2
votes
1answer
13 views

Consider a symmetric matrix $X$ with eigendecomposition $X=UVU^T$, how to call $\sum_{v_{k,k}>0}v_{k,k}u_ku_k^T$?

Consider a symmetric matrix $X$ with eigendecomposition $X=UVU^T$ How do people call $\sum_{v_{k,k}>0}v_{k,k}u_ku_k^T$? Sum of positive components of $X$? The positive semi definite part of $X$? ...
1
vote
0answers
27 views

In regards to metric spaces, does $d^\star$ have an accepted name, or notation? Do any authors use it?

(I write $\omega$ for the set $\{0,1,2,\ldots\}$.) Let $X$ denote a metric space with metric $d$. Define a function $d^{\star} : X^\omega \times X^\omega \rightarrow [0,\infty]^\omega$ by writing ...
0
votes
0answers
50 views

Alternatives to the notation $\|x\|$ for the norm of $x$?

For aesthetic reasons, I don't like the notation $\|x\|$ for the norm of $x$. Have any alternatives been proposed?
0
votes
1answer
28 views

Cone of convex solutions

I have been reading a paper, on monge-ampere type equations, and the existence of a unique convex solution has been proven to exist in $C^{3,\alpha}(\Omega)\cap C^{2,\alpha}(\overline{\Omega})$, for ...
7
votes
2answers
631 views

History of notation: “!”

Does anyone know where the factorial "!" symbol came from? I can't decide if it is my favorite or least favorite notation in mathematics...
2
votes
0answers
59 views

Question about definition of some classes of bimodules.

Suppose that we have a ring $R$ and a $R$-$R$ bimodule $M$ such that: For every $r\in R$ and $m\in M$ there exists $r'\in R$ such that $m\cdot r=r'\bullet m.$ Examples of this bimodules can be seen ...
7
votes
1answer
107 views

First usage of the symbol ∈

Concerning a book [1] I am reading the symbol $\in$ was first used by Giuseppe Peano and is the first letter $\epsilon$ (epsilon) of the word ἐστί (means "is"). Does anyone know in which work of Peano ...
16
votes
2answers
269 views

What is the Coxeter diagram for?

I understand that Coxeter diagrams are supposed to communicate something about the structure of symmetry groups of polyhedra, but I am baffled about what that something is, or why the Coxeter ...
7
votes
1answer
124 views

Commutator subgroup - or?

If $G$ is a group and $X, Y \subseteq G$ then the commutator subgroup of $G$ is defined as $[G, G] = \langle [x, y] \mid x, y\in G \rangle$, where $[x, y] = x^{-1}y^{-1}xy$ and the group generated by ...
1
vote
0answers
41 views

Notation to refer to all the n element subsets of a set?

Is there a notation to refer to all the n element subsets of a set? I know the power set denotes all of the subsets.
-1
votes
1answer
35 views

What is P(Y) here?

A multivalued map, f: X -> Y, from a set X to a set Y, is a map f: X -> P(Y). Multivalued maps will be also called multimaps. I don't understand what a multimap is in category theory and I think the P ...
1
vote
1answer
34 views

Notation for a function that is invertible only in 1 variable

I have not yet seen something similar before. Let $f:D\times [0,1]\to \mathbb{R}$ be a function, and for every $x\in D\subset\mathbb{R}^n$ the function $f(x,s)$ is increasing in $s$. In other words: ...
3
votes
1answer
42 views

Where can I learn more about these subcategories of functor categories?

Note: I've substantially edited the definition; $f$ is now allowed to be a functor. Given categories $\mathcal{C}$ and $\mathcal{D}$, we can form the functor category $[\mathcal{C},\mathcal{D}]$. Now ...
2
votes
3answers
75 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 ...
2
votes
1answer
109 views

Category of topological pairs

Is there a standard abbreviation for the category of topological pairs? I have searched for it in vain.
0
votes
0answers
26 views

“Anti-cumulative” Relation Image using Intersection

Given a binary relation $R \subseteq X \times Y$, the familiar image of some $A \subseteq X$ is defined as $R[A] = \{y\ |\ (x, y) \in R, x \in A\}$. Naturally we have the property $R[A] = \bigcup_{x ...
5
votes
0answers
261 views

Paul Erdős Joke.

I was watching the great documentary "$N$ is a Number" and in it Erdős tells a joke where he writes: PGOM LD AD LD CD Which means poor great old man, living dead, archeological discovery, legally ...
2
votes
1answer
47 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
24 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 ...
1
vote
2answers
56 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
53 views

Reference request: Integrals involving $[x]$, $\{x\}$, $d[u]$, $d\{u\}$

I would appreciate reference suggestions to learn how to deal with integrals involving $[x]$, $\{x\}$, $d[u]$, and $d\{u\}$. Thanks!
0
votes
3answers
164 views

How can i learn to use $\LaTeX$? [closed]

Could you recommend any websites or books or resources for me to learn how to use $\LaTeX$? I've asked many real analysis question recently by uploading images and some people suggested I should use ...
2
votes
1answer
55 views

What does $K^{1/p}$ for a field $K$ mean?

In the proof of the finite generation of the invariant ring of a finite group acting on $k[x_1,\dots,x_n]$, at one time there is a symbol I don't understand. The situation is as follows. $k$ is a ...
5
votes
2answers
285 views

Modern approaches to mathematical notation

I'm interested if there is literature on projects which try to improve formal notation, especially for doing mathematics on an advanced level. For example, I'm thinking along the lines of diagrammatic ...
5
votes
1answer
102 views

Is the variant direct image mathematically significant?

Preimages have the property that for an arbitrary function $f : X \rightarrow Y$ and all $B \subseteq Y$ it holds that $$f^{-1}(B^c)=[f^{-1}(B)]^c.$$ However, the analogous statement for direct ...
2
votes
3answers
113 views

Should I put interpunction after formulas?

I am presently doing my first substantial piece of mathematical writing, hence this, probably somewhat silly, question. How does display-style mathematics interact with punctuation? More ...
1
vote
0answers
68 views

Is there a common notation for the labelled degree of a vertex?

Let $G$ be an undirected graph with labelled edges. The labelled degree of a vertex $v \in V(G)$ is the number of edges incident to $v$ with distinct labels. The definition of the labelled degree ...
6
votes
1answer
323 views

What are $\Sigma _n^i$, $\Pi _n^i$ and $\Delta _n^i$?

Sometimes reading on wikipedia or in this site (and in very different context like topology, arithmetic and logic) I have found these symbols $\Sigma _n^i$, $\Pi _n^i$ and $\Delta _n^i$. They are ...
6
votes
1answer
230 views

Is there a rigorous theory of context, whereby sets can gain additional structure within a context?

Consider sets $G$ and $H$ and a function $f : G \rightarrow H$. So far, it doesn't really make sense to ask whether $G$ and $H$ are groups (technically, the answer is "no, they're not groups"), and ...
3
votes
1answer
156 views

Diagrammatic (Postfix) Composition of Functions

Consider the functions $f : X \to Y$ and $g : Y \to Z$. According to the Wikipedia articles on Function Composition, the application of $f$ to an input $x$ can be written as $xf$ (as opposed to the ...
2
votes
2answers
68 views

Why is $S/R$ a ring extension?

If $S$ is a ring and $R \subset S$ is a subring it's common to write that $S/R$ is an extension of rings. I frequently find myself writing this and read it quite often in textbooks and lecture notes. ...
1
vote
0answers
46 views

Enumeration of symbols in grammatical expressions or vertices in tree graphs

I have expressions (type of a function) like e.g. $$f:(A\to B)\to C \to (D\to E)\to F.$$ (Where I understand $A\to B\to C$ as $A\to (B\to C)$, in case that is relevant.) There might be information ...
3
votes
2answers
202 views

Generalization of a product measure

Let $(X,\mathfrak B(X))$ and $(Y,\mathfrak B(Y))$ be measurable spaces and further let $\mu$ be a measure on $\mathfrak B(X)$ and let $K$ be a kernel, i.e. for any $x\in X$ we have $K_x$ is a measure ...
4
votes
1answer
223 views

Inverse function notation

Suppose $f$ and $g$ are functions that fail to be one-to-one, but $f+g$ is one-to-one. Has anyone ever seen the notation $(f+g)^{-1}$ for the inverse function in that situation? (I find myself ...
1
vote
1answer
126 views

Logic about systems?

In Godel's Incompleteness Theorem, his theorem is about a system of logic. Where can I find more about this study, especially the notation? EDIT I mean logic about systems in general. I worded the ...
3
votes
2answers
133 views

Original papers on the subject of group actions

Does anyone if there are any original paper(s) that first introduced the notion of group action or permutation representation, and who the author(s) were? Any references I have found so far on e.g. ...
3
votes
2answers
102 views

Relation-preserving maps as morphisms of a category

What is the canonical name for the category whose objects are all pairs $(X, \rho)$, where $X$ is a set and $\rho$ is a binary relation on $X$, and whose objects are relation-preserving maps? That is, ...
0
votes
2answers
175 views

Weak Partial Complete Lattice and Homomorphisms

What is the proper nomenclature for a generalization of a lattice $L$ such that not all subsets of $L$ may have a join/meet, sometimes not even for finite subsets? This paper calls it a "weak partial ...
3
votes
2answers
128 views

Textbook determinant convention

My text book is called "Linear Algebra and its applications" by David C. Lay. I am just wondering why the textbook uses the absolute value symbol when it wants us to compute determinants. For ...
4
votes
2answers
384 views

Resources to learn the meaning of any math symbol

There is lots of symbols and may be operators like || in this expression $$ 3^k||n$$ that I would like to be able to quickly find the meaning of. I tried Wolfram|Alpha but I think it expects the ...
10
votes
1answer
220 views

Why do they print the number of Illustrations on some mathematics books? [closed]

Why do they print the number of Illustrations on some mathematics books? For example: In my whole life, I've seen this only on maths books and I can't figure out why they do this or why it's ...
3
votes
2answers
114 views

What letter should I use to denote an ideal?

In commutative algebra, there seem to be two rather different notational conventions for ideals: either $I,J, \dots$ or $\mathfrak{a}, \mathfrak{b}, \dots$. By itself, it is hardly surprising - after ...
5
votes
1answer
205 views

History of Lie algebra notation (in Fraktur)?

Does anyone know how it has become the standard to express Lie algebras in fraktur? I'd also like to know how it's established for each era and region, not only the origin. It doesn't seem that ...
7
votes
2answers
522 views

Notation to work with vector-valued differential forms

What it the standard notation used while working with vector-valued differential forms? I tried using abstract index notation, for example denoting a $1$-form valued $2$-form as $P_{i[bc]}$, but I'm ...
1
vote
5answers
647 views

Mathematical notation for computer science

Can anyone point me in the direction of good introductory material on the use of mathematical notation in the field of computer science? I often come across notation in research papers that I don't ...
10
votes
2answers
214 views

Articles on ideas in the history of mathematics notation?

I'm teaching a course this term on the history of scripts (writing systems) and rather than talking interminably about Semitic and Chinese and their spawn, I'd like to give students a more varied ...
4
votes
4answers
161 views

Why Does Finitely Generated Mean A Different Thing For Algebras?

I've always wondered why finitely generated modules are of form $$M=Ra_1+\dots+Ra_n$$ while finitely generated algebras have form $$R=k[a_1,\dots, a_n]$$ and finite algebras have form ...
1
vote
1answer
279 views

Index notation clarification

Previously, I have seen matrix notation of the form $T_{ij}$ and all the indices have been in the form of subscripts, such that $T_{ij}x_j$ implies contraction over $j$. However, recently I saw ...
2
votes
1answer
140 views

Is the notation for the definition of a set stricly formalized?

In the 800 pages set theory book by Jech, he uncommented starts using $$Y=\{ u\in X : \phi(u)\}$$ as equivalent to $$Y=\{ u:u\in X \wedge\phi(u)\}$$ on the first few pages. The fact that, in the ...
6
votes
3answers
2k views

A digital notebook for Mathematics?

When I studied math 15 years ago, I was dreaming of having a math repository with tags to navigate between the different entries. I imagined it would come eventually to the market, and was hopefull ...