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

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3answers
134 views

What is the correct English name of these lines?

Hello. I'm looking for the English name of these two lines in a two dimensional plane: they go through the origin they make angles of 45° and 135° with the $x$-axis, dividing the plane in two parts ...
1
vote
0answers
22 views

What can I use as the generic term for “a function that is composed with another”?

Suppose I am talking about the composition $g \circ f$ (or more generally $f_n \circ \cdots \circ f_1$). Is there a generic term for the functions $f$ and $g$ (the functions $f_i$)? "Compositand"?
7
votes
1answer
128 views

Is this a misuse of the word “evaluate”?

I have found the following use of the word "evaluate" in several math books: "To evaluate the continued fraction, start at the bottom and work your way up:" $\huge \underbrace{2 + ...
1
vote
1answer
48 views

Is there a name for magmas with $[x+y]+[x'+y'] \equiv [x+x']+[y+y']$?

Is there a name for magmas (written additively) satsisfying the following identity? The square brackets have no particular signifance, but will hopefully promote readability in what follows. ...
3
votes
1answer
31 views

What do we call a model of the empty signature?

Consider a model of the empty unsorted signature. Equivalently, a model of the signature having a single sort, and no function or relation symbols. Intuitively, such a model should be called a "set." ...
1
vote
1answer
105 views

Name for a category

Is there any name or notation for this category? Let $U$ be a set. By "function" I will mean a function $U\rightarrow U$. objects are functions; morphisms from a function $A$ to a function $B$ are ...
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0answers
23 views

Definition of Range as Minimal Interval Containing Codomain

I am studying continuous functions where the domain is some interval (which may or may not be bounded, closed, etc). I am thinking about how continuity is related to other function properties, ...
3
votes
0answers
78 views

Etymology of the term “weight vector”

I am writing a work on the representation theory of $SU(3)$ in basque and I would like to know the etymology of the term $\textbf{weight vector}$ in order to properly translate it.
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vote
1answer
42 views

What is the name of this factor-algebra?

In the polynomial algebra $k[x_1,x_2,\ldots, x_n]$ consider an ideal $I$ generated by the polynomials of the form $x_i^k-x_i$, $i=1 \ldots n$ and $k=2,3,\ldots.$ Consider the quotient algebra ...
2
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2answers
258 views

What are the names in English for Alterando, Invertendo, Componendo and Dividendo?

I am writing an article in English but don't want to use the Latin names. What are their English equivalent?
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2answers
47 views

Naming general objects in more than 3 dimensions

In a paper I am writing, I need to talk about a general "object" formed by the points of a connected set in an $n$-dimensional euclidean space. I have found some suggestion here, but none fit my ...
2
votes
2answers
45 views

Analytic methods vs Monte Carlo (terminology)

What's the correct terminology to say "We can calculate the probability exactly using pure math, as opposed to Monte Carlo simulation"? Analytically sounds like we need Calculus, which we may not ...
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2answers
141 views

Chern classes are not numbers, are they?

Let $X$ be a smooth projective algebraic variety, say over $\mathbb C$. Let $E$ be a rank $r$ vector bundle on $X$. We can associate with $E$ its Chern classes $c_i(E)$. When I read "$c_i(E)$", the ...
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vote
2answers
64 views

Terminology: Groups, rings, fields, etc.

Groups, semigroups, fields, rings, integral domains, vector spaces, R-modules... these are all approximately the same sort of "stuff", but each one refers to a slightly different combination of ...
6
votes
7answers
553 views

Fundamental Theorem of Trigonometry

This is a pretty open ended question and I apologize, in advance, if this is not the place for it. But what do you recommend should be given the title of the Fundamental Theorem of Trigonometry and ...
2
votes
1answer
122 views

Meaning of 'small real parameter'?

Consider the family of functions on $[a,b]$ given by $y(x; \epsilon) : = u(x) + \epsilon \eta (x)$, where the functions $\eta = \eta (x)$ are twice continuously differentiable and satisfy $\eta(a) ...
2
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0answers
46 views

What is the name for the topology where every point is in the boundary of an open set?

Is there a name for topological spaces in which every point is in the boundary of an open set?
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3answers
174 views

Why are group theory and ring theory a part of abstract algebra?

I have followed the courses Algebra 1, which was about group theory and Algebra 2, which was about ring theory. I don't think I really understand why those subjects are part of abstract algebra. What ...
2
votes
1answer
48 views

Notation for translating vectors

I'm completely new to vector geometry and recently encountered some new notation (and wholly unfamiliar) for the translation of vectors. $$T:Z \mapsto A + Z$$ The above is described as A ...
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2answers
47 views

Names for certain numbers.

I am wondering if there is names for numbers with the following characteristics: Numbers that end with 0. Numbers divisible by 5. If there are names for numbers with similar characteristics, I ...
4
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1answer
90 views

What's the real name for these things? Categories whose morphisms have “length.”

A fairly obvious "categorification" of metric spaces is as follows. First, let us agree to view $\mathbb{R}_+$ as an ordered Abelian monoid, where by "Abelian monoid" we really mean a category whose ...
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2answers
45 views

Terminology regarding elements of monoids

In what follows, the symbols $a,b$ and $n$ implicitly range over $\mathbb{N} = \{0,1,2,\cdots\}.$ Are there names for the following properties that an element $x$ in a monoid may or may not possess? ...
0
votes
1answer
49 views

The name of certain permutations.

The permutations I'm looking at are 2341, 2413, 3412, 3421, 4123 and 4312. I'll explain the property with the example 2413: I start with the first digit (2) and go to the position 2. There I see the ...
3
votes
3answers
184 views

Terminology re: continuity of discrete $a\sin(t)$

This question is specifically about the terminology used to explain a particular problem and its solution, not the math itself. I am a programmer, I am not really a math person, but I have at least an ...
1
vote
1answer
36 views

The term “maximal solution” for PDE

A solution $x(t)$ of the ODE is called maximal if it is defined on an open interval and cannot be extended to any larger open interval. from "Ordinary Differential Equation". Alexander ...
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3answers
4k views

What does it mean to solve a math problem analytically?

I'm reading a Calculus book for my own edification and at the beginning the pre-calculus introduction has the problem, $3x+y=7$ They talk about solving the problem graphically, analytically, and ...
2
votes
1answer
322 views

A module as an external direct product of the kernel and image of a function

If $f:A\rightarrow A$ is an R-module homomorphism such that $ff=f$, show that $$A=Ker\,\,f\oplus Im\,\, f$$ Here is a part of what I made as a proof. Let $a\in A$. $f(a)\in Im\,\,f$, ...
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1answer
93 views

Inductive hypothesis vs induction hypothesis

I'm doing a proof by induction. Should I refer to induction hypothesis or to inductive hypothesis in the proof?
58
votes
14answers
3k views

Are “if” and “iff” interchangeable in definitions?

In some books the word "if" is used in definitions and it is not clear if they actually mean "iff" (i.e "if and only if"). I'd like to know if in mathematical literature in general "if" in definitions ...
5
votes
1answer
52 views

Etymology of 'finite place'

In study of algebraic number theory one often comes across the terms 'infinite' and 'finite' places, referring to the archimedean and non-archimedean valuations of your field, respectively - but I ...
2
votes
4answers
68 views

What does “non-decreasing” mean in relation to this definition about the prime factorization of numbers?

I'm reading a text on discrete math and came across a theorem which states: "Every integer greater than 1 can be written uniquely as a prime or as the product of two or more primes where the prime ...
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0answers
45 views

First Homomorphism Theorem and terminology

I am not finding clear terminology in my abstract algebra book to be clear at least and my questions are simple. Consider the construction of a quotient group G': \begin{equation} G/K = G' ...
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2answers
48 views

Matrix with Functions as Entries

What do we call a matrix with functions as entries? $$\textbf{f(x)}=\begin{bmatrix} f_{11}(x) & f_{12}(x) \\ f_{21}(x) & f_{22}(x) \end{bmatrix} $$
0
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1answer
29 views

Equality among multiple numbers?

When I have two numbers and they are the same, we can say that they possess "equality". Let's say I have three or four numbers and they are all the same. What do we call the quality that they ...
2
votes
0answers
65 views

Fields of sets in which, if the l.u.b. of a subset exists at all, it is the union of the subset

I am learning about boolean algebras and how they can be represented as fields of sets. Stone's representation theorem tells us that every boolean algebra is isomorphic to a field of sets. Consider an ...
3
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0answers
97 views

Why tensors are called tensors and how this relates to the rigorous definition?

The algebraic motivation for tensors is fairly good: we know how to deal with linear maps, we must deal with multilinear maps, so we want to reduce them to linear maps. The name tensor however seems ...
1
vote
1answer
64 views

If $Y=\sum_{n=1}^\infty X_n$ diverges, is $Y$ a random variable?

Let $X_n$ be random variables. By definition, a random variable is a function from the probability space to $\mathbb{R}$. If $Y=\sum_{n=1}^\infty X_n$ diverges, is it correct to call $Y$ a random ...
0
votes
1answer
105 views

What's the name of $x^x$?

I know that $$f{(x)} = a^x$$ is called exponential function and $$f{(x)} = x^a $$ is the power function. But what is the name of $f{(x)} = x^x$?
2
votes
1answer
64 views

Adjoint of a Matrix Definition

Tom M. Apostol in his book "calculus Vol. 2" page 122 (see image below) defines adjoint of a matrix as the transpose of the conjugate of the matrix. Is this definition always correct ? Does it agree ...
0
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1answer
91 views

How to make a ghost manifold [closed]

How does one mathematically define a manifold that can pass through another manifold? A "ghost" passing through a "wall" type construction. I understand that this may be done by creating a copy of the ...
2
votes
2answers
174 views

What is the meaning of “integral point”?

While reading this paper (http://cowles.econ.yale.edu/P/cd/d04b/d0473.pdf) I encountered the concept of "integral point", used first in definition 5.1, on page 34. Does anybody know more details about ...
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2answers
89 views

What is this mathematics sub-field called?

I would love to answer another question on this site, but I am totally unfamiliar with the required technique. I mean, I don't even know the sub-field's name. The field I am looking for is one that ...
1
vote
1answer
91 views

Some basic questions about matrix rings and reversibility.

Neither commutative rings nor division rings are viable approaches to studying rings of matrices. However, there is a very cool notion of a reversible ring, which looks like it can fill this void. I ...
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vote
1answer
50 views

In the context of algorithms what does “bookkeeping scheme” mean?

In this paper, begining of page 5 is written: The partial costs are then equivalent to [...] with the bookkeeping entities [...] The bookkeping scheme enable fast evaluation of the cost ...
3
votes
1answer
80 views

Monoids as categories; does this construction have a name?

We can view a monoid $M$ as a category with a single object. However, there is another way to make $M$ into a category. Take the elements of $M$ as objects, and define $\mathrm{Hom}(x,y)$ to be set of ...
0
votes
1answer
50 views

We refer to X for standard notations and definitions from Y

I'm having problems with my mathematical English, so I'd like to ask for your help! Is it correct to write something like "Unless stated otherwise, we refer to [1] and [2], respectively, for standard ...
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votes
2answers
194 views

Is there a proper term and/or symbol for an “agnostic” conclusion?

My question stems from the material conditional: $p \rightarrow q\\p\\\therefore\space q$ However, if $\bar p$ then the conditional is silent. I would like a way to represent this fact using, if ...
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0answers
22 views

A name for a linear map from primary to dual space

Is there a standard name for a linear function $f:\mathbb R^n\to(\mathbb R^n)^*$ defined on the standard basis $e_i, i=1,\dotsc, n,$ of $\mathbb R^n$ by $f(e_i)=e^i$, where $e^i, i=1,\dotsc,n,$ is the ...
2
votes
1answer
60 views

What is the technical term for the $n$-dimensional generalization of the unit interval?

What is the technical term for an $n$-dimensional generalization of the unit interval $[0, 1]$? Would we call an $n = 1,2,3,...$ dimensional generalization of the unit interval an $n$-cube?
1
vote
1answer
45 views

A measure having no point masses.

What does it mean for a measure to have no point masses? Is this sort if like saying that individual points have measure zero?