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

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3answers
186 views

Is an unit-cube polyhedron? What about other platonic solids?

Definitions According to my linear programming course and screenshot here (Finnish), a polyhedron is such that it can be constrained by a finite amount of inequalities such that $$P=\{\bar x\in ...
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1answer
108 views

Divisible abelian $q$-group of finite rank

What does "finite rank" mean in the context of divisible abelian $q$-group? A divisible abelian $q$-group of finite rank is always a Prüfer $q$-group or it can be also a finite product of Prüfer ...
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3answers
9k views

What is the meaning of equilibrium solution?

What are the equilibrium solutions for the differential equation $\dfrac{\mathrm{d}y}{\mathrm{d}t} = 0.2\left(y-3\right)\left(y+2\right)$ My Question: What does equilibrium solution mean in this ...
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1answer
94 views

Does this object have a category-theoretic name?

I have morphisms: $$ f : A \to B \\ g : B \to C $$ The composition is: $$ g \circ f : A \to C $$ In the function $(g \circ f)$ we call $A$ the domain and $C$ the codomain (or range). I'm ...
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1answer
124 views

System, dynamic system and feedback system

Can a system be defined as a mapping from a set of mappings, called input signals, to another set of mappings, called output signals, where the two sets of mappings may or may not have the same ...
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2answers
460 views

Injection and surjection - origin of words

Can anyone give me a good explanation of how and why words surjection and injection came into use in mathematical community? What do they exactly mean? Who introduced them? I have a feeling students ...
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2answers
3k views

Difference between root, zero and solution.

Can somebody precisely tell me what is the difference between a root, a zero and solution ? Is it correct to say that an equation has solutions, and a polynomial has zeros or roots?
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3answers
2k views

Embedding, immersion

Could someone please explain what "embedding" means? (Maybe a more intuitive definition) I read that the Klein bottle and real projective plane cannot be embedded in ${\mathbb R}^3$ but is embedded in ...
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1answer
503 views

Summation formula name

What is the name of the following summation formula? $$\sum_{k = 1}^n f(k)) = \int_1^{n + 1} f - \frac{f(n + 1) + f(0)}2 + \int_1^{n + 1} f'w,$$ where $w$ is the “sawtooth” function, defined by ...
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1answer
405 views

Is “monotonous” ever used as a synonym for “monotonic” in math?

I saw a few questions and answers recently that wrote "monotonous" instead of "monotonic." Then I Googled and see a ton of usages of "monotonous" in M.SE instead of monotonic. It occurred to me this ...
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9answers
2k views

Are all integers fractions?

In a college class I was asked this question on a quiz in regards to sets: All integers are fractions. T/F. I answered False because if an integer is written in fraction notation it is then ...
6
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3answers
271 views

Does “monotonic sequence” always mean “a sequence of real numbers”

When we say a sequence is monotonic, does that imply the sequence is Real Number Sequence? And other propositions about monotonic, all real-valued? When I see some ...
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5answers
797 views

What exactly is “approximation”?

There are a lot of great "approximations" that exist in the mathematical field:$$\dfrac{22}{7} \approx \pi$$ $$e \approx \left(1 + \dfrac{1}{n}\right)^n$$But the fact that I have yet to know what ...
6
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2answers
2k views

opposite of disjoint

Sets whose intersection is the empty set are called disjoint. What is the opposite of a disjoint set? For example the sets $\{1,2\}$ and $\{2,3\}$ satisfy this condition. I know that you can just say ...
6
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1answer
260 views

What are “Lazard” sheaves?

Early in Categories, Allegories, by Freyd and Scedrov (p.12, in the section on basic examples) there appears the following example: Let $\mathcal{LH}$ be the category whose objects are topological ...
6
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2answers
159 views

History of the vocabulary for group extensions

In regular everyday English if you say something like "A was extended by B to get C", to me it means that A was in existence, B was added onto it, and now there is a larger object C. For example, ...
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3answers
76 views

Name for introducing negation with quantifiers

The rewriting of $\varphi\to \psi$ into the logically equivalent $\neg \psi\to\neg \varphi$ is called contraposition. Is there a similar word for rewriting $\forall x.\varphi$ into $\neg\exists ...
5
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1answer
106 views

What word means “the property of being holomorphic”?

As in the title, I am looking for a single word meaning "the property of being holomorphic". The obvious candidates are "holomorphy" and "holomorphicity" but both look wrong to my eye. ...
5
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0answers
106 views

Is there a name for graphs with the following property?

The property of the graph is the following: For any vertex, there is a hamiltonian path starting with this vertex, but the graph is not hamiltonian. The following graph is a small example: ...
5
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1answer
194 views

On proving $n = \sum_{d\mid n}\varphi(d)$

$\def\nset{\{1,\dots,n\}}$ I'm trying to work out my own proof1 of Euler's classic formula $$n = \sum_{d\mid n}\varphi(d)\;.$$ I'm looking for some pointers to the standard terminology and/or ...
5
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1answer
109 views

Is the variant direct image mathematically significant?

Preimages have the property that for an arbitrary function $f : X \rightarrow Y$ and all $B \subseteq Y$ it holds that $$f^{-1}(B^c)=[f^{-1}(B)]^c.$$ However, the analogous statement for direct ...
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3answers
2k views

Can the word “derive” be used to mean “take the derivative of”?

Back when I was in high school, the usage of the word "derive" to mean "take the derivative of" was really widespread. It always bothered me because I felt that the proper verb should be ...
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3answers
129 views

What does “lower density” mean in this problem?

If $\mathscr{U}$ is a ultrafilter on $\omega$, then $\mathscr{U}$ contains a subset $A$ of lower density zero. This is an exercise on page 76 of Problems and Theorems in Classical Set Theory, ...
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4answers
409 views

Is it proper to say that two infinite sets are the “same size” if there is a bijection between them?

I get the fact that a set can be called countably infinite if it can be bijected with $\mathbb{N}$, but it feels wrong on many levels to say that they are the same size. Example: $A=\{x \in ...
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1answer
282 views

What is the last index of a third-order tensor called?

In a third-order tensor I guess the first and second index would be called row and column respectively but is there a name for the third index?
4
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1answer
126 views

What is the correct terminology to say that $\small f(x)=a+bx+cx^2+…$ can be expressed by $\small g(x)=A(1-x)+B(1-x)(2-x)+C(1-x)(2-x)(3-x)+… $

Hm, I do not even know the best formulation for my question in the header. It is not for the math but for the proper writing/terminology. I've come across the term "base change" recently but the ...
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1answer
120 views

“belongs to” versus “contained in”

Let us consider a set $A$. let $B$ be an element of the set. Now what I want to know is that whether saying $B$ is contained in $A$ and $B$ belongs to $A$ means the same? Could anyone here cite any ...
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5answers
582 views

How can I succinctly but correctly say that a set is finite?

If I want to say that a set $A$ is numerable but infinite, I can do so like this: $$|A| = \aleph_0$$ What should I use instead to say that a set is finite? $|A|\in\mathbb{N}$? $|A|< \infty$? ...
4
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0answers
212 views

Is there some official name for this function?

$$\sqrt{1 - (-1 + x\bmod 2)^2}\cdot\operatorname{sign}(-2 + x\bmod 4)$$ Like half-circles connected to each other to look like waves: Its plot looks smooth, but the function is actually not ...
3
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1answer
145 views

The functor $\mathbf{D} \rightarrow \mathbf{Prof}$ obtained by “splitting” $F : \mathbf{C} \rightarrow \mathbf{D}$ at each object of $\mathbf{D}.$

(Write $\mathbf{Prof}$ for the category whose objects are categories and whose arrows are profunctors.) I'm pretty sure that every functor $F : \mathbf{C} \rightarrow \mathbf{D}$ yields a ...
3
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2answers
673 views

Validity vs. Tautology and soundness

I see that valid formula (proposition or statement) is the one that is valid under every interpretation. But this is a tautology. Is there any difference between tautology and valid formula? They also ...
3
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3answers
7k views

In graph theory, what is the difference between a “trail” and a “path”?

I'm reading Combinatorics and Graph Theory, 2nd Ed., and am beginning to think the terms used in the book might be outdated. Check out the following passage: If the vertices in a walk are ...
3
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0answers
909 views

Support of a vector

What is the support of a signed vector? By signed vector, I mean a vector which is determined by considering the signs of the coefficients of the entries of another vector.
3
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3answers
104 views

What are the sets $S_n=\omega-n$ called?

What are the sets $S_n$ where $S_n:=\omega-n$ called? I explain better: if ordinals are defined in this way $0=\varnothing$ $1=\{\varnothing\}=\{0\}$ $2=\{0,1\}$ $n=\{0,1,..,n-1\}$ ...
3
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2answers
260 views

Zorn's Lemma $\equiv$ Axiom of Choice

I'm confused a little bit about this, I've been told many times that Zorn's lemma is equivalent to the axiom of choice. Is it an axiom or is it lemma, I mean is there a proof of Zorn's lemma or we ...
3
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2answers
4k views

What is the difference between an axiom and a postulate?

I here about axioms is set theory and postulates in geometry, but they seem like the same thing. Do the mean the same thing but then are used in different instances or what? Is one word more ...
3
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1answer
82 views

Formalizing the idea of a set $A$ *together* with the operations $+,\cdot$''.

I will illustrate my question in the case of the definition of vector spaces. It is custom to define a vector space in the following way: "Let $K$ be a field. Then a $K$-vector space is a set $V$ ...
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2answers
2k views

Relation between Interior Product, Inner Product, Exterior Product, Outer Product..

Following my previous question Relation between cross-product and outer product where I learnt that the Exterior Product generalises the Cross Product whereas the Inner Product generalises the Dot ...
3
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2answers
214 views

Is a 2-dimensional subspace always called a plane no matter what the dimensions of the space is?

Is a 2-dimensional subspace in a 7-dimensional space still called a plane? I know that a 6-dimensional space in 7-dimensional space is called a hyperplane because the difference in the number of ...
3
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2answers
291 views

Stronger condition lead to weaker result?

Suppose there are two theorems $A \Rightarrow B$ and $C \Rightarrow A$. Then we have $C \Rightarrow B$. Now comparing $A \Rightarrow B$ and $C \Rightarrow B$, we know that $C \Rightarrow A$ means C ...
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1answer
2k views

What does “relatively closed” mean?

Let $S\subset U$. What does it mean to say that $S$ is relatively closed in $U$? Also $U\subset\mathbb{R}^{n}$ is open and bounded, but I don't know if that's essential. Here follows an example from ...
3
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2answers
302 views

What is the name of the function $f(k) = \text{max}(k,0)$ on $\mathbb Z$? [duplicate]

Let $k\in \mathbb Z$; is there a name for the function $f(k)$ below? $$ f(k) = \text{max}(k, 0) $$
3
votes
2answers
376 views

Square for $x^2$, Cube for $x^3$, Quartic for $x^4$, and what's for $x^1$?

What's the general form for $x^y$? What's the specialized form for $x^1$ and $x^0$?
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0answers
34 views

Relation of ideals in probability with other kinds of ideals?

It seems that there are at least 5 kinds of ideals in maths: Ideals in number theory (Kummer, Dedekind) Ideals in abstract algebra (Dedekind, Noether), as kernels of homomorphisms Ideals in order ...
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2answers
62 views

Why is Cumulative “Density” wrong?

CDF stands for cumulative distribution function. However, it is "loosely" referred to as Cumulative Density many times. As i write this question, I have a suggestion toolbar on this page that lists ...
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2answers
60 views

Origin of the words arithmetic and geometric progression

Why are arithmetic progression and geometric progression called arithmetic and geometric respectively?
2
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3answers
270 views

Translating text to functions

I am having problems understanding how to extract this information into a formula. ...
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3answers
111 views

What is linear, numerically and geometrically speaking?

For as simple as it is, I never fully grasped what mathematicians and physicists mean with linear . Intuitively anything that looks like a straight line is interpreted as linear, like something in ...
2
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1answer
84 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 ...
2
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4answers
222 views

Are there two conventional definitions of “holomorphic”?

In Walter Rudin's Real and Complex Analysis, second edition, on page 213, two definitions are stated. One of them says the derivative of $f$ at $z_0$ is $$f'(z_0)=\lim_{z\to ...