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Generally one can see that there are infinite number of solutions for this equation $$a^{2}+b^{2}=c^{2}$$ by taking multiples of the solution $3,4$ and $5$.
Can i use this as a fact to prove, that $k^{2} + (k+1)^{2}$ is a perfect square for infinitely many $k \in \mathbb{N}$? Any hints, suggestions would be helpful... |
Again I have trouble with some exercises in Kunen's set theory.
In the following, let $\kappa > \omega$ a cardinal. Then I want to show that
1) $|H(\kappa)| = 2^{<\kappa}$
2) $H(\kappa)=R(\kappa)$ if and only if $\kappa = \beth_{\kappa}$
In 1) the inequality $\geq$ is easy, but I cannot prove $\leq$. In 2)... |
This question is motivated by the following homework problem. I'm trying to explicitly compute the homeomorphism $f:S^2 \rightarrow \mathbb{CP}^1$ by using stereographic projection and considering $\mathbb{CP}^1 = \mathbb{C}\cup \{\infty\}$. I'll want to prove that this is an isometry, where $S^2$ has the standard an... |
Does projectivizing always fix problems at infinity? (Or, am I making a mistake somewhere?) |
This problem is also considered in the book [Fifty challenging problems in probability with solutions][FCP] by Frederick Mosteller.
[FCP]: http://books.google.co.uk/books?id=QiuqPejnweEC&pg=PA10&lpg=PA10&dq=%22how+thick+should+a+coin+be%22&source=bl&ots=uI9vTy3Egy&sig=S7SjnQTJ-SSwygsV2fTfm4lEOTo&hl=en&ei=516gTP_rHM-... |
If this is homework, it's a pretty strange one. :-? Anyway, do you like the following system?
$$
\begin{align}
x &= 1 \\\
y &= 2 \\\
z &= 3
\end{align}
$$
-Too much easy, I see.
So, try this: your system must fulfil the following equation:
$$
A
\begin{pmatrix}
1 \\\
2 \\\
3
\end{pma... |
*How do I mathematically determine if a number is prime?*
If the number is n, then dividing it by every prime number less than or equal to sqrt(n) and showing that there is a remainder.
There are a number of different [sieve][1] solutions for finding prime numbers, the oldest and most famous of which is the [Siev... |
Why doesn't small perturbations of a matrix decrease its rank? |
"Am I wrong in my philosophical understanding of projective space?"
I think your understanding is incomplete in some ways.
E.g. your remark "resolve singularities by replacing a point with a copy of projective space" doesn't make much sense to me.
It may help in your understanding of "projective spaces" to r... |
I'm looking for some FOSS/GPL programs (or Python libraries) for dealing with sparse matrices. I haven't found much online about these. Can someone please point me in the right direction? |
I am reading a Set Theory book by Kunen. He presents first-order logic and claims that if a set of sentences in inconsistent, then it proves every possible sentence. Since he does not explicitly specify the inference rules, I became curious as to how fundamental this property of inconsistent systems is.
So my questi... |
Why an inconsistent formal system can prove everything? |
<a href="http://mathworld.wolfram.com/CatalansConstant.html"> Quite a few </a> mathematical constants are known to have an infinite continued fraction (see below). On wikipedia (I'm sorry, I can't post anymore hyperlinks because of my low rep.), a condition for irrationality of that continued fraction is given (see ... |
Why are some mathematical constants irrational by their continued fraction while others aren't ? |
<a href="http://mathworld.wolfram.com/CatalansConstant.html"> Quite a few </a> mathematical constants are known to have an infinite continued fraction (see below). On wikipedia (I'm sorry, I can't post anymore hyperlinks because of my low rep.), a condition for irrationality of that continued fraction is given (see ... |
<a href="http://mathworld.wolfram.com/CatalansConstant.html"> Quite a few </a> mathematical constants are known to have an infinite continued fraction (see below). On wikipedia (I'm sorry, I can't post anymore hyperlinks because of my low rep.), a condition for irrationality of that continued fraction is given (see ... |
This question is motivated by the following homework problem. I'm trying to explicitly compute the homeomorphism $f:S^2 \rightarrow \mathbb{CP}^1$ by using stereographic projection and considering $\mathbb{CP}^1 = \mathbb{C}\cup \{\infty\}$. I'll want to prove that this is an isometry, where $S^2$ has the standard an... |
I'm looking to characterize the set
{$a:\exists m,n|60a=5n^2+5n=3m^2+3m$}
that is, <a href="http://oeis.org/A180926">A180926</a>. Any ideas? |
I was maths student but now I'm a software engineer and almost all my knowledge about maths formulas is perished.
One of my client wants to calculate optimal price for perishable products. He has come across an article named **Optimal Pricing Policies for Perishable Products** by Miguel F. Anjos¤ Russell C.H. Cheng ... |
In blind experiments subjects are randomly assigned to one of groups. The most commonly used solution is to use (equivalent of) a coin toss, with the same probability to be assigned to each group.
I have read about two modifications of a procedure of random assignment in blind tests:
* **fair / just coin**, wher... |
According <a href="http://oeis.org/A180926">A180926</a>, the elements of the set
{$a:\exists m,n|60a=5n^2+5n=3m^2+3m$} satisfy the following recurrence relation:
$$a_{n}=\frac{62a_{n-1}+1+\sqrt{(48a_{n-1}+1)(80a_{n-1}+1)}}{2}$$
$$a_1=0$$
Any idea how this can be derived? |
@@$$##&&&%%%%????????????
There are 4 words and the & represents a whitespace.
@,#,$,% represent one letter each (4 total letters).
Each ? represents one of the other 22 letter of the alphabet (12 total letters).
16 different letters used.
ciphertext also needs to be rearranged - it is 'scrambled.'
... |
<a href="http://mathworld.wolfram.com/CatalansConstant.html">Catalan's Constant</a> and quite a few other mathematical constants are known to have an infinite continued fraction (see below). On wikipedia (I'm sorry, I can't post anymore hyperlinks because of my low rep.), a condition for irrationality of that continue... |
(Wade's rewording)
There is a phrase that contains 4 words and 22 total letters (not counting the three spaces between the words). One letter appears four times, no letter appears three times, three letters appear twice, and 12 letters appear once each. What is this phrase?
======================================
... |
Since we have
$$ 5(n^2 +n) = 3(m^2 +m)$$
multiplying by $4$ gives us
$$5(4n^2 + 4n + 1) - 5 = 3(4m^2 + 4m + 1) -3$$
if $x = 2n+1$ and $y = 2m+1$ we get
$$5x^2 - 3y^2 = 2$$
This is a generalized Pell's equation of the form $ax^2 - by^2 = c$ and I believe this is a solved problem.
For instance, see t... |
(Wade's rewording)
There is a phrase that contains 4 words and 22 total letters (not counting the three spaces between the words). One letter appears four times, no letter appears three times, three letters appear twice, and 12 letters appear once each. What is this phrase?
======================================
... |
(Wade's rewording)
There is a phrase that contains 4 words and 22 total letters (not counting the three spaces between the words). One letter appears four times, no letter appears three times, three letters appear twice, and 12 letters appear once each. What is this phrase?
======================================
... |
Suppose I embed a manifold-with-boundary $M$ in some $\mathbb{R}^n$. Are there conditions (necessary, sufficient, or both) that can help determine when the topological boundary of $M$ is equal to the manifold boundary?
By "topological boundary," I'm referring to $\text{Bd } M$, which is the closure minus the interi... |
When does the topological boundary of an embedded manifold equal its manifold boundary? |
I recently learned about the Classification Theorem for compact 2-manifolds. Is there a similar classification theorem for ALL 2-manifolds, not just the compact ones?
Moreover, is there a theorem which classifies the 2-manifolds with boundary? |
Classification Theorem for Non-Compact 2-Manifolds? 2-Manifolds With Boundary? |
I don't think this answer is correct. I'm posting it with the hope that somebody will be kind enough to point out the mistakes to me.
Let $G'_n$ be the group $\mathbb R\times\mathbb R^n$, and $G'$ the inductive limit of the $G'_n$. Assuming the addition of $G'$ is continuous, I'll show that
> there is a $V$ as ... |
I don't think this answer is correct. I'm posting it with the hope that somebody will be kind enough to point out the mistakes to me.
Let $G'_n$ be the group $\mathbb R\times\mathbb R^n$, and $G'$ the inductive limit of the $G'_n$. Assuming the addition of $G'$ is continuous, I'll show that
> there is an open n... |
This is a neat little problem that I was discussing today with my lab group out at lunch. Not particularly difficult but interesting implications nonetheless
Imagine there are a 100 people in line to board a plane that seats 100. The first person in line realizes he lost his boarding pass so when he boards he decide... |
The continued fraction expansion of any possible real number falls in one of 2 possible categories:
* the continued fraction expansion has a finite length if and only if the number is rational.
* the continued fraction expansion never ends if and only if the number is irrational.
*"quite a few other mathematical... |
I'm reading Jech's Set Theory, and in the chapter about measurable cardinals there is a theorem that if $\kappa$ is real-measurable but not measurable then it is $\le 2^{\aleph_0}$ and so and so. (Corollary 10.10)
How can a cardinal number be real-measurable without being measurable? Can't a measure be "destructed" ... |
Can you help me realize why this is true? I can tell you that $P_i$ and $P_j$ are probabilities, i.e. $0 \leq P_i, P_j \leq 1$.
$\displaystyle \sum_{i=1}^\infty \sum_{j=1}^\infty ijP_iP_j \leq \sum_{i=1}^\infty \sum_{j=1}^\infty j^2P_jP_i$. |
Suppose one wants to prove that a 1-dimensional integral proper scheme over an algebraically closed field is projective. This is a step in how Hartshorne has you prove that any 1-dim proper scheme (over an algebraically closed field) is projective.
The method is that first note the normalization is non-singular and ... |
Since this looks like homework, I will give a hint.
Which is the largest among $\sqrt{k^2+1}, \sqrt{k^2+2},\dots, \sqrt{k^2+k}$?
|
Since this looks like homework, I will give a hint.
Which is the largest among $\sqrt{k^2+1}, \sqrt{k^2+2},\dots, \sqrt{k^2+k}$?
For your next step:
I think the mistake you are doing is substituting
$\frac{\sqrt{k^2+k}}{k}$ with $\sqrt{\frac{k^2+k}{k}}$ instead of $\sqrt{\frac{k^2+k}{k^2}}$.
|
Amazingly, the complete classification of noncompact 2-manifolds with boundary was not completed until 2007. Here's the reference:
A. O. Prishlyak and K. I. Mischenko, [Classification of noncompact surfaces with
boundary][1], Methods Funct. Anal. Topology 13 (2007), no. 1, 62–66.
[1]: http://www.ams.org/mat... |
Recently, I came across this problem in my textbook. Any hints on how to solve this using dynamic programming?
"Professor Stewart is consulting for the president of a corporation that is planning a company party. The company has a hierarchical structure; that is, the supervisor relation forms a tree rooted at the pr... |
"Am I wrong in my philosophical understanding of projective space?"
I think your understanding is incomplete in some ways.
E.g. your remark "resolve singularities by replacing a point with a copy of projective space" doesn't make much sense to me.
Edit: I think I didn't understand the question correctly. But... |
Greetings, dear Mathematicians!
How to implement base-9 to base-3 conversion in O(log(n)) on 4-tuple ((stateFrom, encounterWhat, Command, StateTo), where Command can be < (turn left), > (turn right), # (stop execution), or symbol from an alphabet) Turing machine, finite from the left side and infinite from the right... |
Helping a buddy with his "intro to math" course for CompSci. I'm afraid my 'leet math skills are already giving out, so any help would be appreciated!
If *f* and *g* are both surjective functions, then so is the composition *f* o *g*. (so far so good).
I would expect the converse to be true too: If *f* o *g* is a... |
Surjection on composed function? |
It is important to consider the codomain when talking about the `surjectivity' of a function. Let us give a real-valued example of functions for now.
Let $g(x)$ be any function that maps the real number line to the interval $(-1, 1)$ (eg. sin) and let $f(x) = \frac{x}{x^2-1}$ be considered only over the interval $(-... |
By «connecting maps» you mean the maps that appear in the short exact sequence you wrote?
In any case, you will find the details of the construction of the sequence in Hilton-Stammbach's book *A Course on homological algebra*, GTM4. The result is theorem VI.15.1, but the meat of the construction is in section §V.3.
... |
I think this is really weird.
I want to solve the following equation:
$$(x^2+3x+5)^3(x^2+3x+5)^2=0$$
This is a polynomial equation of 6 degrees. It should have 6 roots.
But [wolframalpha][1] only returns 2 roots! Anything I miss?
This is the command I use:
solve ((2x^2+3x+5)^3)*((2x^2+3x+5)^2)=0
... |
I think this is really weird.
I want to solve the following equation:
$$(2x^2+3x+5)^3+(2x^2+3x+5)^2=0$$
This is a polynomial equation of 6 degrees. It should have 6 roots.
But [wolframalpha][1] only returns 4 roots! Anything I miss?
This is the command I use:
solve ((2x^2+3x+5)^3)+((2x^2+3x+5)^2)... |
I'm trying to convey to some folks some logical and bitwise operators such as AND, OR, XOR, &c in programming languages such as C &| Java. Though these I guess could be considered "boolean" operators there is no "Boolean Algebra" in here, just the results of various operation, yes?
I've never heard of Boolean **Ari... |
Arithmetic? Logical/Boolean/Bitwise operations? |
The fact that $S$ is a union of conjugacy classes of $N$ should be immediate, so the real meat of the problem is showing that the *number* of conjugacy classes is the index of $NC_G(x)$ in $G$.
Whenever you see a problem in which you need to show that the number of something or other in a group equals the index of a... |
The fact that $S$ is a union of conjugacy classes of $N$ should be immediate, so the real meat of the problem is showing that the *number* of conjugacy classes is the index of $NC_G(x)$ in $G$.
Whenever you see a problem in which you need to show that the number of something or other in a group equals the index of a... |
A simple question here, which likely demands a somewhat complex answer... Or rather, a set of related questions.
* What are the advantages of typed lambda calculus over untyped lambda calculus in terms of proof theory?
* Specifically, Church's original Lambda Calculus was untyped and allows arbitrarily high-order... |
The fact that $S$ is a union of conjugacy classes of $N$ should be immediate; also, the sizes are the same: let $x\in S$, and let $U$ be the $N$-conjugacy class of $x$. If $T$ is any other $N$-conjugacy class contained in $S$, then let $y\in T$. There exists $g\in G$ such that $gxg^{-1}=y$. Then $|T|=|gUg^{-1}|=|U|$.
... |
A common way to define a group is as the group of structure-preserving transformations on some structured set. For example, the symmetric group on a set $X$ preserves no structure: or, in other words, it preserves only the structure of being a set. When $X$ is finite, what structure can the *alternating* group be sai... |
I'm going to ask a very silly question, so I'm begging you to be understanding if it is absolutely trivial, or if it's an exercise in some Bourbaki. I'm afraid of asking you, because the question entails this particular case: is the addition of (a finite, but variable quantity of) real numbers a continuous function?
... |
Is addition continuous? |
1. What are the advantages of typed lambda calculus over untyped lambda calculus in terms of proof theory? *and also*
2. Specifically, Church's original Lambda Calculus was untyped and allows arbitrarily high-order functions. What are the limitations with respect to constructing a proof calculus from it?
* Simply, ... |
<a href="http://mathworld.wolfram.com/CatalansConstant.html">Catalan's Constant</a> and quite a few other mathematical constants are known to have an infinite continued fraction (see the bottom of that webpage). On wikipedia (I'm sorry, I can't post anymore hyperlinks because of my low rep.), a condition for irrationa... |
I want to show that there is a finite conjunction $\phi$ of axioms of $ZF$, such that every transitive class $M$, which satisfies $\phi$, is already a model of $ZF$.
This is an exercise in Kunen's set theory. There is a hint, it seems to be useful to apply the Reflection principle to the union $M = \cup_{\alpha} M ... |
I've been trying to understand how
$\{x^3-12x+9}$
factors to
$\(x-3) (x^2+3 x-3)$
What factoring rule does this follow? The net result seems to be similar to what is attained through the sum/difference of cubes factoring pattern, but the signs are different.
Additionally, what type of problem is this... |
How would you find the image of $\{ x+yi : -\pi < x < \pi, y > 0 \}$ under $\sin(z)$? I see that $\text{Re } \sin(x+yi) = \sin x \cosh y$ and $\text{Im } \sin(x+yi) = \cos x \sinh y$, but I'm not sure what to do next since $\sinh$ and $\cosh$ are unbounded. |
If $h(x,t)$ is measurable and the sigma-algebra's involved are $\sigma$-finite, does there exist a sequences of functions
$$h_n = \sum_{j=1}^{N_n} f_{j,n}(x) 1_{F_{j,n}}(t)$$
where the sets are pairwise disjoint (in $j$) such that $h_n \to h$?
I know that if we fix $x$ we can get a sequence like that but we ca... |
One way to see that is that $x=3$ is a root of $x^3 - 12x + 9$. So $(x-3)$ will be a factor.
Now you can try to get $x-3$ somehow.
One way you can do that is to rewrite
$$x^3-12x + 9 = (x^3 - 3^3) + 3^3 -12(x-3) - 36 + 9$$
$$ = (x-3)(x^2+3x+9) - 12(x-3) = (x-3)(x^2+3x-3) $$
Here we used the fact that $x^... |
If $h(x,t)$ is measurable and the sigma-algebra's involved are $\sigma$-finite, does there exist a sequences of functions
$$h_n(x,t) = \sum_{j=1}^{N_n} f_{j,n}(x) 1_{F_{j,n}}(t)$$
where the sets are pairwise disjoint (in $j$) such that $h_n \to h$?
I know that if we fix $x$ we can get a sequence like that but ... |
One way to see that is that $x=3$ is a root of $x^3 - 12x + 9$. So $(x-3)$ will be a factor (by the [Factor Theorem][1]).
Now you can try to get $x-3$ somehow.
One way you can do that is to rewrite
$$x^3-12x + 9 = (x^3 - 3^3) + 3^3 -12(x-3) - 36 + 9$$
$$ = (x-3)(x^2+3x+9) - 12(x-3) = (x-3)(x^2+3x-3) $$
H... |
What shape do you get when you fix y and consider all values of x? Then see how the shape depends on y. Of course, you could try the reverse as well (fix x and vary y first), but my way, you don't have to deal with the "y > 0" condition until the end.
In fact, draw a picture showing the images of the lines x=consta... |
I want to show that there is a finite conjunction $\phi$ of axioms of $ZF$, such that every transitive proper class $M$, which satisfies $\phi$, is already a model of $ZF$.
This is an exercise in Kunen's set theory. There is a hint, it seems to be useful to apply the Reflection principle to the union $M = \cup_{\al... |
Looking at a 2<sup>nd</sup>-order Taylor series approximation of the function f, I have this:
f(t1) = f(t0) + h*f'(t0) + 1/2*h^2*f''(t0) + O(h^3)
Now say I approximate f''(t0) with a 2<sup>nd</sup>-order central difference method:
f''(t) = 1/2 * (f'(t+h) - f'(t-h))/h + O(h^2)
What's the resulting er... |
How do I combine "error of order" terms in numerical analysis? |
If $h(x,t)$ is measurable and the sigma-algebra's involved are $\sigma$-finite, does there exist a sequences of functions
$$h_n(x,t) = \sum_{j=1}^{N_n} f_{j,n}(x) 1_{F_{j,n}}(t)$$
where the sets are pairwise disjoint (in $j$) such that $h_n \to h$ pointwise?
I know that if we fix $x$ we can get a sequence like... |
Looking at a 2<sup>nd</sup>-order Taylor series approximation of the function $f$, I have this:
$$f(t1) = f(t0) + h*f'(t0) + 1/2*h^2*f''(t0) + O(h^3)$$
Now say I approximate $f''(t0)$ with a 2<sup>nd</sup>-order central difference method:
$$f''(t) = 1/2 * (f'(x+h) - f'(x-h))/h + O(h^2)$$
What's the resu... |
Martin: Just some hints: You know from a class being transitive and proper that a few axioms already hold, right?
First, recall that classes are definable in ZF.
The main problem is ensuring that replacement and comprehension hold (since these are the axiom schema). Replacement amounts to showing that any class-... |
Looking at a 2<sup>nd</sup>-order Taylor series approximation of the function $f$, I have this:
$$f(t_1) = f(t_0) + hf'(t_0) + {h^2\over 2}f''(t_0) + O(h^3)$$
Now say I approximate $f''(t0)$ with a 2<sup>nd</sup>-order central difference method:
$$f''(t) = {1\over 2}{f'(t+h) - f'(t-h)\over h} + O(h^2)$$
... |
I made a picture of the problem here:
http://img84.imageshack.us/img84/1845/screenshot20100927at111.png
If the link does not work, read this:
Let $a$ be a non-zero vector in $\mathbb{R}^n$. Let S be the set of all orthogonal vectors to $a$ in $\mathbb{R}^n$. I.e., for all $x \in \mathbb{R}^n$, $a\cdot x = 0$
... |
I was reading about Euler's totient function on wikipedia, and it eventually led me to this book on google:
http://books.google.com/books?id=1MTcYrbTdsUC&pg=PA74&lpg=PA74&dq=n+has+r+distinct+odd+prime+factors+euler+totient+function&source=bl&ots=9Eb-MaPpZ7&sig=gDwcPAK_u7QFy0wAqf8q1zwdkdQ&hl=en&ei=eGKiTLCoLof2tgPo7fm... |
One way to see that is that $x=3$ is a root of $x^3 - 12x + 9$. So $(x-3)$ will be a factor (by the [Factor Theorem][1]).
Now you can try to get $x-3$ somehow.
One way you can do that is to rewrite
$$x^3-12x + 9 = x^3 - (3^3 - 3^3) - 12(x - 3+3) + 9$$
$$ = x^3 - 27 + 27 - 12(x - 3) - 36 + 9 = x^3 - 27 - 12(... |
I was reading about Euler's totient function on wikipedia, and it eventually led me to this book on google:
[Page 74 of the book, Prime numbers: the most mysterious figures in math
By David G. Wells][1].
Anyway, the books lists many assertions without proof, only references which I can't find. One I could not s... |
From the comments to [A005277][1] :
If p is prime then the following two statements are true. I. 2p is in the sequence iff 2p+1 is composite (p is not a Sophie Germain prime). II. 4p is in the sequence iff 2p+1 and 4p+1 are composite. - Farideh Firoozbakht (mymontain(AT)yahoo.com), Dec 30 2005
This covers most o... |
I was in the middle of writing the same old geographic distance calculation using the [Haversine formula][1] when it occurred to me: shouldn't there be simpler way to do this? Haversine is of course derived from the Law of Cosines. But in thinking about this problem, I ran across [Napier's Rules][2] for right-angled sp... |
I made a picture of the problem here:
![(statement of the problem, rendered as an image)][1]
If the link does not work, read this:
Let $a$ be a non-zero vector in $\mathbb{R}^n$. Let S be the set of all orthogonal vectors to $a$ in $\mathbb{R}^n$. I.e., for all $x \in \mathbb{R}^n$, $a\cdot x = 0$
Prove that ... |
I would do it this way. First of all, by choosing $\frac{a}{\|a\|}$ as the last vector of an orthonormal basis of $\mathbb{R}^n$, you may assume that $a = (0,\dots , 0,\alpha), \alpha \in \mathbb{R}, \alpha \neq 0$. Hence
$$
S =\left\\{ (x_1 , \dots , x_n) \in \mathbb{R}^n\ \vert \ x_n = 0 \right\\} \ .
$$
... |
You are right, it's just the triangle inequality. $B=b_n+(B-b_n)$. So $|B|\leq |b_n| + |B-b_n|$.
By the way, shouldn't it be $n \geq N$ in your second para? |
Me and a team of programmers are programming a robot to play a game in a competition. We did not create the hardware, and all teams will use the same type of robot.
<hr />
**GAME DESCRIPTION**
The game has two players opposing each other and are both trying to achieve an objective. This objective is to move to a... |
By looking at an integral and bounding the error? |
The standard approach should be along the lines of $\sum_{i\le n}\sqrt i=\sqrt n\sum_{i\le n}\sqrt{\frac{i}{n}}$, so $\frac1{n\sqrt n}\sum_{i\le n}\sqrt i=\sum_{i\le n}\sqrt{\frac{i}{n}}\frac1n\to\int_0^1\sqrt x{} dx=\frac23$, or $\sum_{i\le n}\sqrt i\sim\frac23 n\sqrt n$. What techniques do you know to estimate the er... |
**EDIT 3 (SEP 29, 2010)**
The following "argument", if correct, would contradict Lemma 5.5 page 64 in Milnor and Stasheff's **Characteristic Classes**. So the probability of it being correct is almost nil. Thank you very much for helping me catch the mistakes.
Let $\mathbb R^\infty$ be the inductive limit of th... |
**EDIT 3 (SEP 29, 2010)**
The following "argument", if correct, would contradict Lemma 5.5 page 64 in Milnor and Stasheff's **Characteristic Classes**. So the probability of it being correct is almost nil. Thank you very much for helping me catch the mistakes.
Let $\mathbb R^\infty$ be the inductive limit of th... |
The way you've restated the definition is fairly common in contemporary high school books in the U.S. (perhaps changing "links between two sets" to "ordered pairs"). What I've seen a lot of in middle school and earlier algebra settings is the idea of a "function machine." The function machine graphic below is from [F... |
Maybe you mean the number of elements of order dividing $p$ (so that you are including the identity)? (Think about the case $p = 3$ --- there are two three cycles, not three of them.) For the general question, think about the possible cycle structure of an element of order $p$
in $S_p$.
You can go from the formul... |
first show you only need to consider squares of functions as f.g = 1/4 [(f+g)sqr - (f-g)sqr]. show then that you only need to consider only positive valued functions becuase f(x).g(x)=|f(x)|sqr. then , if 0 <=f(x) <= M on [a,b] show that f sqr(x) - f sqr(y) <= 2M (f(x)-f(y)).
does anyone know how i would answer this... |
I think Cauchy's integral formula and the Hilbert transform can be used to prove one direction, but is this an equivalence or only an implication? |
Is a function analytical on C iff it has no negative frequencies? |
Is a function analytical on C iff its Fourier-transform vanishes for negative frequencies? |
I presume you want to colour the *faces* of the cube?
If so there are $2^6$ colourings as there are six faces.
But do you want two colourings which are equivalent under
rotations (or rotations and reflections) to count as as the same?
If so then this is a classic application of
[Burnside's lemma][1].
[1]: h... |
I think Cauchy's integral formula and the Hilbert transform can be used to prove one direction, but is this an equivalence or only an implication?
**edit** for clarification:
Is a function $f : \mathbb C \to \mathbb C, z\mapsto f(z)$ analytical $\Leftrightarrow$ The Fourier-Transform $\mathcal F\\{f\\}(\omega) = N ... |
I think Cauchy's integral formula and the Hilbert transform can be used to prove one direction, but is this an equivalence or only an implication?
**edit** for clarification:
Is a function $f : \mathbb C \to \mathbb C, z\mapsto f(z)$ analytical $\Leftrightarrow$ The Fourier-Transform $\mathcal F\\{f\\}(\omega) = N ... |
If $h(x,t)$ is measurable and the measures involved are $\sigma$-finite, does there exist a sequences of functions
$$h_n(x,t) = \sum_{j=1}^{N_n} f_{j,n}(x) 1_{F_{j,n}}(t)$$
where the sets are pairwise disjoint (in $j$) such that $h_n \to h$ pointwise?
I know that if we fix $x$ we can get a sequence like that b... |
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