Tag Info

Hot answers tagged

15

Conventionally $a^{b^c}$ means $a^{(b^c)}$. The other way of parsing it, $(a^b)^c$, yields a result equal to $a^{bc}$. In particular $(2^3)^4 = 2^{3\times 4} = 2^{12} = 4096$.


7

There are a number of type-setting situations with algebraic expressions that cause problems when you try to enter them into a one-line input calculator. For example $$\frac{2+3}{4+5}$$must be entered into a calculator as $$(2+3)/(4+5)$$The horizontal line in the fraction implies brackets around the expression in the numerator and denominator. Similarly, ...


5

For real $a$ and $b$ (and in general for any $a$ and $b$ belonging to some order), $\max\{a, b\} = a$ if $a \geq b$; otherwise, $\max\{a, b\} = b$.


4

The notation means that "$11$ divides $a^2$." In other words it means there exists an integer $n$ such that $11n=a^2$.


4

Yes, it is bad style. Not everyone are used to distinguishing mentally between these variants of lower-case phi, so making a distinction will make the paper harder to read. It would be akin to making a distinction between loop-tailed $g$ and open-tailed $g$ in otherwise the same typeface/style. It will also make it harder for someone to quote and discuss ...


4

In some contexts, when direct products are defined (e.g. vector spaces, groups, modules...) given two maps $$f: A \to B \\ g : C \to D$$ one defines $$f \times g : A \times C \to B \times D$$ as $(a,c) \mapsto (f(a), g(c))$. Sometimes (when you work with modules or vector spaces) products are denoted with the symbol $\oplus$ and are called direct sums. This ...


3

For your objection, I prefer: if $u = f(x)=x^2$, then $$ \frac{du}{dx} = x^2 \qquad \text{and}\qquad f'(x)=x^2 $$ Or even $$ \frac{d}{dx}\big(x^2) = 2x $$ Then (if you like) you can avoid both $$ \frac{df}{dx} = x^2\qquad\text{and}\qquad\frac{df}{dx}(x) = 2x $$


3

It is not uncommon to use the same letter from different fonts in the same paper to denote different things. For example I could see easily somebody using $A_n$ for some matrix and $\mathfrak{A}_n$ for the alternating group on $n$ symbols. Thus, I do not think there is something wrong in principle with using $\phi$ and $\varphi$ in the same paper. For ...


3

I'm not aware of this being a common notation anywhere, especially since it can be confused with a decimal point. I think some people use it because they aren't aware that the \cdot macro exists.


3

In standard mathematics the notation $A \mathrel{^\wedge} B$ is not used; one does $A^B$ instead. In some computer science (programming) applications the infix circumflex operator notation $A \mathrel{^\wedge} B$ is used (or rather the ASCII version A^B). What you point out is that this operator $\mathrel{^\wedge}$ is not associative. So does $A ...


2

My guess would be the following: $$ f \ll g \iff f \in \mathcal{O}(g) \iff \limsup_{X \to \infty} \left | \frac{ f}{g} \right | < \infty$$ The next I'm not sure, but I'd guess he means $$ f \ll_{\epsilon} g(\epsilon) \iff \limsup_{X \to \infty} \left | \frac{ f}{g} \right | = M(\epsilon) < \infty $$ Then $$ f \asymp g \iff f \ll g\quad \& \quad ...


2

$\tan^{-1}$ denotes the inverse tangent function, AKA the arc tangent (the angle the tangent of which is the given number). When applied to an argument, you spell $$\tan^{-1}(x)=\arctan(x).$$ As far as I know, $$\tan(x)^{-1}$$ can be interpreted as the reciprocal of the tangent, i.e. the cotangent $$\frac1{\tan(x)},$$ and it is safer to write ...


2

You can't write out the elements of an interval on $\mathbb{R}$, since it is uncountable. However, one can represent an interval using set builder notation like so: $$ [a, b) = \{ x \mid a \leq x < b \} $$


2

There's the usual &. And there's: $\land$, but this is only used in propositional logic; so to use that you should make your statement too formal in shape in order for it not to look weird. It might confuse some as well because it may mean different things in different situations. Also, if you have been doing advanced math for too long, "," might ...


2

The subsets of $\{a\}$ are $\{\}=\varnothing$ and $\{a\}$, no matter what $a$ is. Here, we simply have $a=\varnothing$. On the other hand, if the question asked you for the subsets of $\varnothing$, you'd only have one subset. REMEMBER THAT $\{\varnothing\}\ne\varnothing$!!


1

One is the empty set, and other is the set containing the empty set


1

Assuming that $\mathbb{Z}_{+}$ has the standard meaning of $$\mathbb{Z}_{+}=\{\text{positive integers}\}=\{x\in\mathbb{Z}:x>0\}$$ then your answer is not quite right, since $$\begin{align*} \mathbb{Z}_{+}\mathrel{\triangle}E&=\{x\in\mathbb{Z}_{+}:x\notin E\}\cup\{x\in E:x\notin\mathbb{Z}_{+}\}\\\\ &=\{x\in\mathbb{Z}_{+}:x\not\equiv0\bmod ...


1

Judging from the RHS side of your integral I see believe you are trying to notate a standard multiple integral. In general it is fine to write $f(x)dx$ when $x$ is an element of $\mathbb{R}^n$. However, then care must be taken with how you write the "limits" of integration. First, you don't want to specify a lower and upper limit of integration if you are ...


1

I would write something like $\int_{\prod_{i=1}^{n}[z_i+\Delta z_i,z_i]}f(x) dx $ or define $Q := \prod_{i=1}^{n}[z_i+\Delta z_i,z_i]$ and write $\int_{Q}f(x) dx $.


1

Yes, this is what he means. Let $(X, \mathcal{S})$ be our measurable space, and let $\mathcal{M}$ denote the space of all signed measures on $X$ (e.g., $\mathcal{M} := \{ \mu:\mathcal{S} \to \mathbb{R} : \mu \text{ is countably additive} \}$; I'm ignoring the possibility that $\mu$ takes on one of $+ - \infty$ for conveniene). We have a natural embedding ...


1

For $f:A \rightarrow B$ $f(A)= \{ f(x) | x \in A \}$


1

These are all completely standard notations, and are used often in analytic number theory. (I've used them all at some point or another.) I would say that mathematicians working in analytic number theory would all be pretty comfortable with these notations and would not expect them to be defined in a paper. In a textbook, on the other hand, they'd probably ...


1

The semicolon separates variables from parameters. Think of it as there being a set of pdfs, each of which is defined in terms of a variable $x$, but which differ from each other in their parameter $\theta$. By setting the parameter you are basically picking out one specific pdf of all these possible pdfs. A function does not necessarily need to use its ...



Only top voted, non community-wiki answers of a minimum length are eligible