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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1answer
52 views

What is a “component function” of a matrix that is a linear map?

The matrix is: $$ \pmatrix{ 1&2\\ 3&4} $$ and this matrix is an $\Bbb R^2 \to \Bbb R^2$ linear map. I am asked to explicitly write the component functions of $A$. What does "component ...
3
votes
1answer
31 views

Terminology regarding property of ideals

Is there a name for a property that only needs to be checked for either prime or maximal ideals in order to show that it holds for all ideals? An example would be being a principal ideal for which ...
3
votes
2answers
72 views

Reality, realizing, affinity, etc.: A terminology question

In Snapper & Troyer's book Metric Affine Geometry the word "reality" at one point occurs in the phrase (IIRC) "reality problems", meaning questions about whether the imaginary part of a complex ...
3
votes
1answer
97 views

Name/significance of integral of the square of a probability density function

Background/Motivation Given a probability density function $f(x)$, the mean of the corresponding random variable is the $x$-coordinate of the centroid of the region under the graph of $f$. I ...
0
votes
1answer
49 views

What's the name of the mathematical structure with is an abstraction of things like linear Independence?

This is a terrible question, I know. I can't remember the details for some reason, but I think (hope?) that anyone who's familiar with this object will immediately know what I'm talking about.
1
vote
1answer
57 views

In a dagger category, what do we call an arrow $f$ such that $f \circ f^\dagger \circ f = f$?

In a dagger category, what do we call an arrow $f$ satisfying $f \circ f^\dagger \circ f = f$? In $\mathrm{Rel}$ (and, more generally, an allegory) this is straightforwardly equivalent to $f \circ ...
0
votes
1answer
58 views

On Vanishing Riemann curvature tensor

If a manifold $\mathcal{M}$ has a vanishing Riemann curvature tensor, then what i) does this imply for the manifold? and ii) What is such a manifold called?
2
votes
4answers
103 views

Values of square roots

Good-morning Math Exchange (and good evening to some!) I have a very basic question that is confusing me. At school I was told that $\sqrt {a^2} = \pm a$ However, does this mean that $\sqrt {a^2} ...
2
votes
1answer
75 views

Name for numbers expressible as radicals

What is the name (there must be one!) for real (or perhaps complex) numbers expressible as radicals? Radical numbers? Solvable numbers? (following the same logic as ‘solvable group’). In other ...
0
votes
1answer
28 views

Name for percent “distance” of the “length” of interval?

Suppose we have an interval $[x_{min}, x_{max}]$ (where $x_{min} \neq x_{max}$) and a quantity $x$ that is a member of this interval. Is there a name for the following quantity? $\frac{x - ...
4
votes
1answer
72 views

How to pronounce the partition relation

The partition relation $$ \kappa \to (\alpha)^m_\lambda $$ says that for any $f:[\kappa]^m \to \lambda$, there is a $X\subseteq \kappa$ such that $f$ is constant on $[X]^m$ and the order type of $X$ ...
1
vote
1answer
47 views

In a dagger category, is there a name for morphisms $f : X \rightarrow Y$ with $\mathrm{id}_X = f^\dagger \circ f$?

In a dagger category, is there a name for morphisms $f : X \rightarrow Y$ with $\mathrm{id}_X = f^\dagger \circ f$? Clearly, every such arrow is a split monomorphism; further, if such an $f$ is ...
1
vote
0answers
32 views

“In between” for Lagrange's theorem for series

Everywhere I look, Lagrange Remainder Theorem is stated as Let $f$ be $k +1$ continuously differentiable, …, then there is a number $c$ in between $x$ and $a$ such that $$R_k(x) = ...
1
vote
1answer
107 views

Graph nomenclature

This concerns graphs that are sets of vertices and edges G={V,E}, not graphical depiction of functions. Imagine a graph that is a 2D square mesh of vertices. Such a graph can be constructed, for ...
2
votes
1answer
57 views

In the transition from set theory to order theory, what is the appropriate generalization of “group”?

Given a set $X$, the collection of all bijective endofunctions on $X$ forms a group, called the symmetric group on $X.$ Furthermore, Cayley's theorems says that every group embeds into a subgroup of ...
0
votes
0answers
97 views

Identity relation of many variables

The identity relation on a set $A$ is $\operatorname{id}_A = \{(x;x) \,|\, x\in A\}$. This can be generalized for any (possibly infinite) index set $N$ as $\{(\lambda i\in N: x) \,|\, x\in A\}$ (here ...
2
votes
3answers
386 views

Path components or connected components?

Can anyone explain the difference between these two terms? Are they basically different names for the same thing or totally different things?
2
votes
2answers
106 views

Why are left/right adjoint functors not called up/down?

I am studying category theory and I recently learned about adjoint pairs of functors. It seems to me that they are called left and right adjoints because if we have categories $\mathcal{C}$ and ...
0
votes
1answer
93 views

What are functions with the property $f(f(x)) = x$ called?

Do functions which, when composed with themselves, are equivalent to the identity function (i.e. functions for which $f(f(x)) = x$ in general) have a name and if so, what is it? Additionally, am I ...
0
votes
0answers
25 views

How do i visualize plots illustrating complex functions?

Here's an example : http://en.m.wikipedia.org/wiki/Essential_singularity What is this colored diagram illustrating complex function called? And how should i understand this diagram? What do colors ...
6
votes
2answers
172 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
35 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
117 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
51 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, ) = ...
2
votes
0answers
44 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
63 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 ...
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vote
0answers
55 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
47 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
653 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
88 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
54 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
106 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
46 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 ...
5
votes
1answer
91 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
52 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
39 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
80 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
74 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 ...
1
vote
1answer
55 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
53 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
57 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
145 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
39 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
26 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
34 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
60 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. ...
12
votes
3answers
646 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
2k 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
30 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
72 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 ...