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1

When working with fractions, it's important to understand two separate but interrelated issues: expression in lowest terms, and canonical expression. If both the numerator and denominator are positive, the fraction represents a positive number, but depending on your calculations to get that fraction, you might or might not have it in lowest terms. To give a ...


2

In his famous paper How to write mathematics, P.R. Halmos says the following about "where" "Where" is usually a sign of a lazy afterthought that should have been thought through before. "If $n$ is sufficiently large, then $|a_n| < \varepsilon$, where $\varepsilon$ is a preassigned positive number"; That being said it is common to write $x = ...


0

The general answer is "no", and the reason is more or less for human readability. Proofs are by and large written in paragraph form. Sure there will be the occasional conditional or equality chain, but such things are still often surrounded by introductory and followup text. It turns out that it's often easier to transmit a series of thoughts when they are ...


2

There's not really anything that makes a symbol "official" or not, and there are probably authors out there somewhere who do use a symbol of their own devising for this. However, there is certainly no widely used and understood symbolic way to write what you want. Mathematicians in general seem to think that using prose (such as the word "where") for this ...


0

The use of natural language is often more effective when presenting an idea. In my opinion the less often a symbol is used where a few words can go, the better. That said, there are many places where symbols are useful and simplify matters. The word "where" can often be replaced with "such that", and corresponding to this we have a few regularly used ...


2

This symbol is used to indicate a line integral along a closed loop. if the loop is the boundary of a compact region $\Omega$ we use also the symbol $ \int_{\delta \Omega} $ we can generalize such notation to the boundary of a region in an n-dimensional space and, if $\Omega$ is an orientable manifold we have the generalized Stokes' theorem $$ \int_{\delta ...


1

$\mu^2 (n)$ denotes the squared Moebius function. The typical Moebius function returns values of $-1, 0, 1$; the squared Moebius function is obtained by simply squaring this result. So, we can break it down like this: $\mu^2(n) = 1$ if $n$ is square free. $\mu^2(n) = 0$ if $n$ is not.


1

Depends on your style guide. MLA Style requires the zero, as does US GPO style. APA Style uses the zero before the decimal point if and only if it's possible for the thing being measured to be greater than one. So a child could be “0.99 m” tall, but a probability could be “.99”. Wikipedia requires the zero except for sports ...


0

Basically what you are telling means a norm. So what is a norm, A norm is basically a function on a given vector space $V$ over the field $F$ of complex numbers. So a norm on $V$ is a function $p:V\rightarrow \Bbb{R}$ having the following properties: 1) $p(av)=|a|p(v)$ 2) $p(u+v)\leq p(u)+p(v)$ 3) If $p(v)=0$ $\iff$ $v$ is the zero vector. Where $a \in F ...


2

In regular vectors, they're generally equivalent. (note: "regular" here isn't a special subset of vectors, it's referring to the common meaning of "vector") However, there are vector spaces that aren't just a list of numbers in the way that regular vectors are. And in some of these vector spaces, the absolute value of something can be more vague. For ...


3

The conclusion from the back-and-forth discussion in the comments appears to be that with regard to your concepts of "already squared" and "should be squared in the future" the superscript $2$ in expressions such as $$\sin^2 \theta, \quad 5^2,\quad \mbox{or}\quad h^2$$ means "should be squared in the future," with no exceptions. As explained in comments and ...


2

I think your confusion comes from this notion of time that you are applying to expressions. There is no 'already squared' or 'will be squared', only 'squared'. When one says $\sin^2 \theta$, they mean the value of the sine of theta times the value of the sine of thetea and nothing more. If I am understaning you correctly, the $^2$ notation never signifies ...


1

You're almost correct. $$\|u\|_{C^k(\bar\Omega)} = \sum_{|\alpha|\le k} \|\partial^\alpha u\|_\infty$$ Or (equivalent as a norm) $$\|u\|_{C^k(\bar\Omega)} = \sup_{|\alpha|\le k} \|\partial^\alpha u\|_\infty$$ where $\alpha\in\mathbb N_0^n$ is a multiindex and $\partial^\alpha$ is defined as usual.


3

You might want to ask yourself what is meant when you put a minus sign in front of something. The answer could be, for example, that $-\frac ab$ is the solution to the equation $$x+\frac ab = 0$$ where the unknown is $x$. Now, you want to know whether $\frac {-a}b$ is the same as $-\frac ab$. Just check if it solves the equation! $$\frac {-a}b + \frac ab = ...


1

As vadium123 remarks in the comments, these are all the same number. To say that it has a "negative denominator" or a "negative numerator" or that the "whole fraction" is negative is not mathematically meaningful. A number is either negative or it isn't, and the number in question is $- \frac{1}{2}$. Since dividing $-1$ by $2$ and dividing $1$ by $-2$ both ...


3

The value is the same for all three versions, although the value might be seen as the result of a different operation: \begin{align} \frac{-a}{b} &= \frac{(-a)}{b} = (-a) / b = -(a / b) \\ -\frac{a}{b} &= -\left(\frac{a}{b}\right) = -(a / b) \\ \frac{a}{-b} &= \frac{a}{(-b)} = a /(-b) = - (a / b) \\ \end{align} The middle version is the most ...


17

The reason some authors choose to use $\| \cdot \|$ notation instead of $| \cdot |$ is to better distinguish between vectors and scalars. For example, writing the identity $|kv|=|k||v|$ is somewhat ambiguous whereas $\|kv\| = |k|\|v\|$ is not.


5

That's a common symbol for a norm in mathematics. I guess in your case it means the length of the vector. For more information about norms in general, see Normed vector space (Wikipedia).


2

The notation $\{0,1\}^*$ refers to the space of finite strings in the alphabet $\{0,1\}$, including the empty string.


1

The random variable $X$ is a function whereas the Expectation is a functional. Mathematicians adopt the use of square brackets for functionals. This is why.


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 ...


0

A linear map $f$ is continuous, and so is the function $y\mapsto\|y\|$. Therefore the real-valued function $\phi(x):=\|f(x)\|$ is bounded on the compact set $S^{d-1}\subset{\mathbb R}^d$ (the unit sphere): There is an $a>0$ such that $\bigl|\phi(x)\bigr|\leq a$ for all $x\in S^{d-1}$. From the linearity of $f$ it then follows that $\|f(x)\|\leq a\|x\|$ ...


1

When working with linear maps, one frequently omits brackets, for brevity. This shouldn't surprise you, after all we also write $\log x$ or $\sin x$ instead of $\log(x)$ and $\sin(x)$.


0

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.


1

Since an average is a quotient of a sum and the number of samples (which we can itself write as sum), one way to encode this to incorporate the condition into the sum. For example, if we denote the month by $a$ and the value for the month $a$ by $v_a$, we can write $$\text{(average)} = \frac{\sum_{v_a \neq 0} v_a}{\sum_{v_a \neq 0} 1}.$$ The denominator here ...


0

It's a little awkward, but perhaps you could use $\not\perp$, because, according to this Wikipedia page: the notation $a \perp b$ is sometimes used to indicate that $a$ and $b$ are relatively prime a usage which could be extended to a general lattice (unless the symbol $\bot$ was already in use to denote the bottom element, which would not be a problem ...


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, ...


0

2^-2 can only be interpreted one way, because the minus sign is next to the second argument, and the exponentiation sign isn't. It's not a matter of operator precedence. It's only when a parameter has an operator on each side that we have to use precedence to decide. For a concrete example, FORTRAN has an exponential operator built into the language, and ...


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 ...


2

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 ...


0

You can use Knuth's up-arrow notation: $a\uparrow n=a^n$ $a\uparrow \uparrow n=a^{a^{a^{\dots}}} $ n times $a\uparrow \uparrow \uparrow n= a\uparrow \uparrow a\uparrow \uparrow a\uparrow \uparrow \dots$ n times and in general $a\uparrow ^b n=a\uparrow ^{b-1} a$ iterated n times


4

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 ...


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 $$\text{For example, } \langle a,b,c \rangle=\langle a,b \rangle \text{ (symbol) } \langle \rangle \text{ (symbol) } \langle c \rangle$$


0

Answered in the comments: http://en.wikipedia.org/wiki/Adjoint_representation_of_a_Lie_group – Qiaochu Yuan Apr 23 '11 at 18:27 I'm pretty sure that @Qiaochu's right. I'd even go so far as to say that the motivation stems from the finite-dimensional case $A=M_n(\mathbb C)$, where the unitary group $U(n)=\mathcal U(A)$ leaves the subspace of ...


1

$\newcommand{\Reals}{\mathbf{R}}\newcommand{\Param}{\mathbf{x}}$You're perfectly correct that $(x_{1}, \dots, x_{n})$ is being used to denote coordinates both in $M$ and in $U$ by identifying a point $q$ in $U$ with its image $\Param(q)$ in $M$. Keep in mind there are similar (but distinct) abuses of notation throughout calculus that are probably ...


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


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 $x$ ...


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.


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.


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 ...


-2

Say you have a program p that returns the input plus one. it could be defined like this: p(int i){ i=i+1; return i; } p(int i) is like f'(x): it's a function that takes one input. Now, say you have another program that calls p at some point. for instance, q(int i, int j){ s=i+j; return p(s); } p(s) is like f'(a). It's a ...


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 ...


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 $n,m,k$ ...


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 $f$).


12

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


0

Personally, I would use a notation $\mathbb{C^0} \times \mathbb{H}^{ \frac{1}{2}}$ because the function has two arguments.



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