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

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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
votes
1answer
101 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
105 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
187 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
107 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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0answers
209 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 ...
4
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4answers
404 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 ...
4
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1answer
272 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 ...
4
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3answers
1k 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 ...
4
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1answer
119 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
573 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? $|A|\in\mathbb{N}$? $|A|< \infty$? ...
3
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1answer
144 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
567 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 ...
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3answers
5k 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 ...
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0answers
745 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 $0=\varnothing$ $1=\{\varnothing\}=\{0\}$ $2=\{0,1\}$ $n=\{0,1,..,n-1\}$ ...
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3answers
123 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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2answers
250 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 ...
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2answers
3k 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
78 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
202 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 ...
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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 ...
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2answers
296 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
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2answers
373 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 $x^0$?
2
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3answers
268 views

Translating text to functions

I am having problems understanding how to extract this information into a formula. ...
2
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3answers
103 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
votes
1answer
82 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
votes
2answers
707 views

Precise definition of epsilon-ball

My textbook gives the following definition: "For each $\epsilon>0$, the $\epsilon$-ball about a point $x$ in a metric space $M$ is the set $\{y\in M:d(x,y)<\epsilon\}$." Is this correct? ...
2
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1answer
110 views

Definition of “totient”

I had always taken the term "totient" to be defined by saying that the totient of a positive integer $n$ is the number of positive integers less than $n$ that are coprime to $n$. Thus, for example, ...
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2answers
135 views

What does it mean to “identify” points of a topological space?

I was recently reading about circle rotations (a basic example in dynamical systems) and got confused by some notation. It said consider the unit circle $S^{1} = [0,1]/{\sim}$, where $\sim$ indicates ...
2
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2answers
142 views

Terminologies related to “compact?”

A set can be either open or closed, and there can either be a finite or infinite number of them. A "compact" set is one where every open cover has finite subcover. Is there such a thing as a set ...
2
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1answer
44 views

What is the names of $A\vec{x}=\vec{b}$ linear equation system components?

Having $A\vec{x}=\vec{b}$ . What is the names of $A\vec{x}=\vec{b}$ linear equation system components?
2
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1answer
38 views

Every subset of A is f-saturated

Let $f:A\rightarrow B$ be a function such that $\forall X\subseteq A[ f^{-1}[f[X]]=X]$ (In other words, every subset of $A$ is $f$-saturated). Does the property of the function $f$ have a name ? I ...
2
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1answer
214 views

What is the purpose of defining the notion of inflection point?

What is the purpose of defining inflection point? I know that it is defined to be the point where the second derivative is zero and the second derivative sign changes. It has to have some purpose ...
2
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2answers
126 views

Terminology: $H$ and $K$ are subgroups. What is $HK$ called?

Let $H, K\leq G$. I was wondering what you call the "product" $HK$ of $H$ and $K$. I was trying to verbalise the steps of showing $G$ is a semidirect product: Normality of $H$: $H\unlhd G$. Trivial ...
2
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1answer
91 views

Name of corresponding objects in equivalent categories

This question is only about terminology. Inside a category we have the standard wordings: An arrow $f: X \rightarrow Y$ is an isomorphism if there is another arrow $g: Y \rightarrow X$ such that $g ...
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3answers
1k views

Is there a special name for the operands of a multiplication?

Sometimes operands for a specific operation are given a special name. For example, in division the first operand is a quotient, the second is a divisor. Is there a word that means "one of the operands ...
2
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4answers
593 views

Does the word “integer” only make sense in base 10?

Does the word "integer" only make sense in base 10? I've always wondered this and have never seen it really discussed anywhere. We all understand the typical definition of an irrational number, ...
2
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1answer
117 views

Terminology for a property that holds in the finite but not infinite case?

(I apologize if this is a duplicate, but I don't know what terms to search for. Please feel free to close this if this has already been asked.) There are some properties of finite objects that don't ...
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1answer
2k views

What do [] mean and what does it mean if it is used in an equation?

What do the square bracket symbols mean? Are they what I hear are "sets"? And when it is in an equation, how is it interpreted? Here is an example: $$\dfrac{dy}{dx}[2x2+y(x)2]=50x+2y(dy/dx)=0$$
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vote
1answer
56 views

Is there a name for this result in planar geometry?

I found out that the following statement is fairly easy to prove: Let $A$, $B$ and $C$ be thee distinct points in the plane. Let $S_{AB}$ be the circle that has the line segment $AB$ as a ...
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2answers
85 views

Shapes bounded only by lines

What is a term for the set of geometric shapes in the plane, that are bounded by one or more continuous closed curves? This set contains simply-connected polygons and circles but also polygons with ...
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1answer
158 views

Terminology: Delta vs… absolute?

Delta is the change in a value. Using the term "delta" on the one hand, how, on the other hand, would you refer to the base value from which the given delta is derived? Is there a more precise term ...
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2answers
77 views

Physical significance of knot vector in B-spline.

A B-spline blending curve formulation is: $P(u)=\sum_{k=0}^np_k B_{k,d}(u)$ Given $n+1$ control points, B-spline blending functions are polynomials of degree $d-1$, $(1<d<=n+1)$. ...
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1answer
29 views

Need help in understanding $ord_p{a}$ as used in Theorem 1.1 from “On Some Exponential Equations Of S. S. Pillai”

I have a question about very early argument in the proof of Thereom 1.1. Theorem 1.1 of On Some Exponential Equations of S.S. Pillai states that if $a,b,c$ are nonzero integers with $a,b \ge 2$, then ...
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1answer
53 views

Correct Terminology in the Context of Rings

Suppose I have a ring $(A, ◦, •)$ where $A$ is a set of elements $\{α, β, γ,\ldots\}$. Can $◦$ and $•$ with which the ring is equipped be properly termed, in English, its "internal laws of ...
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0answers
43 views

A map from one set of words to another that is not a morphism

A word is a concatenation of letters from a non-empty set called an alphabet. For example, if the alphabet is $\{a,b\}$, then $bba$ is a word from that alphabet. Let the set of all finite words made ...
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2answers
63 views

how do you call a function that breaks down on y?

How do you call a (linear) function (or the point), which breaks down to 0 on ordinate (axis y), as soon as you breach a certain x1 value?