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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1answer
109 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 ...
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1answer
54 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
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0answers
78 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 ...
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1answer
127 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^{\#}: ...
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1answer
57 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: ...
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2answers
124 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 ...
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4answers
145 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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2answers
136 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 ...
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2answers
162 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
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4answers
162 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 ...
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1answer
106 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 ...
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2answers
103 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 ...
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3answers
120 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 ...
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1answer
81 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
95 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 $n \ge 3$)?
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3answers
173 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 ...
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0answers
42 views
What is $Pic(S)^G?$
Let $S$ be a projective surface and $\text{Pic}(S)$ its Picard group, $G$ is some group (in fact, it consists of automorpisms of $\mathbb P^2$). I came across a notation "$\text{Pic}(S)^G$". Could you ...
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1answer
146 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 ...
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3answers
53 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 $\vec x$?
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4answers
65 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
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1answer
97 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) = ...
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0answers
105 views
What is the correct definition for an imaginary number?
The following is taken from Wikipedia's definition.
An imaginary number is a number whose square is less than or equal to
zero.
But I also heard that
An imaginary number is a number whose ...
2
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1answer
62 views
What does “overdetermined” mean
When we say a problem is an overdetermined system, what do we mean by that in a rigorous fashion?
Thanks.
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0answers
65 views
Where can I find a description of math language symbols?
I am reading math articles. I meet math symbols.
For example $\exists$ or $\forall$.
For example for "For any a exist e that" can be rewriten as: $\forall a \exists e$
Where can I find full ...
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1answer
53 views
Suppose I said “$X$ spans $W$”…
So I've seen two definitions of this:
Let $V$ be a vector space with subspace $W$. We say that $X \subseteq V$ spans $W$ if and only if
(Definition 1): Every $\vec{w} \in W$ can be written as ...
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0answers
82 views
How to prove the definition of arctangent by G. H. Hardy through integral?
From introduction to analysis,by Arthur P. Mattuck,problem 20-1.
I am stuck in the sub-problem (d) of this problem,especially the magic number 2.5,please help,thanks.
Problems 20-1
One way of ...
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2answers
65 views
Exponentation vs Power
What definition of $a^b$ operation is the most popular and standartized: Exponentation or Power?
Is any difference between them?
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1answer
134 views
Epsilon delta definition: $\lim _{x\to-2} (2x^2+5x+3)=1$
I'm trying to use the epsilon delta definition to prove that $$\lim _{x\to-2} (2x^2+5x+3)=1$$
evaluating: $|(2x^2+5x+3)-1|\lt \epsilon$
under the condition: $0\lt |x-(-2)|\lt\delta$
I arrived at: ...
2
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1answer
68 views
What is the meaning of $K/F$ is a cyclic extension?
I have it that $K/F$ is a (finite) field extension, what is the definition
of when $K/F$ is called cyclic ?
I heard it while I studied Galois theory and it was defined as
$K/F$ is called cyclic ...
1
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1answer
273 views
Bounded Set: definition
I'm having some trouble with the definition of "Bounded Set". I have a pretty good idea of what "Limited" means: a Set with a Upper and a Lower bounds.
Now i have a quiz in which I must choose the ...
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0answers
67 views
Definition of matrix derivative
Let $A$ be an $m \times n$ matrix. Let $a_{ij}$ be an element of $A$. What does the notation $\frac{\partial A}{\partial a_{ij}}$ mean?
3
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1answer
144 views
Question about definition of regular projection of a knot
One can define a knot in two ways:
(1) A knot is a closed polygonal curve in $\mathbb R^3$
(2) A knot is an equivalence class of embeddings $S^1 \hookrightarrow \mathbb R^3$
And perhaps also:
(3) ...
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5answers
392 views
Question about the definition of a category
I am confused about the definition of a category given in the Wikipedia article on Category theory:
It seems to me that the structure being described (the "arrows" between objects in some class) is ...
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2answers
182 views
Is there an idea of “primeless isomorphism” studied somewhere in finite group theory?
What I mean by "primeless isomorphism" is essentially a relation on finite groups by identifying groups whose structure differs only in which primes divide the groups' orders. The groups aren't ...
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1answer
43 views
Legal actions on a PDA and terminology
I am unsure about the following so I would like to verify if my statements
are true:
We can remove at most a single character ($Z\in\Gamma$) from the
PDA (top ?) of the stack with one step of the ...
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2answers
141 views
Measure and Outer Measure Definition
I would like to find exemples to show and demonstrate that each of the statements of the definition of:
-measure
$\mu\left(\emptyset \right)=0$
$\mu \left( \bigcup A_n\right)=\sum \mu \left( ...
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1answer
58 views
Definition of a deterministic Pushdown automaton
According to my book the definition of a deterministic Pushdown automaton
allows for $\delta(q,\epsilon,Z)$ to be non-empty if $$\forall\sigma\in\Sigma:\,\delta(q,\sigma,Z)\neq\emptyset$$
Can someone ...
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2answers
60 views
Pushdown automata - definition and definition of $\vdash$
I am reading about pushdown automata and I don't understand the definition
of $\vdash$.
My book writes that $$(q,aw,Z\alpha)\vdash(p,w,\beta\alpha)$$ if $$(p,\beta)\in\delta(q,a,Z)$$
Can someone ...
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2answers
135 views
I need to disprove an alternate definition of an ordered pair. Why is $\langle a,b\rangle = \{a,\{b\}\}$ incorrect?
So we know that the an ordered pair $(a,b) = (c,d)$ if and only if $a = c$ and $b = d$. And we know the Kuratowski definition of an ordered pair is: $(a,b) = \{\{a\},\{a,b\}\}$
...
5
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4answers
261 views
What is the correct definition of the absolute value of $x$, $|x|$?
What is the correct definition of the absolute value of $x$, $|x|$?
Option A
$$
|x|=
\begin{cases}
-x&\text{if } x < 0\\
0& \text{if } x=0\\
x&\text{if } x>0
\end{cases}
$$
Option ...
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0answers
29 views
Definition of “eventually dominates”
What is the definition of the term "eventually dominates"? I guess it's either
$f$ eventually dominates $g$ if for large enough $n$, $f(n) > g(n)$
or the same with $\ge$. A quick Google search ...
0
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1answer
94 views
About the positivity of the inner product on $L^2[0,1]$
My textbook on Hilbert space theory claims that the map $$\langle f,g\rangle=\int_0^1 f(x)\overline{g(x)}~\mathrm{d}x$$ is an inner product on $C[0,1]$. But I am not sure whether ...
6
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2answers
68 views
Alternate definition for boundedness in a TVS
Let $X$ be a topological vector space over $\mathbb R$ or $\mathbb C$. A subset $B\subset X$ is defined to be bounded if
for any open neighborhood $N$ of $0$ there is a number $\lambda>0$
...
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1answer
81 views
Definition of tautological section.
Reading Barth, Peters: Compact complex surfaces, i stumbled across the following:
Let $Y$ be an algebraic surface over $k =\mathbb{C}$, and $\mathcal{L}$ an invertible sheaf on $Y$. Denote by $p: L ...
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2answers
223 views
Check my workings: Prove the limit $\lim\limits_{x\to -2} (3x^2+4x-2)=2 $ using the $\epsilon,\delta$ definition.
Prove the limit $\lim\limits_{x\to -2} (3x^2+4x-2)=2 $ using the $\epsilon,\delta$ definition.
Precalculations
My goal is to show that for all $\epsilon >0$, there exist a $\delta > 0$, ...
3
votes
1answer
85 views
What is meant by `element $x\in H$ of minimal norm'
I do not seek a proof of the following exercise. I just want to understand this question in order to solve it myself.
Let $H$ be a Hilbert space over $\mathbb R$ and let $a, b\in H$ be such that ...
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1answer
62 views
Question about piecewise linear paths
I'm having trouble with a concept about piecewise linear paths. I've been looking on the net but I haven't found a definition of it.
Consider a finite family of hyperplanes in a finite-dimension real ...
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1answer
52 views
What is the “spectrum of $L^1(G)$”?
If $G$ is a locally compact abelian group, what does "the spectrum of $L^1(G)$ mean?" This comes from Folland's A Course in Abstract Harmonic Analysis. As I understand it, $L^1(G)$ is the integrable ...
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1answer
75 views
Why is $X^- = -\min\{X, 0\}$?
Why is $X^- = -\min\{X, 0\}$ ? This is how it is defined in a probability book I am self-studying.
For the function $X:\Omega \to \mathcal{R}$,
The positive part of $X$ is the function $X^+ = \max ...
2
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1answer
133 views
What is a rational character?
Let $G$ be the group of $F$-points of a connected, reductive group over a $p$-adic field $F$. The unramified character of $G$ are $\chi\circ\psi$ where $\chi$ is an unramified character of ...
