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9

The "d$x$" is best regarded as a mnemonic symbol, and it reminds the reader (though somewhat misleading for novices) how an integral is carried out. In Russell's term, "d$x$" is called an incomplete symbol, which has no meaning itself when escaping from a given context. I do not see apparent reasons keeping an author away from writing ...


5

Well, that's just the way the notation works, but there are very good reasons for it. We could have used something like $$\int_{x=1}^{\infty} \frac{1}{x^2}$$ to keep it consistent, in a sense, but the meaning and placement of the differential $dx$ is a bit more subtle than just an index of a summation, and makes perfect sense. The notation $\int f(x) dx$ ...


6

The $dx$ you see in an integral is cosmically related to the $\Delta x$ you see in a Riemann sum: $$ \sum_{i=1}^n f(c_i^*) \Delta x_i $$ where $c_i^*$ is the sample point in the $i$-th interval, and $\Delta x_i$ is the width of the $i$-th interval. The $dx$ is often thought of as the "infinitely small" version of $\Delta x_i$. This is similar to the $dx$ ...


-2

The $dx$ or $dn$ parts are used in integrals to show which variable we're integration over and so we can abuse the notation. For sums and products the variable we're summing over is written below the sum or product sign, there's no reason to write which variable we're summing over twice.


3

I think it is called topological invariant. By definition, a topological invariant is is a property of a topological space which is invariant (i.e., preserved) under homeomorphisms, see here.


0

Elegance of notation, even at the expense of some easy-to-resolve ambiguity is important. To see why everybody is OK with using $f(x)$ and also $(c+d)(a+b)$ even though $c(a+b)$ becomes ambiguous, consider that a mathematician steeped in analysis would write something like $$ f : \Bbb{R} \rightarrow \Bbb{R} | \forall x \in \Bbb{R} f(x) = x^2 $$ where some ...


1

The symbol $\supset$ (called "horseshoe") was used a century ago for the conditional connective "if-then"(see The Notation in Principia Mathematica , and after replaced (mainly) by $\rightarrow$ or $\Rightarrow$. It does not mean that the sentence $A$ is "included" into the sentence $B$. Of course, there is a "relation", through The Algebra of Logic ...


0

We want to call the above mentoided formula $\varphi(x)$ with $x'$. Therefore, we transform it into an equivalent formula $\varphi'$. $$ \varphi'(x') :=\exists x ( x=x' \land \varphi (x) )$$


4

The easiest and most direct way to say that $f$ is a function is by stating it as such. Actually, even if you use the letter $f$, you should always explicitly state that $f$ denotes a function. For example, this statement: If $f$ is a differentiable function, then $f$ is continuous is, by my oppinion, much better than If $f$ is differentiable, ...


2

I think that there is no "compact" formalization of the required double substitution. Assuming that [using $v_0, v_1, v_2$] : $\varphi (v_1) := \forall v_0 \psi(v_0, v_1)$ we have : $\forall v_0 \psi(v_0, v_1)[v_0 ← v_2] \equiv \forall v_2 \psi(v_2, v_1)$ with $v_2$ a new variable. This condition licences us to prove the equivalence. Now we can ...


1

Wolfram Mathworld gives it as from Hastad et al. 1988 though I haven't seen it before. It seems like sensible notation due to the possible confusion of $[x]$ with square brackets. http://mathworld.wolfram.com/NearestIntegerFunction.html Edit: After closer inspection & looking at Hastad papers from 1988, I can't find the specific notation. I may be ...


1

There are no standard rules. Writers of maths are allowed to use whatever letters they like for things, and different writers are allowed to use different letters. Most people follow some unwritten conventions, but different users of maths use different conventions. Users in the same discipline area who write to each other a lot tend to use the same ...


1

You follow conventions for the thing you're reading/writing. Vectors are often bolded, matrices are often capital letters, random variables are often capital letters, etc. While writing, if you're deviating from conventions, make a note of it. And you can use words in mathematical writing to help detail what the meaning of things are.


1

You can write down the solutions both explicitly or implicitly, the latter uses the notation $\{x|\ldots\}$ or $\{x:\ldots\}$ which both means “the set of numbers $x$ such that $\ldots$ is true”. Here are some examples: $$\begin{array}{c|c|c} \text{Equations} &\text{(Implicit) Set of solutions} & \text{(Explicit) Set of solutions} \\ \hline ...


0

Well, you can list the elements of the set of solutions: For your two examples, you can use $\lbrace 4\rbrace$ and $\lbrace 0,1\rbrace$.


0

If the solution to your equation is $a_{1},...a_{m}$ then in set notation you simply write $$ \{a_{1},...,a_{m}\} $$ in set notation


4

I've seen the notation $\mathsf{CAlg}_R$ in many places. I prefer $\mathsf{CAlg}(R)$ in order to stress the functoriality. This is also used in Yves Diers' work. Many papers restrict to commutative rings and algebras in the first place and therefore just write $\mathsf{Alg}_R$ (which might be confusing - but this is just a local notation). More generally, ...


2

Seeing the way you wrote it, I thought of basis. A natural number $a$, written in base $b$ (also natural), is denoted by: $$a =(a_na_{n-1}\ldots a_1 a_0)_b = a_nb^n + a_{n - 1}b^{n-1} + \ldots a_1b + a_0$$ where $0 \leq a_i < b$. For example, when we write stuff like $124$ or $1675$ or $223$, we actually mean $$(124)_{10} = 1 \times 10^2 + 2 \times 10 + ...


1

If you don't care what order the terms come in, it is a partition If you do care, it is a composition


3

Not quite, because you use the same $x$ to mean different things in the formula. This would be a bit better ($x_i$ means the dose corresponding to quantity $i$). $${\overline{x}=}{\frac{1}{n}\sum_{i\in {S}}^n\frac{30}{i}x_i}$$ But this still isn't right because your $S$ is not really a set. A set cannot contain the same number (e.g., $5$) twice; it either ...


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 $y=1$ ...


2

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 ...



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