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0

With regard to using brackets in your notation as Iverson brackets, it's fine so long as you make it clear that that is what they are. The most important thing is that the meaning of the equations/expressions in your mathematics is clear to the reader, and even though you could use Iverson brackets to express something, there is often a clearer way without ...


0

Yes, $\max(1,3)$ means the biggest number from both and $\min(1,3)$ means the smallest number.


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, $\max\{a, b\} = b$.


0

I will just provide short answers and a link to wikipedia. 1.Antiderivative and Integral: $\int$ is used to denote integration. Usually written as $$\int f(x)\, \text{d}x$$ the so called antiderivative. Or as: $$\int_a^b f(x)\, \text{d}x,$$ this is the so called definite case. $\int\int\int$ is used to denote multiple(3) applications of this. see above ...


0

It usually means the image of A. Set-theorists use a different notation. The problem is that a member of a set A can also be a subset of A. So if $B \subset A=$ domain$(f)$ and $B \in A$ then "$f(B)$" is ambiguous.And "$fB$" for the image of $B$ is inadequate because if you want the image of $B \cap C$, then $fB \cap C$ and $f(B \cap C)$ are both ambiguous. ...


0

$f(A)$ is the image of the function. Sometimes also called the range of the function in high school. It is the set of element $b\in B$ so that $b=f(a)$ for some $a\in A$. for example, if $f(x)=x^2$ is defined with domain $\mathbb R$ and co-domain $\mathbb R$ then $f(\mathbb R)$ is $[0,\infty)$, that is, the image of $\mathbb R$ is the set of non-negative ...


1

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


5

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


0

$\tan^{-1}$ would normally denote the FUNCTION $\mathbb{R}\longrightarrow (-\frac{\pi}{2}, \frac{\pi}{2})$ given by sending $x$ to the unique $y$ satisfying $\tan(y)=x$, while $\tan(x)^{-1}$ presumably stands for the real NUMBER, which is the multiplicative inverse $\frac{1}{\tan x}$ and is well-defined whenever $\tan(x)\neq 0$, i.e. whenever $x$ is not an ...


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 ...


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 ...


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 ...


0

Given $$|z-y|\leq |1-\bar{y}z|\Rightarrow |z-y|^2\leq |1-\bar{y}z|^2$$ Now Using the formula $\bullet \; |z|^2 = z\bar{z}$ and $\displaystyle \bar{\bar{z}} = z$ So we get $$(z-y)\cdot (\bar{z}-\bar{y})\leq (1-\bar{y}z)(1-y\bar{z})$$ So we get $$z\bar{z}-y\bar{z}-\bar{y}z+y\bar{y}\leq 1-\bar{y}z-y\bar{z}+y\bar{y}z\bar{z}$$ So we get $$|z|^2+|y|^2\leq ...


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 ...


0

"tends to" is different. I prefer the 'equivalency' sign with an arrow beneath. Not only does it convey 'tends to', it specifies 'what' is approaching 'what.' The 'equal' sign with a scroll beneath does not.


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, ...


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 $(2^3)^4 = 2^{3\times 4} = 2^{12} = 4096$.


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 ...


0

The authors should have explained their notation, so shame on Bourgain and Fuchs. I've never seen something like $\leq_{\epsilon}$.


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 ...


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.


-2

$x \ll y$ means $x$ is much less than $y$ $x \gg y$ means $x$ is much greater than $y$ $x \ll \varepsilon$ means $x$ is much less than $\varepsilon$ $x \gg \varepsilon$ means $x$ is much greater than $\varepsilon$ $≍$ -not sure about this $\sim$ stands for an poor approximation. $\approx$ stands for a close approximation Basically, the symbol is used ...


0

$$[a,b]=\{a,a+1,\cdots b\}.$$ $$(a,b]=\{a+1,a+2,\cdots b\}.$$ $$[a,b)=\{a,a+1,\cdots b-1\}.$$ $$(a,b)=\{a+1,a+2,\cdots b-1\}.$$


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: $$ [a, b) = \{ x \mid a \leq x < b \} $$


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 $$ \frac{df}{dx} = x^2\qquad\text{and}\qquad\frac{df}{dx}(x) = 2x $$


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 ...


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 ...


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 ...


0

Edit. You can find two notations in the literature. The notation $V^{\leqslant n}$, suggested by Zhen Lin and the notation $(1 + V)^n$ (or $(1 \cup V)^n$ if you prefer the union symbol). The latter one does not need to introduce any definition, but the first one is rather intuitive.


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 $.


0

Generally, I see something like the following: The identity element is $0^n$, a string of $n$ consecutive zeros. The important thing here is not the $0^n$, which is an arbitrary choice of notation (and could be any of the other choices you gave, or still others), but the exposition, which explains, more or less clearly, what the identity element ...


0

Wedge product of two $0$-forms (AKA smooth functions $X \to \mathbb{R}$) is ordinary multiplication. This could be what they mean.


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 ...


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 ...


0

That depends on the probability you are considering (I know, typical useless mathematician-answer). Usually one has $f: V \to [0, +\infty)$ or at least $f: V \to [0,+\infty]$. However, for this to work we would need our probability measure to be continuous. If you are more on the physical side of life, then you probably want to also consider the delta ...


2

It seems that it is Einstein's sum notation. So you have actually $A = (a_{ij})_{1\le i, j \le 3}$ with $$ a_{ij} = \sigma_{ij} + \sum_{k=1}^3 \left( \sigma_{ik} w_{kj} - w_{ik} \sigma_{kj} \right), $$ or $$ A = \Sigma + \Sigma W - W \Sigma, $$ with $\Sigma = (\sigma_{ij})_{1\le i, j \le 3}$ and $W = (w_{ij})_{1\le i, j \le 3}$.


2

I don't know about the specific equation in hand but a common way to represent matrix elements is to using $\sigma_{ij}$ to mean the element of a matrix, say $\Sigma$ at the $i$th row and the $j$th column. Then $\sigma_{ik}\omega_{kj}$ is the product of the element of $\Sigma$ matrix at the $i$th row and the $k$th column and that of $\Omega$ matrix at the ...


1

I'm not sure if this is what you want but perhaps you want to the codomain to be: $ \Delta(\Pi_{t\in T} F(x,y(t)))$ or $\Pi_{t\in T}\Delta (F(x,y(t)))$ The first is more general since it allows correlation across time. The second is more restrictive because assumes independence (and in this case we may have to assume $T$ discrete as there are problems in ...


0

I haven't encountered such truncating function, but you're basically getting a submatrix of a matrix. So let me say the following. In Horn & Johnson's Matrix Analysis, they introduced the notation: Given $A\in M_{n}$, if $\alpha,\beta\subseteq\{1,\ldots,n\}$, then $A[\alpha, \beta]$ denotes the submatrix of entries that lie in the rows of $A$ indexed by ...


1

In the context where I have encountered this notation, it meant that $n$ is a lot smaller than $m$. As opposed to, for example, smaller but arbitrarily close to $m$.


2

It depends on context (just ask a C or C++ programmer) but in mathematics and physics it usually denotes that $n$ is much smaller than $m$. It is a vague notion, but useful at times.


2

This symbol means "proportional to". Generally it might be taken to include the case of a negative constant of proportionality, but here the minus sign would seem to indicate that the intended meaning is with a positive constant of proportionality. The right-hand side of the implication then follows in view of equation $(2.20)$ in the book, taking into ...


5

Let $E_k = \{x \in X: \vert f_k(x) - f(x) \vert \geq q\}$ for all $k \geq 1$.The set you ask is simply $$\bigcup_{k \geq n} E_k$$ Note that $x \in \bigcup_{k \geq n} E_k$ if and only if $x \in E_k$ for some $k \geq n$ if and only if $$\vert f_k(x) - f(x) \vert \geq q \text{ for some } k \geq n$$ Therefore the set $\bigcup_{k \geq n} E_k$ is simply ...


0

The general rule for mathematical notation is that a variable should be denoted by a single italic letter, possibly with embellishments. Letters used for purposes other than denoting variables, such as descriptive subscripts (as in your example 3) or the names of standard mathematical functions, should be in roman type. The convention is that juxtaposition ...


0

I've learned in elementary school that $\mathbb{R}_{*}$ means the set without the zero, so $\mathbb{R}^{+}=[0,\infty)$ and $\mathbb{R}^{+}_{*}=(0,\infty)$.


1

I think it depends a lot on the context. For a one-off formula, you're often better off spelling it all out. You don't have to explain what all of the terms mean. But in a situation where various values, especially in different formulas, are related to each other (as, say, $p_\text{market}$ is to $p_\text{trade}$), the use of subscripts on a common ...


0

First you can/should define the variables, which you want to use in a formula. In general you can define a variable as you like. Here are my suggestions: $p$=price of the good $q$=quantity of the good $r$=revenue Now you can simply write down the formula for the revenue. $r=p\cdot q$ For the price I would superscript the abbreviations for the market ...



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