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1

Commonly I use the following letters as indices: $$\text{Discrete}: i,j,k,l,m,n,p,q,r,s\\ \text{Continuous}: \alpha, \beta, t, \epsilon$$ And these as sizes of discrete sets: $$K,l,L,m,M,n,N,P,Q,r,R,s,S$$ That's of course opinion based, but these are my favorites and $m,n$ only when not in use as an index. $r,s$ occur as sizes mostly in numerical context ...


0

The letters $\alpha$ and $\beta$ are common for indices.


1

The problem with $\pm$ is that it only refers to two points, not the range between them. Although $x = \{A \pm B \}$ is not commonly seem, I would take it mean that $x$ is $A+B$ or $A-B$. One popular abuse of $\pm$ is when it's used to mean "approximately". Someone might say that something will happen in "plus or minus three days", and while that precisely ...


1

They are just different conventions.   They don't signify any different meaning. I personally find the $\Pr$ notation most useful when the discussion involves combinatorics.   It distinguishes probability somewhat from permutation. (Unless you use ${^n{\rm P}_r}$ ...) It also has that convenient LaTeX command \Pr which renders it in times roman ...


1

They are just different notation. Some authors even use the blackboard bold font: $\mathbb{P}$. What matters is what's inside of the subsequent parentheses (or sometimes brackets, [].) Several notation species exist for expectation ($E, \text{E},\mathbb{E}$) and variance ($V, \text{V},Var, \mathbb{V}$) too but they all have the same definition.


0

In France, students are taught to use the notation to mean $\{1,2,\dots,n\}$. We can use the notation for $\{m,\dots,n\}$. It is not an international convention. Also, it is not listed in the ISO 80000-2, which is an international standard that defines mathematical signs and symbols.


0

The bar notation is often used in Lie Group theory. For example, the $n \times n$ unitary matrices are defined as $$U(n) := \{ X \in \mathrm{Mat}(n,\mathbb C): \overline{X}^{\top}\!X = E\}$$ where $\overline{X}^{\top}$ is the conjugate matrix of $X$ transposed, and $E$ is the identity matrix. Just as the (real) orthogonal matrices $X^{\top}\!X=E$ ...


2

The symbol $\vee$ usually denotes the logical disjunction (the OR truth-functional operator), just as $\wedge$ usually denotes logical conjunction (the AND truth-functional operator). The empty operator probably refers to logical conjunctions as well, and the upper bar should refer to logical negation. As suggested by @Joffysloffy, you can map these to ...


0

Read it inside out. First, suppose you had picked some $t$: then you could scan your available $t_j\in T_N$, and see which of these is closest to $t$. You could repeat that for a different $t$, and presumably get a different minimum distance in each instance. Among all these $t$ and their minimum distances to any of the $t_j$'s, there's at least one $t$ ...


0

Maybe define $f_x(\gamma)$ instead of $f(x, \gamma)$ and go with $x\mapsto \int_\Gamma f_xdS$, unless the $dS$ part itself causes confusion.


3

A Wikipedia page that seems to be relevant: $\ldots$ blackboard bold in fact originated from the attempt to write bold letters on blackboards in a way that clearly differentiated them from non-bold letters, and then made its way back in print form as a separate style from ordinary bold, possibly starting with the original 1965 edition of Gunning and ...


3

I have no references here, just stories my professor told us. As far as I know, the double lines were not double lines in the beginning, but rather boldened lines, so people wrote R with a slightly thicker vertical line to denote the real numbers. The laziness of mathematitians and difficulty of writing a bolder line that students will recognise on a ...


0

If you interpret $x,y,z$ to take values $0$ (false) and $1$ (true), then you can see the logical AND operator (here logical AND of $x$ and $y$ is written as $xy$) as regular multiplication: $xy=1$ if and only if $x=y=1$. Then the logical OR ($\vee$) can be seen as the maximum function. So $x\vee y=\max\{x,y\}$. Hence $x\vee y=1$ if and only if $x=1$ or $y=1$ ...


0

A Venn diagram might make it clearer how to imagine $\lor$ as the logical or: The diagram can be used to visualize boolean values as sets. An element may either be member of a set or not. Three boolean variables lead to eight different areas in theVenn diagram or eight corresponding cases as shown in the following truth table: In the Venn diagram ...


1

I see there is some disagreement, but the notation $R^\ast$ or $R^\times$ with $R$ a commutative ring means the group of units of $R$ to me. With this interpretation, we have $\Bbb Z^\ast = \{\pm1\}$.


6

@dwalke: This means all integers except for 0. Edit: I'm all the more convinced now that you said it's a "vestibular"-type question. This notation is taught in the standard high-school curriculum in Brazil. See http://www.infoescola.com/matematica/numeros-inteiros/ for example.


0

I believe there is a error in the book. Note that the book performs the expansion in two steps: $$ a_{ij} x^i x^j = a_{1j} x^1 x^j + a_{2j} x^2 x^j + \cdots + a_{nj} x^n x^j \\ = a_{11} x^1 x^1 + a_{22} x^2 x^2 + \cdots + a_{nn} x^n x^n $$ The first step has the cross products, as you correctly expected, but the second step ...


1

Your definition is perfectly valid. Another way of expressing it using standard set theory notation would be: $$ f(e) = \sum_{j=1}^n |\{e\}\cap a_j| $$ Where $\{e\}$ is a single-element set consisting of the element $e$ only. $\{e\}\cap a_j$ is the intersection of $\{e\}$ with the subset $a_j$. So if $e\in a_j$ then $\{e\}\cap a_j=\{e\}$ otherwise ...


0

$a$ is a vector of vectors, not necessarily from the same space. Thus $a_i$ has it's conventional meaning of the $i$th component. Then $a_{i,j}$ naturally has the meaning you want it to. You could also use $a_{ijk\dots}$. Also the two sets you mention are not equal, but isomorphic as many spaces eg. as real vector spaces. They are probably homeomorphic ...


0

If I needed a vector of row-minima, I'd write $$ u_i = \min_{j\in J} x_{ij}, ~i = 1, \ldots, m. $$ In other words, almost exactly what you've written. In general, I like to denote the $ij$ element of a matrix $A$ by $a_{ij}$, so my real preferred answer would be $$ u_i = \min_{j\in J} a_{ij}, ~i = 1, \ldots, m. $$


1

Usually we just say that $X$ is a random variable taking values $3$, $4$ and $5$. Of course you also have to specify the distribution, so you could say for example that $X$ is uniformly distributed on $\{3,4,5\}$. Various distributions also have names, and then you can use the notation $X \sim N(0,1)$, to for example indicate that $X$ is distributed like a ...


2

You are asking for a number to be expressed in base $100$. That means the first "place" after the "decimal" point is in units of $1/100$, the second "place" after the "decimal" point is in units of $(1/100)^2$, and so on. Each "place" is populated by a whole number ("digit") from 0 to 99. Because we do not have distinct symbols for each of those possible ...


0

The notation means that the function $g$ is not identically zero. This means that $g(z) = 0$ for all $z$ does NOT hold.


2

In Lee's 'Intro to Smooth Manifolds', $\Lambda^k(V)$ refers to the space of alternating $k$-tensors on a vector space $V$, as you mentioned. However, the space $\Omega^k(M)$ is the space of smooth $k$-forms on a smooth manifold $M$. That is, an element of $\omega \in \Omega^k(M)$ is a smooth map $M \to \Lambda^k(T^* M)$ (called a smooth section of ...


1

In general for a given set $S$ which is nonempty and a subset of an ordered field we define the smallest element in the set to be the element $x \in S$ such that $x\leq y, \ \forall y \in S$. Since you said in a set, I will not introduce the notion of inf. I hope this helps.


7

The notation you looking for is: $$\min$$ Suppose you have a ordinary finite set $A=\{a_1,\ldots,a_k\}$, then you can write the minimum notation as follows: $$\min\{a_1,\ldots,a_k\}$$ In your case, $$\min\{2,1,3,4,8,10\}=1$$ In case of functions, you can represent its minimum over a set as follows: $$\min_{x\in S}f(x).$$ An example: $$S=\mathbb{R},\ ...


0

I've seen explanations of number systems in which the decimal point is followed by infinitely many digits, which in turn are followed by more digits, but (aside from not understanding these systems well enough to say anything useful about them) I don't think I've ever seen a number system in which there is both an infinite number of digits after the decimal ...


0

Contrary to some of the answers in this thread, your idea of an infinite sequence of 9s followed by an 8 is perfectly coherent. Normally when we consider infinite sequences, we imagine the positions in the sequence indexed by positive integers, and we that each infinite decimal is associated with a function that takes one of these positions and tells us ...


0

We can use the exact same proof as the one for $0.\bar{9}=1$ to prove that $0.\bar{9}8=1$. (Except that $0.\bar{9}8$ doesn't exist. The simple reason that $0.\bar{9}8$ doesn't exist: The infinite sequence of decimals is too long to be a number. The more advanced reason: Real numbers can be defined as a sequence of integers optionally including one decimal ...


1

$$.99999.....8=\lim_{n \to \infty}\left( \sum_{k=1}^n {9\over 10^k} \right)+{8\over 10^{n+1}}$$ $$=\lim_{n \to \infty} 1-10^{-n}+.8*10^{-n}$$ $$=\lim_{n \to \infty} 1-.2*10^{-n}$$ So $.99999...8$ is actually five times closer to one than $.9999...$ (yes, I'm aware that I'm being inconsistent with my infinities, but the result is the same) and therefore if ...


1

$0.\bar 98$ is nonsense, since $0.\bar 9$ informs us that $9$, and $9$ only repeats indefinitely (infinitely), and hence does not change to another digit, ever. In order to have an $8$ tagged on to a series of $9$, the series of nines would have to be finite (terminate). With respect referencing "the largest decimal less than $1$," there is no such number. ...


0

When you write $.\bar 9$ you are indicating that there is nothing more to the decimal expansion than $.99999999...$. Placing anything after the bar doesn't make sense with how the repeating part is defined. Particularly, this is because it does not indicate where the "8" will go.


1

There is not such thing as the real number closest to another real number, this is because suppose $a'$ is the number closest to $a$. Then define $a''=\frac{a'+a}{2}$ then $a''$ is "between" $a$ and $a''$ so $a'$ was not the closest number to $a$. Since there is no number "between" $.\overline{9}$ and $1$ we have $1=.\overline{9}$


1

It's pretty ambiguous. I believe the standard would be: $$ \left(\prod_i P_i\right)(x)=(P_1\cdots P_n)(x)=P_1(P_2(\cdots P_n(x)\cdots))$$ We'll see what the votes of my answer reveal. I even recall seeing this notation mean $\left(\prod_i P_i\right)(x)=\prod_i(P_i(x))$ once before.


2

I think your use of $\pm$ is just fine. It seems to be widely used. I was told by one professor that writing things like $$i=1,2,3,4$$ was also improper and that I should rather use $i\in\{1,2,3,4\}$, but I have noticed that a lot of books and professors at my university use the former; including your use of the $\pm$ symbol. However, if your teacher ...


5

The second one is simply wrong, that's not how things should be written. It's unambiguous here because there is only one quantifier, but if had been something like $\exists yP(x,y)\forall x$ you wouldn't know whether $\forall x\exists yP(x,y)$ or $\exists y\forall xP(x,y)$ was meant. In my experience, the second option is what's usually meant. But I've ...


3

Translate the two statements into English: For all v, either $f(x_v) = 0$ or $g(x_v) = 0$. Either $f(x_v) = 0$ or $g(x_v) = 0$, for all v. The two statements are equivalent. The difference is that the emphasis that this is is true for all $v$ in the set $\{1,\dots,n\}$ has been moved.


2

Rather than directly answer the question, I prefer to address the seeming implication that the Leibniz notation is bad. Maybe a really good expository defense of that notation has yet to be written. For now I shall link to this answer I wrote a while ago: What is $dx$ in integration?


3

$$\frac{\partial f}{\partial x}$$ Partial derivative notation is misleading in that it suggests "how does $f$ vary as $x$ varies?" is a meaningful question, and is an outright abomination in any context where one might consider making a change of variable. What this notation secretly means is to ask how $f$ varies as $x$ does... while certain other ...


4

Big $O$ notation: $f(n)=O(g(n))$: $\exists \text{ constant } c>0 \text{ and } n_0 \geq 1 \text{ such that, for } n \geq n_0: \\ 0 \leq f(n) \leq c g(n)$ $$$$ Little $o$ notation: $f(n)=o(g(n))$: $\text{ if for each constant } c>0 \text{ there is } n_0 \geq 1 \text{ such that } \\ 0 \leq f(n) < c g(n) \ \forall n \geq n_0$ In other words ...


1

≈ is for numerical data, homeomorphism ≃ is for homotopy equivalence ≅ is for isomorphism, congruence, etc These are just my own conventions.


5

The notations $\cong$ and $\simeq$ are not totally standardized. Both are usually used for "isomorphic" which means "the same in whatever context we are." For example "geometrically isomorphic" usually means "congruent," "topologically isomorphic" means "homeomorphic," et cetera: it means they're somehow the "same" for the structure you're considering, in ...


0

There are three notations for matrices, (),[ ] and ll ll . Use the one with which you are comfortable with.


2

As I mentioned in the comment (or see Asaf Karagila's post), "$\backslash$" indicates set difference. However, if your question was what is the significance of the set difference appearing in the definition of the transition function for Turing machines, I can also elaborate a bit: I assume the $q_Y$ and $q_N$ refers to something like the accept and ...


1

Recall that $X\setminus Y=\{x\in X\mid x\notin Y\}$. The last line says that $\delta$ is a function with takes pairs $(q,\gamma)$ such that $q\in Q\setminus\{q_Y,q_N\}$ and $\gamma\in\Gamma$, and its result is a triplet $(q',\gamma',i)$ where $q'\in Q,\gamma'\in\Gamma$ and $i\in\{-1,1\}$.


0

"Not practical " or "unusual", because that "all args first then all ops" notation would only work if the operators used all have a unique arity. So what about a - operator? Also one would pile up much memory, no intermediate reductions.


4

The most likely explanation seems to be that you're misremembering the meaning of the notation, and that the interviewer actually was using Reverse Polish notation. This notation, as given, is not used anywhere, while RPN is quite common for a lot of computer science operations. In RPN, 1 2 9 8 4 6 + * / + * would mean ...


2

This is almost a version of Polish notation called Reverse (or postfix) polish notation, but with the operators in inverted order.


1

There is also an "historical" motivation behind. Functions in mathematics originated form the idea of "recipe" or "procedure" which, taking an input $x$ "produce" an output $y$. Paradigmatic examples are the simple mathematical functions like : "double of __" (i.e. $y = 2 \times x$), "square of __" (i.e. $y = x^2$). This origin explains the "old" ...


0

I don't think there's anything inherently bad about either notation (used properly). The less common notation is useful in some cases, e.g.: Let $\alpha$ be a permutation. Let $\alpha'$ be the map defined by $\alpha(i) \overset{\alpha'}{\longmapsto} i$. We see that $i \overset{\alpha}{\longmapsto} \alpha(i) \overset{\alpha'}{\longmapsto} i$ and ...



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