
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3answers
26 views

Ideal generated by an element

Let $R\left [ x \right ]$ denote the set of all polynomials with real coefficients and let A denote the subset of all polynomials with constant term 0. Then A is an ideal of $R\left [ x \right ...
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1answer
32 views

Notation: rotation matrix with a condition

I'm building a space simulation & am using this resource for converting Keplerian Orbit Elements to Cartesian Co-ordinates. The notation for step 6 has me slightly confused: Is the top part ...
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1answer
35 views

is Cov(X) and Var(X) same? when X is random vector

i'm studying with hogg. introduction to mathematical statistics. and i learned about random vector but i wonder whether Cov(X) and Var(X) is same or not. as intuitive thinking , if X is a random ...
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1answer
22 views

Index notation in vector functions

Suppose I have a vector $\vec u$ that depends on a vector $\vec x$ and a scalar t, so each component of $\vec u$ depend on all components of $\vec x$. How can I show this relationship with index ...
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2answers
26 views

Notation for the subset of a set without the last elements

I have an ordered set $S=\{23,45,32,56\}$ I would like to make an ordered subset $M=\{23,45,32\}$ which contains the elements of $S$ except the last one. In this case. How do I represent this formally ...
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1answer
18 views

ordered set notation in functions

Do please forgive me, if this question is a duplicate. How does one correctly notate a function $f$, which takes a ordered subset $S$ from the field $\mathbb{K}$ and returns an other (ordered) subset ...
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1answer
14 views

Asymptotic notation basics

Say that we have the function $$ f(n)=kn, \, k>0 $$ does that imply the following? $$f(n) \in O(n), \, f(n) \in \Theta(n) \text{ and } f(n) \in \Omega(n)$$ I'm fairly new to these notations and am ...
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1answer
55 views

What does $(0:x)$ mean?

The following excerpt is from Eisenbud's "Commutative Algebra with a view toward Algebraic Geometry" on pg. 424 We can decide whether an element $x\in R$ is a nonzerodivisor from the homology of ...
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1answer
39 views

Question on the Notation of an Abstract Algebra Question

The following is a question that I came across in a textbook I'm reviewing for self-study. The book is "Introduction to Abstract Algebra", 4th Edition, by W. Keith Nicholson. I have a question both ...
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1answer
25 views

Concerning the notation $\chi (U)$ in one of the hypothesis for some properties of curl and divergence

I have the following exercise: Let $U \subset \mathbb{R}^3$ be open, $X \in \chi (U)$ and $f \in C^{\infty}(U)$, prove the following: $$curl(\nabla f)=0 \\ div(curl(X))=0 \\ curl(f.X)= f.curl(X)+(\...
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1answer
22 views

Partitions: reading skew diagrams?

Consider the following two partitions. Partition $\lambda=4^4=(4,4,4,4)$ o o o o o o o o o o o o o o o o And partition $\mu=(4,2,1,0)$ ...
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2answers
75 views

$A^\dagger$ - how to handwrite this?!

In one book I came across the notation $A^\dagger := \overline{A}^T$. But how does one usually handwrite it? When I try to do it, it seems so similar to $A^+$
0
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1answer
44 views

Is there a blackboard bold letter for the set of Boolean numbers? [duplicate]

Is there a symbol (e.g. $\mathbb{B}$) for the special set of Boolean numbers or values; ${0,1}$ or ${True,False}$?
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3answers
55 views

What is $\mathbb{Z_{n}}\left [ x \right ]$

Question: Show that $\mathbb{Z_{n}}\left [ x \right ]$ has characteristic $n$. What does $\mathbb{Z_{n}}\left [ x \right ]$ stands for? I'm very sure this is not the gaussian ring.
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1answer
28 views

Enunciating utility maximization problem using set theory

I need to enunciate a problem using set theory and I am not sure how to start. The problem goes like this: You are a car manufacturer and need to decide how many colours to use in your next bash of ...
0
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1answer
13 views

How does variable ordering in expressions work when creating functions from an equation?

I'm having a really hard time understanding some aspects of functions, i've tried looking around on Khan academy and haven't quite found something to answer my question, i'm sure i'm overlooking ...
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1answer
54 views

What mathematical notation can use for this formula

I just played around with archimedes $\pi$ formula and ended up with $\pi = \lim\limits_{n \to \infty} 6 \cdot 2^n \cdot \sqrt{2 - \sqrt{2 + \sqrt{2 + ...n times... \sqrt{2 + \sqrt{3}}}}}$ I want to ...
2
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2answers
52 views

$\ln(x)$ and Big O notation

I have tried to assert that $\ln(x)=O(x^0)$ a few times, but it seems fairly obvious that this statement should be false, and so I've been faced with some rightful speculation. My reason is that $$\...
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0answers
32 views

What does the middle dot in parentheses mean?

The below picture is from section 3.3.1 in this paper; I'm not familiar with the notation of using middle dots in square parenthesis. Are the authors using this simply to indicate that any two ...
2
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1answer
97 views

What does $ f ^ {n} (x ^ {1/n}) = … $ mean?

I was asked to check whether the sequence of functions $ \{ x_{n} (t) \} $ defined as $$ x ^{n} _{n}(t ^ \frac{1}{n}) = \begin{cases}n, & t \leq \frac{1}{n} \\\frac{1}{n},& t > \frac{1}{n}...
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2answers
70 views

Is this notation on the restriction of a function in group theory common?

If $f: X \rightarrow Y$ is a function between sets $X$ and $Y$, then a common notation to use when we want to restrict $f$ to a certain domain $X' \subset X$ is $f|_{X'}: X' \rightarrow Y$. I'm doing ...
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1answer
45 views

What the sign ' | ' stand for?

Going through a proof of a theorem, I encountered the following statement: $e\mid a/d$, $e\mid b/d$ Then, $a/d = ex$, $b/d = ey$ where $x,y$ belongs to $\mathbb{Z}$. However, my question ...
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1answer
42 views

What does $X_j \approx X$ mean when used in this blog post?

I was trying to learn disjoint union topology and used the following blog : https://drexel28.wordpress.com/2010/04/02/disjoint-union-topology/ The second theorem about disjoint topology says that if ...
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0answers
27 views

2 dimensional Real space set notation

Is there such notation as: $\Bbb R^2_+$ Like $\Bbb R^2$ but with strictly positive numbers? Or is there better notation for such a set?
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1answer
103 views

Who introduced the notation $\lesssim$?

Who in history introduced the notation $X\lesssim Y$ for meaning $X\leq CY$ for some constant $C$? I've seen this notation in modern literature in PDE a lot. (See for instance the notation section of ...
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1answer
51 views

What does $q\%p$ mean

Sorry for such a simple question. In notes on quadratic reciprocity there was a phrase replace $q$ by $q$%$p$. $p$, $q$ are odd primes. What does that mean? Thanks
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0answers
38 views

Applying Bayes Theorem to find the probability of a finite intersection of events

I have a notation problem. Let $A$ and $B$ be events such that $P(B)\ne 0$. The by Bayes' theorem, $$ P(A \cap B) = P(A|B)P(B)~~~~~~~~~(1) $$ Similarly, $$ P(A\cap B \cap C) = P(A|B \cap C)P(B\cap C)...
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0answers
14 views

Nested conditional function

I have a number of equations that I am solving in steps so that the answer to preceeding steps affects the answer to the current and future steps. This leads to a number of conditional functions, but ...