Definitions are the core of mathematical precision; they come to answer "what is X" in mathematics. Into this category fit questions regarding equivalence of definitions; clarifications regarding complicated definitions; as well questions with purposed definitions for mathematical notions with ...

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Graph Theory: Help with a definition

I need some help to see if the definition I found of Cycles and Cycles Decomposition is right, here it is: A graph is a Cycle if it is isomorph to another graph $G=(V,E)$ with the following ...
5
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1answer
265 views

What is “algebra” in $\sigma$-algebra (or “field” in $\sigma$-field)?

I know that $\sigma$ in $\sigma$-algebra stands for the closure under countable union property. What about "algebra"? Surely it cannot algebra over a field or a ring as defined in algebra textbooks. ...
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0answers
243 views

different kind of convergence in Real analysis

Now I am studying real analysis, I wonder if anyone can sum up all the relations between all kinds of convergence, convergence in measure, convergence a.e., convergence a.u., and convergence in ...
2
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1answer
371 views

bipartite graph - sufficient and necessary conditions

Sorry for a silly question, I got confused with the definition of bipartite graph. What is a necessary and sufficient condition for a bipartite graph. ...
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3answers
162 views

If there is only one one-sided limit, the limit exists?

Question on theory: If $\lim_{x\to a^+}F(x)=L$ and $\lim_{x\to a^-}F(x)$ doesn't exist, then does the $\lim_{x\to a}F(x)$ exists and is $L$? Thanks.
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3answers
356 views

Why does the definition of limits of a function have strict inequality?

Definition (As written in Michael Spivak's Calculus) The function $f$ approaches a limit $l$ near $a$ means: for every $\epsilon >0$ there is some $\delta > 0$ such that, for all $x$, if ...
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1answer
121 views

What is a “set-like class”?

Just / Weese contains the following theorem (p 126): Theorem 5: Let $\mathbf Z \neq \varnothing$ and let $\mathbf W \subseteq \mathbf Z \times \mathbf Z$ be a wellfounded set-like class. Let $\mathbf ...
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2answers
97 views

Number of Vertices of Graphs

So, I was looking at some graph theoretical stuff, more specifically Topological Graph Theory, and I had a question about the definition of graphs: is there usually a condition in the definition ...
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0answers
227 views

What is derived number definition? ( in Vitaly covering)

In Vitali covering definition i see "derived number" word, but I dont know what that mean. Example for vitali covering: If $f$ is strictly increasing and $$E=\{x: \text{ there is a derived number } ...
2
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0answers
194 views

Explanation of Mixed Strategy Definition in Game Theory

Definition: Let $(N, A, u)$ be normal-form game, and for any set $X$ let $\Pi(X)$ be the set of all probability distributions over $X$. Then the set of fixed strategies for player $S_i=\Pi(A_i)$. ...
10
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2answers
427 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 ...
2
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1answer
113 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 ...
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1answer
96 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 ...
2
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1answer
796 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 ...
2
votes
1answer
255 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
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3answers
76 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
1answer
299 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
63 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 ...
3
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2answers
950 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. ...
3
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1answer
2k 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
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1answer
96 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
444 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, ...
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1answer
48 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 ...
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0answers
76 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, ...
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0answers
41 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 ...
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1answer
27 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 $\mathbb{R}$ ?
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4answers
425 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 ...
2
votes
1answer
111 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 ...
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1answer
161 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 ...
2
votes
1answer
439 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 ...
3
votes
2answers
149 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 ...
1
vote
1answer
959 views

What does “refinement” mean?

I was reading a book and it had the following sentence: $A$ is a refinement of $B$ where $A$ and $B$ are sets. What does this mean? Perhaps $A \subseteq B$ ?
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1answer
112 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 ...
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1answer
91 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 ...
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votes
4answers
189 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. ...
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0answers
23 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 $D$ ...
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2answers
78 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.
0
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1answer
79 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
1answer
388 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^{\#}: ...
3
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0answers
118 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 ...
2
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4answers
305 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
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2answers
642 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 ...
8
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2answers
1k 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 ...
3
votes
2answers
179 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
151 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 ...
4
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1answer
108 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
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2answers
113 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 $n \ge 3$)?
2
votes
1answer
175 views

define simultaneous substitution recursively

Can you help me with my approach to the following task: Define simultaneous substitution $\phi[\psi_1,...,\psi_k/p_1,...,p_k]$ recursively. Usually we have recursive definitions about formulas, but ...
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1answer
250 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
138 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 $\vec x$?