For requesting, clarifying, and comparing definitions of mathematical terms.

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15
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2answers
3k views

What happens if we remove the requirement that $\langle R, + \rangle$ is abelian from the definition of a ring?

Ever since I learned the definition of a ring, I've wondered why the additive group is required to be abelian. What happens if we allow $\langle R, + \rangle$ to be nonabelian as well as $\langle R, ...
3
votes
1answer
139 views

Why is the term “composition” used to mean a certain binary operation on the set of relations on a given set?

I looked for material relating to compositions of equivalence relations, and was surprised to find the claim (here) that the composition of equivalence relations is not necessarily again an ...
14
votes
3answers
726 views

True Definition of the Real Numbers

I've found lots of resources that say this is a real number if it's not rational, but what is a real number, really? I mean what is the definition of a real number? If nothing else, anyone know of a ...
31
votes
12answers
3k views

What exactly is infinity?

On Wolfram|Alpha, I was bored and asked for $\frac{\infty}{\infty}$ and the result was (indeterminate). Another two that give the same result are $\infty ^ 0$ and ...
0
votes
1answer
117 views

Meaning of index in matrices

Question is, what does "index" mean? For systems of order greater than the number of characteristic roots of $C$ of index one Also, can anyone explain why is $u_1 + u_2 + n -1 =0$ and what ...
3
votes
1answer
2k views

What is the difference between a function and a map? [duplicate]

Possible Duplicate: Is there any difference between mapping and function? I am an aspiring mathematician who just started out. What is the difference between a function and a map? Or are ...
0
votes
1answer
375 views

The Definition of Consistency and Compactness in FOL

First order logic: "consistency," "compactness"? Consistency: A set $\Sigma\subseteq\text{WFF}$ is consistent iff there is no $\varphi\in\text{WFF}$ such that $\Sigma\vdash\varphi$ and ...
2
votes
1answer
506 views

What is the intuitive meaning of the adjugate matrix?

The definition of the adjugate matrix is easy to understand, but I have never seen it used for anything. What is the intuitive meaning of this matrix? Are there examples of applications which may ...
3
votes
3answers
86 views

Definition of a tangent

I've been involved in a discussion on definition of a tangent and would appreciate a bit of help. At my high school and at my college I was taught that a definition of a tangent is 'a line that ...
2
votes
2answers
188 views

Defining $0^0=1$

I've read in several places that defining $0^0=1$ is convenient in several (primarily discrete) settings. One argument on Wikipedia in favor of this definition was the need of a special case for the ...
2
votes
1answer
481 views

finding nearest perfect square for a fractional number

Is it possible to have a clear definition for the nearest perfect square number for a fractional number? For example, let us consider a number 0.004. What is another decimal number closest to it, that ...
2
votes
1answer
68 views

Using the notation $m^{O(1)}$

$m^{O(1)}$ denotes the set of functions $\mathbb{N} \to \mathbb{N}$ which are polynomially bounded. (Is that what is usually means?) Now it used as follows: $$ f(m) \leq m^{O(1)}$$ To express that ...
5
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2answers
2k views

Direct sums and direct products

This question has been in my head for a while. And today it appears again when I am reading Arveson's book on $C^*$-algebras. He says Countable direct products of Polish spaces are Polish. ...
5
votes
1answer
3k views

Why are vector spaces sometimes called linear spaces?

I have never come across the term 'linear space' as a synonym for 'vector space' and it seems from the book I am using (Linear Algebra by Kostrikin and Manin) that the term linear space is more ...
6
votes
1answer
106 views

Can we give a definition of the cotangent based on a functional equation?

I've recently learned that the cotangent satisfies the following functional equation: $$\dfrac1{f(z)}=f(z)-2f(2z)$$ (true for $f(z)\neq 0$). Can we solve this equation for real or complex ...
3
votes
2answers
701 views

Is zero a multiple of any number?

nooby question. I heard many times that 0 is a pair number. I'm fairly sure that the definition of pair is multiple of 2. Yet I heard too that multiples of a prime number p are only 1 and p, ...
0
votes
1answer
57 views

definition of $l$-equivalence

In the following paper http://www.math.ucsd.edu/~ronspubs/74_01_van_der_waerden.pdf, just in the first paragraph the author defines what $l$-equivalence for two m-tuples $\in [0,l]^m$ means. Can ...
2
votes
0answers
45 views

Name for the initial object of a connected component of a category

I was just wondering whether there is already a name for an object of a category as described below. Recall that an initial object in a category $\mathcal C$ is an object $I\in\mathcal C$ such that ...
6
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7answers
463 views

The definition of “number”

Maybe my question is very trivial. I would like to have the definition of "number". Can anyone advise me some documents online? thank you very much
1
vote
1answer
35 views

Definition domains of the pochhammer symbols?

What are the definition domains for $n$ and $x$ that gives $x^{(n)}$ (upper pochhammer symbol) and $(x)_n$ (lower pochhammer symbol) in $\mathbb{R}$ ?
1
vote
1answer
282 views

Meaning of no explicit time dependence

What does "no explicit time dependence" mean in this context? : A symmetry of the KdV is given by $$\tilde x=x, \tilde t=t+\epsilon, \tilde u =u$$ as there is no explicit time dependence in the ...
1
vote
0answers
98 views

Defining things in a non-recursive manner

In this question I asked, if it was possible to define certain functions without the use of the recursion theorem. The answer, among other things, indicated that it theoretically would be possible, ...
2
votes
1answer
116 views

Algebraic Structures : Compositions and More

I am working through the these notes (pdf) for my enumeration class, and I am having some trouble understanding some of the definitions and theorems. Could someone try and provide me with an ...
2
votes
1answer
779 views

Functional independence

Definition confusion: I wish to show that $$f(x,y)={-y\over x}$$ and $$g(x,y)=\log |x|$$ are functionally independent on some domain. What does that mean? What do I have to show? And how does one ...
8
votes
3answers
491 views

What is straight line?

I have found the definition of line in metric space. It is general but has two problems. Considering about $\mathbb R^2$ equipped rectilinear distance, every line by this definition contains a ...
1
vote
4answers
2k views

what is total order - explanation please

sorry for the dumbest question ever, but i want to understand total order in an intuitive way, this is the defition of total order: i) If $a ≤ b$ and $b ≤ a$ then $a = b$ (antisymmetry); ii) If $a ...
1
vote
1answer
166 views

Question about definition of pullback as a smooth bundle map.

In Lee, there is an exercise involving the pullback that I can't understand. If $M,N$ are smooth manifolds and $F:M\to N$ is smooth, I am asked to show that the pullback $F^*$ is a smooth bundle map ...
1
vote
1answer
133 views

Understanding revenue and profit math stuffs in labor theory of value

In http://wrongarithmetic.wordpress.com/2010/08/22/keen-i/, it talks about how economists Steve Keen's argument against Labor Theory of Value (LTV) is wrong. What I do not get is from This ...
1
vote
0answers
24 views

Definition of Chromatic class digraph of a m-coloured digraph

I have to prove that if $D$ is a m-coloured digraph, then $K(D) = K\big(C(D)\big)$ Where $C(D)$ is the chromatic transitive closure of $D$, which is a multidigraph with the same nodes and arcs of $D$ ...
1
vote
2answers
122 views

what does it mean to say a space is norm separable?

I came across in my textbook the term: norm separable. I looked in the textbook and online and could not find a definition.
3
votes
2answers
6k views

The degree of a polynomial which also has negative exponents.

In theory, we define the degree of a polynomial as the highest exponent it holds. However when there are negative and positive exponents are present in the function, I want to know the basis that we ...
1
vote
1answer
115 views

analogue of diag operator for functions

If $x\in{\rm I\! R}^n$, then diagonal matrix $\mathop{\rm diag}(x)$ is a linear operator $\mathop{\rm diag}(x): {\rm I\! R}^n \to {\rm I\! R}^n$. I am curious if there is some analogy for infinite ...
3
votes
0answers
140 views

Steps toward making the Feynman integral rigorous

I would like to know if there has been any progress recently toward giving the Feynman integral a rigorous definition. I heard that Richard Borcherds is working on giving quantum field theory a ...
4
votes
1answer
673 views

Definition of the Ideal Sheaf

Let $Y$ be a closed subscheme of a scheme $X$ and let $i:Y \rightarrow X$ be the inclusion morphism. Then the ideal sheaf of $Y$ is defined to be the kernel of the morphism of sheafs $i^{\#}: ...
2
votes
2answers
144 views

Separated and Finite Type Scheme over an Algebraically Closed Field

Let $(X,\mathcal{O}_X)$ be a separated scheme and of finite type over an algebraically closed field $k$. The fact that $X$ is separated means that the image of $X$ under the diagonal morphism $\Delta: ...
0
votes
2answers
1k views

Definition of denumerable (countable) set

When we say that a set $S$ is denumerable, that is, there is a bijection $S \to \omega$, do we mean that there exists such a bijection or do we mean that we have one and are talking about a pair ...
3
votes
4answers
206 views

What does she mean by star?

I was looking at this video http://www.youtube.com/watch?v=ygqIfLHGTu4&feature=g-all-f#t=06m33s33 and i wondered what she means by star. How is this number defined and where does it come up. ...
3
votes
2answers
156 views

“Good” closure conditions

[Attention! This question requires some reading and it's answer probably is in form of a "soft-answer", i.e. it can't be translated into a hard mathematical proposition. (I hope I haven't scared away ...
10
votes
2answers
2k views

*Recursive* vs. *inductive* definition

I once had an argument with a professor of mine, if the following definition was a recursive or inductive definition: Suppose you have sequence of real numbers. Define $a_0:=2$ and ...
2
votes
4answers
498 views

What's the definition of $\omega$?

This is a follow up on a comment to one of my previous questions. What's the definition of $\omega$? Are the following equivalent definition of $\omega$: $\omega$ is the initial ordinal of ...
0
votes
1answer
628 views

Dot products in commutators

Suppose $\hat r$ is an position operator, $\hat p$ is a momentum operator and $\vec c$ is a constant vector. What does the commutator $[\hat p, \vec c\cdot\hat r]$ mean? I see that you can expand ...
3
votes
2answers
236 views

Understanding of extension fields with Kronecker's thorem

In the book Contemporary Abstract Algebra by Gallian it defines an extension field as follows: A field $E$ is an extension field of a field $F$ if $F\subseteq E$ and the operations of $F$ are ...
2
votes
3answers
170 views

Is the zero of a commutative ring not a zero divisor or is it “undefined?”

In the Contemporary Abstract Algebra book by Gallian it defines zero-divisors as follows: Definition 1) A zero-divisor is a nonzero element $a$ of a commutative ring $R$ such that there is a ...
5
votes
1answer
143 views

What is the definition of the well-founded part of a model of set theory?

I've been trying to understand John Steel's various notes on inner model theory, but the one thing that trips me up is what he calls the well-founded part of a model of set theory. What exactly is the ...
3
votes
2answers
120 views

Is every triangle a quadrilateral?

I can imagine a quadrilateral where one of the angles is $180^\circ$. Is this still considered a quadrilateral? More generally, is every $n$-gon also a $(n+1)$-gon (for $n \ge 3$)?
1
vote
4answers
1k views

Partition of a set, definition not clear

From wikipedia: Equivalently, a set P is a partition of X if, and only if, it does not contain the empty set and: The union of the elements of P is equal to X. (The elements of P are said ...
1
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1answer
288 views

$\epsilon - \delta$ definition of a limit

Where can I find a good explanation of the $\epsilon - \delta$ definition of a limit. I have tried looking at my textbook and it doesn't make much sense, and I have also looked on Google as well ...
1
vote
3answers
233 views

Taylor series of a modulus argument

What is the definition of a Taylor series of the function $F(|\vec a -\vec x|)$ about the point $\vec a$ in $\vec x$?
1
vote
4answers
75 views

Acummulation point

Which one of these points is accumulation point, which not and why? I read the definition x-times but I'm quite confused :-/ I also found this post which is relevant to my question but it seems to me ...
3
votes
1answer
209 views

Minimal set of trig identities to define all the trig functions

What are a minimal set of trig identities that can uniquely define the trig functions? I know that you can define, for example, $\sin(x)$ as the unique solution to the differential equation $f''(x) = ...