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

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32 views

Is there always a unit in topological rings?

I'm reading Qing Liu's Algebraic Geometry and Arithmetic Curves, http://176.58.104.245/ALGANT/TONG/Liu-1-4.pdf . On the first page he wrote that "Unless otherwise specified, all rings is this book ...
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1answer
60 views

Difference between locally and globally defined function

What is the difference between a function defined locally at $0$ and globally at $0$ on a set $S$? My textbook keeps referring to these things, but I couldn't find any definition about it anywhere. ...
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0answers
18 views

Partition of unity and coordinate patches

I have a question related to terminology. Assume that $M$ is a $k$ manifold. What does it mean to say that the partition of unity $f_1,f_2,...f_n$ on $M$ is dominated by the collection of all ...
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1answer
33 views

Math terminology: What are rules regarding hyphens? (Nonzero vs. non-zero)

This question is geared toward clarifying terminology in writing math. Which terms are correct and why? A set $E$ is non-empty. A set $E$ is nonempty. The number $x$ is non-negative. The ...
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0answers
37 views

Why “cylinder sets”?

If $I$ is any set of indexes, we define $E^I=\{(x_i)_{i\in I}:x_i\in E\,\,\forall i\in I\}$, $E$ being any set. Subsets of $E^I$ of the form $C_J=\{x_i\in B_i\,\,\forall i\in J\}$, where ...
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1answer
28 views

How do we say in English that pyramid is not skewed?

I would like to know if there is a term kind of "straight pyramid"? I mean, is there a word for a pyramid such that the line segment from the apex to the center of the base is perpendicular to the ...
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0answers
51 views

Intuitive meaning of benign subgroup

I've been studying a proof of Higmann's Embedding Theorem which makes use of the notion of a "benign subgroup". The definition is quite straight-forward: $G\leq H \ is\ benign\ in\ H \Leftrightarrow$ ...
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4answers
432 views

What does it mean to solve or find solutions in mathematics?

Something that has been really confusing me lately is that this equation has four solutions $$3x(x+1)(x^2+x+2)=16x(x+1)(2x+1)$$ But what does that mean? Until now solutions to me has meant, what are ...
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1answer
19 views

Need help with some regular polygon terminology.

I'm looking for the names of three different measures of a regular polygon. The name for the line between the centerpoint of the polygon, and any of its vertices. The name for the line between the ...
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2answers
40 views

What's the difference between an axiomatization and a characterization of a structure?

Regarding structures such as the natural numbers, complex numbers, groups, etcetera. Would it make sense to say that a collection of properties is a characterization of sets? I know that there are ...
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3answers
40 views

Basic question about terminology, notation and definitions in calculus

When reading stuff about differential equations I'm coming across some strange (for me) notations/terminology. For example, when coming across something like this: $$\frac{dy}{dt}=f(y,t)$$ or ...
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2answers
125 views

Why is Zorn's Lemma called a lemma?

By convention a Lemma is a technical intermediate step which has no standing as an independent result. Lemmas are only used to chop big proofs into handy pieces. (Quoted from here) I am wondering why ...
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0answers
19 views

Is there a term to designate the “complexity” of a complex number

The simplest way to explain my question is using an analogy: When classifying integers as odd and even, we speak of parity. Is there an equivalent term when speaking of the "complexity" of a number? ...
2
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0answers
58 views

What is a fewnomial?

I came across the theory of "fewnomials" (by Khovanskii), which (I guess) are related to polynomials. However, I was surprised that there is no single question on stackexchange concerning fewnomials, ...
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3answers
77 views

How is the relation between $y$ and $x$ called in $y = 1/x$?

Probably a very simple question for most of you, but how is the relation called between $x$ and $y$ if $y = 1/x$? As in, if I want to say: $y$ is .... related to $x$, what should go on the dots? ...
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1answer
22 views

Difference between cofinal map and cofinal

In this note I read, the cofinal map is defined as, Let $\mathbb D´$ and $\mathbb D$ be two directed sets, $f: \mathbb D´ \to \mathbb D$ is a cofinal map, if $M$ is cofinal in $\mathbb D´$, then ...
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1answer
36 views

Equivalent definitions of a quasi-affine variety?

I have a concern about a definition of a quasi-affine variety. I had a professor who defined a quasi-affine variety to be an intersection of an open set and a closed set in some affine space ...
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0answers
21 views

Name of this theorem? (Generalization of the class equation)

Let $X$ be a nonempty finitr set and $G$ be a finite group acting on $X$. Let $G.x_1,...,G.x_n$ be the distinct orbits of $G$. Define $F(X)=\{x\in X : \forall g\in G, g.x=x\}$. Then ...
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1answer
26 views

Let $f: \mathbb R \times \mathbb R^n \rightarrow \mathbb R^n$ given as $f(t,y)$. How I understand:“$f$ has continuous partial $y$-derivatives”?

Suppose I have a function $f: \mathbb R \times \mathbb R^n \rightarrow \mathbb R^n$ given as $f(t,y)$. Then how should it be understood "$f$ has continuous partial $y$-derivatives" ? Should it be ...
2
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0answers
22 views

Exponents for Hölder functions on metric spaces

Sometimes people talk about Hölder functions on metric spaces without mentioning the allowed range for the exponent. On manifolds, it's traditional to assume an exponent in $[0, 1]$, since all ...
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2answers
37 views

What's the name of this law in Boolean algebra?

I forgot the name of a law in Boolean algebra, and I can't think of how to ask this question to a search engine. It's the law that states that the disjunction of a variable with the conjunction of its ...
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1answer
24 views

Difference between finite and discrete set of discontinuities

I am reading this paper. In page 395 and 396, theorems $1$ and $2$ use the terms 'finite' and 'discrete' to refer to sets, in this case sets of discontinuities. What I don't understand is: what is the ...
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1answer
54 views

Is there a name for this special, “most parallel” ultraparallel line in hyperbolic geometry?

Suppose you're in the hyperbolic plane, and you have a line L and a point P not through L. There are an infinite number of lines parallel to L that go through P. However, there's one line M which is ...
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0answers
12 views

What is the proper name of a model that takes as input the output of another model?

Thanks in advance for the help. I am writing a paper and for the life of me can't remember the proper term for a model that works as follows. rawData -> model1 -> outputModel1 -> model2 -> ...
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2answers
27 views

What's the term for antisymmetry where equal elements are not in the relation?

The most common definition of antisymmetry of a relation $R$ on a set $S$ is $$ \forall a, b \in S, R(a, b) \land R(b, a) \to a = b. $$ However, this doesn't cover a relation such as $<$, for ...
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1answer
162 views

What is the line of greatest slope on a plane? [closed]

Let $P$ be a plane in $\mathbb{R}^3$ that is inclined (neither horizontal nor vertical). When considering lines lying on $P$, it is sometimes said "$L$ is a line of greatest slope of $P$". What is ...
3
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2answers
18 views

What's the name of a solid that results from extruding an area straight along an axis?

If you have any kind of 2D shape and move it up into the third dimension, what do you call it, because it seems like prism is used only if the base is a polygon. It also seems like extrusion is a ...
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0answers
23 views

Name for problems where the constraints are on inner products

I have a problem with a lot of dot-product constraints like $V_1 \cdot V_2 = 0$ or $V_1 \cdot V_3 = V_2 \cdot V_4$. However, I don't know what these types of problems are called so I can't look up ...
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0answers
36 views

“Absolute retracts” in arbitrary category

Is there a standard notion of something like "absolute retract" in arbitrary categories that generalizes absolute retracts in topology? I am mostly interested in categorical approach to Hausdorff ...
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1answer
41 views

What does the term “perturb” mean?

I've been studying Calculus of Variations and I came a cross with the term "perturb" in my study material, but the term was not defined. The sentence where I read it from was: "Rigid extremals are ...
2
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0answers
48 views

What abstract structures allows us to describe “nets that converge toward each other”?

Topological spaces are equipped with a bare minimum of structure to allow for a formalization of the statement "the net $a$ converges to the point $x$." Actually this isn't strictly true, but its true ...
2
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3answers
69 views

What is a conventional name for a set of values having no properties except that values are distinct?

I know essentially nothing about math but I'm interested in very low-level concepts. I'm thinking of something like a finite or infinite set (although I'm not married to consider sets per se, maybe ...
3
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1answer
54 views

Wide equalizers

If $(f_i : A \to B)_{i \in I}$ is a family of morphisms in a category, we may declare their wide equalizer as a universal morphism $\iota : E \to A$ which satisfies $f_i \iota = f_j \iota$ for all ...
3
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1answer
44 views

Name of Inequality

Let $x_i, y_i$ be complex numbers for all $i$. Is there a name for the following inequality? $$\left| \sum_{i=1}^n x_i \right| \leq \sum_{j=1}^n |x_j| $$ In particular, is it a special case of this ...
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2answers
39 views

All the squares in the multiplicative group $\mathbb{Z}_n^*$ [closed]

I just want to know what this statement means: all the squares in the multiplicative group $\mathbb{Z}_n^*$.
1
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1answer
70 views

Can somebody explain with one example the concepts: Lemma-Hypothesis-Theorem-Assumption-Proof-Axiom-Thesis-Determination-Definition-Proof [closed]

It would be great if someone can give me for each concept a simple explanatory example ! What is the difference between: Lemma Hypothesis (Hypothese) Theorem (Satz) Assumption (Annahme) Proof ...
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1answer
46 views

What is the difference from a theorem and a meta-theorem?

I'm confused about what a meta-theorem exactly is and if a meta-theorem can be used to prove a theorem. To illustrate my confusion i give an example. Given the three statements: Every vector space ...
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0answers
31 views

Is there a name for the inequality $\min(a+b,c+d) - \min(a,c) \ge \min(b,d)$?

Is there a name for the inequality $$\min(a+b,c+d) - \min(a,c) \ge \min(b,d)$$? And does anyone have any nice examples or applications, especially with an economic flavor? The transposed multivariate ...
2
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0answers
57 views

Does this family of special matrices have a name?

These are the bisymmetric matrices that are "pyramid" shaped as follows: $$f(14) =\begin{bmatrix}1&1&1& 1& 1& 1& 1& 1& 1& 1& 1& 1& 1& 1\\1& ...
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0answers
24 views

Is there a name for the relation between Menger Sponge and Vicsek Fractal?

Both the Menger Sponge and the Vicsek Fractal in 3D can be constructed by starting with a cube, dividing it into 27 smaller cubes (3x3x3 grid), removing some of these cubes, and then applying the ...
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2answers
72 views

What is the difference between a counter-intuitive statement and a paradox?

In mathematics and logic, what is the difference between a counter-intuitive statement and a paradox? For example, what differs something like the Banach-Tarski theorem or Gabriel's horn from ...
1
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1answer
41 views

What is an omega model?

I went to a seminar and a side question was if a theory had an omega model, however from the context I could not deduce the exact meaning. Does an omega model have a general meaning in mathematical ...
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1answer
36 views

What does “$C^{\infty}$” convergence mean?

I'm studying first notions about several complex variables. As a consequence of the (generalized form) of the Cauchy esteem for holomorphic functions, the book says that in the space $\mathcal ...
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1answer
63 views

Meaning of the term “Sledgehammer”

I know a Sledgehammer is a special type of hammer, but I still do not quite get the exact meaning of the word in such a paragraphs as: The computational sledgehammer par excellence is the spectral ...
2
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1answer
168 views

What is the difference between coordinates transformation and change of coordinates?

In the context on 3D computer graphics, what is the difference between coordinates transformation and change of coordinates? It can just be a matter of notation, but my book makes a clear distinction ...
2
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3answers
85 views

How many $n$th roots does $0$ have?

Do we say that $0$ has $n$ $n$th roots, all nondistinct, or only one? I don't think it makes any difference, but I'm curious what the convention is.
3
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0answers
34 views

Terminology question : “half smooth, half topological” fibre bundle

First, I know (or I think I know...) the definition of fiber bundle, be it in the smooth or topological category. Here is my situation, which is kind of between the two: I have a smooth manifold $E$, ...
2
votes
2answers
125 views

Is the standard scalar product in a coordinate space basis independent?

Would you say that the standard scalar product in $K^n$, $\left< x,y \right>=\sum_i x_i y_i$, is basis-independent or not ? I would argue that it is, because we don't use the components of the ...
2
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1answer
67 views

Question about the wording of a topology problem.

I was asked to show that the topology $\mathcal{T}_{X\times Y}$ is the smallest topology for which the functions $$f_X:X\times Y \rightarrow Y , f_X((x,y))=x $$ and $f_y$ are continuous (where $f_Y$ ...
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1answer
59 views

Given a field extension $K\colon F$, $K$ is an $F$-vector space

I'm having a hard time understanding fields. Could someone help with the following I need to show that if $F$ $\subseteq$ $K$ are both fields and addition and multiplication on F are the ...