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

Does the Statement $\lim_{f(x)\to a}k(x)$ Make Sense

In a formal mathematics context does the statement $$\lim_{f(x)\to a}k(x)$$ where $f(x)\neq c$, where $c$ is a constant, make sense? For example does $$\lim_{x^2\to 0}x$$ make any sense in a formal ...
16
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4answers
1k views

Are there dictionaries in math?

Consider the following dictionary in the programming language Python: D = {'A': 1, 'B': 2, 'C': 3} It is saying that the value of A is 1, the value of B is 2, ...
0
votes
0answers
28 views

Clarification on super script notation

I'm following an algorithm for computing sparse singular vectors for an SVD decomposition. a part of the algorithm states: set initial values $u^{0}$ and $v^{0}$ and set $i=0$ update $u$: $u^{i+1} ...
3
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4answers
13k views

“such that” logical symbol

So, in the definition of what is a square root: Sqrt(x) are all numbers y such that y*y = x Are there any logical mathematical symbols so that the above ...
0
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1answer
44 views

The notation $ \otimes $ (Tensor product) [duplicate]

Related to the question Riemannian manifolds isometry, could anyone be able to explain to me what the notation $ \otimes $? Some examples were given in the question : $dx\otimes dx+dy\otimes dy = \...
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1answer
25 views

Notation check, argmax of values in a set [on hold]

Among C different possible values for each y and m total ...
1
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0answers
60 views

When to use “:=”?

I know that "$:=$" is used in definitions. For example, one can write: $A\times B:=\{(a, b)\mid a\in A, b\in B\}$ But would you use "$:=$" in the following example: Let a function $f\colon A\...
0
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1answer
44 views

Big O Notation in a book

Can somebody explain to me how the "Big O Notation" or the "Landau symbols" work when applied to the Taylor expansion? I'm currently reading a book about physics and I came across this— —what does ...
0
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3answers
36 views

How to write an equation that's nested?

Let's say I have an equation that's nested: $$ \displaystyle x = f\left(f\left(f\left(f\left(f\left(i, a_4\right), a_3\right), a_2\right), a_1\right), a_0\right) $$ If I wanted to write this ...
1
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2answers
31 views

Notation for binomial coefficient set

I've been searching for a way to express "the set of all combinations generated by taking $\binom{n}{k}$ items". For example, if I have the set $\{3,7,6,5,9\}$, and I want the set of all sets that ...
1
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0answers
31 views

What is this space called in general topology? [duplicate]

I am self-studying the general topology these day and find that the third axiom of the topological space $(X,\tau)$ defined by open set is: For any finite collection of $U_i \in \tau$, the ...
0
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2answers
24 views

I need a better understanding of notation for quantifiers

The statement: (∀x ∈ Z) ((∃y ∈ Z) x = 2y) or ((∃y ∈ Z)x = 2y+1) says that every integer is even or odd. I can break down the statement into each part (∀x ∈ Z) means for all x in set Z, (∃y ∈ Z) x = ...
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1answer
27 views

Function Application and its Notation

For years and years and years I've always been taught that in mathematics, functions are applied as $f(x)$. But in my university textbook they also use three other notations: $$f\ x,$$ $$fx,$$ $$\...
2
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0answers
64 views

Unclear passages in the paper “On a New Class of Theorems in Elimination Between Quadratic Functions” by J. J. Sylvester

I'm writing an essay about the origin of some mathematical terms in the work of J. J. Sylvester. He first used the word matrix in his paper Aditions to the Articles "On a New Class of Theorems" and "...
3
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1answer
69 views

Writing Math: Is using both $e^x$ and $\exp(x)$ ok for longer works?

I wanted to know what you guys think about mixing both notations for the exponential function $e^x$(for simple argument and to save space for larger equations) and $\exp(x)$ (for more complicated ...
2
votes
1answer
33 views

What does the notation $\overline{\mathbb R}$ mean in that context?

In an old question, it can be read that "the finiteness of $\text{Gal}(\overline{\mathbf R}/\mathbf R)$" is one of the "impressive finiteness results in mathematics". I commented the question to know ...
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2answers
67 views

How could we formalize the introduction of new notation?

What I am thinking about is how in a textbook/proof/theorem/discussion/definition one states that from now on a new notation will be used in the appropriate scope. Example: Let $V^*$ denote the ...
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1answer
11 views

Notation clarification in Schilling's Brownian Motion

In Chapter 1's Problems, Problem 1(b), we are given that $X,Y\sim \beta_{1/2}:=\frac{1}{2}(\delta_0+\delta_1)$ are Bernoulli random variables. How am I to interpret this? That the probability of ...
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1answer
24 views

Need help with inductive proof of Binomial Theorem

I'm new to math and trying to learn about the Binomial Theorem, by following this tutorial. I got stuck trying to read the Induction Proof. They give an example of using the Sum notation: $$ (x + y)^...
0
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1answer
16 views

Proper Use / Interpretation of homogeneous notation

I'm trying to understand the proper use or interpretation of homogeneous notation . I understand the concept of a linear transform $T$ such that $T(c_1\cdot\mathbf{x_1}+c_2\cdot\mathbf{x_2}) = c_1\...
0
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0answers
35 views

How to define a union(ish) operation on tuples

Let us say we have 2 tuples whose elements are all distinct (relative to the given tuple they are apart of) and have some ordering relation already defined on them, given as: $t_1$ =($\psi_1$, $\...
4
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3answers
62 views

On notation: is it better to say $A^B = \{f| f:B \to A\}$ or $A^B = \{f :B \to A| f \text{ is a function}\}$

The title says it all, let $A^B$ denote the set of all functions from $B$ to $A$, then it is better to write in set notation $A^B = \{f\mid f:B \to A\}$ or $A^B = \{f :B \to A\mid f \text{ is a ...
5
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3answers
268 views

Difference between | and / [duplicate]

"If we find a prime $p$ such that $p\mid n$ , then $n/p$ is a positive integer that's smaller than $n$." I understand $n/p$ is $n$ divided by $p$ but what is $n\mid p$?
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1answer
30 views

How to show continuity of a function with $n-1$ exponentiations?

Say we are given a function $$\Gamma(x)=f_1(x)^{f_2(x)^{\cdot^{\cdot^{f_n(x)}}}}$$ where $f_i,i\in[1;n]$, are continuous functions in their domains. Also assume that the function makes sense, e.g., ...
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3answers
42 views

Why the notation $A\equiv C \pmod B$ instead of $A \text{ mod } B = C$?

Why is, for the modulus operation, the notation $A\equiv C \pmod B$ used instead of $A \text{ mod } B = C$? Or alternatively $\text{mod}(A,B) = C$ or, as in many programming languages, $A\text{ } \% \...
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0answers
9 views

Is this correct for an explicit represenation of the $\Vert f \Vert_{C^2(\mathbb{R}^2)}$ norm that uses multi-index notation?

I'm not familiar with multi-index notation so I'm not sure if I have this correct. Say we have (taken from here) $$ \Vert f \Vert_{C^2(\mathbb{R}^2)} = \sum_{j=0}^2 \sup_{x \in \mathbb{R}^2} |\nabla^j ...
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0answers
14 views

Function notation ??

From the Euler-Lagrange equation $$ I(\alpha) = \int_{x_1}^{x_2}F(Y,Y',x)dx $$ $$ \frac{dI}{d\alpha} = \int_{x_1}^{x_2}[\frac{\partial F}{\partial Y}\frac{\partial Y}{d\alpha}+\frac{\partial F}{\...
2
votes
1answer
32 views

Simple set-builder notation for counting pairs

Today I was curious about writing a simple expression using set-builder notation. The expression is the number of integer pairs $(a, b)$ such that $a \mid n$, $b \mid n$, and $a \mid b$. My attempt is ...
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2answers
46 views

How do I write $B = \left\{\left[\begin{smallmatrix} x \\ y \end{smallmatrix}\right] \in \boxed{?}| \ldots\right\}$ with proper notation

Let $x \in X \subset \mathbb{R}^n$, then I define a set: $$A = \{x \in X| 1^Tx = 0\}$$ Now supose I have another element $y \in Y \subset \mathbb{R}^n_{+}$ I concatenate $x,y$ in to a single vector ...
93
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6answers
11k views

What did Alan Turing mean when he said he didn't fully understand dy/dx?

Alan Turing's notebook has recently been sold at an auction house in London. In it he says this: Written out: The Leibniz notation $\frac{\mathrm{d}y}{\mathrm{d}x}$ I find extremely difficult ...
0
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4answers
82 views

$E(X)$ versus $E(X|Y)$

Why is $E(X)$ considered a constant but $E(X|Y)$ considered a random variable? Seems like confusing notation since I'd assume the latter is a fixed constant "the expected value of random variable $X$ ...
0
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0answers
56 views

Double integral, $dx$ before the integral sign

Suppose we have $f(x)g(x,y)$, In some books (Rudin Real and Complex Analysis) the double integral is written as $$\int_X f(x) dx \int_Y g(x,y) dy$$ instead of $$\int_X \int_Y f(x) g(x,y) dy dx.$$ If ...
1
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1answer
33 views

Hartshorne: Definition of $K^*$ where $K$ a function field of scheme.

Let $X$ be a noetherian integral separated scheme which is regular of codimension one. Let $K$ be the function field of $X$. Now let $f \in K^*$, (I am interpreting $K^*$ to be the set of field ...
3
votes
1answer
50 views

Is there a meaning to the notation “\arg \sup”?

When $f$ is a function on a set $A$, the notation: $\arg\max_{x\in A} f(x)$ denotes the set of elements of $A$ for which $f$ attains its maximum value. This set may be empty, for example, if $f(x)=x$ ...
1
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2answers
44 views

Given the function $f:[0,1]→[0,1]$; $f(x)=x^2$, check which one(s) of the properties it has.

This homework is past due, but I am still fiddling trying to figure this out. question: I do not understand what the heck the notation of $f:[0,1] \to [0,1]$; means. I thought I did, but my ...
0
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0answers
19 views

Notation of maximization in pseudocode

I'm working on the implementation of a simple algorithm. See this small excerpt: S is a set of points from which I have to choose point p. I am confused as how to find p. I know how to calculate ...
1
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1answer
31 views

set notation for vectors of unequal elements

I am looking for a compact way to represent a group of vectors for which each vector contains no two elements that are the same. $$\textbf{y} \in R^n \quad | \quad y_i \neq y_j \quad \forall \quad ...
2
votes
3answers
32 views

Converting from standard to functional, Polish and Reverse Polish notation

I wanted to convert the following expression to Functional, Polish and Reverse Polish notation. $$Y =A + \frac{B+ \dfrac{BA}{B+CA}}{A - \dfrac{BC}{B-C+A}}$$ I know how to do Standard -> Functional ->...
8
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2answers
18k views

What are the common abbreviation for minimum in equations?

I'm searching for some symbol representing minimum that is commonly used in math equations.
3
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2answers
74 views

If $\lim a_n$ does not exist, then does this mean that $\lim a_n\neq 0$?

If $\displaystyle\lim_{n\to\infty} a_n$ does not exist or if $\displaystyle\lim_{n\to\infty}a_n\neq0$, then the series $\displaystyle\sum_{n=1}^{\infty} a_n$ is divergent. From Stewart's Early ...
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2answers
36 views

Notation of the square (or other power) of a function $f(x)$

How do you notate the square (or other power) of a function $f(x)$? Is it $f^2(x)$ (similar to $\sin^2(x)$ for example), $f(x)^2$ or do you have to use $(f(x))^2$? Thanks in advance.
5
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2answers
196 views

⋇ “Division Times” operator in Unicode (U+22C7)? [duplicate]

I found this maths operator in Unicode: ⋇ It is called "Division Times" (U+22C7). Does it behave like ±? For example: 3 ± 2 means it is an ∈ {1, 5}. So 3 ⋇ 2 means it is an ∈ {1.5, 6}?
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0answers
22 views

Notation for subring of quotient ring

Let $S$ be a subring and $I$ an ideal of the ring $R$. Is there some standard notation for the subring of $R/I$ given by $\{ s + I: s \in S \}$. Is it appropriate to write $S/I$ even though $I$ is not ...
1
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1answer
425 views

Element of a sequence notation

I'm writing some equations dealing with sets and sequences. I have a sequence $S$ and want to show that $x$ is an element of $S$, however I am hesitant writing $x \in S$ because I don't want to ...
5
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1answer
104 views

Why isn't it mathematically rigorous to treat dx's and dy's as variables? [duplicate]

If I do something like: $$\frac{dy}{dx} = D$$ $$dy = D \times dx$$ People would often say that it is not rigorous to do so. But if we start from the definition of the derivative: $$\lim_{h \to ...
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1answer
62 views

Question about the notation $X/A$ in topology

In Hatcher, the notation $X/A$ as appearing in the following text is never defined: If $(X,A)$ is a CW Pair consisting of a cell complex $X$ and a subcomplex $A$, then the quotient space $X/A$ ...
0
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0answers
36 views

notation: meaning of $\frac{dB}{dr}$

I'm wondering what $\frac{dB}{dr}$ could be? Is it a common notation for $\mid\vec\nabla B\mid$? This would the only thing that would make sense for me! (Occurrence in a text about some physics 'in ...
0
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1answer
35 views

Summation Notation (Discrete Mathematics) [closed]

I am currently studying sequence which I think will lead up to my next topic induction. My question is if $$\sum_{k=0}^n \frac{k+1}{n+k}= \frac{1}{n}+\frac{2}{n+1}+\frac{3}{n+2}+\cdots+\frac{n+1}{2n}$...
1
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0answers
26 views

How is functions with “universal” or “unknown” domains written

Related to $f$ equals <fill in blanks> Is there a way to write a "function" (without lambda calculus) without a domain? For instance, when differentiating, one often just considers $x^3$ as a "...
4
votes
0answers
58 views

What are these S,Z,M short for in linear algebra?

In the text Linear algebra (Hoffman), there are notations S, Z, M. What are these short for -- that is, why are these particular three letters used for the following concepts? (i) S. Let $W$ ...