Tagged Questions
9
votes
6answers
229 views
Should every group be a monoid, or should no group be a monoid?
Question: What is more convenient/useful? Writing mathematics as if every group is a monoid, or as if these two classes are disjoint?
Additional discussion. Define a monoid as follows.
Defn 1. A ...
2
votes
1answer
30 views
Which expressions in English should I use for a morphism having a certain source and target?
Say that $f: A \rightarrow B$ is an arrow in a category $\mathcal C$. Which verbs or expressions do we use to express in an alternative way that $A$ is the source of $f$ and $B$ its target? E.g., ...
6
votes
4answers
142 views
Are there rules in the useage of prepositions in Math?
It is often to use prepositions in various expressions.
E.g.
$2$ is in the set of natural numbers $\mathbb N$
The symmetric group on 3 letters $S_3$ is the group consisting of all possible ...
1
vote
1answer
44 views
Correct way of saying that some value depends on another value x only by a function of x
I would like to know what good and valid ways there are to say (in words) that some value f(x), which depends on a variable x, in fact only depends on x "through" some function of x.
Example: For ...
5
votes
3answers
134 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 ...
14
votes
5answers
586 views
Is 'no solution' the same as 'undefined'?
Today in class my teacher wrote something along the lines of:
$6^x = 0$
And proceed to heed a response from the class. A few people shouted undefined.
So the teacher then writes:
no solution ...
4
votes
2answers
111 views
Does this problem have a name?
Recently our lecturer told us that it is an unsolved mathematical problem if the following while loop aka iteration ever terminates. Unfortunately I forgot to ask him what it is called. If someone ...
5
votes
5answers
226 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 ...
0
votes
1answer
130 views
What other rules are there in mathematics?
I'm reading Conceptual Mathematics: A First Introduction to Categories.
a set $A$, called the domain of the map;
a set $B$, called the codomain of the map;
a rule assigning to each element ...
4
votes
2answers
82 views
Why saying that “$x$ is an indeterminate real number” is misleading?
I'm reading: Behnke's Fundamentals of Mathematics, Vol.1
On page 23, he says:
In order to indicate that a variable $x$ has the real numbers for its
range, mathematicians often say that $x$ is ...
3
votes
1answer
103 views
What does arithmetic actually mean (as an adjective)
Ok so I've seen the adjective 'arithmetic' (stress on the e) bandied about from time to time in reference to the "arithmetic theory of some subject" (elliptic curves for instance), or the "arithmetic ...
6
votes
2answers
700 views
Difference between “intercept” and “intersect”
What is the difference between intercept and intersect? Can they be used interchangeably? For example, intersecting lines and intercepting lines.
3
votes
2answers
164 views
The usage of ad hoc vs a priori in mathematical papers
I was reading the paper Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems, where the author phrased something as this:
The analyses of these methods are ad hoc but ...
11
votes
4answers
370 views
Short forms like “haven't”, “don't”, “let's” [closed]
Is it true that short forms like "haven't", "don't", "let's" should not be used in serious mathematical texts?
0
votes
4answers
159 views
What sets satisfy $V = V^V$?
I'm not sure if said set exist or whether it is unique, but what name could I use to find more about it and what kind of interesting properties does it have?
Clarification edit: I meant a set $V$ ...
6
votes
1answer
126 views
When do modifiers denote sub or super? Pseudo-, quasi-, ultra-, strong-, well-, pre-, c0- …
One only needs to search MMA.SE, math journals, wikipedia, or god-forbid, n-cat lab, for keywords listed in the title, which can be extended with: uniform-, regular-, complete-, local-, partial-, non- ...
1
vote
1answer
69 views
What preposition to use when fitting data?
Please could some real experts give an opinion on this question on English.SE: What preposition to use when fitting data?
Do we fit data with, by, or as a linear function?
Forgive me as I do ...
6
votes
1answer
304 views
“So That” vs. “Such That”
In definitions and exercises, I notice that "so that" and "such that" are seemingly used interchangeably. Are they in fact interchangeable, or is one more appropriate for a specific context?
Note: ...
10
votes
2answers
265 views
Elementary Geometry Nomenclature: why so bad?
A long-ish wall of text, and I apologize.
Some background: when I was a first-year university student, my chemistry professor was lecturing and was trying to find the word to describe a shape. A ...
6
votes
3answers
525 views
What does a “convention” mean in mathematics?
We all know that $0!=1$, the degree of the zero polynomial equals $-\infty$, the interval$[a,a)=(a,a]=(a,a)=\emptyset$ ... and so on, are conventions in mathematics. So is a convention something that ...
7
votes
1answer
176 views
Spectrum of a real number
Whilst reading Concrete Mathematics, the authors mention something which they refer to as the "spectrum" of a real number (pg. 77):
We define the spectrum of a real number $\alpha$ to be an ...
3
votes
2answers
125 views
Number of elements vs cardinality vs size
I have been wondered the definition of cardinality and number of elements. One mathematician told me that one can't said that the cardinality or size of the set $\{1\}$ is one, it should be said that ...
6
votes
0answers
166 views
How to read equations and expressions out loud? [duplicate]
Possible Duplicate:
Is there a definitive guide to speaking mathematics?
This may be an incredibly stupid question, but I was wondering how would one pronounce simple mathematical equations ...
3
votes
2answers
153 views
Question on mathematical writing
I am now writing my graduate thesis, it includes some basics mathematical theorems/propositions.
I got a trouble in writing, more concretely, I do not know when can I state a mathematical claim as a ...
5
votes
3answers
559 views
What does it really mean for something to be “trivial”?
I see this word a lot when I read about mathematics. Is this meant to be another way of saying "obvious" or "easy"? What if it's actually wrong? It's like when I see "the rest is left as an exercise ...
1
vote
0answers
82 views
I'm interested in different meanings of “normal”~ [duplicate]
Possible Duplicate:
What is it to be normal?
I've learned in algebra class that
"normal" means a linear operator is commutative with its adjoint;
also we say that $H$ is a normal ...
0
votes
1answer
117 views
Name of probability distribution
Does this distribution have a name:
$f(x) = yx^{y-1}$ for $0 < x<1$ and $y>0$?
It looks like an exponential distribution. Or is it a nameless distribution?
11
votes
2answers
301 views
When does “pairwise” strengthen and when does it weaken?
"Pairwise disjoint" is stronger than "disjoint"; it sometimes happens that $\displaystyle\bigcap\limits_{i\in I} A_i=\varnothing$ but for every $i,j$, or at least for some, one has $A_i \cap ...
6
votes
2answers
199 views
terminology: what is meant if someone writes “calculus of ..”?
This question might be a little soft as it does not have a definite answer, so I hope I do not break the conventions of this forum by posting it here.
I have now come across the term "calculus of ...
6
votes
2answers
514 views
Why filters and nets in topology?
I am wondering why do mathematicians categorizes some structures and called them filters , Nets?
In English, filter means: A porous material through which a liquid or gas is passed in order to ...
1
vote
1answer
585 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:
$$\dfrac{dy}{dx}[2x2+y(x)2]=50x+2y(dy/dx)=0$$
1
vote
0answers
130 views
Pronouncing $V(E)$ in Zariski Topology
In Atiyah-Macdonald the Zariski Topology is a topology defined on Spec$(R)$, $R$ a commutative ring as follows. The Zariski-closed sets are those of the form
$V(E) = \{P \in \text{Spec}(R) : E ...
1
vote
1answer
153 views
What is an honest basis?
In a comment to this question, the commentator stated that
"the monomials form an honest basis for your vector space".
To be honest, I never heard of that. Is this something elementary?
7
votes
4answers
2k views
Is there any difference between mapping and function?
I wonder if there is any difference between mapping and a function. Somebody told me that the only difference is that mapping can be from any set to any set, but function must be from $\mathbb R$ to ...
2
votes
1answer
123 views
The reason for different terminologies
Different authors seem to have different conventions when they define the term affine variety (similarly projective variety). For the purposes of this question let us stick with the affine case, and ...
2
votes
2answers
144 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 ...
2
votes
1answer
105 views
Eccentricity of a vertex
Eccentricity of a vertex $v$ in a graph $G$ is defined as max $\{d(v,w):w\in V(G), w\ne v\}$. My question is why is the word eccentricity used, what is the reason?
Thanks
7
votes
1answer
259 views
Why is $i$ called “imaginary”?
I was reading this question, and, after reading the responses, I felt like I had a much better understanding about how they're just another type of number definition.
Why, then, are they called ...
1
vote
2answers
287 views
Confused about Wikipedia definition of NP
I've been checking my understanding of the definitions of NP and NP-complete and
I am confused by some of the definitions given on Wikipedia;
for example, the
article about NP-complete describes NP ...
16
votes
3answers
832 views
Why algebraic topology is also called combinatorial topology?
I remember reading somewhere(at least more than once) that algebraic topology is also known by the name "Combinatorial Topology" which essentially tags the subject fundamentally with some counting ...
6
votes
3answers
516 views
What does it mean to “count (some number) of (some finite set of objects)”?
I'm not a native speaker of English. I usually pride myself of my proficiency, but I think I may be stumped here. My problem arises out of this question, which among other things asked for a ...
4
votes
1answer
316 views
Motivation for the term “separable” in topology
A topological space is called separable if contains a countable dense subset. This is a standard terminology, but I find it hard to associate the term to its definition. What is the motivation for ...
14
votes
5answers
672 views
What does “formal” mean?
I know the definition of formal power series, power series and polynomials. But what does the adjective "formal" mean? In google English dictionary, does it mean "9. Of or relating to linguistic or ...
7
votes
4answers
656 views
What is the name of the vertical bar?
I've always wanted to know what the name of the vertical bar in these examples was:
$f(x)=x^2+1\mid_{x = 4}$ (I know this means evaluate x at 4)
$\int_0^4 (x^2+1) dx = (\frac{x^3}{3}+x+c) ...
8
votes
2answers
407 views
What's the difference between “duality” and “symmetry” in mathematics?
Motivated by the answer to this question--"What kind of “symmetry” is the symmetric group about?", I read the article about dual graph. It is said in this article that "the term 'dual' is used because ...
6
votes
1answer
298 views
What is u-substitution?
In answers on this site often the word "u-substitution" appears. Why this is called u-substitution? Is it the same as integration by substitution?
(I never heard this term before and I'm not sure if ...
7
votes
1answer
238 views
When can we say that a theorem has been proven?
I'm taking a Data Structures and Algorithms course for a CS program. The introductory material was all mathematics, mostly a series of formulas that we are to remember. I can work through the formulas ...
0
votes
1answer
258 views
What is the correct way to write product or sum in capital pi or capital sigma notation when you wish to exclude an index?
I have a series of terms $\{t_n : t_n = a_n x_n\}$, and I want to talk about the product of each term except $t_j$.
Would any of these be an appropriate way to say that? I like this:
$$\prod_{i \ne ...
3
votes
2answers
163 views
Relationship between torsion modules and topology
I was reviewing my class notes and found the following:
"The name 'torsion' comes from topology and refers to spaces that are twisted, ex. Möbius band"
In our notes we used the following definition ...
14
votes
8answers
3k views
Why does “convex function” mean “concave *up*”?
A function $f : \mathbb{R} \to \mathbb{R}$ is convex (or "concave up") provided that for all $x,y \in \mathbb{R}$ and $t \in [0,1]$,
$$f(tx + (1-t)y) \le tf(x) + (1-t)f(y).$$
Equivalently, a line ...
