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

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6
votes
2answers
147 views

Why 'closed differential forms' are called 'closed'?

As is well known a differential form $ \omega $ is called closed differential form if it satisfies $ \mbox{d} \omega = 0 \, \, (\ast) \, $ where $ \mbox {d} $ is the exterior derivative. I think the ...
0
votes
1answer
20 views

Partial order up to equivalence

In certain contexts one runs into something like a partial order, but the antisymmetry property is weakened as follows: if $x \preceq y$ and $y \preceq x$ then $x \simeq y$, where $\simeq$ is a given ...
1
vote
2answers
87 views

Business math: how to say increasing faster and faster

I have to make a presentation to business directors and I want to explain that for a particular item, the function of cost in effort is beyond "linear growth". Here's where it gets hairy. My function ...
0
votes
2answers
38 views

What is called the property of function that it does not change value when you transform arguments.

My question is probably rather simple, but I cannot find appropriate name or am too stupid to find a definition of such property. I'd like to give an example: $f({\bf r}_1, ..., {\bf r}_N, ) = ...
1
vote
0answers
35 views

What are the names for the structures obtained when we drop some topological space axioms?

Motivation: If I start with the group axioms and drop the requirement that I have inverses, I get the monoid axioms. If I proceed to drop the requirement that I have an identity, I get the semigroup ...
2
votes
1answer
40 views

Hypergraph terminology

Suppose I have a hypergraph with vertices V and hyperedges H, where each hyperedge is a subset of V. I want to form a normal graph with vertex set V, where two vertices are adjacent if they lie in ...
1
vote
0answers
43 views

Name of rational numbers of the form $p/q$ with $p,q$ prime

For the life of me, I cannot think of whether there is a name for fractions of the form $\frac{p}{q}$, where $p,q$ are both prime. Fractions such as $\frac{4}{5}$ are sometimes said to be in "reduced ...
5
votes
0answers
37 views

Is there a name for the algebra of substructures?

Let $X$ denote an entropic algebra (see here), which just means that all the operations of $X$ are homomorphisms $X^n \rightarrow X.$ Abelian groups are the classic example. Then for any operation of ...
0
votes
2answers
57 views

Probability of a Min/Max

I am studying probability for an exam and I am finding hard to understand the notion of $P(\min(X_1,X_2))$ and $P(\max(X_1,X_2))$, where $X$ is a discrete or a continuous variable. I have found in my ...
0
votes
3answers
54 views

How can I mathematically read this map $f:A\longrightarrow B$?

In Group Theory I found a "$f:A\longrightarrow B$" but I don’t know how to pronounce this term in English. I know there is a mathematical term for ":" and "$\longrightarrow$" in the map ...
2
votes
0answers
47 views

Is there a word for a number that can be expressed as an exponential with the same base and exponent?

Some examples: \begin{align*} 1 &= 1^1 \\ 4 &= 2^2 \\ 27 &= 3^3 \\ 256 &= 4^4 \\ 3125 &= 5^5 \\ \end{align*} and so on. Is there a name for these types of numbers? It seems like ...
4
votes
2answers
98 views

Have action/predicate systems (or similar) been considered in the literature?

Question. Has the following concept, or anything similar, been considered in the literature? Definition. An action/predicate system consists of sets $A$ (the actions) and $X$ (the predicates) such ...
1
vote
1answer
44 views

Convex Function Help and Counterxample

Given $g: \mathbb{R}^n \to \mathbb{R}$ is convex and $f:\mathbb{R} \to \mathbb{R}$ is convex and increasing. Show that $(f \circ g): \mathbb{R}^n \to \mathbb{R}$ is convex. I had no problem proving ...
4
votes
1answer
73 views

Weakened associativity axiom: $(x * y) * z \leq x*(y*z).$

Call a partially ordered magma $(X,*)$ sub-associative iff it satisfies the following axiom. $$(x*y)*z \leq x*(y*z).$$ Basically, this is saying that we may shuffle brackets right to get a larger ...
0
votes
1answer
48 views

Is there a name for this theorem about the convergence of a function?

Let $f(x)$ be a continuous function over $\mathbb{R}$ such that for all $a < b$, we have $a < f(a) < b$. Then, for any $x < b$, the sequence $\{t_n\}$ defined by $t_0 = x, t_n = ...
3
votes
0answers
26 views

How to call this homomorphism-like property?

In my communication theory work I derived a property that is essentially \begin{align} f(x)\cdot f(y) = f(x-y) \cdot A^N \end{align} where $A^N$ is some quantity from the technical context that is ...
3
votes
3answers
68 views

What is the term for whatever is being differentiated?

When we integrate a function: $ \int^b_a {2\over x^2} dx$ The expression to be integrated (is this case $ {2\over x^2} $) is referred to as the integrand. When we differentiate a function: $ {d ...
1
vote
2answers
38 views

What does the term “distinguished basis” mean?

I know what a basis is (talking about vector spaces here), but I don't know what a distinguished basis is. Can you please explain the difference to me? I did not grow up in an English-speaking ...
0
votes
0answers
18 views

Is there an accepted definition of the “coefficients” of a multivariate rational function?

I'm reading Complexity and Real Computation by Blum, Cucker, Shub, and Smale. In in defining "machine constants" of a Blum-Shub-Smale machine, they talk about the "coefficient" of a multivariate ...
1
vote
0answers
37 views

Definition of a binary operation is the same as definition of a closed binary operation?

I'm reading Wikipedia about operations and binary operations . Intuitively I always thought that a binary operation is a operation that takes two arguments. But Wikipedia defines a binary operation as ...
2
votes
0answers
50 views

Terminology problem

I am not a mathematician, please forgive my incorrect language. My question involves terminology. If a finite non-abelian group G is represented by a set of unitary operators ${\mathbf G}_r, r = ...
2
votes
0answers
55 views

Why do we say a linear space is “over” a scalar field?

This terminology has puzzled me for a while and I haven't seen it actually discussed anywhere. Why does the language indicate relative positions of some space or operator and the objects they deal ...
1
vote
2answers
44 views

What is meant by a formal statement in mathematics/computer science

While reading books in Mathematics and Theoretical Computer Science, usually the term Formal Statement props up. What is meant by that?
2
votes
1answer
33 views

Why do we want to define a $k$-scheme to be birational if the rational map (and its inverse) to $\Bbb A_k^n$ is over $k$?

Two varieties $X,Y$ are said to be birational if there exist rational maps in each direction such that either composition is the identity on a open dense subset. Note that here the morphisms aren't ...
1
vote
0answers
20 views

What is the name of this type of stochastic processes?

I've seen someone briefly define a continuous time stochastic process $X$ on $\mathbb{N}$ as the (a?) solution to $$X(t)=X(0)+Y\left(\int_0^t f(X(s))ds\right)$$ where $Y$ is an inhomogeneous ...
3
votes
0answers
28 views

Terminology Regarding Basic Properties of Functions

Is there a cultural difference between saying that a function is 1-to-1 or injective, onto or surjective and a 1-to-1 correspondence or bijective?
1
vote
1answer
36 views

Meaning of 'Isomorphism (with respect to inclusion)'

This is the first time that I see this phrase. I'm reading Commutative Algebra by N.Bourbaki. I'll extract 2 propositions that use this phrase. The first one is on page 68 of the book. ...
0
votes
0answers
12 views

How is affine space analogue for lattices called?

Lattices are so like vector spaces that it seems natural to have an affine space construction for them. Unfortunately I could not find how such a construction is called. Could you please help me? ...
11
votes
0answers
275 views

Why is an orthogonal matrix called orthogonal?

I know a square matrix is called orthogonal if its rows (and columns) are pairwise orthonormal But is there a deeper reason for this, or is it only an historical reason? I find it is very confusing ...
2
votes
1answer
279 views

into function vs injective function

In many mathematical books that I have read and from lectures from professors, the words 'into' and 'injective' were used interchangeably, but in Patrick Suppes book Axiomatic Set Theory he gives a ...
0
votes
1answer
27 views

degenerate plane in $\mathbb{C}^3$

Does someone know about "degenerate plane"?when do we say a plane in $\mathbb{C}^3$ namely $a z_1 + bz_2 + c z_3 = d$ where $a,b,c,d$ are complex constants , to be degenerate? Is there any reference ...
1
vote
2answers
49 views

Basic Cartesian prodcuts

I am having some issues grasping basic ideas of Cartesian products. I am reading a PDF my professor gave us explain sets/Cartesian products. If $\mathbb{R}\times \mathbb{R}$ can be written as ...
4
votes
1answer
68 views

Have arrows in a category with this property a special name?

Studying posets I encountered the notation $a\prec b$. It means that $a<b$ and no $c$ exists with $a<c<b$. If $a\prec b$ then in words $a$ is covered by $b$. Looking at a poset $P$ as a ...
5
votes
1answer
83 views

Difference between graded ring and graded algebra

Wikipedia says that a graded $A$-algebra is just a graded $A$-module that is also a graded ring. Question: when one says then "finitely generated graded $A$-algebra", does one mean that every element ...
1
vote
2answers
28 views

Terminology for maximize/minimize choice

I'm writing optimization software where the user needs to decide whether they want to minimize or maximize the value the objective function (where the output will contain many putative optimal ...
2
votes
1answer
128 views

Is there a name for this property of a binary relation? $\forall x\forall y(x\mathsf{R}y\to\exists z(x\mathsf{R}z\land y\mathsf{R}z)))$

Consider a binary relation $\mathsf{R}$ such that $x\mathsf{R}y$ is the case only if there is some $z$ such that both $x\mathsf{R}z$ and $y\mathsf{R}z$ are the case. Is there a well-known name for the ...
37
votes
19answers
4k views

What is the difference between a point and a vector

I understand that a vector has direction and magnitude whereas a point doesn't. However, the course note that I am using states that a point is the same as a vector. Also, can you do cross product ...
3
votes
1answer
29 views

Families of graphs where the shortest path between vertices uniquely determines vertex pairs

Imagine a graph $G$ with unlabeled vertices and unlabeled edges, and where we have an arbitrary vertex pair $(v_1,v_2)$. Let $k$ be the length of the shortest edge-wise path between $v_1$ and $v_2$. ...
2
votes
2answers
32 views

Comparison of almost planar graphs

I have multiple graphs all of which are almost planar. Is there any existing terminology / method which compares them, such that one can say which one is more planar? This could simply be the required ...
0
votes
1answer
47 views

What does “possible to define” mean?

What does "possible to define" mean in general? First I thought it means that "can not lead to a contradiction", but such seems to be hard to prove. Then for the proof I was looking at, involving ...
3
votes
2answers
105 views

Definition: Theorem, Lemma, Proposition, Conjecture and Principle etc.

Definition: Theorem, Lemma, Proposition, Corollary, Postulate, Statement, Fact, Observation, Expression, Fact, Property, Conjecture and Principle Most of the time a mathematical statement is ...
1
vote
1answer
87 views

Why is it called 'discrete' mathematics?

I understand why you would refer to mathematics which concerns itself with all of the numbers on the number line as 'continuous' but why would you refer to countable or finite mathematics as ...
0
votes
1answer
38 views

In the context of vectors is there a difference between the terms “magnitude” and “length”?

I noticed vectors are usually said to have "length and direction" but then it is said that people want to find the "magnitude". Is this just a difference in terminology or is there something more to ...
2
votes
0answers
66 views

Do subhomomorphisms / subfunctors have a standard name, and where can I learn more?

Sorry for all the mistakes in the original! I think they're mostly fixed now. Thank you for your patience. Part 1. If $A$ and $B$ are models in $\mathrm{Pos}$ of an algebraic signature $\sigma$, ...
2
votes
1answer
55 views

The Empty Relation?

In elementary set theory, a relation on sets $A,B$ is usually defined as a subset of $A\times B$. We know that there are $2^{|A\times B|}$ subsets of $|A\times B|$. One of these subsets is the empty ...
1
vote
1answer
106 views

Pythagoras Theorem - Why a Theory? [closed]

Why is Pythagoras Theorem a Theory but not a Law? I mean we use it many times in School and to build stairs etc. and it has been proven however it is called a Theory.
2
votes
0answers
29 views

Is there a source that explains or defines terms like Theorem, Proposition, Lemma and Corollary?

I will write the German names of the following English terms in brackets as it is important that I use the correct terms. This is a question about terms and I'm not completely sure if I've ...
0
votes
1answer
67 views

Is it OK to say “for almost every natural number”?

Suppose you want to say that a predicate $P(x)$ holds for every sufficiently large natural number $n$. I have seen several authors write "for almost every $n \in \mathbb N$, $P(n)$" or "$P(n)$ a.e." ...
0
votes
1answer
36 views

Notation for permutation corresponding to the action of a group element

Let $G \times X \to X,\ \ (g,x) \mapsto g.x$ be an action of $G$ on $X$, i.e., $e.x = x$ for all $x \in X$; $gh.x = g.(h.x)$ for all $g \in G$, $x \in X$. For a fixed $g \in G$, how should I refer ...
2
votes
2answers
227 views

Fundamental theorem of linear algebra

When I studied linear algebra we (our books, our professors) used to call Fundamental theorem of linear algebra the theorem that says: Fundamental theorem of linear algebra: A linear ...