Various aspects of writing mathematics such as style, notation, grammar, frequently used phrases and common mistakes.

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Soft Question: Good strategies for writing a technical book or textbook

Most people would agree that doing technical work, whether it be pure or applied, and learning the background knowledge necessary to do this work comprises most of the literature and curriculum in ...
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3answers
41 views

How to write a set with an index

I'd like to write a set $\{x_1, x_2, ..., x_n\}$ in a simple way. What is a popular way? In my high school, I wrote it as $\{x_i\}_{i=1}^{n}$. Is it a correct way?
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4answers
103 views

Is there an alternative way to represent the $\operatorname{diag}$ function?

In optimization, it is common to see the so called $\operatorname{diag}$ function Given a vector $x \in \mathbb{R}^n$, $\operatorname{diag}(x)$ = $n \times n$ diagonal matrix with components of $x$ ...
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1answer
59 views

Can someone come up with a better way to write $V = \operatorname{diag}(x_1,x_2)(Y-\mathbf{1}X^TY)$

$\newcommand{\diag}{\operatorname{diag}}$Let $X = \begin{bmatrix} x_1 \\ x_2 \end{bmatrix}$, $Y= \begin{bmatrix} y_1 \\ y_2 \end{bmatrix}$ I have a vector: $$V = \begin{bmatrix} x_1(y_1 - \...
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6answers
4k views

How to represent “not an empty set”?

I'm writing a academic paper and need to represent "A is not the empty set". What is usual way for professional mathematicians? My idea is: $|A| > 0$ However, using the emptyset $\emptyset$ ...
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2answers
41 views

What is the formal way to denote a fraction of a fraction?

I need to write a paper and it's not clear which is correct: $\frac{x/y}{z}$ or $\frac{\frac{x}{y}}{z}$, or possibly some other alternative.
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3answers
49 views

Punctuation in mathematical writing.

I am trying to collaborate with a fellow student on a piece of work. Unfortunately we have now reached a creative impasse. I am under the impression that to include punctuation marks in equations ...
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0answers
20 views

Constant value of a sequence

I am writing a article about sequence number. Is it acceptable to write two words "constant value" for limitation of two successive terms of a sequence. I mean, assume Fibonacci number, I used "...
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0answers
15 views

Acceptable notation? $\lesssim_{n} n^{-\beta}$ for constants NOT depending on $n$

I am preparing a paper and found it convenient to write things like $$ |\text{Expression of a lot of variables}|\lesssim_{n} n^{-\beta} $$ when an inequality is true up to a multiplicative factor that ...
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1answer
52 views

How to briefly mention that you answered someone's question without seeming dismissive?

I'm writing my first paper. I would like to mention at the end that I have answered someone's question that is related to the subject of the paper. The proof is really short and doesn't use any ...
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0answers
29 views

representing a periodic circular shift for a vector formally

I'm writing an algorithm in which the operation of circular shift to a given vector $x=[1 \ 0 \ 0\ ... 0]$ is needed on periodic basis i.e. every $\Delta t$ a circular shift will occur. How this ...
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2answers
76 views

Let $f\colon A\to B$ be a function

The following sentence is often used in mathematical articles and writings: Let $f\colon A\to B$ be a function. I wonder, why this senctence is used often, since it is redundant. It is like I ...
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0answers
9 views

Simple visualisation tool for small graphs

I am looking for a way to add illustrations to a paper I am currently writing. I want to display very basic graphs (not more than 8 vertices, a few edges). The focus is on the edge crossings - I need ...
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1answer
64 views

Proper way to write a mathematical theorem

Consider the Bernoulli Inequality, for instance. Basically it says the following: For all $x\in\mathbb{R}$ such that $x\geq -1$ and all $n\in\mathbb{Z}^+$, we have that $(1+x)^n \geq 1+nx$. I'm ...
2
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3answers
138 views

Is it acceptable to use “But” in a proof that doesn't use contradiction?

I have recently read a lot of proofs that like to say "But..." right before the punchline. I feel that the word "But..." should be used if what follows is contradictory in some way, as in proofs by ...
2
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1answer
32 views

If a word starts with a number, does capitalization apply to the first letter?

If a mathematical word begins with a number and a hyphen, such as "4-dimensional" or $3-manifold," and this word occurs at the beginning of a sentence, should you capitalize the first letter? For ...
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0answers
27 views

Ordinal numbers

I have a sequence of coefficients. I can obviously write "i-th" or "h-th coefficient", but I'm referring to the coefficient with number n-h. Can I write "(n-h)-th coefficient"?
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0answers
19 views

line breaking for group actions in latex

When the group $G$ acts on a manifold $M$, we typically use expressions as $M$ is a $G$-manifold, or $G$-action on $M$. If we write those sentences in latex, is it recommended to put $G$ ...
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1answer
44 views

Is this correct math notation?

Is the following correct? $$ x_{ap} = x_{ij} \quad \forall a,i \in [1,4]\ \forall p,j \in [1,5] $$ I want to write that $x_{ap}$ is equal to $x_{ij}$. And $i$ and $a$ can only be values between $1$ ...
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4answers
54 views

Terminating each branch of a proof with $\square$

My question is a variation on this one. I have a proof which divides at the top level into a number of mutually exclusive cases, with further partitioning within that. Is it reasonable to place a $\...
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0answers
31 views

Symbol for exactness in diagram

When making commutative diagrams there are many symbols one can use and even omit in favours of others. For example in the following short exact sequence $$0\to A\to B\to C \to 0$$ we can rewrite it ...
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1answer
52 views

Analytic vs. Analytical [closed]

I am trying write an article. Little Summary: I have developed some tools to analyze derivative of some function $f$. This characterization leads to better results than previous works that only ...
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0answers
25 views

How to abbreviate “Check that assumption is not violated for the solution that we got through the assumption”

Note that this is not a math question, but a language one. Nonetheless it requires some understanding of math and I thus hope it is appropriate to be asked here. Also be warned of non rigorous ...
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1answer
42 views

Comma placement inline math

In the following sentence If all $f(x_j), j=1,\dots,N$ are positive, then ... Do you think there should be a comma after the $N$? I always considered the math as a single unit and read the ...
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0answers
42 views

The following question is about the expression “The following.”

In a mathematical document, it is common to have some English prose leading into a mathematical display and continuing after it. The display is some computation or equation, written symbolically, ...
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1answer
44 views

How do you say in a correct way using some technique?

Sorry, I know it's not really related to Math itself, but I still need a native English Mathematician to help me through that and don't know where else to ask... So please don't close it because it ...
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2answers
55 views

Mathematical usage of “$\dots$” during enumeration, is it ok to be imprecise?

I am guilty of writing things like the following in proofs: so by lemma 1.2 we have that for $k<n$ all of the integers $k,k+1,k+2,k+3,\dots ,n$ are pompous. I really like how this looks, the ...
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1answer
32 views

“The following are equivalent” for only two statements

I often see "The following are equivalent" in theorems that proceed to only have two statements, (a) and (b) be equivalent to each other. When is this appropriate to do rather than say "if and only if"...
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0answers
19 views

Additive and Multiplicative Effects: Terminology

I am trying to explain the relationship between several variables. The outcome of interest is $Y \in \mathbb{R}$, and the predictors of interest are $A \in \mathbb{R}$ and $B \in \mathbb{R}_{>0}$. ...
2
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0answers
94 views

When should we use `\frak{…}` in LaTeX

I saw in a note that when writing a paper some notations must be written by \frak{ }, in LaTeX, I want to know what the meaning of some notations is, and where I ...
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3answers
560 views

Is it acceptable style to mix equalities and inequalities in one line

Is this considered bad style $$2 = \sqrt{4} < \sqrt{16} = 4?$$ It seems as though this is not strictly correct, since $2 = \sqrt{4}$ is a logical proposition which represents boolean value (true ...
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22 views

In mathematical writing, what does the phase “Class of nonlinear system” mean?

In many engineering and applied mathematics resources, author specify his work or theory for "a class of Y , Y={Nonlinear systems, Hybrid systems, discontinuous systems .. etc}" what does this mean? ...
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1answer
35 views

Difference between “Let x be a …” and “Let x … be arbitrary”

What is the difference between "Let x be a ..." and "Let x ... be arbitrary"? Consider the following example: Let $R$ be an equivalence Relation on $A$. Then $\forall x, y\in A.\ [x]_R=[y]_R.$ ...
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1answer
41 views

Is it ok to make an operation over some definition at very the first time you are defining it?

Suppose we have an expression and want refer to it with some symbol. Usually one can write something like "$X:= \text{Expression}$" to mean the symbol $X$ is defined to be the expression after the ...
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1answer
36 views

When is it appropriate to use \rightarrow or \leftrightarrow

When writing papers on maths, when is it appropriate to use \rightarrow or \leftrightarrow, and when should one refrain from using them. For example, would this be appropriate use of \leftrightarrow....
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1answer
51 views

Tail driven inequality [closed]

In this empirical model, Lj is a measure of left-tail driven inequality and Rj a measure of right-tail driven inequality. It represents what in this article? What exactly is the meaning of ''left-tail ...
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3answers
66 views

Good way to state, in english, the negation of “$f$ equals zero almost everywhere”.

Let $X = (X, \mathcal{E}, \mu)$ be a measure space and let $f$ be a measurable function on $X$. Consider the statement: $f$ equals zero almost everywhere Is there an concise, unambiguous way of ...
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1answer
22 views

Best practice for naming variables to distinguish chosen upper bound from computed maximum.

Say I have a vector of a matrix's singular values $[s_1, s_2, \cdots, s_N]$, and would like to define two variables, one which holds the maximum value in the vector, and another which holds an ...
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0answers
8 views

Tree decomposition (Citation needed)

Recently, I read the statement "Fix $k\geq 1$, Any tree with at least $k$ edges may be decomposed as a union of edge-disjoint subtrees, each having between $k$ and $3k$ edges" Now I was wondering ...
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2answers
80 views

How widely known are Iverson brackets?

Here on the Mathematics Stack Exchange I recently learned about the Iverson bracket notation: $[P] = \begin{cases} 1 & \text{if } P \text{ is true;} \\ 0 & \text{otherwise} \end{cases}$ It ...
2
votes
1answer
50 views

End of step symbol

This is more of a style question. We all know to end a proof with the good old QED (I use LaTeX's $\qed$ $\square$). I have a proof that is kind of long, and I was ...
2
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0answers
18 views

Changing Notation in Direct Quote

Tried asking this on Academia.SE but the mathjax doesn't render. Direct quote: "...the estimator for the expected logarithmic return, $\hat{\mu} \equiv [\sum_1^n X_k]/h$, will have the properties ...
8
votes
1answer
90 views

English possessive in Mathematics

As a non-native English speaker, I wonder what is the correct way to use the English possessive, also known as Saxon genitive, when dealing with mathematical objects. It seems to me that there is not ...
0
votes
1answer
60 views

The use of English language in mathematics

My question is about the use of English language in mathematics. Should I write "Given a basis $b_n$ of the linear space $B_n := span\,b_n$..." or "Given a basis $b_n$ of a linear space $B_n := span\,...
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1answer
58 views

On the mathematical convention used to talk about biconditional proofs

Given $P \Longleftrightarrow Q$, the following does apply: $P \Rightarrow Q$ is equivalent to: $P$ is a sufficient condition for $Q$, $Q$ is a necessary condition for $P$. $Q \Rightarrow P$ is ...
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0answers
15 views

writing rolling formulas -e.g. average-

Question about writing a formula. Can someone verify if they are correct? Assume Rt = one-period simple return on asset in month t. The average return, Mu, is calculated over the sample from t = 1, .....
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1answer
27 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
32 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 ...
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1answer
79 views

How to simplify the equation of combination?

If there are three random variables and three related thresholds, how to simplify the following expression by summation or multiply or other operators? Thank you. $\mathbb{E}[f_1(\cdot)|H_1>\...
2
votes
2answers
137 views

Concise description of why rotation quaternions use half the angle

I'm currently writing the report on my master thesis project, where I use Android sensors to perform inertial navigation in a heavy industrial environment. In my application, I make use of quaternions ...