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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3
votes
3answers
175 views

$x^{y^z}$: is it $x^{(y^z)}$ or $(x^y)^z$?

Of the following, why is a usually considered true, and for what reason other than "tradition" and "more convenient"? a: ${x}^{y^z} = x^{(y^z)} \neq {(x^y)}^z$ b: ${x}^{y^z} = {(x^y)}^z \neq ...
3
votes
2answers
65 views

degree of commutativity

What is the exact definition of the degree of commutativity of a $p$-group? When we use notations $d(G)$ and $c(G)$ for other concepts, what is the best notation for degree of commutativity of $G$?
2
votes
1answer
48 views

Can objects repeat in commutative diagrams?

Are objects allowed to repeat in commutative diagrams? This seems to be necessary when representing endomorphisms such as the morphism $f : X \to X$ in the category $\mathbf{Set}$, such as when $f$ is ...
0
votes
0answers
25 views

Restricting binary relations by composing with an “inclusion binary relation”

If $X' \subseteq X$ then we may define an inclusion map $\iota : X' \to X$ where $\iota(x) = x$. One use of $\iota$ is that we can express the restriction of some $f : X \to Y$ to $X'$ as $f|_{X'} = f ...
0
votes
2answers
122 views

Notation of logarithm and its exponent

I am little confused about this notation, $\log^3 n$. Does it mean $(\log n)^3$ or $\log (\log (\log n))$?
1
vote
1answer
142 views

Big Omega Notation Proof

Can someone show me how to start this off? I need to prove it, but I'm not sure how I would prove Big Omega. Prove that $f(n)=\sum_{i=1}^ni^k\in\Omega(n^{k+1})$. Thank you for the help!
1
vote
2answers
61 views

“Preimage” of a binary relation

Consider the binary relation $R \subseteq X \times Y$. Is there a standard name and notation for the set $X' = \{x\ |\ (x, y) \in R\}$? ProofWiki calls $X'$ the preimage of $R$, denoted as ...
1
vote
1answer
37 views

A notation problem of partial derivatives

I have a notational problem of partial derivatives: Let $z=f(x,\phi(x))$,and let $\phi(x),f$ be a differentiable functions.What is the difference between $\frac{\partial z}{\partial x}$ and ...
2
votes
3answers
65 views

A notation for a morphism in a thin category

Consider a thin category with objects $A\leq B$. There exists a unique morphism $A\rightarrow B$. Is there a standard notation for this morphism (given $A$ and $B$)?
1
vote
0answers
22 views

Clear notations when working with random variables

I have a set of random variables $X_1,X_2,...,X_n$ each variable $X_i$ has a set of possible values $\Omega_{X_i}$ and a function associated with it $f(X_i)$. An assignment $\bf x$ for a subset of ...
0
votes
3answers
44 views

Expressing how many of $a,b,c$ can be zero

My scenario is that I need to express with mathematical syntax the following condition: There are three integers: ${a, b}$ and ${c}$. Case 1: two of the three can be zero. Case 2: only one can be ...
1
vote
1answer
36 views

Notation for Permuting Sets

If I have some arbitrary sets $A_i : i \in I$ and I want to permute their intersections pairwise, how would I write such a permutation? Would I use some permutation tensor? Essentially I want to ...
2
votes
2answers
99 views

How to prove $x-y = x+(-y)$ in ring theory.

Okay, I have talked with a lot of people about this silly question. And I have thought about this way longer than is good for me. Everybody seem to disagree with me, and that is the reason I think I ...
0
votes
1answer
296 views

Mathematical notation to represent all combinations of three variables?

This is a fairly simple question involving mathematical notation which is outside of my expertise. I'm looking for the correct representation of a matrix M which shows all combinations of three ...
1
vote
1answer
36 views

Vector field of functions equalling real numbers

We consider the set $\mathbb{N}^{\mathbb{R}}$ (i.e. all functions $f\colon\mathbb{N}\to\mathbb{R}$). I've been asked to prove that this forms a real vector space (that is, a vector space over ...
0
votes
2answers
40 views

Is this notation correct?

I am writing a paper and I have an expression something like this: $$\begin{equation} \notag x = \text{large_expression} + \begin{cases} y & \text{if } a<b \\ z ...
1
vote
1answer
93 views

Symbol for “if any”

I am looking for a symbol if any for the following equation in my algorithm This is to find closed pattern where $p_i$ is longer than $p$ and $p$ is a sub-pattern of $p_i$ and $support(p) = ...
3
votes
3answers
86 views

What are the sets V and ON?

Are there any known classes called "V","ON" in the subject of ordinals? I have seen it in a few places but can't find the definition in wikipedia.. Thank you!
0
votes
1answer
34 views

notation for numeral concatenation

I have a set, I need to represent in a form: $\lbrace f(x) \ | \ x \in A \ \& \ P(x)\rbrace$. However, have not been able to find the right mathematical notation for numeral concatination. What is ...
0
votes
1answer
59 views

$\sigma$-algebras, containment, and the notation used.

I'm proving the following proposition taken from Royden 4th Edition, Let $\mathcal{F}$ be a collection of subsets of a set $X$. Then the intersection $\mathcal{A}$ of all $\sigma$-algebras of subsets ...
1
vote
0answers
42 views

Notation for linear transform where only a subspace of the domain is used

I posed this question (sort of) here: A 2x2 matrix $M$ exists. Suppose $M^3=0$ show that (I want proof) $M^2=0$ Suppose I have a linear transformation over a field $K$ $T:K^3\rightarrow K^3$ ...
4
votes
4answers
809 views

A 2x2 matrix $M$ exists. Suppose $M^3=0$ show that (I want proof) $M^2=0$

I'm sure I've done this before in abstract algebra. Regardless it's escaped me now. I have proved that for $T:U\rightarrow V$, with $dim(U)=m$ and $dim(V)=n$ that $rank(T)\le m$ which is obvious, ...
0
votes
2answers
76 views

Let $R$ a commutative ring and let $a\in R$. What does $aR$ mean?

Let $R$ a commutative ring and let $a\in R$. What does $aR$ mean ? I would think it means $\{ar : r \in R \}$ as that was the meaning in group theory. The thing that confuses me is that in group ...
1
vote
1answer
125 views

understanding 'p∈ (n, succ)'

I understand that this may be a stupid question to some, but I've come to my wit's end trying to understand this condition: if p ∈ (n, succ) then I keep ...
7
votes
3answers
948 views

What is the symbol $\triangleq$?

I came across this new symbol while reading a document about writing proofs, and I have never seen it before.
1
vote
2answers
91 views

can someone explain this notation to me?

$$ dz_t \sim O\left(\sqrt{dt}\,\right) $$ $z$ is a Brownian motion random variable, for reference. I just don't understand what the $\sim O$ part means. I've looked up the page for Big O notation ...
1
vote
0answers
29 views

Nested log notation

As a complement to this question: What's the correct notation for log squared? What is the correct notation for $\log (\log \log ( ... \log((n))))$? By analogy, with the $n$th derivative of ...
0
votes
2answers
133 views

Meaning of NOT all but finitely often

Can someone clarify for me the meaning of the statement "NOT all but finitely often"? It's driving me crazy. I'm not able to break it up. Thanks.
1
vote
1answer
68 views

Is this absolute value notation or something else?

In this document, in Figure 1 (second to last page) there are several uses of $\| \;\;\|$: Is this another notation for absolute value, or is this a notation for something to do with ...
2
votes
0answers
54 views

Seeking advice on copying Peano's notation

Can anyone give me advice, or a URL for advice, on simulating the notation of Peano's Formulario in LaTeX? or in Word?
0
votes
1answer
107 views

What is the dot product between a vector of matrices?

There is a notation used in many sources (e.g. Wikipedia: http://en.wikipedia.org/wiki/Exponential_family) for the natural parameters of exponential family distributions which I do not understand, and ...
1
vote
3answers
222 views

Standard notation for the set of integers $\{0,1,…,N-1\}$?

I was wondering if there exist a standard notation for the set of integers $\{0,1,...,N-1\}$. I know for example $[N]$ could stand for the set $\{1,2,...,N\}$ but what about the former, i.e. ...
5
votes
3answers
182 views

How to write $\frac{k}{k}$ using $\sum$ notation?

I want to use $\sum$ notation for this:$$\underbrace{\frac{1}{k}+\frac{1}{k}+\ldots +\frac{1}{k}+\frac{1}{k}}_{k\text{ times}}$$ I guessed$$\sum_1^k\frac{1}{k} ,$$but it equals ...
0
votes
1answer
54 views

Question about notation in differential equations.

In general, an ordinary differential equation is in the form $$ \begin{cases} x'(t) = f(t, x(t)) \\ x(t_0) = x_0 \end{cases}. $$ When proving the existence and uniqueness theorems, an operator $T$ was ...
1
vote
1answer
47 views

Limit Comparison Test Defined entirely in symbolic notation

Is it possible to define the limit comparison test entirely with symbols (no textual explanation), or with as little textual explanation as possible? How? My latest best attempt: ...
0
votes
1answer
64 views

Notation question in linear algebra problem

I am confused by the symbol $\mathcal{L}$. What do $\mathcal{L}(V)$ and (especially) $\mathcal{L}(\mathcal{L}(V),\mathcal{L}(W))$ in b) mean? Let $T:V\to W$ be an isomorphism. For each ...
0
votes
1answer
91 views

What does $R^+$ mean?

I'm not sure if it's statistics related but I came over this in my stats related computing assignment. Does $R^+$ (looks like R to the power of plus) mean all positive real numbers? Does it include ...
1
vote
1answer
81 views

What is the definition of $\sum\limits_{0\leq i\leq m,\text{ }0\leq j\leq n}a_{ij}$

I understand the concept of double summations, at least intuitively, but I'm trying to understand it formally. So, to begin with, I have a question: Is this double summation equality true by ...
1
vote
2answers
37 views

Simple notation question

Let A = {2, 3, 4, 6, 7, 9} and define a relation R on A as follows: For all x, y ∈ A, x R y ⇔ 3 | (x − y). Then 2 R 2 because 2 − 2 = 0, and 3 | 0. What does the ...
3
votes
2answers
98 views

Proving $\bigcup \mathcal{P}(\mathcal{A})=\mathcal{A}$

I'm working on operations on collections of sets and I've run aground. I'm trying to prove that if $\mathcal{A}$ is a collection of sets $\mathcal{A}_i, i=1,2,...$, then ...
2
votes
1answer
86 views

Difference between colon and membership symbol?

For my discrete math class, my instructor has told us that the following notation is incorrect: $\exists x \in \mathbb{N} | x > 0 \bullet x < 3$ . And we should instead write: $\exists x : ...
2
votes
1answer
70 views

Style when typesetting functions and operators

I've been making an effort to type all function names and operators in roman font. For example $$\int \operatorname{f}(x) \, \operatorname{d}\!x$$ This was all well and good until I tried to write ...
0
votes
1answer
56 views

Is this an accurate way to represent n! using Π?

I recently learned of the $\Pi$ symbol, and was wondering if the following is an accurate way to represent $n!$: $\Pi_{i=0}^{n-1} n - i$
2
votes
0answers
73 views

Is this symbol $\supset\kern-1.7pt\rightarrow$ commonly used in mathematics?

In Multidimensional Real Analysis I by J.J. Duistermaat and J.A.C. Kolk, the symbol $\supset\kern-1.7pt\rightarrow$ is commonly used. For example, $f: A\supset\kern-1.7pt\rightarrow B$ would mean a ...
4
votes
2answers
122 views

Sum Notation with restrictions

I understand normal sigma notation but what does it mean when we place under a sum the restriction that $i + j + k = n$, for example? Is this simply $3$ sums in disguise or is it something else?
1
vote
1answer
462 views

Identities for Kronecker delta and alternating unit tensor

How I can prove this equations? Please help me... I can solve it. $$\sum_j\sum_k \varepsilon_{ijk} \varepsilon_{hjk} = 2\delta_{ih}$$ $$\sum_k \varepsilon_{ijk} \varepsilon_{mnk}= \delta_{im} ...
1
vote
2answers
40 views

What does $2^H$ mean, where H is finite group?

From Henry Cohn paper: Definition 6.5. Let $H$ be a finite abelian group. An $H$- chart $\mathcal{C} = (Γ, A, B, C)$ consists of a finite set of symbols $Γ$, together with three mappings $A, B, C: ...
1
vote
2answers
73 views

An expression which has $10$ different meanings by using some brackets appropriately

My friend taught me the followings without his memory of the answer: He said that it is known that there exists an numerical expression which satisfies the following two conditions : Condition 1 : ...
1
vote
0answers
118 views

Measure theory, notation

This is from Avner Friedman's "Foundations of Modern Analysis": Let $\mu$ be a measure with domain $A$ and let $E_n$ ($n=1,2...$) be sets of $A$. Then $\mu (\underline{\mathrm{lim}}_{n \rightarrow ...
0
votes
2answers
52 views

Are these statements in logic form correct?

Let M represent the set of all Mathematics courses and S represent the set of all students. Predicates: ...