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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3
votes
1answer
79 views

What does the notation $\mathbf{R}^\mathbf{R}$ mean?

I was reading the Princeton Review of GRE math subject test (4th edition), and one question was (page. 251) Example 6.24 Is the ring $\mathbf{R}^\mathbf{R}$ an integral domain? ...
1
vote
2answers
55 views

Notation issue - Asymptotic behaviour: is $\sim$ too restrictive?

As a student I am completely unable to understand unambiguously what is meant by a notation such as $$f \sim g $$ when in Physics the behaviour of two functions at infinity is evaluated. I found a ...
1
vote
2answers
75 views

Differential $dx$

I have some trouble understanding a thing. I will reproduce two texts from two different books. In the first, the author defines the function $T:\mathbb{R}\longrightarrow \mathbb{R}$, ...
0
votes
1answer
40 views

Equation that defines multi-dimensional polynomial

In two-dimensions a complete n-th degree polynomial is given by $P_n(x,y) = \sum_{k=0}^{n}\alpha_kx^iy^j \qquad i+j \leq k \qquad (1)$ . However, now I am dealing with the following two-dimensional ...
5
votes
1answer
101 views

What does a single-line superscript left arrow mean?

I'm pretty sure it's a limit but I haven't been able to find any page explaining this notation (see below). It's from a paper on block maxima. 3 out of 5 occurences: $V=(-1/logF)^\leftarrow$ (p.4) ...
3
votes
2answers
27 views

Set Notation help?

If $A=\{a_1,a_2,\dots,a_7\}$ and we want to know how many $3$ element subsets exist in $A$, would we simply use ${7\choose3}=35$ on a calculator, or does this notation not account for the empty set, ...
2
votes
2answers
59 views

I've been told that writing $x\equiv a,b,c \pmod d$ is abuse of notation, is it really?

I've been told that writing $x\equiv a,b,c \pmod d$ is abuse of notation, and that I should always write: $$ x\equiv a\pmod {d}\text{ or }x\equiv b\pmod {d}\text{ or }x\equiv c\pmod {d} $$ How true ...
0
votes
1answer
18 views

Argmax as the parameter of a function?

Let $P = \{p_1,\dots,p_n\}$ and $Q = \{q_1,\dots,q_n\}$ be two sets of points and $d(p,q)$ a distance function between points. Given an element $p_k$ I would like to know which is the maximum distance ...
1
vote
1answer
29 views

Notation for a statistic, or function of a random variable

A statistic is a function of random variables, so it is also a random variable. Suppose we have a collection $X = (X_1, X_2, \dots, X_n)$, where $X:\Omega \to \mathcal{X}^n$. There are two common ...
1
vote
3answers
42 views

Strict ceiling and floor notation

The normal ceiling and floor functions, denoted $\lceil x \rceil$ and $\lfloor x \rfloor$ respectively, refer to the smallest integer greater than or equal to $x$, and similar for the floor function. ...
1
vote
1answer
43 views

Why using $v^T \cdot u\text{ instead of simply } v \cdot u$?

I know they are equivalent, but why and when should we prefer using $v^T \cdot u$ instead of simply $v \cdot u$, when $v$ and $u$ are vectors of $\mathbb{R}^m$, for example?
3
votes
2answers
66 views

Why do they use absolute value symbols for $|z|=r$ considering any number squared is positive?

Am I right in saying that the absolute value symbols act like a function such that if $x$, for example, is $x<0$ then $x=-x$ In other words $x$ will be positive regardless of what value you give ...
0
votes
2answers
39 views

How should trigonometric expressions be simplified?

I have been learning trigonometric identities and I am having trouble understanding how they should be applied to simplify expressions. For example, the expression $2\sin{x} \cos{x}$ is equal to ...
0
votes
2answers
53 views

How should this equation be read, $|z+1+3i|=|z-5-7i|$

$$|z+1+3i|=|z-5-7i|$$ $z$ represents a complex number right? Then if $$|z+1+3i|=0$$ $${\implies}|z|=|-1-3i|$$ In which sense does this $$|z+1+3i|=|z-5-7i|$$ imply, $$\implies|-4-4i|$$ But $z$ has ...
1
vote
3answers
49 views

What's the difference between $\sum_{r=1}^n(ar+b)$ and $\sum_{r=1}^nar+b$

Does $$\sum_{r=1}^n(ar+b)=\sum_{r=1}^nar+b$$ or does $$\sum_{r=1}^n(ar+b)=\sum_{r=1}^nar+\sum_{r=1}^nb$$ If I'm given $u_r=ar+b$ how would I substitute that into $$\sum_{r=1}^nu_r$$ Does that mean ...
0
votes
1answer
27 views

Homomorphism Notation

I have a question on notation. The question in my text: A function $f:\mathbb{R}\rightarrow\mathbb{R}^{\times}$ is a homomorphism if and only if $f(x+y)=f(x)+f(y)$ for all $x,y\in \mathbb{R}$. My ...
0
votes
3answers
40 views

What does this notation mean: $\tiny\left(\begin{matrix} y2-N \\ d_f -c, d_f-c \end{matrix}\right)$?

Again, the notation is: $p_{split} = \left(\begin{matrix} y_2-N \\ d_f -c, d_f-c \end{matrix}\right)\left(\begin{matrix} 2N-y_2 \\ c,c \end{matrix}\right)\left(\begin{matrix} N \\ d_f, d_f ...
0
votes
1answer
12 views

Notation to define a function mapping from a vector to a two-dimensional matrix

I have a set $\mathcal{D}$, and I'm trying to define a mapping from that set to a two-dimensional matrix where each location contains either a $1$ or $0$. The notation I am using is ...
0
votes
2answers
65 views

How to remember various set operations very easily?

I need an way to remember the set operations very easily. Does anybody have any idea? For example, how do you remember the distinction between Set-Intersection and Set-difference? I regularly mess ...
3
votes
2answers
75 views

Why do we think of group compositions as multiplication?

This has bothered me for some time: The composition in a group is usually denoted $xy$ or $x\cdot y$. Powers (note the word) are denoted by $x^n$, inverses by $x^{-1}$, and the neutral element by $1$. ...
1
vote
2answers
22 views

Notational question

We are in the framework of measurable transformations, i.e. let $(X,\mathcal{B},m)$ be a measure space and let $T:X\to X$ be a measurable transformation. In your opinion, what does the following ...
2
votes
1answer
39 views

What it means SO* (2N)?

I'm puzzled about the $"*"$ in the following notation for Lie groups: $SO^* (2N)$ or $SU^* (2N)$. I don't understand what is the meaning of this notation. It is introduced for example in Gilmore ...
2
votes
0answers
16 views

How to refer a block in an image?

I have an image $I$ with size $X,Y$. I want to refer to a varticle stripe of the image between column $S_1$ and $S_2$. In matlab we write it as, $B = I(:,S_1:S_2)$ But how to write in mathematical ...
1
vote
0answers
21 views

What is the origin of the use of $\Pi$ and $\Sigma$ for dependent function and dependent product types in type theory? [duplicate]

In the type theory I have read (e.g. homotopy type theory) I have seen the following notion used for dependent function types: $$\prod_{x : A} B(x)$$ and the following for dependent product types: ...
0
votes
1answer
15 views

How to report significant digits in coefficient of determination?

Say that I fit some data with some model, for instance a linear function $y = mx+b$. What is the proper way to report the fitted coefficients and the goodness of fit? Specifically, if I do the fit in ...
0
votes
1answer
38 views

Is there a name for the integral used to define square integrable functions?

A square integrable function on $\mathbb{R}^n$ would be defined as a function $f(\mathbf{x})$ satisfying the constraint $$ \int |f(\mathbf{x})|^2 d^n\mathbf{x} < \infty, $$ for $\mathbf{x} \in ...
22
votes
7answers
3k views

Is $\exp(x)$ the same as $e^x$?

For homework I have to find the derivative of $\text {exp}(6x^5+4x^3)$ but I am not sure if this is equivalent to $e^{6x^5+4x^3}$ If there is a difference, what do I do to calculate the derivative of ...
0
votes
1answer
17 views

Terminology for when a variable is implicitly a member of some set?

I have sets $N = \{1, \ldots, n\}$ and $M = \{1, \ldots, m\}$. When referring to a generic element of these sets, I typically use variables $i \in N$ and $j \in M$. Is there any standard ...
1
vote
0answers
14 views

How to create a set with the results of a function?

Let $A$ and $B$ be sets of elements and $f()$ be a function that according to a condition return one or more associations $(a_k,b_k)$ that comply that condition otherwise if there is not any ...
1
vote
1answer
34 views

Mixing definite and indefinite integrals

If I have the differential equation $\frac{d^2 y}{dx^2} = f(x)$ and integrate once using indefinite integrals $\frac{d y}{dx} = c_1 + \int f(x) dx $ then apply the boundary condition $\frac{d ...
2
votes
1answer
36 views

Notation help abstract algebra

If $R$ is a ring and $x$ is an indeterminate, what does the notation $R(x)$ mean? I've seen $R[x]$ but not $R(x)$.
0
votes
0answers
22 views

Switching between functions based on last values.

I have a problem in which I have three possible states "Heating", "Off", and "Cooling". Now my system can always only have one of these states at a time but may switch between them over time. To know ...
0
votes
0answers
28 views

What is the meaning of Lagrange Multiplier in this formula?

Consider $$F = L'VL + (L' X − K' )\Phi$$ $L$, $V$ and $K$ are matrices. $F$: function notation $\Phi$: Lagrange Multiplier, imposes the restriction $L' X = K'$ What does Lagrange Multiplier mean in ...
1
vote
0answers
32 views

Question on notation (topology & fiber bundles)

This is a very elementary question but I can't quite seem to track down a worthwhile source, so I was hoping someone more knowledgeable than I could lend their superiority. In Moore & Schochet's ...
0
votes
2answers
38 views

Scalar notation to vector notation for a system of equations

I have a ($1 \times n$) row vector $\boldsymbol{x}$, an ($n \times n$) matrix $\mathbf{F}$, and an ($n \times n \times n$) tensor $\mathbf{Q}$. I also have a system of equations that reads ...
0
votes
0answers
26 views

Equality notation query

I want to say that $a=b$ and $c=d$ but $a\ne c$. Is this a valid expression: $a=b\ne c=d$?
2
votes
3answers
82 views

What does this notation mean? $x \mapsto f(x)$

What does this notation mean? $x \mapsto f(x)$ I've seen it at the beginning of functions but don't know what it is.
0
votes
1answer
32 views

Set notation - products of subsets

In terms of conventional set notation, a set and it's corresponding power set of cardinality $2$ can be defined: \begin{align} &A&=\quad&\{a,b,c,d\}&\\ ...
2
votes
2answers
30 views

Which rule is applied to define the operator precedence for factorial

Please apologize the question, I struggled with finding a good formulation in the first place: Looking at $\binom{2n}{k}$ it is very clear that for n,k integer and n>k we can solve it by calculating: ...
0
votes
1answer
35 views

Exponentiation for hash function & associativity

Some cryptographic papers use $H^n(x)$ to mean $H(H^{n-1}(x))$ where $H^0(x) = x$ and $H$ is a cryptographic hash. So $H^3(x)$ would be $H(H(H(x)))$. Is this definition formally correct? It seems to ...
0
votes
0answers
34 views

How to denote the minimal bounding curve of two intersecting function curves?

If there are two functions $y=f(x)$ and $y=g(x)$, $x,y\in\mathbb{R}$, how could I denote the minimal bounding curve of these functions? (See the green dotted line on the figure.) I'm looking for an ...
0
votes
1answer
42 views

Notation for a derivative

I am interested if there is notation for a derivative that is in between a total derivative and partial derivative. The total derivative of $f(t,x,y)$ with respect to $t$ is $$ ...
0
votes
0answers
53 views

'Is true' in maths notation?

I want to write the following statement in only maths notation without any words: $$\sum^n_{r=1}r=0.5n(n+1)$$ is true for all positive integers n. This is what I have got so far. ...
1
vote
0answers
37 views

Product over real interval? Is there a better way of putting this?

In my amateur interest, I have arrived at this (nothing rigorous here at all):$$\prod_{a\in [1,2]} \prod_{b=0}^\infty f(a,b) \neq 0$$ For starters, there might be a more intuitive way about doing ...
0
votes
1answer
49 views

Is there a notation for “open subset”

I very often have to write something like: $\exists U,V\subseteq M$ where $U,V$ are open, but there's no short hand for it. On my written notes, I do tend to write something like: $\exists ...
2
votes
3answers
35 views

If $f$ is a uniformly continuous function show that $g = f(x) - f(y)$ is uniformly continuous on all of $\mathbb{R}^2$

Problem statement: Let $f: \mathbb{R} \to \mathbb{R}$ be a uniformly continuous function on $\mathbb{R}$ and let $g: \mathbb{R}^2 \to \mathbb{R}$ be defined by $f(x) - f(y)$. Then $g$ is uniformly ...
0
votes
4answers
110 views

Which is more preferable to write $\log(x)$ or $\ln(x)$ [duplicate]

Which one is more preferable to write when you are writing an exam. Is it $\log(x)$ which denotes the natural logarithm or is it $\ln(x).$
2
votes
1answer
86 views

What does the sign “$=$” exact meanings?

How can I understand the sign "$=$" from the following expression: $$\mathcal{o}f((x))=\mathcal{o}f((x))+\mathcal{o}f((x));$$ $$\mathcal{o}(kf((x)))=\mathcal{o}(f(x));$$ ...
0
votes
0answers
33 views

Is there a notation for a set of angles?

Suppose that $f:[0,180]\to[-1,1]$ given by $\theta\mapsto\cos\theta$ I wondered whether there was a notation for this domain, the principle values of a trig function? For example we would use ...
4
votes
3answers
123 views

Integration by substitution notation question

Often with integration by substitution I see (and use) the notation $ x \to \frac{\pi}{2} - x $, for the simple reason that I don't have to rename the variable that I am integrating with respect to, ...