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1

In the example you provide, it seems pretty clear that $ab$ denotes standard concatenation of letters. The symbol $\Sigma$ is usually used to designate an alphabet (a non-empty set of letters). You defined $\Sigma$ to be the set of the two letters $blue$ and $green$. Thus $bluegreen\subset L$. If $a^R$ is defined the usual way (the reversal of a word is ...


1

I've usually seen that notation mean "restricted to", which isn't too far from "evaluated at" in meaning, but it is more general. For instance, $$ \left. f\right|_{[0,1]} $$ could be interpreted to be a function with values that agrees with $f$'s values, but is only defined on the interval $[0,1]$. One advantage is that you're not dependant on a function ...


2

Formally, elements of $A^*$ are finite sequences of $A$, that is, functions $\{1,2,\dots,n\}\to A$ for some $n\in\Bbb N$. Then, if $w=(a_1,\dots,a_{n})$ then we define $$w^R:= (a_n,\dots,a_1):=i\mapsto a_{n+1-i}$$


2

If $w=a_1a_2\cdots a_n$, then $w^R=a_na_{n-1}\cdots a_2a_1$


1

There is another viewpoint different from the answer of @Travis, in which there is no ambiguity up to conjugacy. Namely, it is common to put the base point into the notation for the map $f$ itself, like this: $$f : (X,x_0) \to (Y,y_0) $$ In this notation, one regards $f$ not just a morphism in the category of topological spaces and continuous maps, but as ...


0

You have misunderstood how permutations work: If we have the following, from the set $S=\{1,2,3,4\}$ $f=\begin{pmatrix}1&2&3&4\\3&4&1&2\end{pmatrix}, g= \begin{pmatrix}1&2&3&4\\2&1&3&4\end{pmatrix}$ Then $fg=\begin{pmatrix}1&2&3&4\\3&4&2&1\end{pmatrix}$, since we take the elements and ...


0

Your notation really doesn't make sense. Perhaps you can clarify it? Note that the quantity $\{x:F_X(x)\geq \alpha\}$ is frequently written as $$F_X^{-1}([\alpha,\infty))$$ so the quantity in question is just $$\inf F_X^{-1}([\alpha,\infty))$$ The infimum is used because the set may not contain its lower bound (if it does contain this lower bound, then the ...


1

The reason why one writes "inf" in this case is that there may not be any x satisfiying $F_X(x) = \alpha$. In this case the "equals" condition would yield an empty set and be useless. Therefore you need to define it as the infimum. Consider for example $\alpha=0$, $x \in M = \{\frac{1}{n} | n \in \mathbb{N}\}$ and $F(x)=x$. Then there exists no $x$ such ...


1

To my knowledge there is no common notation for this. This is perhaps because the conjugacy class of a fundamental group of a connected manifold is actually independent of the base point: Given two base points $x_0, y_0 \in X$ and any path $\alpha: [0, 1] \to X$ such that $\alpha(0) = x_0, \alpha(1) = y_0$, we get a map $$\Phi: \pi_1(X, x_0) \to \pi_1(X, ...


2

$$a\equiv b\pmod p$$ means that $a-b$ is a multiple of $p$, or equivalently that the remainder of dividing $a$ by $p$ is the same as the remainder of dividing $b$ by $p$. This notation is nearly the first topic discussed in the Wikipedia article on modular arithmetic. Have you tried reading that?


1

$[n]$ is a common notation for the set $\{1,2,\dots,n\}$.


2

Looking at formula (4) at the top of page 3 of the paper makes it clear that Gessel uses the usual convention, in which summation over $i_1+\cdots+i_j = n$ is a summation over all non-negative integers $i_1,\ldots,i_j$ which sum to $n$, rather than all partitions. In the case of (4) we also have a lower bound on $i_1,\ldots,i_j$. If he wanted to sum over all ...


2

The answer is "no, it is not logically reasonable to interpret $dy/dx$ as a quotient, or at least not until you take a much more advanced course". Sorry if that's confusing. (By the way, when you say "And, when $x \rightarrow 0$", you probably mean $\Delta x \rightarrow 0$). Before we begin, remember the basic intuition is the following. Differentiable ...


3

In many concrete situations, you may have some specific expression on hand. Consider the density of the normal distribution, $f(x,\mu,\sigma)=(2\pi\sigma^2)^{-1/2}\exp(-|x-\mu|^2/2\sigma^2)$. If you simply write $$ \left((2\pi\sigma^2)^{-1/2}\exp(-|x-\mu|^2/2\sigma^2)\right)', $$ it will of course be completely unclear what you mean. If you want, you can ...


1

You can use the dot product with the vector $(1, \dots, 1)^T$.


0

I do not think so. Why not use $$ s \, e_k =<v,e_k>\,e_k $$ with a canonical vector $(e_k)_i = \delta_{ki}$.


3

The statement that is being proved is a biconditional statement. The key phrase in the theorem is "if and only if". The biconditional statement $A$ if and only if $B$ can be written symbolically as $A\Longleftrightarrow B$. You prove a biconditional statement by proving $A\Longrightarrow B$ and $A\Longleftarrow B$. That is what the symbols are trying to ...


4

The inconsistency comes from different mathematicians doing different branches of mathematics. In exterior algebra, a wedge product is written with an upward-pointing wedge, like this: $\alpha \wedge \beta.$ The wedge sum in topology is written with a downward-pointing wedge. Only one of those symbols can be called \wedge in LaTeX. Evidently the ...


3

A wedge refers to the shape of this object: It doesn't matter if it's facing up or down, it's still a wedge. But since there was a need for two commands to produce $\wedge$ and $\vee$, and since $\vee$ conveniently looks like the letter V, it was probably decided that one should get the command "vee" and the other "wedge" (instead of something like ...


-2

The thing is that 'wedge' may also refer to the wedge product in the exterior algebra (or differential forms), which is also writen as $\wedge$. I'm unaware of the real reason, you may try on TeX.SE, but coming from a geometrical background I believe this is how it should be, even though some topologists may disagree.


0

The $d$ sticks to its argument. It's not a multiplication. Just like the trigonometric function $\mathrm{sin}\, x$ is not a multiplication either. So $dx \cdot dx = (dx)^2$. You cannot separate the $x$ and the $d$. So it's nonsense to write it as $xd^2x$. The applicable rules are defined by the differential algebra. So in your example ...


7

It’s probably a multiset coefficient; $$\left(\!\!\binom{n}k\!\!\right)=\binom{n+k-1}k=\binom{n+k-1}{n-1}$$ is the number of multisets of cardinality $k$ that can be chosen from a set of $n$ distinct types of object, and the notation has become moderately standard.


1

In the context of sets, that operation is set difference. It is defined as: $$x\in A\backslash B \iff{} x \in \{t\, | \, t\in A \land t \notin B\}.$$ So, if we have the sets $A = \{1, 2, 3\}$ and $B = \{2, 3, 5\}$, then $A\backslash B = \{1\}$. Note: Another notation for set difference is $A - B$.


2

It means set subtraction...all the things that are in the set to the left but NOT in the set to the right. An equivallent definition is $X \setminus Y=X\cap Y^C$ Your equality is false. The left hand side says things that are in A or B, but not C. The right hand side says things in A or (In B but not C) So nothing from C can be in the left hand side, ...


3

The $\setminus$ operand is called set difference. $X\setminus Y$ is defined to be the set which is comprised of elements in $X$ that are not in $Y$. For example, if $X = \{1,2,3,4\}$ and $Y = \{0,1,2\}$, then $X\setminus Y$ is the set of elements in $X$ which are not in $Y$. $1$ and $2$ appear in both $X$ and $Y$ so we remove these elements and we are left ...


1

It means that you subtract C from the combination of A and B


2

There are different conventions: the sum is $0$; the expression is undefined; the expression is interpreted as $\displaystyle\sum_{i=-123}^1x_i$. In my experience the first is the most common, and the last is quite uncommon.


0

$$\sum_{i=1}^\text{row} \mathbb{1}_{ \sum_k \text{img}_{i,k} > 0 } = \text{card}\{i \in 1\ldots \text{row}: \sum_k\text{img}_{i,k} > 0\}$$ ? Where $\mathbb{1}_\mathcal{A}$ is the indicator function for the set $\mathcal{A}$.


0

Firstly, one should distinguish between the capital $X$ used to denote the random variable and the lower-case $x$ used as the argument to the density function or the cumulative distribution function, etc. This makes it possible to understand an expression like $\Pr(X\le x)=F(x)$ (with lower-case $x$ in two places and capital $X$ in one). Then $x\mapsto ...


2

In set theory we use $\bigcup H$ to denote the union of all the elements of $H$. Sometimes we write it explicitly, in one of several ways: $\bigcup_{h\in H}h$, or $\bigcup\limits_{h\in H}h$, $\bigcup\{h\mid h\in H\}$ (this is useful when $H$ is not assigned a variable, but defined via a formula), $\{x\mid\exists h\in H:x\in h\}$, $\bigcup H$, as I remarked ...


0

I would do $f(t_1) > f(t_0) \forall t_1 > t_0$ or something similar. It is not that hard to write out and is quite clear.


1

If the elements of $H$ are sets, it would make sense to write $$\bigcup_{h\in H} h$$ However, if the elements of $H$ are not sets, taking the union of them would not make sense. In this case, the totality of all elements of $H$ is just $H$.


1

Okay I think I've got enough responses to answer this question myself. I might add more to this list if more things come up, or modify it if what is written is incorrect. I've tried not to be too biased in this list, although I must admit that I'll start using this notation. Pros Already in use The Wolfram language already uses notation like this to ...


2

If you define that the "long"-hand notation is: $\dfrac{\partial^n f}{\partial x_1 \partial x_2 \dots \partial x_n}$ then a shorthand can be $\partial_{x_1\,x_2\,\dots\,x_n} f$, where you deduce the degree of the derivative by the number of footers of "$\partial$".


8

Prime notation for multivariate functions is very confusing. Two common notations for partial derivatives are$$\frac{\partial}{\partial x} f(x,y) = f_x(x,y).$$ Similarly $$\frac{\partial}{\partial y} f(x,y) = f_y(x,y).$$


2

"closed under finite intersections" means that if $A_1,A_2,\ldots, A_k$ are each in the set, then their mutual intersection $A_1\cap A_2\cap \cdots \cap A_k$ is in the set. However this must be a finite collection of elements, i.e. $k<\infty$. "closed under arbitrary unions" means something stronger. If $\mathcal{A}$ is a collection of sets, i.e. ...


1

A set being closed under some operation means that you can perform this operation, and the result will still be part of your set. As an example: You have your set $X$, which is closed under finite intersections. This means that for every finite set $\left\{x_1,...,x_n\right\}\subseteq X$ we have $\cap_{i=1}^n{x_i}\in X$


0

$x$ is a factor of $y$ is the same thing as $y$ is a multiple of $x$. You can write $$y=kx \ \text{ for some integer $k$}$$


2

Well $|$ is used to describe divisibility so I think it meets our requirements. For example, $3|6$. $x$ is a factor of $y$ $\Leftrightarrow$ $x|y$.


1

Both the left and the right are commonly used notations for the directional derivative of $f$ in the direction of the vector $\mathbf a$. When $f$ is differentiable, this is, of course, given by dotting the gradient of $f$ with $\mathbf a$.


1

There is no "more correct way". It all depends on convention. Some people prefer $\vec i, \vec j, \vec k$, some people prefer $\hat x, \hat y, \hat z$, some people prefer $e_1, e_2, e_3$... No choice is truly objectively better than the others. My preference tends towards $e_1, e_2, e_3$ becauses it easily generalizes in higher dimension, for example, but in ...


2

As far as I've seen, $z = a + ib$ is the most common with $(a, b)$ being used sometimes and $(a, 0) = a$.


2

Hint: If André Nicolas' meaning for $p\#$ is correct, and $q$ is a prime such that $q \le p$, do you see why $q | p\#$? Then the stated result should be clear.


2

The notation $p\#$ is sometimes used for the product of all the primes $\le p$. Please see, for example, this article. For more, search using the key word primorial. If $q\le p$, then $q$ is one of the primes that got multiplied together to form $p\#$. It follows that $q$ divides $p\#$, and therefore $q$ divides $p\#-q$.


1

$\overline{30} = \left[30\right] =\left\{54m+30 \ : \ m\in\mathbb{Z}\right\}$


6

$a + bi$, or occasionally $a + ib$, is preferred nowadays. It's best to mostly use the lowercase Roman letters for arbitrary variables and functions and leave the Greek letters for special constants and special functions. The use of $\sigma + it$ is now mainly of historical interest, as it has been attributed to Riemann and his contemporaries. The earliest ...


4

The rules are actually incomplete not only for cubic rings and beyond, but also for quadratic rings. Your rules address the ordering of factors, but the problem I'm talking about is a non-problem for factorization in $\mathbb{Z}$, and that's the ordering of "nomials" within factors. For example, consider this factorization of $4$ in $\mathbb{Z}[\sqrt{65}]$: ...


1

A complex number is usually written as $z=a+bi$ where $a,b$ are real. There is no essential difference between the $a+bi$ and $\sigma + it$ notation you mentioned.


2

Found it! http://en.wikipedia.org/wiki/Transpose 'properties' section, point no.8:


2

How should a half primorial be notated? $\dfrac{p\#}2$ Is there a name for a half primorial? No, there is no established term for this notion.



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