Questions on the usage and meaning of words in mathematics, the names for mathematical entities, and other such questions.

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2
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2answers
85 views

Why the word “projective” for $PGL_n(\mathbb{F})$?

I wrote the title for this question exactly as I had it exactly in my mind. Let me denote by $G=GL_n(\mathbb{F})$ for simplicity; I was working throughout the previous years many times with the ...
1
vote
0answers
18 views

Are there established names and/or symbols for these orderings?

Consider the following orderings on $\mathbb{Z}^2$. Say $(a, b) \leq_1 (c, d)$ if $a \leq c$ or if $a = c$ and $b \geq d$. So for instance $$(1,3) <_1 (1,2) <_1 (1,1) <_1 (2, 3) <_1 (2,...
7
votes
1answer
90 views

What does the word “norm” stands for in linear algebra?

I know that "norm" is the formal name for length, but where did this name came from? or from what language is came from? Thank you in advance.
2
votes
0answers
19 views

Is there a name for this acyclic quiver?

Sorry for the trivial question, but I don't know much about the subject and don't seem to be able to come up with much by Googling. Is there an established name for quivers of the form $$\require{...
0
votes
1answer
22 views

Characteristics and Mantessa

I've just heard about these terms. Could someone elaborate on what's their use is? And plus could you explain it using a few examples?
1
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2answers
61 views

Explain Example on Maximal Element with sets

I am trying to understand maximal element and I cannot understand this example from Wikipedia As an example, in the collection $$S = \{\{d, o\}, \{d, o, g\}, \{g, o, a, d\}, \{o, a, f\}\}$$ ...
3
votes
0answers
66 views

Monads in monoids

This question is almost a duplicate of this one, but not quite. There the person asked about examples and intuition, I am asking about terminology and applications, and I am addressing my question ...
2
votes
1answer
61 views

What are all the different classes of functions upon real numbers and what do they mean, exactly? [closed]

I have been hearing terms like "piecewise C1", "continuous", "linear", "piecewise constant", "trigonometric", "logarithmic", "exponential", "elementary", etc. functions for many years. I know what ...
1
vote
1answer
32 views

Can someone please offer a simple definition of “derived net”?

I was looking up a term called "derived net", however the Google search seems to conflate "net" with .NET programming language. (And filter with electronic filters, and "derived net from a filter" ...
0
votes
2answers
47 views

Hatcher basic terminology/phrasing

I'm trying to self-study some algebraic topology, reading Hatcher. His questions seem much less straightforwardly worded than Munkres - with Munkres it was always clear that you weren't expected to ...
21
votes
5answers
3k views

Is a proof also “evidence”?

Can I use the terms proof and evidence synonymously or is there a difference? You usually see mathematicians writing about "proof" while other sciences instead discuss "evidence" - is there a ...
1
vote
0answers
24 views

Arc-bases and Point-bases: when are they different for finite digraphs?

Definitions Given a digraph $D=(X,U)$, a point-basis of a digraph $D$ is a minimal vertex set from which a dipath exist to every vertex in $D$. An arc-basis is a minimal arc-reaching set. A subset ...
4
votes
1answer
46 views

Hamiltonian Mechanics and the Symplectic Category

Are canonical transformations (in the sense of Hamiltonian mechanics) morphisms for a certain category? They seem to fit the archetypal description of morphisms being "structure-preserving maps". ...
1
vote
0answers
24 views

What is the opposite of “sparsity” in a matrix?

If a sparse matrix has only 1% non-zero entries, I find it weird to speak of "1% sparsity". In particular, "increasing sparsity" goes along with a smaller percentage of non-zero entries, so this is ...
1
vote
0answers
22 views

How is called 1D and 2D analogy to saddle point

Consider n-dimensional function with multiple local minima (e.g. http://ac6la.com/aeopt5.png). There exist some basins of attraction for which particle fall to one or the other minimum. Boundary ...
1
vote
3answers
83 views

Is “Connected Component” unique for each graph?

Definition A connected component of an undirected graph $G$ is a subgraph where any two vertices are connected by paths. A connected component is a maximal connected subgraph in $G$. Consider a ...
2
votes
1answer
52 views

What does it mean to “identify” two mathematical objects?

There is an informal notion of "identifying" two mathematical objects that I have run into several times, and I'm am wondering how to formally express this idea. A case of this I ran into long ago ...
1
vote
1answer
42 views

What is the difference between accumulation point and $\omega$ accumulation point?

The title says it all. Accumulation point has a widely known definition: a point in $X$ is accumulation point if every open set containing $x$ contains infinitely many points of $X$ Sometimes I ...
2
votes
1answer
59 views

What's the numerator and the denominator of a fraction called?

Just a quick question, is it right to call the numerator and the denominator of a fraction by "terms"? I don't think that "terms" is the right word here, but i don't know any alternatives. Can any ...
1
vote
2answers
39 views

Is there an adjective to describe systems of equations which is neither underdetermined nor overdetermined?

What might I call a system of equations in which the number of equations equals the number of free variables? In other words, if a system of equations is neither underdetermined nor overdetermined, ...
0
votes
0answers
8 views

What is the term for a general set of objects whose higher dimensional analogs have hyper- in front of them?

Strange terminology question: we tend to name things in low dimensional space and then generalize after a certain point. For instance, we have point, line, plane, and then hyperplane (there is no 4 ...
1
vote
1answer
29 views

What is a polynomial with infinite number of terms?

My instructor commented that a structure function $\phi(G)$ of a graph is a polynomial if a finite number of terms. So what is the thing with infinite number of terms? Why not polynomial?
0
votes
0answers
6 views

Set intersection with margin: terminology

I implemented an algorithm that calculates the intersection of two sets with a certain margin and returns the matched tuples: Let A, B be sets. $C = \{ (a \in A, b \in B) | lowerbound <= a - b &...
1
vote
1answer
25 views

Properties of Infinite set on co-finite topology and Countable set on co-countable topology

I am trying to verify some of the properties of infinite set on co-finite topology and countable set on co-countable topology but it is proven to be very tricky because I cannot visualize the spaces. ...
2
votes
1answer
81 views

Are there any “default” properties which hold for almost all topological spaces in analysis?

What is the simplest/most general commonly used (type of) topological space in analysis? For instance, every example I can think of in analysis is first-countable. I don't think it can be metric ...
3
votes
0answers
54 views

Is the infinitesimal generator for Lie groups the same as the infinitesimal generator of a Markov semigroup?

Is the infinitesimal generator for Lie groups related to the infinitesimal generator of a Markov semigroup? Or are they totally different concepts? https://en.wikipedia.org/wiki/Lie_group#...
2
votes
0answers
16 views

Is there a name for the relation of a directed set?

If $(X,R)$ is an ordered set then we say $R$ is an order in $X$, how do we call the relation $R$ if $(X,R)$ is a directed set? A direction in X?
3
votes
0answers
54 views

Is there a name for a topological space $X$ in which every proper closed subset is compact? [duplicate]

Is there a name for a topological space $X$ in which every proper closed subset is compact$^{(*)}$? It is well known that in a compact topological space, every closed set is compact. Hence, the class ...
1
vote
0answers
20 views

Help understanding proof of Frostman's Lemma - issue technical or termonological?

I was reading Hochman's proof of Frostman's lemma in his online lecture notes here and got hung up. I'm not sure if I'm missing a part of the proof or I'm misunderstanding the theorem itself. The ...
1
vote
2answers
38 views

Is this the correct definiton of $T_1$ space?

I found this in a handwritten note: Def: A topological space $X$ is $T_1$ if $\forall x \neq y \in X$ there exist a neighborhood of $y$ such that s.t. $x \not\in V$. I was almost certain that ...
0
votes
0answers
24 views

Why is the typewriter sequence named as such?

I refer to the typewriter sequence (see https://terrytao.wordpress.com/2010/10/02/245a-notes-4-modes-of-convergence/) defined by: $$f_n := {\mathbb 1}_{\left[\dfrac{n-2^k}{2^k},\dfrac{n-2^k+1}{2^k}\...
12
votes
2answers
315 views

Can Path Connectedness be Defined without Using the Unit Interval?

Can path connectedness be defined without using the unit interval or more generally the real numbers? I.e., do we need Dedekind cuts or Cauchy convergence equivalence classes of the rational ...
1
vote
0answers
40 views

What does “factoring out an (group) action $\tau$ of a group $G$ acting on some set $E$” mean?

I am reading a survey article where they define the following objects: $\Gamma:=\mathbb{Z}^{n}$ seen as a group of translations. $\mathbb{T}:=\mathbb{R}^{n}/\Gamma$ is the $n$-dimensional ...
1
vote
1answer
35 views

inverse limit as a functor

I have a question about inverse limits based on https://en.wikipedia.org/wiki/Inverse_limit . In the section "general definition" there is noted that an inverse system is a contravariant functor $I\...
0
votes
1answer
26 views

What's the definition of being zero to infinite order at some point?

I'm reading a paper and it says The differential form $\psi$ is zero to infinite order at all points of $F$.That is,all coefficients of $\psi$ are zero to infinite order at points of $F$.(Here $\psi$...
1
vote
1answer
58 views

Can someone reconcile the two definition of Suslin's condition?

I am given two definitions of the so called "Suslin's condition" and I need to reconcile them. I am an undergrad and this is just for exploration. Definition 1. A partially ordered set $X$ is said ...
0
votes
2answers
65 views

what do curly brackets mean in this number theory equation?

$$(\lfloor y\rfloor-1)(\lfloor x\rfloor-1)=\{x\}+\{y\}+1$$ From looking at the LaTeX, I can see the left-hand side symbols mean the floor of the variable, but the right-hand side doesn't give much ...
0
votes
2answers
129 views

When are quantities outside of the real numbers considered equal, and when do they exist?

I know of the complex number $i$ and it's existence as the result of invalid square rooting (the square root of negative one does not exist inside the real numbers), but other than complex numbers, ...
0
votes
1answer
36 views

Topology: What is a quick way to check whether a subset $D$ is dense in $(X, \mathcal{T})$?

Def $1$: Let $(X, \mathcal{T})$ be a topological space, then $D \subseteq X$ is dense if $\overline {D} = X$ Def $2$: $x \in \overline D$ iff for all $U \in \mathcal{T}, x \in U \implies D \cap U \...
2
votes
1answer
38 views

Name of the distribution with density $P(x) e^{-x/\theta}$, where $P$ is a polynomial with positive coefficients.

The Gamma distribution of shape $k$ and scale $\theta$ has density $\frac1{\Gamma(k)\theta(k)} x^{k-1} e^{-x/\theta}$. Consider the more general distribution with density (up to a normalizing constant)...
0
votes
0answers
52 views

Covering maps of schemes.

A curve $X$ is modular if there is a finite covering $X_0(N)\rightarrow X$. What does covering mean in this context, and for more general morphisms of schemes? Just covering as topological spaces?
0
votes
0answers
12 views

Quasi-Isomorphism of Sheaf complexes terminology

Let $F$ and $G$ be sheaves of $R$-modules over a topological space $X$. Suppose we have isomorphisms $H^i(F)\simeq H^i(G)$ for all $i\in\mathbb{N}$. What is the proper term for this? Can one just say ...
1
vote
1answer
43 views

Complementarity and Substitutability

I am reading a paper in the international journal of game theory entitled Unequal Connections by Goyal and Joshi (2006) and it has the following sentences: "If strategic complementarity obtains... In ...
1
vote
1answer
28 views

Proposition about rings of fractions

This is taken from Atiyah-Macdonald's Commutative algebra book page 41. Someone please explain to me what is the meaning of "$a$ meets $S$". This is the first time I'm seeing this in the book.
0
votes
1answer
48 views

Are there any examples of consistent proper axiomatic extensions of classical logic?

By a proper axiomatic extension, I mean a logic with the same set of well formed formulas as classical logic, but with the set of theorems of the logic a proper superset of the theorems of classical ...
0
votes
2answers
80 views

What is the difference between Definite Integral & Indefinite Integral on the basis of their connection with derivatives?

I'm reading Denlinger's text on Real Analysis.There is a difference mentioned b/w definite integrals and indefinite integrals as-"The definite integral is a fundamentally significant concept,existing ...
1
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0answers
24 views

On a stronger property than being an Armendariz ring

A ring $R$ is said to be Armendariz if $f(x), g(x) \in R[x]$ are such that $f(x)g(x) = 0$, where $f(x) = a_nx^n + \dots a_0, g(x) = b_mx^m + \dots + b_0$, then $a_ib_j=0$ for all $i,j$. In other ...
1
vote
1answer
57 views

What does “pcr” stand for?

For example, as shown in this OEIS page: $P(n,4)=\frac{2n^3+6n^2-9n-13+(9n+9)\text{pcr}\{1,-1\}(2,n)-32\text{pcr}\{1,-1,0\}(3,n)-36\text{pcr}\{1,0,-1,0\}(4,n)}{288}$
1
vote
0answers
22 views

Are these correct extension of injective and surjective functions?

Recall the definition of injective and surjective functions: 1.a A function $f: X \to Y$, is injective if for all $a,b \in X$, if $a \neq b$, then $f(a) \neq f(b)$ 1.b A function $f: X \to Y$,...
1
vote
0answers
22 views

Subgraph with “dangling edges”?

I was wondering if there is a notion in graph theory where one can have a subgraph such that the endpoints of all of the edges in the subgraph are not necessarily included in the vertex set of the ...