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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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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1answer
21 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)= ...
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2answers
22 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
16 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
13 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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0answers
28 views

Series with Markov Chains Probabilities

Notation For each $t \in \mathbb{N}$, let $h_t \in H$ be a random variable that follows a Markov chain, and $h^t \equiv \{h_0,h_1,\dots,h_t\} \in H^t$. Let $\Pi(h^{t})$ be the probability that a ...
3
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1answer
52 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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3answers
49 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
37 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 ...
38
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5answers
3k views

What does the symbol |_ mean?

For example, (6) The sequence of primes is endless. For, if $p$ is any prime, the number ${\begin{array}{|c}\color{red}p\\\hline\end{array} + 1}$ is greater than $p$ and is not divisible by ...
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3answers
81 views

Sequence Notation in Analysis

If real sequence is a function from the set $\mathbb N$ to the set $\mathbb R$ and function is represented by $(a,b)$, where $a$ is domain and $b$ is range, then why do we represent sequence only by ...
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1answer
25 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 ...
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1answer
21 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
71 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 $A^+$
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1answer
41 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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1answer
51 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 ...
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2answers
69 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
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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5answers
6k views

Scientific notation and negative numbers

My daughter is learning scientific notation in school, and her textbook says something to the effect of this: Scientific notation is a method of writing numbers as the product of two factors ...
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2answers
49 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 ...
2
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1answer
90 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 > ...
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0answers
23 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
40 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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1answer
44 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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452 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. ...
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1answer
31 views

Notation for separating out factors of a number

I have an integer (let's call it $n$), and I want to define it as the product of two values: one that's a pure power of two, and another that is odd. Obviously, these two values are unique for a ...
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0answers
26 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?
5
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0answers
81 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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0answers
32 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 ...
0
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1answer
50 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
2
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1answer
51 views

A little help on mathematical notation and on the definition of a manifold

I was going through the mathematical definition of manifold and it defined it as follow : " A topological space, $\ M $ is an n-dimensional real manifold if there is a family of subsets $\ U _\alpha ...
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0answers
12 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 ...
5
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5answers
208 views

Where could (do?) we go after exhausting greek letters?

I'm still in high school but after all my various math and science classes including calculus, statistics, geometry, and physics, I think that we've pretty much run the course of both upper and lower ...
0
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0answers
34 views

Basis of all real polynomials?

I am studying the book Topics in Algebraic Graph Theory by Beineke et all and the page 12. By the book, the set of all real polynomials can be generated by the set $\{1,x,x^2,\ldots\}$ which I ...
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1answer
44 views

Gröbner basis is not a vector basis?

We use the same notation for Gröbner basis and vector basis. I recall that $\langle 1\rangle_{GR}$ is the largest Gröbner basis while $\langle 1\rangle_{vector}$ is the smallest vector basis. So for ...
22
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3answers
4k views

Is there a sign for 'not less than', 'not greater than', etc.?

I was wondering about this, just now, because I was trying to write something like: $880$ is not greater than $950$. I am wondering this because there is a 'not equal to': $\not=$ Not equal to is an ...
3
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0answers
29 views

Why does (h,k) generally represent the center of a circle?

Why are h and k generally used to denote the coordinates of the center of a circle? After a bit of research, we found that h may represent "horizontal shift" or "horizontal translation", but we're ...
2
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1answer
72 views

Notation i.i.d sample

I am learning measure theory and sometimes I am not sure if I am using the correct notations, especially with respect to distributions of random variables. In the following I try to formulate the ...
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0answers
41 views

The notation of 'greater than or equal to'.

I've known that the following marks are equal. However, both marks are used in the same book. I was wondering whether there is some difference between them.
0
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0answers
34 views

Why “deformation retracts to” instead of “deformationally retracts to”?

First, correct me if I'm wrong: I grammatically understand "X deformation retracts to Y" as "X(subject) deformation(adverbial) retracts to(verb) Y(object)". But I just don't understand why we don't ...
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1answer
31 views

Is $\forall x \in U: f(x) \in V$ the same as $x \in U \implies f(x) \in V$?

As the title says, do the following two statements have the same meaning? $$\forall x \in U: f(x) \in V \text{ (for all $x \in U$, $f(x) \in V$)}$$ $$x \in U \implies f(x) \in V \text{ ($x \in U$ ...
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votes
2answers
65 views

Is there any difference between $\sum_{ij}$ and $\sum_i\sum_j\;?$

I was reading Tensors by Feynman, where he said: [...]$$u_P=\tfrac{1}{2}\sum_i\sum_j\alpha_{ij}E_iE_j\;.\tag{31.7}$$ The energy density $u_P$ is a number independent of the choice of axes, so ...
3
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2answers
44 views

What's the notation of the one-sided differential of a function?

Assume $$f(x)=\left \{\begin{matrix} x-1 & x \leq 0\\ x+1 & x > 0 \end{matrix} \right.$$ Obviously, $df(x)/dx$ do not exists at $x=0$ but the one-sided differential exists. What's the ...
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2answers
32 views

How to express that one interval is included in another interval?

Would the symbol "$\in$" work for denoting that one interval is included in another interval? Like this: $(x>2) \in (x>0)$
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2answers
214 views

Math formula with $\mathcal N$ symbol. What is it?

Occasionally I stumble across formulas that use exotic letters, that I do not know how to read or pronounce or what they mean, even. Also it is hard to google for them, since there is no way of typing ...
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2answers
29 views

Quantum group notation

I was jumping into the deep end and reading a few papers and lectures on quantum groups. My knowledge on Lie algebras is a bit thin but I was just wondering the notation used in the starting of this ...
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2answers
15 views

Name for opposing (complementary) multisets (bags)

I am looking for the accepted mathematical name for opposing multisets (or complementary multisets). I have done a Google search and a Stack Exchange search, and have come up empty. These are the ...
0
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0answers
25 views

Show that $(\log(n))! = \Omega(n^3)$

I want to prove that $(\log_{2}(n))! = \Omega(n^3)$. I Think that $(\log_{2}(n))! = \Theta({\log_{2}(n)}^{\log_{2}(n)})$ But, I dont know how to continue. Someone has an idea ? Thanks
4
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2answers
169 views

What is the notation for the empty matrix?

My question: What is a notation for an empty 0x0 matrix (i.e. the matrix for the only linear map $f:\{0\}\to\{0\}$)? Is it written $()$? How can I distinguish the 0x0 matrix with for example the 0x3 ...
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2answers
35 views

How are these different types of notation called?

My teacher used to give questions on integrals of vector fields with different notation to get us used to seeing things in different forms. I'm just wondering what these three different forms of ...