# Tagged Questions

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

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### What are “words”?

Related but not duplicate. I am reading Classical Mathematical Logic by Richard L. Epstein, page $3$: B. Types When we reason together, we assume that words will continue to be used in the ...
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### Names for related pairs of angles

I seek the names (if they exist) of two relationships between angles. Two angles are complements of each other if they add up to a quarter circle. $\sin\alpha=\cos\beta$ and vice versa. Two angles ...
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### Unclear passages in the paper “On a New Class of Theorems in Elimination Between Quadratic Functions” by J. J. Sylvester

I'm writing an essay about the origin of some mathematical terms in the work of J. J. Sylvester. He first used the word matrix in his paper Aditions to the Articles "On a New Class of Theorems" and "...
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### Nichtnegativstellensatz the same as Handelman's Theorem?

Wikipedia on "Handelman's theorem: If $K$ is a compact polytope in Euclidean $d$-space, defined by linear inequalities $g_i ≥ 0$, and if $f$ is a polynomial in $d$ variables that is positive on $K$, ...
15k views

### What's the difference between stochastic and random?

What's the difference between stochastic and random? I've read in the Portuguese Wikipedia that there's a difference, but I still didn't see this point on English Wikipedia.
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### Is the phrase “random number function” contradictory? [on hold]

If a Function is something that produces an output from an input(s) and is consistent, then the phrase "Random Number Function" should not be allowed, right?
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### Vector fields (on a manifold) and terminology

I read in several books (Do Carmo, Riemannian Geometry or John M. Lee, Smooth manifolds) that a vector field $X$ on a smooth manifold $M$ is a mapping which associates to each point $p \in M$ a ...
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### What does it mean to say a point is uniquely mapped?

I am looking at space filling curves. Essentially their is a mapping $f: I \to \mathcal{Q}$ where I is an interval in $\mathbb{R}$ such as $[0,1]$ and $\mathcal{Q}$ is a square $[0,1]^2$. For the ...
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### What do mathematicians mean when they say “form”?

As in differential form, modular form, quadratic form? I'm sorry if this is a really silly question.
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### All directed paths between any two vertices have the same length

Is there a term for the condition that, given some directed graph $G = (V, E)$, for all $v, w \in V$ every directed path from $v$ to $w$ has the same length as every other?
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### Are there more proper terms for “between, inclusive” and “between, exclusive”?

I searched for this, but all I found was an English usage result. However, I am strictly asking about ranges of numbers, not "normal" English. So, are there a terse terms for: inclusive between, ...
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### Specifying from the general in probability: Does it work? [closed]

If the average classroom AC holds 30 students, and 1 in 10 students throughout the US has a probability of having condition A, does that mean there's a 300% chance there's a student in classroom AC ...
134 views

### What is a constant?

The word "constant" is used in such expressions as "The derivative of a constant is $0$." What does it mean? (I will post my own answer here, but I'm sure lots of others can have fun with their own ...
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### Names for the vector spaces $T(V)$ and $S (V)$

Are there any names for the vector spaces $T(V) = \bigoplus_{n\geq 0} V^{\otimes n}$ and $S(V)= \bigoplus_{n\geq 0} V^{\otimes n}/\Sigma_n$? The best thing I could come up with is "the underlying ...
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### Difference between Ordering and Order?

I am confused by the two terms order and ordering. I am learning on Ideals, Varieties and Algorithms by Cox et all. The context is monomial orderings and Gröbner basis on polynomial rings. How are ...
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### Is there such a thing as a matrix of functions?

Do we ever put functions as entries of a matrix? If so, are these matrices used in linear algebra or do they have some other special use? There have been minor not neccessarily conflicts per se, but ...
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### What does it mean in general to show something is well defined? [duplicate]

There is another post that addresses this but quickly fix the problem to be something in arthmetics, and in turn what it means for that arithematics problem to be well defined. I have never ...
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### What does “Borel space”, unqualified, refer to?

For examples of use, Google "in Borel space", without the quotes. I'm thinking it means either ℝ equipped with its Borel σ-algebra, or to Borel spaces in general (that is, topological spaces with a σ-...
27 views

### difference between variance and covariance

What exactly is the physical interpretation of variance (in terms of a data set) and the difference between variance and co-variance matrices?
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### Alternative Definition of Contravariant Functor

Given two categories, $C$ and $D$, a covariant functor is usually defined as a regular functor $C \to D$, whereas a contravariant functor is usually defined as a regular functor $C^{op} \to D$. ...
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### What does “coefficients from all of $\mathbf{F} _q$” mean

I was reading Wikipedia's page on Ring Learning with Errors, and came to wonder what is meant by "with coefficients from all of $\mathbf{F} _q$" which is a requirement for the set of known polynomials....
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### What is $x \mapsto f(x)$ called?

A function is defined by either A function $f: A\to B$ is defined by $\color{green}{x\mapsto f(x)}$ or \begin{align}f:\quad&A\to B\\&\color{green}{x\mapsto f(x)}\end{align} Is ...
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### What are the relations between sums or products of numbers that accounts are so good at called? [closed]

Accountants are good at seeing the relations between sets of numbers to see how they are combined together to form a particular total or sum. What is the skill of being able to do this called?
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### A question of terminology regarding exceptional curve or is it divisor.

So I kept on reading the book by Griffiths and Harris called Principles of Algebraic Geometry and I've seen a definition of exceptional divisor of the first kind. On page 487: A smooth rational ...
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### Relationship Between Variables Both Growing/Decreasing, not Proportional

So proportional is when the variables are equal to one another when multiplied by a constant. What is the term for something like weak proportionality that when one variable increases the other will ...
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### Is “closedness” a proper word?

In one of my papers I had to prove a list of properties of a set, say, $S=\{a,b,c\}$. Among them we have a fact that $S$ is downward closed with respect to a binary relation $R$. I found it awkward to ...
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### Does “Doing a thing to both sides of an equation” have a name?

A two part question. 1 True or False: when working with an equation or inequality, everything that you do is either: a substitution, or an operation performed on each side Note that algebraic or ...
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### Ring->module->$R$-algebra, Field->Vectorspace->algebra

I haven't done any mathematics for a long time, and I have forgotten some things. I want to try to remember some of the words and how they interact. A module is a 'vectorspace over a ring' rather ...
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### Is Site Percolation with Bernoulli variables i.i.d. independent and identically distributed?

I cannot understand the identically distributed part in the i.i.d assumption. Consider a site percolation where each event is a Bernoulli variable. Does this mean ...
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### How to call the region bounded by a hyperbola

Given an hyperbola, is there a mathematical name that describes the region/area bounded by one arm of the hyperbola? In this image the area is marked grey. To clarify my question: I'm looking for a ...
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### What is the definition for totally ramified extension for a global field?

What is the definition for totally ramified extension for a global field? For local fields it means the maximal prime ideal generated from the uniformizer totally ramifies. But what is the definition ...
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### Is there an equivalent word for '3/4?'

It's already known that the most of the quarter fractions have a single word equivalent that correspond with its numerical counterpart, such as '1/4' is a quarter, '1/2' is half, and '4/4' is the same ...
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### Binary tree traversal with fixed final node

What type of traversal is this called? Given a root node A and final node B: ...
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### Names of higher-order derivatives

Specific derivatives have specific names. First order is often called tangency/velocity, second order is curvature/acceleration. I've also come across words like Jerk, Yank, Jounce, Jolt, Surge and ...
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### Are Free Groups the “Smallest Group” Containing their Generators

I apologize if this is a duplicate; I was not sure how to search for this. When I say "the smallest group" I mean unique up to isomorphism of course. Specifically, is "the smallest group containing ...
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### Terminology for a process with subcritical, critical, and supercritical cases?

I've noticed that, in a number of domains in pure and applied mathematics, there are processes or structures involving exponential growth or decay where the process splits into three cases: a ...
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### Proving a theorem, what is meant by sufficiency and necessity?

I am looking at the proof of a theorem and the proof begins by saying ...is the proof of the sufficiency part of this theorem so we just need to establish the necessity of the condition. What ...
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### Terminology Question: Precompose vs Compose?

I was wondering if there was a standard convention on what 'precompose' means compared to 'compose', as I am often confused between the two when all sorts of text casually use both terminologies. For ...
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### Have these (extremely simple) classes of algebraic structures been considered in the literature? If so, what are they called?

Questions. Have the following kinds algebraic structures been considered in the abstract algebra literature etc.? If so, what are they really called? (I have used made-up terminology for the sake of ...
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### How is the threshold probability $p_c$ defined for oriented site percolation?

The threshold probability for unoriented site percolation is such that \begin{eqnarray*} \mathbb{P}_{p} & = & \underset{v\in\mathbb{\mathbb{L}}^{d}}{\prod}\mu_{v}\\ \theta(p) & = & \...