Question on the meaning, history, and usage of mathematical symbols and notation. Please remember to mention where (book, paper, webpage, etc.) you encountered any mathematical notation you are asking about.

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3answers
71 views

Tensor notation about $A^Tx$

I can express $x=x^ie_i$ and $x^T$ by $x_ie^i$. But how to express $A^Tx$ where $A=a^i_je_i\otimes e^j$? I don't think I can write as $a^j_ix^i$ or $a^j_ix^j$.
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1answer
49 views

Proper way to use universal quantification with multiple arguments

Which one is correct? $\forall x \in X, y \in Y:$ some expression or $\forall x \in X \wedge y \in Y:$ some expression or $\forall x \in X \forall y \in Y:$ some expression
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1answer
64 views

What does “$A \cap B$ is a algebra. Moreover, the product map is continuous” mean?

Let $A$ and $B$ be (Sobolev) vector spaces. $A \cap B$ is a algebra. Moreover, the product map is continuous. What does this mean precisely? Is it: if $c_i \in A \cap B$ then $c_1c_2 \in A \cap ...
2
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1answer
70 views

A notational problem of a double integral

What does the following double integral mean?I don't quite understand the notation. $$\int_0^1dy\int_\sqrt{y}^\sqrt{2-y^2}f(x,y)~dx~.$$ Does it equal to $$\int_0^1\int_\sqrt{y}^\sqrt{2-y^2}f(x,y)~...
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2answers
81 views

What does this function mean?

$$f(x) = \frac{x}{e^{x^2}}$$ Differentiate $f(x)$. How should the above function be interpreted? Is the function equivalent to: a)$$f(x) = \frac{x}{e^{x^2}} = \frac{x}{{(e^x)^2}} = \frac{x}{e^{...
0
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3answers
272 views

Notation for the union of all the sets in a family.

Does anyone know how to write the following: If $\mathcal{F}$ is a family of sets, I want the union of all $X$ in this $\mathcal{F}$. Thanks in advance!
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1answer
57 views

Please explain this notation of mapping into a set and product space (related to Sobolev spaces)

So does this mean that I can say that, for example, $\gamma \frac{\partial u}{\partial \nu}$ has a unique continuous extension as an operator from $W^s_p(\Omega)$ onto $W^{s-1-{\frac 1 p}}_p(\Gamma)$, ...
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2answers
42 views

How to repesent n x m multiplication into symbol notation?

I am not a mathematician and so I might not be using the right terms. I have a vector of n components and another vector of m components ...
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2answers
38 views

Applying a function to a set rather than a value

I do apologize about the title, I dont understand the question so I couldnt come up with a better title, if someone else could edit it to a more meaningful title I would appreciate it. So here is the ...
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1answer
35 views

Numbers-matlab-to which number does it correspond

I have written a code in matlab and got this result for something I calculated: 1.3000e+003 To which number does my result correspond?
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2answers
2k views

For all but finitely many $n \in \mathbb N$

In my book I have the following theorem: A sequence $\langle a_n \rangle$ converges to a real number $A$ if and only if every neighborhood of $A$ contains $a_n$ for all but finitely many $n \in \...
5
votes
1answer
310 views

What does the notation $P[X\in dx]$ mean?

I am studying probability, specifically regular conditional distributions, and came across the notation $P[X\in dx]$. What does this mean? Here, $X$ is a random variable and $P$ is a probability ...
2
votes
0answers
48 views

Preferred Notation for Indexing the Naturals from $0$

Depending on circumstantial factors, I occasionally want to index the natural numbers $\mathbb{N}$ from $0$, and at other times, from $1$: i.e. $\mathbb{N} = \{0,1,2,3,\ldots\} \text{ or } \{1,2,3,\...
3
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3answers
347 views

Notation: for $x$ not much less than $a$

"Everybody knows" that $a\ll b$ means a quite vague thing, something like $a$ is very much less than $b$. (And on math.stackexchange.com, it may be observed that not everybody knows the difference in ...
0
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1answer
31 views

General notation for indicating the last digit of a given power

Let's say I wanted to state, for example, what's the last digit of a power with a base of a number ending with 4 as its last digit. Casually, I'd just write it down as: $$...4^{2n}=...6 \\ ...4^{2n+...
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1answer
30 views

Multiplication formula for Lie derivative

Let $U\in\mathbb{R}^n$ be an open set, and let $f_1,f_2\in C^1(U)$. Prove that $$L_v(f_1f_2)=f_1L_vf_2+f_2L_vf_1$$ Suppose $f_1,f_2:U\rightarrow\mathbb{R}$. Let $p\in U$. We have $$L_v(f_1f_2)(p)=D(...
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2answers
57 views

Difficulty understanding this notation

Under the summation it says $\tau=t$. So $\tau$ take the value of $t$ in the first Beta, $t+1$ in the 2nd beta, $t+2$ in the 3rd beta and so on (Is this right?) If the above statement is right, first ...
5
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2answers
102 views

Question about Algebraic structure?

Mathematicians use parentheses to represent a algebraic structure e.g. $ (G, \times)$. Also parentheses are used to represent ordered pair (or general ordered n-tuple) e.g. $(x,y)$. My question: Is ...
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1answer
74 views

Notation in “proof sketch” of the Banach Tarski paradox on wikipedia

I'm trying to understand the proof sketch here. In step 3 of the proof sketch we have $A_{1} = S(a)M \cup M \cup B$. My understanding is that $S(a)$ and $M$ are both sets. I have failed to understand ...
2
votes
1answer
70 views

How to explain last part of Two-sample Kolmogorov–Smirnov test

Sorry for the dumb question. I am trying to understand the last part of Two-sample Kolmogorov–Smirnov test.. The part is: $$ \sqrt{\frac{n + n'}{nn'}} $$ I am thinking that $n$ and $n'$ are the two ...
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1answer
62 views

Is there a typo on this definition?

This is from Iishi's User's guide to viscosity solution. I don't understand the $\ni$ in the definition 2.6 at the end of its first line, is it a typo? User's guide page 11
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2answers
168 views

Notation for Subspaces

Is there a proper notation for denoting subspaces? For example, if $U$ is a subspace of some vector space $V$. I would usually just write "the subspace $U \subseteq V$" but I'm wondering is there is a ...
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0answers
47 views

Mathematical notation - defining sets

Building on this question, if the cubeful numbers were defined as follows: $$\Bbb Z_{\{3+\}} = \{a \in \Bbb Z \mid \not\exists b \in \Bbb Z \text{ s.t. } a \neq b^3 \}$$ Would it suffice to say that:...
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2answers
517 views

Notation for unordered product of sets

Frequently, when referring to the edges of an undirected graph $G=(V,E)$, I want to write that $E \subset V \times V$, which isn't correct since the Cartesian product is ordered and the edges are not. ...
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1answer
97 views

Mathematical notation

Is there some generally accepted notation for squarefree, cubefree, etc. numbers? And is there also some notation for squareful, cubeful, etc. numbers?
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2answers
99 views

How would one prove $[f,[\nabla^2,f]]=-2(\nabla f)^2$?

How would one prove this equation: $$[f,[\nabla^2,f]]=-2(\nabla f)^2 $$ And I'm confused that $\nabla f\nabla f$ equals $(\nabla f)^2$ or $\nabla(f\nabla f)$.
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1answer
35 views

Notation to describe amount of relevant elements in a tuple?

Say, we have the set $A=\{♠,♣,♥,♦\}^3$ and would like to define the following map: \begin{align} f: A &\to \{0,1,2,3\} \\ a &\mapsto \text{amount of ♥'s in the tuple } a \end{align} For ...
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3answers
70 views

What does $f|_V$ mean?

I want to proof: If $W$ is vector space, $U, V \subseteq W$ linear subspaces, and $f : W \rightarrow Z$ is homomorphism, $\ker f \subseteq U$ and $W = U \oplus V$ then $f|_V : V \rightarrow Im(f|_V)$ ...
3
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3answers
95 views

$\sin^2$ notation and uses of the alternative.

So I was taking my calculus class and I was shocked by the following: Apparently its a convention for $\sin^2(\alpha)=(\sin(\alpha))^2$ As opposed to what I thought made more sense which was $\...
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1answer
47 views

How to correctly write this ring theoretic thing?

Im unsure how to write this thing below in a formal way : For an integer $n>2$ Let $F_n(x) = a_0 + a_1 x + a_2 x^2 + ... + a_{n-1} x^{n-1}.$ Also we have $x^n = 1$ and $1 + x + x^2 + ... + x^{n-...
0
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0answers
277 views

Notation for Kronecker product of a matrix and itself?

What is the notation for the Kronecker product of a matrix and itself? In other words, is there a short-hand way I can express the following: $X⊗X$ $X⊗X⊗X$ $X⊗X⊗X⊗X$ Where $X$ is a matrix? What ...
1
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2answers
119 views

Nice notation for projection maps

Let $X\times Y$ be a product of two object of a category, and consider the natural projections $$ X\times Y \to X \quad\text{ and }\quad X\times Y \to Y. $$ Usually I denote them by $\pi_X$ and $\pi_Y$...
1
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1answer
69 views

What does this set actually look like? (predicates)

I am learning set theory right now and I am struggling to get to grips with definitions of sets involving predicates. For example, can someone tell me what "typical" elements look like in this set? $$...
3
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4answers
81 views

Question of trig formatting

Is there a difference between the following: $$\sin^2x$$ $$x\sin^2$$ How about: $$\sin(x)$$ $$\sin x$$ I'm new to trig and I've been confused on the formatting involved in trig, whether something ...
5
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2answers
80 views

Is there a commonly accepted notation for algebraic numbers?

In this question I needed a way to denote an algebraic number using a polynomial equation it satisfies and its isolating polynomial. Because I am not aware of any commonly accepted notation for this, ...
2
votes
1answer
65 views

Defintion of $\ell^\infty$

I have come across the space of bounded sequences denoted as $\ell^\infty$ in my course, but not a clear, concise definition. I have seen sometimes when these includes sequences in $\mathbb{R}$ that ...
1
vote
1answer
651 views

How to convert parentheses notation for trees into an actual tree drawing?

Trees are usually drawn as a set of objects connected by edges. But sometimes one sees a non-graphical, parentheses-based notation, like on the example below. What does the indentation mean in such ...
0
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2answers
92 views

Definition of $[G:C_G(x)]$

What is the meaning of $[G:C_G(x)]$ in group theory? Is this equivalent to $\frac{|G|}{|Z_G(x)|}$, or to $|Z_G(x)|$?
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1answer
36 views

Notation: $F^{*m}$ for field $F$

Quick question (which is surprisingly hard to Google): if $F$ is a field, what is $F^{*m}$? I suspect it's the $m$-th powers of the elements of the multiplicative group $F^*$, so $\{ x^m : x \in F^* \}...
1
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0answers
23 views

Good notation for many random points approximating an area.

I'm trying to say that as the number of random coordinate points points you plot approaches infinity, it is equivalent to an area integral where each point is an infinitesimally small $\mathrm{d}x \...
0
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1answer
21 views

What does it mean to randomly choose an integer from a constant?

In this paper on pg. 1241 under section 2.3 "The Elect Protocol" 2nd paragraph the author says Each party samples a random value $x_i$ from [n/k]. What does that mean? If there are n parties ...
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1answer
131 views

The ring hom $\mathbf{Z}\rightarrow A$

Let $A$ be a ring with idenity. Is there a standard notation for the canonical ring hom $\mathbf{Z}\rightarrow A$?
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1answer
50 views

summation index notation, specify all variables?

I am reading lecture slides for a logistics course and for one of the Linear Programming contrainsts, the summation is written as follows: $$ \sum_{i \in I} X_{ij} $$ and $$ \sum_{c \in C} Y_{jc} $$...
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0answers
65 views

Skip specifying sets under summation signs

I have a couple of summations that all have the notation below them that s is an element of S, k is an element of K, etc. The set S or K is not given but assumed to be generally known. $$ \sum_{k \...
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1answer
88 views

How to characterize the set of all real functions defined on $X$.

Let $X$ be an arbitrary set. I consider the set of all real functions defined on $X$. I know that this is usually denoted by $\mathbb{R}^X$. However, I am interested in characterizing each point of $\...
1
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2answers
108 views

Notation and terminology for functions, interpreting $f(y)$

It seems to me there are two different interpretations of a symbol $f(y)$. I will explain what I mean: Suppose I have a function $f(x) = x$. (I took the identity map to have a simple example). Also ...
0
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2answers
62 views

What does power of '+' in $\hat{\beta}_j=\operatorname {sign} (\hat{\beta^0_j})(|\hat{\beta^0_j}|-\gamma)^+$means?

I encountered the following equation in a paper. $$\hat{\beta}_j=\operatorname {sign} (\hat{\beta^0_j})(|\hat{\beta^0_j}|-\gamma)^+$$ What does the power of '+' mean? The paper can be viewed at: http:...
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1answer
64 views

Question about Notation. What does this means? $f[0]=1, f[0,1]=-1$

Question about Notation. What does this means? $f[0]=1, f[0,1]=-1, f[0,1,2]=2$ (The values are exact, which is pretty confusing too, if they are refering to intervals) This question is from a ...
4
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3answers
3k views

'Does not necessarily equal' symbol

What symbol would I use if I wanted to express that, in the context of some binary relation $P$ implied from context, that $\exists (a,b)\in P: a\ne b$, but not to the extent that $\forall (a,b) \in P:...
2
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1answer
203 views

Why do we write $f : X \rightarrow Y$ as opposed to $f \in X \rightarrow Y$.

I've always been taught to write $f : X \rightarrow Y$ as opposed to $f \in X \rightarrow Y$. This seems weird though, since $X \rightarrow Y$ can be viewed as the set of all functions with source $X$ ...