Use the convention tag for questions about standard, cultural practices in mathematics.

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Does an integral without an integrand imply the integrand is $1$?

This is a question concerning mathematical convention. If we see something like $$ \int_a^b dx $$ Does this imply that $$ \int_a^b dx = \int_a^b 1\ dx\text{?} $$
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50 views

Prove thoroughly: If the degree of all vertices is greater or equal to $\frac{|V| - 1}{2}$, then the simple graph is connected.

I am struggling to write a good, thorough proof. The proof is supposed to be logically rigorous, correct and complete (e.g. no hidden assumption). Moreover, style is important - the proof should be ...
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2answers
48 views

Should “together with” be taken as slang for an n-tuple?

When an algebraic structure is defined, it is often defined as a set $S$ "along with"/"together with"/"having" operations $\circ_1, \circ_2, \ldots, \circ_n$, and "denoted" by $(S, \circ_1, \circ_2, ...
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2answers
127 views

Usage of $\cdot$ in calculus

I often find myself caught in the dilemma of whether or not to use the symbol $\cdot$ in calculus. Take for example, the chain rule: $$\frac{dy}{dx} = \frac{dy}{du}\cdot\frac{du}{dx}$$ Is the ...
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536 views

Starting sentences with mathematical symbols.

I apologise if this is a duplicate in any way or is too opinion-based. To what extent is it best not to start a sentence with a mathematical symbol? I find that when trying to solve a problem or ...
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222 views

Rationale for a convention: Why use the semiperimeter in Heron's formula?

Heron's formula says that the area of a triangle whose sides have lengths $a, b, c$ is $\sqrt{s(s-a)(s-b)(s-c)}$ where $s=(a+b+c)/2$ is the semiperimeter. It can also be stated by saying that the ...
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Graph diameter of a single vertex?

Convention-wise, given the simple graph with a single vertex, what is the graph diameter? I can see three options, zero, since there is a trivial path from the vertex to itself. infinity, since ...
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1answer
32 views

Is there a symbol/ way to mathematically indicate an answer?

I know this is going to sound rather contrived, but I was wondering if there was a symbol to indicate a solution to a problem. I'm a student studying engineering and constantly shift between notations ...
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1answer
55 views

Game Theory: players' gender convention?

What is the Game Theory convention of using gender terms (male/female) for the players? I found only one reference suggesting that odd-numbered players are male and even-numbered players are female. ...
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1answer
68 views

Combine a rotation matrix with transformation matrix in 3D (column-major style)

I'm trying to concatenate a rotation matrix with a 4x4 homogeneous transformation matrix with column-major convention. I want this rotation matrix to perform a rotation about the X axis (or YZ plane) ...
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35 views

Given a set of rotation matrices, is it possible to determine the rotation conventions employed?

Given three rotation matrices, is it possible to determine the conventions used in defining them? These conventions include: passive or active transformation, pre- or post-multiplication, right- or ...
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44 views

Commas between adjectives [duplicate]

Is it correct to use these expressions without commas (as written below)? A finite simple group. A finitely generated abelian group. A finitely generated torsion-free abelian group. A ...
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1answer
29 views

Name and notation convention for “unnormalized probability”

Given a finite set of non-negative numbers $S={s_1,...,s_n}$, we can divide them by a normalization constant $Z$ (i.e. their sum) to get a probability distribution. Then we typically say (and write) ...
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37 views

shouldn't we specify the topology when we talk about Borel $\sigma$-algebras?

Maybe this is me being pedantic, but this thought has just occurred to me. The Borel $\sigma$-algebra on X is the smallest $\sigma$-algebra containing all open sets. When we talk about Borel ...
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2answers
57 views

Question about secant and cosecant.

Ok so if we take a right triangle and consider an angle $\alpha$ we get the following: From here we can define the fundamental trigonometric functions sine and cosine where ...
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5answers
441 views

What are reasons why some symbols in mathematical logic are not standardized?

Why is so hard to find a standardisation regarding symbolism and/or terminology in Mathematical Logic ? We see again and again students asking if e.g. $\rightarrow$ and $\implies$ means the same ...
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1answer
35 views

Semidirect product notation convention

I was taught that if $G \simeq H \ltimes K$, then (by convention) $H$ is the subgroup that is normal, but I see on Wikipedia and elsewhere that other people use the convention that the above indicates ...
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2answers
90 views

why function argument is on right side $f(x)$ rather than on left side as $xf$

Is there an advantage for writing function arguments on the right side as $f(x)$ rather than on the left side as $xf$? The latter looks more natural if we think about it in diagram as $domain ...
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1answer
45 views

Notation in functions spaces

I'm wondering if there is a convention for the following. Let $g:\mathbb{R}^2\to \mathbb{R}$ be given and u in $C^1(\mathbb{R})$. I'm looking for a notation for $g(x,u(x))$ is in the space of ...
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Wikipedia claims that number theorists tend to prefer $0 \notin \mathbb{N}$. Is this actually true?

According to wikipedia here: $\mathbb{N}$ means either $\{ 0, 1, 2, 3, ...\}$ or $\{ 1, 2, 3, ...\}.$ The choice depends on the area of mathematics being studied; e.g. number theorists ...
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0answers
34 views

Convention on the order of scalar multiplication (Multiplier vs multiplicand)

Is there a convention on the order of scalar multiplication? I know there were questions before mine, but I would like to know if such distinction is culturally dependent. This came from a news in ...
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Category of topological pairs

Is there a standard abbreviation for the category of topological pairs? I have searched for it in vain.
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2answers
98 views

Fourier transform convention: $\frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} f(x)e^{\pm ikx}dx $?

I've come across the Fourier transform being defined as: $$\tilde{f}(k)=\frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} f(x)e^{ikx}dx$$ But this convention is not present in the Wikipedia article. The ...
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1answer
86 views

Normalization of Orthogonal Polynomials?

The generalized Rodrigues formula (Hassani, Mathematical Physics, p. 174) is of the form $$p_n=K_n\frac{1}{w}\left(\frac{d}{dx}\right)^n(wp^n)$$ The constant $K_n$ is seemingly chosen completely ...
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Why are the order-of-operations conventions good?

Children are sometimes taught silly mnemonics like "PEMDAS" to remember conventions on order of operations. (I never heard of "PEMDAS" until long after graduating from college, as far as I can ...
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80 views

Why is $m$ used as the variable for slope in slope-intercept form?

I was wondering if you could answer a question I have on slope intercept form of a linear equation. I know its $y=mx+b$, but why is it $mx+b$? Don't get me wrong. I know that $m$ is the slope and $b$ ...
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2answers
70 views

What is a vector with a single non-zero component called?

What do you call a vector with a single component, e.g. $[a, 0, ..., 0]$ or $[0, b, 0, ..., 0]$, where $a, b$ are any non-zero number? I'd like the language to differentiate from a vector with ...
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472 views

Is there a fundamental reason that $\int_b^a = -\int_a^b$

Is there a fundamental reason that switching the order of the limits in an integral results in the negative, i.e., $$\int_b^af(x)\,dx = -\int_a^bf(x)\,dx?$$ As far as I can tell, this is just chosen ...
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1answer
30 views

Value expressions

What is the value of the expression: $\sum_{k=0}^{d}b^{k}\left[\begin{array}{c} d\\ k \end{array}\right]\prod_{i=0}^{k-1}\left(c-b^{i}\right) $ for $ k = 0 $? In particular, what is then the value ...
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172 views

$x^{y^z}$: is it $x^{(y^z)}$ or $(x^y)^z$?

Of the following, why is a usually considered true, and for what reason other than "tradition" and "more convenient"? a: ${x}^{y^z} = x^{(y^z)} \neq {(x^y)}^z$ b: ${x}^{y^z} = {(x^y)}^z \neq ...
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40 views

Order of operations: Matrix product and hadamard product

Is there any convention of the order of operations in a term with both ordinary matrix multiplication and hadamard (elementwise) multiplication? Obviously, $$ A ( b \circ c ) \ne (A b) \circ c $$ ...
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No radical in the denominator — why?

Why do all school algebra texts define simplest form for expressions with radicals to not allow a radical in the denominator. For the classic example, $1/\sqrt{3}$ needs to be "simplified" to ...
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3answers
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What does a “convention” mean in mathematics?

We all know that $0!=1$, the degree of the zero polynomial equals $-\infty$, the interval$[a,a)=(a,a]=(a,a)=\emptyset$ ... and so on, are conventions in mathematics. So is a convention something that ...
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139 views

The Degree of Zero Polynomial

I wonder why the degree of the zero polynomial is $-\infty$ ? I heard that, it is $-\infty$ to make the formula $\deg(fg)=\deg(f)+\deg(g)$ hold when one of these polynomials is zero. However, if that ...
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A question on notation: $u(x,y)$ vs. $u(x+iy)$.

In Tao's complex analysis notes, he uses the convention that $$f(z)=u(z)+iv(z)= u(x+iy)+iv(x+iy)$$ implying that the functions are $$u,v:\mathbb{C} \to \mathbb{R}.$$ But in other places I see the ...
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201 views

No difference between $0/0$ and $0^0$?

I have seen discussions about both $0/0$ and $0^0$ and they differ a bit in the way that most seem ok with calling $0/0$ "undefined", while the $0^0$ discussion still seems like a dispute. If this is ...
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103 views

Are there two conventional definitions of “holomorphic”?

In Walter Rudin's Real and Complex Analysis, second edition, on page 213, two definitions are stated. One of them says the derivative of $f$ at $z_0$ is $$f'(z_0)=\lim_{z\to ...
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Is “iff” the same as equality if each member is a predicate?

"Iff" - if and only if ($\Leftrightarrow$ or $\leftrightarrow$, although the first usually carries a "meta" meaning, something that is not evaluated) - is used in $2x=3\Leftrightarrow x=\frac32$, ...
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55 views

Polynomials and sign conventions

I can never make up my mind when dealing with polynomials how to "format" them. Are there any general guidelines to follow? Which of the following is the best practice? $-a^2 + b$ $b - a^2$ $-(a^2 − ...
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decimal digit grouping delimiters

I feel a bit silly having to ask this but I just can't seem to find any resources that give an answer to this. When dealing with decimal values that have a large number of digits to the right of the ...
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2answers
39 views

Is there a convention for precedence of operators in an additive category?

The laws for an additive category are that there must be a zero object, binary products, that every Hom-set is an abelian group, and that the morphism addition distributes over composition. My ...
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5answers
546 views

What type of input does trigonometric functions take in

I see in my Book that 45 deg is equivalent of π/4 . Ι also do the conversion if I simply convert degrees into radians like this 45* π/180 = π/4 radians and ...
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1answer
43 views

What's the name of each pseudo-rectangle in a spherical surface?

Consider the common surface of a spherical segment crossed with a spherical wedge. This produces a pseudo-rectangle in the sphere surface, and a perfect rectangle in a mercator projection. What's the ...
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443 views

What is a Neighborhood?

Which of these definitions is more commonly used, and in which contexts? Fix a point $x\in (X, \tau)$. Then a neighborhood around a point $x$ is: a set $N\ni x$ and $N\in \tau$ a set $N$ with $x\in ...
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1answer
158 views

Commas separating adjectives to describe mathematical objects.

Is there a convention which rules that in mathematics an object with some properties should be referred to as a "property 1 property 2 property 3 ...object" rather than a "property 1, property 2, ...
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56 views

Matrix with rank more than $1$

Is there a shorter way of saying that a matrix $A$ has rank more than $1$? I am looking for something like "full-rank", or "rank-deficient."
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115 views

Where should the 2$\pi$ go in the Fourier Transform?

In some lecture notes on Harmonic Analysis from Terence Tao here, he defines the fourier transform by $$\hat{f}(\tau)=\int_{\mathbb{R}}e^{-2\pi i t\tau}f(t)dt$$ and then says This is really the ...
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40 views

Canonical way of denoting the set of all (totally) ordered subsets

Given a finite set $S$, we usually denote the set of all subsets of $S$ by $\mathcal{P}(S)$, i.e. the power set of $S$. I need to denote the set of all totally ordered subsets of $S$, let us call it ...
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1answer
165 views

Convention of writing constraint sets

As I will write constraint sets very often, I would like to make sure that I respect the convention. First, I would like to represent a set of constraints and their relation are conjunction. For ...
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32 views

Definition by Recursion and a Question about Induction

I have some questions to ask. Suppose I want to define some sequence of propositional formulas $\{\varphi_{j}\}_{j\in\mathbb{N}}$. First, I define it this way. Fix an enumeration ...