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18

It's safe to say that the math doesn't mean anything. No branch of mathematics or science or anything else uses the symbol $\{$ for an integral sign, and there are mismatched parentheses.


12

In mathematics, $x$ usually denotes a variable and $a$ denotes a (fixed) constant (however, any constant). The idea that the author wanted to give is that if you can calculate the derivative at any point, then you can consider the function that sends each point $x$ to the derivative of $f$ at $x$ (function known precisely as the derivative of $f$).


10

There's no real difference between $x$ and $a$ used as variable names. We distinguish between variables, parameters, constants - and yet this distinction does not really exist. This hierachy is just a customary order among variables, as it is customary to denote "more variable" variables with $x$ and "more constant" variables with $a$, say. Or to use $n,m,k$ ...


10

Googling fourier "a(y)" "b(y)" yields this first hit:


6

Consider the statement $x \in \bigcap_{m=1}^\infty \bigcup_{n=m}^\infty A_n$. This means exactly that $x \in \bigcup_{n=m}^\infty A_n$ for every $m$. (Here I have expanded the definition of the intersection.) This means exactly that for every $m$, $x \in A_n$ for some $n \geq m$. (Here I have expanded the definition of the union.) This means exactly that $x$ ...


4

The passage typically matrices are printed in bold is only applicable to certain fields of maths. In linear algebra, for example, the standard notation is just capital latin letter for matrices and lowercase latin letter for vectors (opinions divide whether we should add an arrow above a vector). If you want to write in bold, you can use the same approach ...


4

In complex manifold theory, I think the most common convention is to use $\Omega^p(M)$ for the space of holomorphic $p$-forms, and some other notation like $\mathscr A^p(M)$ and $\mathscr A^{p,q}(M)$ (or $\mathscr E^p(M)$ and $\mathscr E^{p,q}(M)$) for smooth forms. But if you want to stick with $\Omega^p(M)$ and $\Omega^{p,q}(M)$ for smooth forms, one ...


3

This is how the plural form of "$e_i$" is denoted, usually (I think) in the writing of European mathematicians. Consider the alternatives: $e_i$s — ugly and very likely to confuse $e_i$'s — looks like grocer's apostrophe $e_i$:s — looks odd if you're unfamiliar with it "numbers/vectors $e_i$" is what I would likely use


3

Sometimes the notation $$\hat v = \frac{v}{\|v\|_2}$$ is used. I've especially seen it on wikipedia and 3D-Graphics related articles containing math.


3

$\mathbb{Z}$ refers to the set of integers (Wikipedia link), $$\large \mathbb{Z}=\{\ldots,-2,-1,0,1,2,\ldots\}$$ $\mathbb{Z}_n$ for some number $n$, in this context, refers to the "integers modulo $n$" (Wikipedia link, notation is here but I recommend reading the full article), the set $$\large\mathbb{Z}_n=\{[0],[1],\ldots[n-1]\}\\[0.1in] {\small\text{also ...


3

Not everyone was taught what you say. I was not, for example. I was never taught how to write expressions with exponents in-line, so I never found out what the canonic meaning of x^a+b actually is. What I was taught is that whenever there is some confusion and there may exist two ways of interpreting an expression, I should use parentheses. And that is ...


3

This designates the $2$-norm of your vector. If $1<p<\infty$, then the $p$ norm of a vector $x=[x_1,x_2,\dots,x_n]\in\mathbb C^n$ is defined as $$||x||_p = \sqrt[p]{|x_1|^p + |x_2|^p + \cdots + |x_n|^p}$$ For the particular case of $p=2$, the $2$-norm of the vector is also called the Euclidean norm (and it is equal to our standard definition of ...


3

It sounds like you are looking for the notation for an iterated function. If $f$ is the function you want to iterate, you generally write $$f^n$$ for the $n^\text{th}$ iterate of $f$. (Notice that this is different from the notation for the $n^\text{th}$ derivative of $f$, which is given by $f^{(n)}$). So, for example, you could write $$f(n) = n!$$ and ...


3

If you have Word version 2007 or later, you can use: Insert > Equation And then you can choose between a lot of symbols. You can also use the command \frakturA or \fraktura, where A and a can be replaced by any letter.


2

To answer shortly: Yes, we can formalize mathematics in a 1st order language. To give lengthy extra information: I had the same questions myself two years ago so I learned how to do it. Basically you just need to study [First Order Logic] (FOL) 1, do some formalization exercises and apply this knowledge to the axioms of ZFC-Set Theory (<-- you shoul ...


2

Usually the first letters $a,b$ are used to indicate constant values, that are not specified but are intended to be fixed. The last letters $x,y,z$ are used to indicate variables, that is a symbol for a number that can have any value. In an equation we usually want to find the value of the variables, considering the other terms as fixed. But this rule is ...


2

Let $C = (1 \ \cdots \ 1) A^T$ (so it's a row vector). Then $B = C^TC$. In other words, $$B = A \mathfrak{I} A^T, $$ where $\mathfrak I$ is the $n \times n$ matrix with all entries equals 1.


1

In the end of the day, they are the same. However, the first one is better when studying the implicit/inverse function theorems... when you write $$f: \Bbb R^n \times \Bbb R^m \to \Bbb R,$$ your function gets two arguments, $f({\bf x},{\bf y})$ with ${\bf x}\in \Bbb R^n$ and ${\bf y} \in \Bbb R^m$. If you write $$f: \Bbb R^{n+m}\to \Bbb R,$$ your function ...


1

For $f=f(x,y)$ the first definition $$ f:\mathbb{R}^m\times\mathbb{R}^n\to \mathbb{R} $$ is correct. The second definition $$ f:\mathbb{R}^{m+n}\to \mathbb{R} $$ drops the grouping information for the arguments: it says $f = f(x)$ which is a function of arity 1 while your given $f(x,y)$ has arity 2. I would consider these to be different type signature ...


1

Note that, perhaps somewhat counterintuitively, there is no actual difference between "a fixed number $z$" and "a variable $z$". Both mean exactly the same thing: for an arbitrary element $z$ in the domain of $f$, $f'(z)$ is defined to be such-and-such real number. Logically "fixed numbers" and "variables" have the same semantics and correspond to universal ...


1

If you think back to the time when you were in 9th grade learning to solve quadratic equations, what you saw was that $$ \text{if } ax^2+bx+c=0\text{ then }x=\frac{-b\pm\sqrt{b^2-4ac\,{}}}{2a} $$ (not to be confused with $\dfrac{-b\pm\sqrt{b^2-4a} c}{2a}$ or $\dfrac{-b\pm\sqrt{b^2-4} ac}{2a}$ or $\dfrac{-b\pm\sqrt{b^2-{}} 4ac}{2a}$, etc., all of which I've ...


1

$B_i$ gives the set of player i's optimal (i.e. utility-maximising) actions in response to a given profile of opponents' actions. ⇒ denotes a (set-valued) correspondence rather than a (single-valued) function. The utility function $u_i$ is one of the primitives of a normal-form game, mapping action profiles (such as $(s_i,s_{-i})$ into real numbers.


1

Presumably $B_i(s_{-i})$ is a best response (or possibly the set of best responses) by player $i$ when the others play $s_{-i}$. In a collection of game theory notation the set is called $BR_i(s_{-i})$. As it is the response to a particular play $s_{-i}$ by the others, it is reasonable for that to be an argument. I suspect $\Rightarrow$ may just be a ...


1

The characteristic function of a set $S$ is $$\chi_S(x) = \begin{cases}1,& x\in S\\0,& x\notin S\end{cases}.$$ So what you're looking for is $$\sup_{x\in\mathcal U}\chi_{A\cap B}(x),$$ where $\mathcal U$ is our universal set (i.e. $A,B\subset\mathcal U$).


1

I do not know of any specific notation for this. But what about $$A\cap B\ne \emptyset$$ This statement is true if $A$ and $B$ have any overlap, false otherwise. This returns true for any overlap, including edges and vertices. If you want to exclude those, use the notation for the interior of the polygons, ...


1

I found it used in Communicating Sequentional Processes, by C. A. R. Hoare. In it, he defines the symbol as (page xvii) (between traces) followed by $$\text{For example, } \langle a,b,c \rangle=\langle a,b \rangle \text{ (symbol) } \langle \rangle \text{ (symbol) } \langle c \rangle$$


1

You should be able to do this using inference rules like: $$\frac{\Gamma \vdash \forall x:X, P(x)}{\Gamma, x:X \vdash P(x)}$$ You'll need to settle on a syntax for making definitions. I don't know the details, but perhaps check out Principia mathematicia, which is written almost entirely in this style.


1

To add to Strants' answer, which indeed gives the most common way of writing "do $f$ to an argument $n$ times", I'll give the "expanded form" of this which is $$f^n(x) \equiv f(f(\stackrel{(n)}{\cdots}f(x))$$ This is cumbersome and I strongly recommend using the compact notation, even if it means defining a new function. However it might suit your purposes ...


1

Let $f : \mathcal{M} \to \{0,1\}$ be defined as $$f(m) = \begin{cases}1 &\text{if } \mathrm{value}(m) \neq 0 \\ 0 &\text{otherwise}\end{cases},$$ then your average is $$\frac{\sum_{m \in \mathcal{M}}\mathrm{value}(m)}{\sum_{m \in \mathcal{M}}f(m)}.$$ When you write it like below (which gives the same result because of how $f$ is defined): ...


1

Let $m_i$ be non-negative integers such that $1 \leq i leq 5$. Let $P$ be the set of $m_i$ such that $m_i > 0$ i.e. $P = \{ m_i | m_i > 0 \}$. The average can be written as $$ A(m) = \frac{1}{|P|} \sum_{m_i \in P} m_i $$ where $A(m)$ is defined to be something for the empty set.



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