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Pick's theorem says that given a square grid (that is, all points in the plane with integer coordinates) and a polygon without holes and non selt-intersecting whose vertices are grid points, its area is given by
i + b/2 - 1
where i is the number of interior lattice points and b is the number of points on its boundary. Theorem and proof may be found on [Wikipedia][1].
Let us suppose that the grid is not square but triangular (or hexagonal). Does a similar theorem hold?
[1]: http://en.wikipedia.org/wiki/Pick%27s_theorem |
Given an initial position and a subsequent position, each given by latitude and longitude in the WGS-84 system. How do you determine the heading in degrees clockwise from true north of movement? |
How to calculate a heading on the earths surface? |
![complex plot of the zeros][1]
(Diagram and setup from UCSMP _Precaluclus and Discrete Mathematics_, 3rd ed.)
Above is a partial plot of the zeros of $p_c(x) = 4x^4 \+ 8x^3 - 3x^2 - 9x \+ c$. The text stops at showing the diagram and does not discuss the shape of the locus of the zeros or describe the resulting curves. Are the curves in the locus some specific (named) type of curve? Is there a simple way to describe the curves (equations)?
The question need not be limited to the specific polynomial given--a similar sort of locus is generated by the zeros of nearly any quartic polynomial as the constant term is varied.
[1]: http://www.imgftw.net/img/572465889.jpg |
I am trying to show the following:
![alt text][1]
but I really can't remember what I am supposed to do to get from the LHS to the RHS. I have tried using integration this way
![alt text][2]
And then use integration by parts, but even though what I get resembles it, it can't be correct (because $e^{-\gamma W}$ is not the distribution of W).
I have also tried using Taylor series expansion, but I think I am way off, and I don't think an approximation here is what I need, I think the equality above is exact.
FYI, this is not homework, I am working through a [paper][3] (page 10) and I would really like to know how every step was derived.
Can anyone at least point me to the right direction?
[1]: http://dl.dropbox.com/u/1885087/Expectation.png
[2]: http://dl.dropbox.com/u/1885087/integ.png
[3]: http://www.princeton.edu/~markus/research/papers/liquidity.pdf |
If W is randomly chosen with the PDF P(x), then the expectation value should be
>  |
If W is randomly chosen with the PDF P(x), then the expectation value should be
> ![$E[e^{-\gamma W}]=\int_{-\infty}^\infty P(x) e^{-\gamma x} dx$](http://mathcache.appspot.com/?tex=%5cpng%5c%5bE%5Be%5E%7B-%5Cgamma%20W%7D%5D%3D%5Cint_%7B-%5Cinfty%7D%5E%5Cinfty%20P%28x%29%20e%5E%7B-%5Cgamma%20x%7D%20dx%5c%5d)
And I think this is correct only when W is normal. |
If W is randomly chosen with the PDF P(x), then the expectation value should be
> ![$E[e^{-\gamma W}]=\int_{-\infty}^\infty P(x) e^{-\gamma x} dx$](http://mathcache.appspot.com/?tex=%5cpng%5c%5bE%5Be%5E%7B-%5Cgamma%20W%7D%5D%3D%5Cint_%7B-%5Cinfty%7D%5E%5Cinfty%20P%28x%29%20e%5E%7B-%5Cgamma%20x%7D%20dx%5c%5d)
And I think that equation (E[e<sup>-γW</sup>] = e<sup>-γ(E[W] - ½γVar[W])</sup>) is correct only when W is a normal distribution. |
Both [Benford's Law][1] (if you take a list of values, the distribution of the most significant digit is rougly proportional to the logarithm of the digit) and [Zipf's Law][2] (given a corpus of natural language utterances, the frequency of any word is roughly inversely proportional to its rank in the frequency table) are not theorems in a mathematical sense, but they work quite good in the real life.
Does anyone have an idea why this happens?
(see also [this question][3])
[1]: http://en.wikipedia.org/wiki/Benford%27s_law
[2]: http://en.wikipedia.org/wiki/Zipf%27s_law
[3]: http://math.stackexchange.com/questions/58 |
Why does Benford's Law (or Zipf's Law) hold? |
in a category i have two objects $a$ and $b$ and a morphism $m$ from $a$ to $b$ and one $n$ from $b$ to $a$. Is this always an isomorphism? Why is it emphasized that this has to be true, too: $m \circ n$ = $id_b$ and $n \circ m$ = $id_a$?
I am looking for an example in which the id-part is not true and therefore $m$ and $n$ are not isomorph. |
There is a general formula for solving quadratic equations, namely the Quadratic Formula.
For third degree equations of the form $ax^3+bx^2+cx+d=0$, there is a set of thee equations: one for each root.
**Is there a general formula for solving equations of the form
$ax^4+bx^3+cx^2+dx+e=0$
?**
How about for higher degrees? If not, why not? |
Is there a general formula for solving 4th degree equations? |
Say there are three jars, $j_1, j_2, j_3$ filled with different binary sequences of length two.
The distribution of the binary sequences in each of the jars is given by the $p_i^k(1-p_i)^{n-k}$, where
$p_i = \frac{i}{m + 1}$ where m is the number of jars, i is the jar index, k is number of 1's and n is the length of the string.
So for three jars we have $p_1 = 0.25, p_2 = 0.5$, and $p_3 = 0.75$ for $j_1, j_2, j_3$ respectively.
Here are the sequences and their probabilities for $j_1$ with $p_1 = 0.25$:
P(00) = 9 / 16
P(10) = 3 / 16
P(01) = 3 / 16
P(11) = 1 / 16
If I tell you that I have selected a binary sequence and the first element is 1 what is the expected value of X over $p_i$?
Well, this can be calculated by looking at each of the jars and adding up the probability of candidate sequences times the value of $p_i$.
**Edit:** I wasn't normalizing this conditionally space properly. I'm skipping a step which I'll explain, someone wants.
E(X) = (4/24 * 1/4) + (8/24 * 1/2) + (12/24 * 3/4) = 14 / 24 = 0.58
So the question is ... what is E(X) when the numbers of jars goes to infinity (or alternatively, when p can take on values between 0 and 1)? Also what happens when the size of the binary strings goes to infinity? Does it have an effect on the outcome? If it does, does the order we take the limits change the answer?
And most importantly what is the general case for when I have $s$ 1's and $r$ 0's?, with a continuous p from 0 to 1 and infinite sequences?
|
The question that follows was inspired by [this][1] question:
When trying to solve for the roots of a polynomial equation, the quadratic formula is much more simple than the cubic formula and the cubic formula is much more simple than the quartic formula.
1. Just how fast is the complexity of each formula growing after each successive degree of polynomial?
2. That the general solutions to various polynomial equations are so complex and difficult to derive seems to suggest a fundamental limitation in the problem solving capabilities of the mathematical machinery. Does this intuition of mine make any sense? What should I make of it?
3. Why is it that with each successive degree in a polynomial equation, the solution becomes so much more complex? Can I gain some intuition about what makes finding the roots so hard?
[1]: http://math.stackexchange.com/questions/785/is-there-a-general-formula-for-solving-4th-degree-equations |
Why is it so hard to find the roots of polynomial equations? |
The question that follows was inspired by [this][1] question:
When trying to solve for the roots of a polynomial equation, the quadratic formula is much more simple than the cubic formula and the cubic formula is much more simple than the quartic formula.
1. That the general solutions to various polynomial equations are so complex and difficult to derive seems to suggest a fundamental limitation in the problem solving capabilities of the mathematical machinery. Does this intuition of mine make any sense? What should I make of it?
2. Why is it that with each successive degree in a polynomial equation, the solution becomes so much more complex? Can I gain some intuition about what makes finding the roots so hard?
3. According to the [Abel-Ruffini theorem][2]: "there is no general algebraic solution—that is, solution in radicals— to polynomial equations of degree five or higher." What is so special about the quintic that makes it the cut-off for finding a general algebraic solution?
[1]: http://math.stackexchange.com/questions/785/is-there-a-general-formula-for-solving-4th-degree-equations
[2]: http://en.wikipedia.org/wiki/Abel%E2%80%93Ruffini_theorem |
Say there are three jars, $j_1, j_2, j_3$ filled with different binary sequences of length two.
The distribution of the binary sequences in each of the jars is given by the $p_i^k(1-p_i)^{n-k}$, where
$p_i = \frac{i}{m + 1}$ where m is the number of jars, i is the jar index, k is number of 1's and n is the length of the string.
So for three jars we have $p_1 = 0.25, p_2 = 0.5$, and $p_3 = 0.75$ for $j_1, j_2, j_3$ respectively.
Here are the sequences and their probabilities for $j_1$ with $p_1 = 0.25$:
P(00) = 9 / 16
P(10) = 3 / 16
P(01) = 3 / 16
P(11) = 1 / 16
If I tell you that I have selected a binary sequence and the first element is 1 what is the E($p_i$)?
Well, this can be calculated by looking at each of the jars and adding up the probability of candidate sequences times the value of $p_i$.
**Edit:** I wasn't normalizing this conditionally space properly. I'm skipping a step which I'll explain, someone wants.
E($p_i$) = (4/24 * 1/4) + (8/24 * 1/2) + (12/24 * 3/4) = 14 / 24 = 0.58
So the question is ... what is E($p_i$) when the numbers of jars goes to infinity (or alternatively, when p can take on values between 0 and 1)? Also what happens when the size of the binary strings goes to infinity? Does it have an effect on the outcome? If it does, does the order we take the limits change the answer?
And most importantly what is the general case for when I have $s$ 1's and $r$ 0's?, with a continuous p from 0 to 1 and infinite sequences?
|
This is still WIP. There are a few missing details, still I think it's better than nothing. Feel free to edit in the missing details.
Given a problem of `SUBSET-SUM`. We have a set of `A`={a<sub>1</sub>,a<sub>2</sub>,...,a<sub>n</sub>} numbers, and another number `s`. The question we're seeking answer to is, whether or not there's a subset of `A` whose sum is `s`.
I think the `SUBSET-SUM` problem is `NP`-hard even if you allow treating each a<sub>i</sub> as a negative number. That is even if `A` is of the form `A`={a<sub>1</sub>,-a<sub>1</sub>,a<sub>2</sub>,-a<sub>2</sub>,...,a<sub>n</sub>,-a<sub>n</sub>}. This is still a wrinkle I need to iron out in this reduction.
Obviously, if there's a subset of `A` with sum `s`, then there's a solution to the `24`-problem for how to reach using `A` to `s`. The solution is only using the `+` sign.
The problem is, what happens if there's no solution which only uses the `+` sign, but there is a solution which uses other arithmetic operations.
Let us consider the following problem. Let's take a prime `p` which is larger than `n`, the total number of elements in `A`. Given an oracle which solves the `24`-problem, and a `SUBSET-SUM` problem of `A`={a<sub>1</sub>,a<sub>2</sub>,...,a<sub>n</sub>} and `s`. We'll ask the oracle to solve the `24`-problem on
> `A`={a<sub>1</sub>+(1/p),a<sub>2</sub>+(1/p),...,a<sub>n</sub>+(1/p)}
for the following values:
> s<sub>1</sub>=s+1/p,s<sub>2</sub>=s+2/p,...,s<sub>n</sub>=s+n/p.
If the solution includes multiplication, we will have a denominator larger than `p` in the end result, and thus we will not be able to reach any s<sub>i</sub>.
What about division? How can we be sure no division will occur. Find another prime `q` which is different than p, and larger than the largest a<sub>i</sub> times `n`. Multiply all answers by `q`. The set `A` will be
> `A`={qa<sub>1</sub>+(1/p),qa<sub>2</sub>+(1/p),...,qa<sub>n</sub>+(1/p)}
We will look for the following values:
> s<sub>1</sub>=qs+1/p,s<sub>2</sub>=qs+2/p,...,s<sub>n</sub>=qs+n/p.
In that case, a<sub>i</sub>/a<sub>j</sub> will be smaller than the minimal element in `A`, and therefor the end result which will contains a<sub>i</sub>/a<sub>j</sub> will never be one of the s<sub>i</sub> we're looking for. |
For $n > 1$ an integer, the volume of the simplex in $\mathbb{R}^n$ is $1/ \Gamma(n+1)$ where $\Gamma$ is the gamma function, or more simply $1/n!$.
What is the analogous statement in a Banach space/Hilbert Space? |
What is the volume of intersection of the three cylinders with axes of length $1$ in $x, y, z$ directions starting from the origin, and with radius $1$?
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How to compute the volume of this object via integration? |
How to prove that the exponential function and the logarithm function are the inverses of each other? I want it the following way. We must use the definition as power series, and must verify that all the terms of the composition except the coefficent of $z$ vanish, and that the first degree term is $1$.
I can write down the proof for the coefficient of $x^n$ for arbitrary but fixed $n$ by explicit verification. But how to settle this for all $n$ at one go?
|
I don't think the people working in higher-order theorem proving really care about Henkin semantics or models in general, they mostly work with their proof calculi. As long as there are no contradictions or other counterintuitive theorems they are happy. The most important and most difficult theorem they prove is usually that their proof terms terminate, which IIRC can be viewed as a form of soundness.
Henkin semantics is most interesting for people trying to extend their first-order methods to higher-order logic, because it behaves essentially like models of first-order logic. Henkin semantics is sound, but it is somewhat weaker than true higher-order logic.
> Where does typed lambda calculus come into proof verification?
To prove some implication `P(x) --> Q(x)` with some free variables `x` you need to map any proof of `P(x)` to a proof of `Q(x)`. Syntactically a map (a function) can be represented as a lambda term.
> Are there any other approaches than higher order logic to proof verification?
You can also verify proofs in first-order or any other logic, but then you would loose much of the power of the logic. First-order logic is mostly interesting because it is possible to automatically find proofs, if they are not too complicated. The same applies even more to propositional logic.
> What are the limitations/shortcomings of existing proof verification systems (see below)?
The more powerful the logic becomes the harder it becomes to construct proofs.
Since the systems are freely available I suggest you play with them, e.g. Isabelle and Coq for a start.
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There is a theorem of Riemann to that effect. How to prove it?
Note: This was asked by Kenny in the beta for "calculus".
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How to prove that a conditionally convergent series can be rearranged to sum to any real number? |
What class of Partial Differential Equations can be solved using the method of separation of variables? |
There are so many available bases. Why is the strange number 'e' preferred over all else?
Of course one could integrate 1/x and see this. But is there more to the story?
|
What's so "natural" about the base of natural logarithms? |
There are quite simple, intuitive and straightforward expressions for evaluating the area or volume of a figure. But why is the expression for the length of a curve so complicated?
|
Why is the integral expression for the length of a curve more complicated than the expression for area or volume of a figure rotated? |
Ok, Ok, I know that in fact the discriminant is defined (up to sign) as a product of differences of the roots of the polynomial.
But why does it then have integral coefficients, if the polynomial you started with had integer coefficients? |
How does the discriminant of an equation capture whether it has multiple roots or not? |
The [Pythagorean Theorem][1] is one of the most popular to prove by mathematicians, and there are [many proofs available][2] (including one from [James Garfield][3]).
What's the most elegant proof?
My favorite is this graphical one:
![alt text][4]
According to cut-the-knot:
> Loomis (pp. 49-50) mentions that the
> proof "was devised by Maurice Laisnez,
> a high school boy, in the
> Junior-Senior High School of South
> Bend, Ind., and sent to me, May 16,
> 1939, by his class teacher, Wilson
> Thornton."
>
> The proof has been published by Rufus
> Isaac in Mathematics Magazine, Vol. 48
> (1975), p. 198.
[1]: http://en.wikipedia.org/wiki/Pythagorean_theorem
[2]: http://www.cut-the-knot.org/pythagoras/
[3]: http://en.wikipedia.org/wiki/James_A._Garfield
[4]: http://upload.wikimedia.org/wikipedia/commons/thumb/1/16/Pythagorean_Proof_%283%29.PNG/800px-Pythagorean_Proof_%283%29.PNG |
What is the most elegant proof of the Pythagorean theorem? |
I have seen and studied this class equation for a finite group acting on itself by conjugations. The only applications I know are Cauchys' theorm and Sylow's theorem. Are there more?
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There is the notion of class number from algebraic number theory. Why is such a notion defined and what good comes out of it?
It is nice if it is $1$; we have unique factorization of all ideals; but otherwise?
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Which mathematicians have influenced you the most? |
If you know some linear algebra, then here is an abstract reason: $e^x$ is the unique eigenvector of eigenvalue $1$ of the derivative $D$ acting on, say, the space of smooth functions on $\mathbb{R}$. Why is this important? The study of solutions of linear differential equations with constant coefficients is equivalent to the study of nullspaces of operators which are polynomials in $D$, e.g. operators of the form $\sum a_k D^k$. Any such operator automatically commutes with $D$, so this nullspace splits up into eigenspaces of $D$. That's why solutions to linear differential equations with constant coefficients can be expressed as sums of complex exponentials. The choice of $e$ makes it particularly easy to see what the eigenvalue is: the eigenvalue of the eigenvector $e^{\lambda x}$ is $\lambda$. |
Look at [this answer](http://mathoverflow.net/questions/32269) on MathOverflow:
> Yes, there is a way to guess a number asking 14 questions in worst case. To do it you need a linear code with length 14, dimension 10 and distance at least 3. One such code can be built based on Hamming code (see http://en.wikipedia.org/wiki/Hamming_code).
> Here is the strategy.
> Let us denote bits of first player's number as *a<sub>i</sub>*, *i* ∈ [1..10]. We start with asking values of all those bits. That is we ask the following questions: "is it true that *i*-th bit of your number is zero?" Let us denote answers on those questions as *b<sub>i</sub>*, *i* ∈ [1..10].
> Now we ask 4 additional questions:
> Is it true that *a<sub>1</sub>* ⊗ *a<sub>2</sub>* ⊗ *a<sub>4</sub>* ⊗ *a<sub>5</sub>* ⊗ *a<sub>7</sub>* ⊗ *a<sub>9</sub>* is equal to zero? (⊗ is sumation modulo 2).
> Is it true that *a<sub>1</sub>* ⊗ *a<sub>3</sub>* ⊗ *a<sub>4</sub>* ⊗ *a<sub>6</sub>* ⊗ *a<sub>7</sub>* ⊗ *a<sub>10</sub>* is equal to zero?
> Is it true that *a<sub>2</sub>* ⊗ *a<sub>3</sub>* ⊗ *a<sub>4</sub>* ⊗ *a<sub>8</sub>* ⊗ *a<sub>9</sub>* ⊗ *a<sub>10</sub>* is equal to zero?
> Is it true that *a<sub>5</sub>* ⊗ *a<sub>6</sub>* ⊗ *a<sub>7</sub>* ⊗ *a<sub>8</sub>* ⊗ *a<sub>9</sub>* ⊗ *a<sub>10</sub>* is equal to zero?
> Let *q<sub>1</sub>*, *q<sub>2</sub>*, *q<sub>3</sub>* and *q<sub>4</sub>* be answers on those additional questions. Now second player calculates *t<sub>i</sub>* (*i* ∈ [1..4]) --- answers on those questions based on bits *b<sub>j</sub>* which he previously got from first player.
> Now there are 16 ways how bits *q<sub>i</sub>* can differ from *t<sub>i</sub>*. Let *d<sub>i</sub>* = *q<sub>i</sub>* ⊗ *t<sub>i</sub>* (hence *d<sub>i</sub>* = 1 iff *q<sub>i</sub>* ≠ *t<sub>i</sub>* ).
> Let us make table of all possible errors and corresponding values of *d<sub>i</sub>*:
position of error -> (*d<sub>1</sub>*, *d<sub>2</sub>*, *d<sub>3</sub>*, *d<sub>4</sub>*)
> no error -> (0, 0, 0, 0)
> error in *b<sub>1</sub>* -> (1, 1, 0, 0)
> error in *b<sub>2</sub>* -> (1, 0, 1, 0)
> error in *b<sub>3</sub>* -> (0, 1, 1, 0)
> error in *b<sub>4</sub>* -> (1, 1, 1, 0)
> error in *b<sub>5</sub>* -> (1, 0, 0, 1)
> error in *b<sub>6</sub>* -> (0, 1, 0, 1)
> error in *b<sub>7</sub>* -> (1, 1, 0, 1)
> error in *b<sub>8</sub>* -> (0, 0, 1, 1)
> error in *b<sub>9</sub>* -> (1, 0, 1, 1)
> error in *b<sub>10</sub>* -> (0, 1, 1, 1)
> error in *q<sub>1</sub>* -> (1, 0, 0, 0)
> error in *q<sub>2</sub>* -> (0, 1, 0, 0)
> error in *q<sub>3</sub>* -> (0, 0, 1, 0)
> error in *q<sub>4</sub>* -> (0, 0, 0, 1)
> All the values of (*d<sub>1</sub>*, *d<sub>2</sub>*, *d<sub>3</sub>*, *d<sub>4</sub>*) are different. Hence we can find where were an error and hence find all *a<sub>i</sub>*.
*Answered by [falagar](http://mathoverflow.net/users/7079/falagar).* |
Another definition of discriminant is as the resultant of the polynomial with it's first derivative (up to a scalar), and the resultant of two polynomials vanishes if and only if they have a common root. So when does a polynomial have a common root with it's derivative? That's when a zero is also a local max or min, precisely a multiple root. |
In [this][1] blog post, I go over how to compute the equation of the dual curve, in terms of discriminants and resultants and whatnot. Is this helpful? If it is, but not completely, post a comment and I'll either write a new post on the topic or otherwise just respond.
[1]: http://rigtriv.wordpress.com/2008/08/04/dual-curves/ |
Suppose you want to put a probability distribution on the natural numbers for the purpose of doing number theory. What properties might you want such a distribution to have? Well, if you're doing number theory then you want to think of the prime numbers as acting "independently": knowing that a number is divisible by $p$ should give you no information about whether it's divisible by $q$.
That quickly leads you to the following realization: you should choose the exponent of each prime in the prime factorization independently. So how should you choose these? It turns out that the probability distribution on the non-negative integers with maximum entropy and a given mean is a geometric distribution, as explained for example by Keith Conrad <a href="https://docs.google.com/viewer?url=http://www.math.uconn.edu/~kconrad/blurbs/analysis/entropypost.pdf">here</a>. So let's take the probability that the exponent of $p$ is $k$ to be equal to $(1 - r_p) r_p^k$ for some constant $r_p$.
This gives the probability that a positive integer $n = p_1^{e_1} ... p_k^{e_k}$ occurs as
$$C \prod_{i=1}^{k} r_p^{e_i}$$
where $C = \prod_p (1 - r_p)$. So we need to choose $r_p$ such that this product converges. Now, we'd like the probability that $n$ occurs to be monotonically decreasing as a function of $n$. It turns out (and this is a nice exercise) that this is true if and only if $r_p = p^{-s}$ for some $s > 1$ (since $C$ has to converge), which gives the probability that $n$ occurs as
$$\frac{ \frac{1}{n^s} }{ \zeta(s)}$$
where $\zeta(s)$ is the zeta function.
One way of thinking about this argument is that $\zeta(s)$ is the partition function of a statistical-mechanical system called the <a href="http://en.wikipedia.org/wiki/Primon_gas">Riemann gas</a>. As $s$ gets closer to $1$, the temperature of this system increases until it would require infinite energy to make $s$ equal to $1$. But this limit is extremely important to understand: it is the limit in which the probability distribution above gets closer and closer to uniform. So it's not surprising that you can deduce statistical information about the primes by studying the behavior as $s \to 1$ of this distribution. |
When I have watched Deal or No Deal (I try not to make a habit of it) I always do little sums in my head to work out if the banker is offering a good deal. Where odds drop below "evens" it's easy to see it's a bad deal, but what would be the correct mathematical way to decide if you're getting a good deal? |
The rational numbers are both a continuum (between any two rationals you can find another rational) and countable (they can be lined up in correspondence with the positive integers).
Mathematician missed it for hundreds (thousands?) of years, until Cantor.
Of course, the proof of that works both ways, and is equally surprising the other way -- there is a way to order the integers (or any countable set) that makes it into a continuum. |
If I remember rightly there are some integrals of real functions which are easier to compute by using complex analysis.
Is this because of properties of the particular function or because of a lack of a known real analysis technique?
Are there functions which would require hypercomplex analysis to integrate? |
In logic, a semantics is said to be compact iff if every finite subset of a set of sentences has a model, then so to does the entire set.
Most logic texts either don't explain the terminology, or allude to the topological property of compactness. I see an analogy as, given a topological space X and a subset of it S, S is compact iff for every open cover of S, there is a finite subcover of S. But, it doesn't seem strong enough to justify the terminology.
Is there more to the choice of the terminology in logic than this analogy? |
Why is compactness in logic called compactness? |
Is there any mathematical significance to the fact that the law of cosines...
cos(angle between a and b) = (a^2 + b^2 - c^2) / (2ab)
... for an impossible triangle yields a cosine < -1 (when c > a+b), or > 1 (when c < |a-b|)
For example, a = 3, b = 4, c = 8 yields cos(angle ab) = -39/24.
Or a = 3, b = 5, c = 1 yields cos(angle ab) = 33/30.
Something to do with hyperbolic geometry/cosines? |
For any prime $p > 3$, why is $p^2-1$ always divisible by 24? |
This may be a poorly phrased question - please let me know of it - but what is the correct way to think of the cotangent bundle? It seems odd to think of it as the dual of the tangent bundle (I am finding it odd to reconcile the notions of "maps to the ground field" with this object). |
<a href="http://en.wikipedia.org/wiki/Nicolas_Bourbaki"><h1>Nicolas Bourbaki</h1></a>
The story had me from the moment I realized it is true. And as I come to understand more mathematics, I find that I frequently ask myself "What would Bourbaki do?" Also, the <a href="http://www.amazon.com/Apprenticeship-Mathematician-Andre-Weil/dp/3764326506">individuals</a> have <a href="http://en.wikipedia.org/wiki/Jean_Dieudonn%C3%A9">done</a> much to expose the <a href="http://www.fermentmagazine.org/home5.html">human</a> side of mathematics. |
I understand what the Hodge dual is, but I can't quite wrap my head around the dual space of vector space. They seem very similar, almost the same, but perhaps they are unrelated.
For instance, in R^3, the blade a^b gives you a subspace that's like a plane, and the dual is roughly the normal to the plane.
Is there a similarly simple example for the dual space of a vector space, or is there a way to describe the vector space dual in terms of the Hodge dual? |
What is the relationship between the Hodge dual of p-vectors and the dual space of an ordinary vector space? |
When drawing an angle line in rectangle from p, a general (x,y) point on the right or the top line of a rectangle.
How can I find the equation that get the end of the line - the target point coordinates?
In other words, I want to write the target point, p2, with my current information: x, y, w, h. (This variables are described in the picture below).
![alt text][1]
[1]: http://img690.imageshack.us/img690/9819/rectangleline.jpg |
How to get an equation that output the end point of an angle line in rectangle? |
I'm not completely sure what you mean by this: "It seems odd to think of it as the dual of the tangent bundle (I am finding it odd to reconcile the notions of "maps to the ground field" with this object)," but maybe the following will help you see why it is natural to consider the dual space of the tangent bundle.
Given a function f on our manifold, we want to associate something like the gradient of f. Well, in calculus, what characterized the gradient of a function? Its the vector field such that when we take its dot product with a vector v at some point p, we get the directional derivative, at p, of f along v. In a general manifold we don't have a dot product (which is a metric) but we can form a covector field (something which gives an element of the cotangent bundle at any point) such that, when applied to a vector v, we get the directional derivative of f along v. This covector field is denoted df and is called the exterior derivative of f. |
<i>The sum of two Gaussian variables is another Gaussian.</i>
It seems natural, but I could not find a proof using Google.
What's a short way to prove this?
Thanks!
Edit: Provided the two variables are independent. |
When drawing an angle line (45 degrees) in rectangle from `p`, a general (x,y) point on the right or the top line of a rectangle.
How can I find the equation that get the end of the line - the target point coordinates?
In other words, I want to write the target point, p2, with my current information: x, y, w, h. (This variables are described in the picture below).
![alt text][1]
[1]: http://img690.imageshack.us/img690/9819/rectangleline.jpg |
When drawing an angle line (45 degrees) in rectangle from `p`, a general (x,y) point on the right or the top line of a rectangle. How can I find `p2` point coordinates, the intersection point of this line in the rectangle?
In other words, I want to write the target point, `p2`, with my current information: `x, y, w, h`. (This variables are described in the picture below).
![alt text][1]
[1]: http://img690.imageshack.us/img690/9819/rectangleline.jpg |
(Edit: I have edited this answer several times because my understanding of the situation has been improving.)
It is always profitable to understand these kind of constructions by understanding exactly what information they depend on. The Hodge dual depends on a surprising amount of information: you need a vector space $V$ which is equipped with both an inner product and an orientation, which is essentially a choice of which bases of $V$ are "right-handed." So let's see what we can say ignoring all this information first.
Any abstract vector space $V$ of finite dimension $n$ has exterior powers $\Lambda^2 V, \Lambda^3 V, ... \Lambda^n V$, the last of which is one-dimensional. The vector spaces $\Lambda^k V$ and $\Lambda^{n-k} V$ always have the same dimension, so we would like to be able to define some sort of "canonical" map between them. What can we say? Well, they are always dual: the wedge product defines a natural bilinear map $\Lambda^k V \times \Lambda^{n-k} V \to \Lambda^n V$, and since the latter is one-dimensional this means (once you've proven nondegeneracy) that the two vector spaces are in fact dual.
But duality does not give you a map between them. When two vector spaces $V, W$ are dual, meaning there is a nondegenerate bilinear map $V \times W \to F$ (where $F$ is the ground field), all you get is an isomorphism $V \simeq W^{\ast}$. Here you get an isomorphism $\Lambda^k V \simeq \Lambda^{n-k} V^{\ast}$, **once you have specified an isomorphism** $\Lambda^n V \simeq F$. This is equivalent to picking out a distinguished vector in $\Lambda^n V$, which there is no way to do in general.
So the answer is to introduce extra data. To identify $\Lambda^{n-k} V^{\ast}$ with $\Lambda^{n-k} V$, we need an inner product. An inner product gives you **two** distinguished vectors in $\Lambda^n V$, as follows: take any orthonormal basis $b_1, ... b_n$. Then wedging together the $b_i$ in any order gets you one of two elements of $\Lambda^n V$, depending on whether the corresponding permutation is even or odd. But without any extra data, there is no way to identify one of these elements with $1$ and one of these elements with $-1$.
The extra data that does this is an orientation on $V$, which tells you which bases are "right-handed" and which are "left-handed." So an oriented orthonormal basis gives you a distinguished element of $\Lambda^n V$, which gives you a distinguished isomorphism $\Lambda^k V \simeq \Lambda^{n-k} V^{\ast}$, which composed with the isomorphism $\Lambda^{n-k} V^{\ast} \simeq \Lambda^{n-k} V$ is the Hodge dual.
Phew.
This is explained in <a href="http://www-users.math.umd.edu/~toni/hodge.pdf">these notes</a> I just found on Google. |
Okay so [this question][1] reminded me of one my brother asked me a while back about the hit day-time novelty-worn-off-now snoozathon [Deal or no deal][2].
###For the uninitiated:
In playing deal or no deal, the player is presented with one of 22 boxes (randomly selected) each containing different sums of money, he then asks in turn for each of the 21 remaining boxes to be opened, occasionally receiving an offer (from a wholly unconvincing 'banker' figure) for the mystery amount in his box.
If he rejects all of the offers along the way, the player is allowed to work his way through several (for some unfathomable reason, emotionally charged) box openings until there remain only two unopened boxes: one of which is his own, the other not. He is then given a choice to stick or switch (take the contents of his own box or the other), something he then agonises pointlessly over for the next 10 minutes.
###Monty hall
[If you have not seen the monty hall 'paradox' check out this [wikipedia link][3] and prepare to be baffled, then enlightened, then disappointed that the whole thing is so trivial. After which feel free to read on.]
There is a certain similarity, you will agree, between the situation a deal or no deal player finds himself in having rejected all offers and the dilemma of Monty's contestant in the classic problem: several 'bad choices' have been eliminated and he is left with a choice between a better and worse choice with no way of knowing between them.
###So???
>**Question:** The solution to the monty hall problem is that it is, in fact, better to switch- does the same apply here? Does this depend upon the money in the boxes? Should every player opt for 'switch', cutting the 10 minutes of agonising away???
[1]: http://math.stackexchange.com/questions/835/optimal-strategy-for-deal-or-no-deal
[2]: http://en.wikipedia.org/wiki/Deal_or_no_deal
[3]: http://en.wikipedia.org/wiki/Monty_Hall_problem |
Deal or no deal: does one switch (to avoid a goat)?/ Should deal or no deal be 10 minutes shorter? |
When drawing an angle line (45 degrees) in a rectangle from a general point `p(x,y)` that located on the right or the top line of the rectangle. How can I find the intersection point `p2` of this line with the rectangle?
In other words, I want to write the target point, `p2`, with my current information: `x, y, w, h`. (This variables are described in the picture below).
![alt text][1]
[1]: http://img690.imageshack.us/img690/9819/rectangleline.jpg |
Answer edited, in response to the comment and a second wind for explaining mathematics:
Let $F(x_0,x_1,x_2)=0$ be the equation for your curve, and take $(y_0,y_1,y_2)$ to be coordinates on $(\mathbb{P}^2)^*$. Also, assume that $F$ is irreducible and has no linear factors.
Then $y_0 x_0+y_1 x_1+y_2 x_2=0$ is the equation of a general line in $\mathbb{P}^2$ (recall, here $y_0,y_1,y_2$ are fixed, and the $x_i$ are the coordinates on the plane) and we look at the open set of $(\mathbb{P}^2)^*$ where $y_2\neq 0$. On this open set, we can solve the equation of the line for $x_2$, and look at $g(x_0,x_1)=y_2^n F(x_0,x_1,-\frac{1}{y_2}(y_0x_0+y_1x_1))$, a homogeneous polynomial of degree $n$ in $x_0,x_1$ with coefficients homogeneous polynomials in the $y_i$. This polynomial has zeros the intersections of our curve $C$ with the line $L$ we're looking at.
So we want to find points of multiplicity at least two. So how do we find multiple roots of a polynomial? We take the discriminant! Specifically, we do it for an affinization, and we get a homogeneous polynomial of degree $2n^2-n$ in the $y_i$.
All that's left is to factor the polynomial, and kill all the linear factors, just throw them away, the reasons are explained in more computational detail on my [blogpost][1], and there I also do explicit examples, but the method for calculating the equation of the dual curve is as above.
[1]: http://rigtriv.wordpress.com/2008/08/04/dual-curves/ |
If it is just 45 degrees, then the answer is not very difficult. Center a coordinate system at the bottom left hand corner of the rectangle. Hence the coordinates of the (???) point are (q,0) for some q<w.
Note that the because theta is 45 degrees, y = w-q . (Isosceles right triangle). Hence q=w-y, and our point is simply (w-y, 0)... |
So the projective plane RP^2 is not a vector space. Is it still isomorphic to its dual? If not, is there at least an invertible map that takes RP^2 to its dual? |
I am trying to show the following:
![alt text][1]
but I really can't remember what I am supposed to do to get from the LHS to the RHS. I have tried using integration this way
![alt text][2]
And then use integration by parts, but even though what I get resembles it, it can't be correct (because $e^{-\gamma W}$ is not the distribution of W).
I have also tried using Taylor series expansion, but I think I am way off, and I don't think an approximation here is what I need, I think the equality above is exact.
FYI, this is not homework, I am working through a [paper][3] (page 10) and I would really like to know how every step was derived.
Can anyone at least point me to the right direction?
**EDIT**: This expectation on the RHS is very similar to the moment generating function formula (with a positive exponent). If you check [here][4], you will see that the moment generating function for the normal distribution is like the LHS (but with a positive sign). So in a way I have my answer, but I still would like to know how to derive it, if there is a way. I know little if anything at all about moment generating functions, so maybe I shouldn't try and derive it but rather just use the result?
[1]: http://dl.dropbox.com/u/1885087/Expectation.png
[2]: http://dl.dropbox.com/u/1885087/integ.png
[3]: http://www.princeton.edu/~markus/research/papers/liquidity.pdf
[4]: http://en.wikipedia.org/wiki/Moment-generating_function#Examples |
I am trying to show the following:
![alt text][1]
but I really can't remember what I am supposed to do to get from the LHS to the RHS. I have tried using integration this way
![alt text][2]
And then use integration by parts, but even though what I get resembles it, it can't be correct (because $e^{-\gamma W}$ is not the distribution of W).
I have also tried using Taylor series expansion, but I think I am way off, and I don't think an approximation here is what I need, I think the equality above is exact.
FYI, this is not homework, I am working through a [paper][3] (page 10) and I would really like to know how every step was derived.
Can anyone at least point me to the right direction?
**EDIT**: This expectation on the RHS is very similar to the moment generating function formula (with a positive exponent). If you check [here][4], you will see that the moment generating function for the normal distribution is like the LHS (but with a positive sign). So in a way I have my answer, but I still would like to know how to derive it, if there is a way. I know little if anything at all about moment generating functions, so maybe I shouldn't try and derive it but rather just use the result? Does it even make sense to try and derive it?
[1]: http://dl.dropbox.com/u/1885087/Expectation.png
[2]: http://dl.dropbox.com/u/1885087/integ.png
[3]: http://www.princeton.edu/~markus/research/papers/liquidity.pdf
[4]: http://en.wikipedia.org/wiki/Moment-generating_function#Examples |
In First Order Logic with Identity (FOL+I), one can express the proposition that there are exactly 3 items that have the property P.
Why is it not possible to express the proposition that there is a finite number of items that have the property P (in FOL+I)? |
Why is the "finitely many" quantifier not definable in First Order Logic? |
I am trying to show the following:
![alt text][1]
but I really can't remember what I am supposed to do to get from the LHS to the RHS. I have tried using integration this way
![alt text][2]
And then use integration by parts, but even though what I get resembles it, it can't be correct (because $e^{-\gamma W}$ is not the distribution of W).
I have also tried using Taylor series expansion, but I think I am way off, and I don't think an approximation here is what I need, I think the equality above is exact.
FYI, this is not homework, I am working through a [paper][3] (page 10) and I would really like to know how every step was derived.
Can anyone at least point me to the right direction?
**EDIT**: This expectation on the RHS is very similar to the moment generating function formula (with a negative exponent). If you check [here][4], you will see that the moment generating function for the normal distribution is like the LHS (but with a positive sign). So in a way I have my answer, but I still would like to know how to derive it, if there is a way. I know little if anything at all about moment generating functions, so maybe I shouldn't try and derive it but rather just use the result? Does it even make sense to try and derive it?
[1]: http://dl.dropbox.com/u/1885087/Expectation.png
[2]: http://dl.dropbox.com/u/1885087/integ.png
[3]: http://www.princeton.edu/~markus/research/papers/liquidity.pdf
[4]: http://en.wikipedia.org/wiki/Moment-generating_function#Examples |
When drawing an angle line (45 degrees) in a rectangle from a general point `p(x,y)` that located on the right or the top line of the rectangle. How can I find the intersection point `p2` of this line with the rectangle?
In other words, I want to write the target point, `p2`, with my current information: `x, y, w, h`. (This variables are described in the picture below).
The point (0,0) is in the top-right corner.
![alt text][1]
[1]: http://img690.imageshack.us/img690/9819/rectangleline.jpg |
There are a few subtleties that will probably effect the final answer.
1) Are we required to find the solution, or merely establish existence? By analogy, determining if a number has a prime factorization is trivial, but finding its prime factorization is hard.
2) Is the runtime being measured in terms of {a_1,...,a_n,s} or {log(a_i),...,log(a_n),log(s)}? By analogy, SUBSET-SUM is in P in the first case, but NP-complete in the second case. |
There are a few subtleties that will probably effect the final answer.
1. Are we required to find the solution, or merely establish existence? By analogy, determining if a number has a prime factorization is trivial, but finding its prime factorization is hard.
2. Is the runtime being measured in terms of {a_1,...,a_n,s} or {log(a_i),...,log(a_n),log(s)}? By analogy, SUBSET-SUM is in P in the first case, but NP-complete in the second case. |
Let $A$ be a commutative ring. Suppose $P \subset A$ is a minimal prime ideal. Then it is a theorem that $P$ consists of zero-divisors.
This can be proved using localization, when $A$ is noetherian: $A_P$ is local artinian, so every element of $PA_P$ is nilpotent. Hence every element of $P$ is a zero-divisor.
Can this be proved without using localization? |
Which is a better textbook - Dugundji or Munkres? I'm concerned with clarity of exposition and explanation of motivation, etc. |
If you are interested in the factorization algorithms employed in modern computer algebra systems such as Macsyma, Maple, or Mathematica, then see any of the standard introductions to computer algebra , e.g. Geddes et.al. "Algorithms for Computer Algebra"; Knuth, "TAOCP" v.2; von zur Gathen "Modern Computer Algebra"; Zippel "Effective Polynomial Computation". See also
Kaltofen's surveys on polynomial factorization [116,68,56,7] in his publications [list][2], which contains plenty of theory, history and literature references. Note: Kaltofen's home page appears to be temporarily down so instead see his [paper \[1\]] [1] to get started (see comments)
[2]: http://www4.ncsu.edu/~kaltofen/bibliography/kaltofen.html
[1]: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.39.7916&rep=rep1&type=pdf
[1] Kaltofen, E. Factorization of Polynomials, pp. 95-113 in:
Computer Algebra, B. Buchberger, R. Loos, G. Collins, editors, Vienna, Austria, (1982).
http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.39.7916&rep=rep1&type=pdf
|
While I feel quite confident that I've inferred the correct meaning of "In the sequel" from context, I've never heard anyone explicitly tell me, so first off, to remove my niggling doubts: What does this phrase mean?
My main questions:
At what points in the text, and for what kinds of X, is it appropriate to use the phrase "In the sequel, X" in a paper? In a book? Is it ever acceptable to introduce definitions via
"In the sequel, we introduce the concept of a "blah", which is a thing satisfying ..."
at the start of a paper or book **without** a formal
"Definition. A "blah" is a thing, satsifying ..."
in the main text of the paper or book? Finally, out of curiosity, I'm wondering how long this phrase has been around, if it's considered out of date or if it's still a popular phrase, and what some good alternatives are. |
Correct usage of the phrase "In the sequel"? History? Alternatives? |
While I feel quite confident that I've inferred the correct meaning of "In the sequel" from context, I've never heard anyone explicitly tell me, so first off, to remove my niggling doubts: What does this phrase mean?
(Someone recently argued to me that "sequel" was actually supposed to refer to a forthcoming second part of a paper, which I found highly unlikely, but I'd just like to make sure. )
My main questions:
At what points in the text, and for what kinds of X, is it appropriate to use the phrase "In the sequel, X" in a paper? In a book? Is it ever acceptable to introduce definitions via
"In the sequel, we introduce the concept of a "blah", which is a thing satisfying ..."
at the start of a paper or book **without** a formal
"Definition. A "blah" is a thing, satsifying ..."
in the main text of the paper or book? Finally, out of curiosity, I'm wondering how long this phrase has been around, if it's considered out of date or if it's still a popular phrase, and what some good alternatives are. |
Slightly more generally: 24 | M^2 - N^2 if M and N are coprime to 6. The proof is easy. Namely, suppose that K is coprime to 6. Then K odd => K = +-1 or +-3 (mod 8) => K^2 = 1 (mod 8). Also, K coprime to 3 => K^2 = 1 (mod 3). Thus 3, 8 | K^2 - 1 => 24 | K^2 - 1. Hence M^2 - N^2 = 1 - 1 = 0 (mod 24).
This is a very special case computation of Carmichael's Lambda function, e.g. see my post
http://groups.google.com/group/sci.math/msg/6783604a14dcc1cd
http://google.com/groups?selm=y8zk5bvq88u.fsf%40nestle.csail.mit.edu |
This is still WIP. There are a few missing details, still I think it's better than nothing. Feel free to edit in the missing details.
Given a problem of `SUBSET-SUM`. We have a set of `A`={a<sub>1</sub>,a<sub>2</sub>,...,a<sub>n</sub>} numbers, and another number `s`. The question we're seeking answer to is, whether or not there's a subset of `A` whose sum is `s`.
I'm assuming that the 24-game allows you to use rational numbers. Even if it doesn't, I think that it is possible to emulate rational numbers up to denominator of size `p` with integers.
We know that `SUBSET-SUM` is NP-complete even for integers only. I think the `SUBSET-SUM` problem is `NP`-hard even if you allow treating each a<sub>i</sub> as a negative number. That is even if `A` is of the form `A`={a<sub>1</sub>,-a<sub>1</sub>,a<sub>2</sub>,-a<sub>2</sub>,...,a<sub>n</sub>,-a<sub>n</sub>}. This is still a wrinkle I need to iron out in this reduction.
Obviously, if there's a subset of `A` with sum `s`, then there's a solution to the `24`-problem for how to reach using `A` to `s`. The solution is only using the `+` sign.
The problem is, what happens if there's no solution which only uses the `+` sign, but there is a solution which uses other arithmetic operations.
Let us consider the following problem. Let's take a prime `p` which is larger than `n`, the total number of elements in `A`. Given an oracle which solves the `24`-problem, and a `SUBSET-SUM` problem of `A`={a<sub>1</sub>,a<sub>2</sub>,...,a<sub>n</sub>} and `s`. We'll ask the oracle to solve the `24`-problem on
> `A`={a<sub>1</sub>+(1/p),a<sub>2</sub>+(1/p),...,a<sub>n</sub>+(1/p)}
for the following values:
> s<sub>1</sub>=s+1/p,s<sub>2</sub>=s+2/p,...,s<sub>n</sub>=s+n/p.
If the solution includes multiplication, we will have a denominator larger than `p` in the end result, and thus we will not be able to reach any s<sub>i</sub>.
Given an arithmetric expression that contains a<sub>i</sub>a<sub>j</sub>=x+(1/p<sup>2</sup>), It is impossible that the denominator p<sup>2</sup> would "cancel" out, since there are at most `n` elements in the summation, and thus the numerator would never reach `p`, since `p>n`.
What about division? How can we be sure no division will occur. Find another prime `q` which is different than p, and larger than the largest a<sub>i</sub> times `n`. Multiply all answers by `q`. The set `A` will be
> `A`={qa<sub>1</sub>+(1/p),qa<sub>2</sub>+(1/p),...,qa<sub>n</sub>+(1/p)}
We will look for the following values:
> s<sub>1</sub>=qs+1/p,s<sub>2</sub>=qs+2/p,...,s<sub>n</sub>=qs+n/p.
In that case, a<sub>i</sub>/a<sub>j</sub> will be smaller than the minimal element in `A`, and therefor the end result which will contains a<sub>i</sub>/a<sub>j</sub> will never be one of the s<sub>i</sub> we're looking for. |
There is an extremely beautiful Lie-theoretic approach to separation of variables,
e.g. see Willard Miller's [book \[1\]][1]. I quote from his introduction:
> This book is concerned with the
> relationship between symmetries of a
> linear second-order partial
> differential equation of mathematical
> physics, the coordinate systems in
> which the equation admits solutions
> via separation of variables, and the
> properties of the special functions
> that arise in this manner. It is an
> introduction intended for anyone with
> experience in partial differential
> equations, special functions, or Lie
> group theory, such as group
> theorists, applied mathematicians,
> theoretical physicists and chemists,
> and electrical engineers. We will
> exhibit some modem group-theoretic
> twists in the ancient method of
> separation of variables that can be
> used to provide a foundation for much
> of special function theory. In
> particular, we will show explicitly
> that all special functions that arise
> via separation of variables in the
> equations of mathematical physics can
> be studied using group theory. These
> include the functions of Lam6, Ince,
> Mathieu, and others, as well as those
> of hypergeometric type.
>
> This is a very critical time in the
> kistory of group-theoretic methods in
> special Iunction theory. The basic
> relations between Lie groups, special
> functions, and the method of
> separation of variables have recently
> been clarified. One can now construct
> a group-theoretic machine that, when
> applied to a given differential
> equation of mathematical physms,
> describes in a rational manner the
> possible coordinate systems in which
> the equation admits solutions via
> separation of variables and the
> various expansion theorems relating
> the separable (special function)
> solutions in distinct coordinate
> systems. Indeed for themost important
> linear equations, the separated
> solutions are characterized as common
> eigenfunctions of sets of
> second-order commuting elements in the
> umversal enveloping algebra of the
> Lie symmetry algebra corresponding to
> the equation. The problem of
> expanding one set of separable
> solutions in terms of another reduces
> to a problem in the representation
> theory of the Lie symmetry algebra.
[1]: http://www.ima.umn.edu/~miller/separationofvariables.html
[1] Willard Miller. Symmetry and Separation of Variables.
Addison-Wesley, Reading, Massachusetts, 1977 (out of print)
http://www.ima.umn.edu/~miller/separationofvariables.html
http://gigapedia.com/items:links?id=64401
|
Scott Aaronson once basically did this for a bunch of theorems in computer science <a href="http://scottaaronson.com/blog/?p=152">here</a> and <a href="http://scottaaronson.com/blog/?p=153">here</a>. I particularly like this one:
<blockquote>Suppose a baby is given some random examples of grammatical and ungrammatical sentences, and based on that, it wants to infer the general rule for whether or not a given sentence is grammatical. If the baby can do this with reasonable accuracy and in a reasonable amount of time, for any “regular grammar” (the very simplest type of grammar studied by Noam Chomsky), then that baby can also break the RSA cryptosystem.</blockquote> |
For a nice introduction to the history of ring theory see the following paper
I. Kleiner. **From numbers to rings: the early history of ring theory**.
Elemente der Mathematik 53 (1998) 18-35.
http://retro.seals.ch/cntmng?type=pdf&rid=elemat-001:1998:53::22&subp=hires
http://www.springerlink.com/content/e7p7kxk2y3a71j0d/fulltext.pdf
|
Alright, I'm not 100% sure I'm understanding this correctly. You say that p can be located on the right or top line and that p2 can be located on the bottom or left line. Do you mean the rectangle can be rotated? If that's the case, the question should say that p can be on the right or bottom line of the rectangle. Also, are you looking for two separate answers or one that works both when p2 is on the bottom and on the left?
If you do mean that the rectangle can be rotated, and want two different answers, it's pretty simple. First I'll deal with when p2 is on the bottom and p is on the right.
Since p2 is on the bottom line we know the y-coordinate is h, according to the diagram. We also know that p is (0,y). Because of the 45 degree angle, we know that the distance between p's y-coordinate and the lower right corner is the same as the distance between the lower right corner and p2's x-coordinate, which in this case is p2's x-coordinate. Therefore, the coordinates of p2 are (h-y, h).
If p2 is on the left and p is on the bottom, it's very similar. Since p2 is on the left, it's x-coordinate is h. Because p is on the x-axis, it's (x,0). Because of the 45 degree angle, the distance between the lower left corner and p is the same as the distance between the lower left corner and p2, which this time gives us p2's y-coordinate. Therefore the coordinates of p2 are (h,h-x).
Hopefully I understood your intentions correctly. If not, I hope you can use my misunderstandings to further improve your question. |
Let $S$ be a set of size $n$. There is an easy way to count the number of subsets with an even number of elements. Algebraically, it comes from the fact that
$$\sum_{k=0}^{n} {n \choose k} = (1 + 1)^n$$
while
$$\sum_{k=0}^{n} (-1)^k {n \choose k} = (1 - 1)^n.$$
It follows that $\sum_{k=0}^{n/2} {n \choose 2k} = 2^{n-1}$. A direct combinatorial proof is as follows: fix an element $s \in S$. If a given subset has $s$ in it, add it in; otherwise, take it out. This defines a bijection between the number of subsets with an even number of elements and the number of subsets with an odd number of elements.
The analogous formulas for the subsets with a number of elements divisible by 3 or 4 are more complicated, and divide into cases depending on the residue of n mod 6 and mod 8, respectively. The algebraic derivations of these formulas are as follows: observe that
$$\sum_{k=0}^{n} w^k {n \choose k} = (1 + w)^n = (-w^2)^n$$
$$\sum_{k=0}^{n} w^{2k} {n \choose k} = (1 + w^2)^n = (-w)^n$$
and that $1 + w^k + w^{2k} = 0$ if $k$ is not divisible by $3$ and equals $3$ otherwise. (This is a special case of the discrete Fourier transform.) It follows that
$$\sum_{k=0}^{n/3} {n \choose 3k} = \frac{2^n + (-w)^n + (-w)^{2n}}{3}.$$
-w and -w^2 are sixth roots of unity, so this formula splits into six cases (or maybe three). Similar observations about fourth roots of unity show that
$$\sum_{k=0}^{n/4} {n \choose 4k} = \frac{2^n + (1+i)^n + (1-i)^n}{4}$$
where $1+i = \sqrt{2} e^{ \frac{\pi i}{4} }$ is a scalar multiple of an eighth root of unity, so this formula splits into eight cases (or maybe four).
**Question:** Does anyone know a direct combinatorial proof of these identities? |
How do I count the subsets of a set whose number of elements are divisible by 3? 4? |
I was recently surprised to discover that it's actually not known. The number of *closed* knight's tours (cyclic) was computed in the 1990s, using [Binary decision diagrams](http://en.wikipedia.org/wiki/Binary_decision_diagram). There are 26,534,728,821,064 closed directed knight's tours, and the number of undirected ones is half that or 13,267,364,410,532. If you count equivalence classes under rotation and reflection, there are 1/8th of that: 1,658,420,855,433.
(Loebbing and Wegener (1996) wrote a paper ["The Number of Knight's Tours Equals 33,439,123,484,294 — Counting with Binary Decision Diagrams"](http://www.combinatorics.org/Volume_3/Abstracts/v3i1r5.html); the number in the title in the mistake, as they pointed out in [a comment](http://www.combinatorics.org/Volume_3/Comments/v3i1r5.html) to their paper. Brendon McKay independently [computed](http://www.combinatorics.org/Volume_3/Comments/v3i1r5.01.ps) the correct number with another method, and the original authors seem to have later [found](http://books.google.com/books?id=-DZjVz9E4f8C&pg=PA369&dq=532) the same answer.)
Finding the exact number of open tours (not cyclic/reentrant) remains [open, but it is estimated](http://www.ktn.freeuk.com/pa.htm) to be about 10<sup>15</sup> or 2×10<sup>16</sup>. |
I think you were being a little too hard on Isaac. The truth is that the real numbers are a sophisticated mathematical construction and that any explanation of what they "are" which pretends otherwise is a convenient fiction. Mathematicians need these kind of sophisticated constructions because they are what is required for rigorous proofs. Before people explicitly constructed the real numbers and used them to define and prove things about other concepts, it was never totally clear what was true or what was false, and everybody was very confused.
For example, Cantor proved that the number of points in the plane is the same as the number of points on a line. Many people thought that this was impossible before he did it; they had an intuition that you couldn't possibly "fit" the plane into the line. More generally, people were pretty sure you couldn't fit $\mathbb{R}^n$ into $\mathbb{R}^m$ if $n$ was greater than $m$. It wasn't until quite a bit later that mathematicians formalized and proved a rigorous mathematical statement which justified this intuition called <a href="http://en.wikipedia.org/wiki/Invariance_of_domain">invariance of domain</a>, which says you can't do this in a _continuous_ way. One of the many mathematical constructions you need to even state this theorem is the construction of the real numbers. (Another is a formal definition of what "continuous" means, but one thing at a time.)
So, what are the real numbers? They are a formal way to fill "holes" in the rational numbers, which is necessary for all sorts of things. The most basic thing they are necessary for is doing geometry. You probably know that the square root of 2 is irrational. What this means is that it is impossible to think about the diagonal of a square as being the same kind of object as the sides of a square while only using rational numbers. But you can rotate a diagonal, and it looks just like the side of a square, only a bit longer. So you'd like a number system in which you can sensibly talk about any number you can construct geometrically. You'd also like to be able to talk about rotation! You can't do that with just rational numbers, either.
So how do you fill in enough holes to do geometry? <a href="http://en.wikipedia.org/wiki/Dedekind_cut">Dedekind</a> came up with a very clever way to do this. It starts by observing that a rational number $q$ is completely determined by the set of rational numbers greater than it and the set of rational numbers less than it. For example, 1/2 is completely determined by the fact that it's always between 1/2 + 1/n and 1/2 - 1/n. (For the initiated, this is a special case of the Yoneda lemma.) But there are "numbers," such as the square root of 2, which aren't rational, and yet have the property that we can always tell what rational numbers are greater than it and what rational numbers are less than it. For the square root of 2, these are precisely the fractions p/q such that 2q^2 < p^2 and such that 2q^2 > p^2, respectively. Dedekind's brilliant idea was the following:
> _Define_ a real number to be a partition of the rational numbers!
In Dedekind's construction, the square root of 2 quite literally _is_ the set of rational numbers that are greater than it, and the set of rational numbers that are less than it. You can define all the usual arithmetic operations on these "numbers," called Dedekind cuts, and prove all the wonderful theorems you'll find in a standard book on real analysis. In particular, the property that guarantees that all the holes are filled is called <a href="http://en.wikipedia.org/wiki/Real_number#Completeness">completeness</a>. |
I don't think the people working in higher-order theorem proving really care about Henkin semantics or models in general, they mostly work with their proof calculi. As long as there are no contradictions or other counterintuitive theorems they are happy. The most important and most difficult theorem they prove is usually that their proof terms terminate, which IIRC can be viewed as a form of soundness.
Henkin semantics is most interesting for people trying to extend their first-order methods to higher-order logic, because it behaves essentially like models of first-order logic. Henkin semantics is somewhat weaker than what you would get with standard set-theoretic semantics, which by Gödels incompleteness theorem can't have a complete proof calculus. I think type theories should lie somewhere in between Henkin and standard semantics.
> Where does typed lambda calculus come into proof verification?
To prove some implication `P(x) --> Q(x)` with some free variables `x` you need to map any proof of `P(x)` to a proof of `Q(x)`. Syntactically a map (a function) can be represented as a lambda term.
> Are there any other approaches than higher order logic to proof verification?
You can also verify proofs in first-order or any other logic, but then you would loose much of the power of the logic. First-order logic is mostly interesting because it is possible to automatically find proofs, if they are not too complicated. The same applies even more to propositional logic.
> What are the limitations/shortcomings of existing proof verification systems (see below)?
The more powerful the logic becomes the harder it becomes to construct proofs.
Since the systems are freely available I suggest you play with them, e.g. Isabelle and Coq for a start.
|
I think you were being a little too hard on Isaac. The truth is that the real numbers are a sophisticated mathematical construction and that any explanation of what they "are" which pretends otherwise is a convenient fiction. Mathematicians need these kind of sophisticated constructions because they are what is required for rigorous proofs. Before people explicitly constructed the real numbers and used them to define and prove things about other concepts, it was never totally clear what was true or what was false, and everybody was very confused.
For example, Cantor proved that the number of points in the plane is the same as the number of points on a line. Many people thought that this was impossible before he did it; they had an intuition that you couldn't possibly "fit" the plane into the line. More generally, people were pretty sure you couldn't fit $\mathbb{R}^n$ into $\mathbb{R}^m$ if $n$ was greater than $m$. It wasn't until quite a bit later that mathematicians formalized and proved a rigorous mathematical statement which justified this intuition called <a href="http://en.wikipedia.org/wiki/Invariance_of_domain">invariance of domain</a>, which says you can't do this in a _continuous_ way. One of the many mathematical constructions you need to even state this theorem is the construction of the real numbers. (Another is a formal definition of what "continuous" means, but one thing at a time.)
So, what are the real numbers? They are a formal way to fill "holes" in the rational numbers, which is necessary for all sorts of things. The most basic thing they are necessary for is doing geometry. You probably know that the square root of 2 is irrational. What this means is that it is impossible to think about the diagonal of a square as being the same kind of object as the sides of a square while only using rational numbers. But you can rotate a diagonal, and it looks just like the side of a square, only a bit longer. So you'd like a number system in which you can sensibly talk about any number you can construct geometrically. You'd also like to be able to talk about rotation! You can't do that with just rational numbers, either.
So how do you fill in enough holes to do geometry? <a href="http://en.wikipedia.org/wiki/Dedekind_cut">Dedekind</a> came up with a very clever way to do this. It starts by observing that a rational number $q$ is completely determined by the set of rational numbers greater than it and the set of rational numbers less than it. For example, 1/2 is completely determined by the fact that it's always between 1/2 + 1/n and 1/2 - 1/n. (For the initiated, this is a special case of the Yoneda lemma.) But there are "numbers," such as the square root of 2, which aren't rational, and yet have the property that we can always tell what rational numbers are greater than it and what rational numbers are less than it. For the square root of 2, these are precisely the fractions p/q such that 2q^2 < p^2 and such that 2q^2 > p^2, respectively. Dedekind's brilliant idea was the following:
> _Define_ a real number to be a partition of the rational numbers!
In Dedekind's construction, the square root of 2 quite literally _is_ the set of rational numbers that are greater than it, and the set of rational numbers that are less than it. You can define all the usual arithmetic operations on these "numbers," called Dedekind cuts, and prove all the wonderful theorems you'll find in a standard book on real analysis. In particular, the property that guarantees that all the holes are filled is called <a href="http://en.wikipedia.org/wiki/Real_number#Completeness">completeness</a>.
---
Figured I might as well add something about the complex numbers. The story here is beautiful, and if you're really interested you should check out Tristan Needham's <a href="http://usf.usfca.edu/vca//">Visual Complex Analysis</a>. Some people say that the point of the complex numbers is to let you solve polynomials, but this is really selling them short. The complex numbers are an inherently _geometric_ construction, and should be understood as such. Their geometry and topology just happens to be responsible for the fact that you can solve polynomials with them, but it's also responsible for much more.
Here is a quick sketch. Now that you've got the real numbers on your hands, you can rigorously talk about plane geometry. In plane geometry, an important notion is that of _similarity_. Informally, two figures are similar if they have the same shape. More formally, two figures are similar if you can rotate, translate, and scale one figure so that it matches up with the other. So similarity is all about a certain collection of transformations of the points in the plane. It was <a href="http://en.wikipedia.org/wiki/Erlangen_program">Klein</a> who first realized that the important features of different flavors of "geometry" are captured in what kind of transformations are allowed. So to do geometry the modern way we should focus our attention on these transformations, which form a <a href="http://en.wikipedia.org/wiki/Group_theory">group</a>.
To make this easier, let's ignore the translations for now. We'll pick an origin for our plane, and we'll only allow rotations and scalings about this origin. Rotations and scalings have the property that they are both linear transformations; this means that if you know what the transformation does to two points $u, v$, you also know what it does to the vector sum $u + v$. In particular, a linear transformation is determined by what it does to the point $(1, 0)$ and to the point $(0, 1)$.
However, rotations and scalings satisfy an extra property: they are, in fact, determined by what they do to the point $(1, 0)$. This is because $(0, 1)$ can be obtained from $(1, 0)$ by a rotation by 90 degrees, and rotations and scalings _commute_ with each other: if you rotate x degrees then y degrees, that's the same as rotating y degrees then x degrees, which is the same as rotating x+y degrees. Similarly, if you rotate x degrees then scale by 2, that's the same as scaling by 2, then rotating x degrees. So if you know what a rotation-and-scaling does to $(1, 0)$, you just rotate that vector by 90 degrees, and you know what it did to $(0, 1)$.
So to every rotation-and-scaling, we can assign two real numbers: the coordinates of the image of the point $(1, 0)$. In general, a rotation by $\theta$ angles followed by a scaling by $r$ sends $(1, 0)$ to $(r \cos \theta, r \sin \theta)$. A different transformation, say a rotation by $\phi$ angles followed by a scaling by $s$, sends $(1, 0)$ to $(s \cos \phi, s \sin \phi)$. And their _composition_ sends $(1, 0)$ to $(rs \cos (\theta + \phi), rs \sin (\theta + \phi))$. In other words, composition of rotations-and-scalings defines a multiplication law on pairs of real numbers. What is this law, exactly? Well, by the angle addition formulas, it's
$$(a, b) * (c, d) = (ac - bd, ad + bc).$$
And this is precisely the rule for multiplication in the complex numbers, where $(a, b)$ corresponds to $a + bi$. You get the rule for addition by observing that not only can you compose two rotations-and-scalings, you can also add their results.
Together, the real numbers and the complex numbers provide a foundation for much of modern mathematics and physics. For example, the complex numbers turn out (for reasons which are still not well understood) to be fundamental in the description of quantum mechanics. |
How do I count the subsets of a set whose number of elements is divisible by 3? 4? |
This is somewhere between an answer and commentary. As others have said, the question is equivalent to showing: for any prime $p > 3$, $p^2 \equiv 1 \pmod 3$ and $p^2 \equiv 1 \pmod 8$. Both of these statements are straightforward to show by just looking at the $\varphi(3) = 2$ reduced residue classes modulo $3$ and the $\varphi(8) = 4$ reduced residue classes modulo $8$. But what is their significance?
For a positive integer $n$, let $U(n) = (\mathbb{Z}/n\mathbb{Z})^{\times}$ be the multiplicative group of units ("reduced residues") modulo $n$. Like any abelian group $G$, we have a squaring map
$[2]: G \rightarrow G$, $g \mapsto g^2$,
the image of which is the set of squares in $G$. So, the question is equivalent to: for $n = 3$ and also $n = 8$, the subgroup of squares in $U(n)$ is the trivial group.
The group $U(3) = \{ \pm 1\}$ has order $2$; since $(-1)^2 = 1$, the fact that the subgroup of squares is equal to $1$ is pretty clear. But more generally, for any odd prime $p$, the squaring map $[2]$ on $U(p)$ is two-to-one onto its image -- an element of a field has no more than two square roots -- so that precisely half of the elements of $U(p)$ are squares. It turns out that when $p = 3$, half of $p-1$ is $1$, but of course this is somewhat unusual: it doesn't happen for any other odd prime $p$.
The group $U(8) = \{1,3,5,7\}$ has order $4$. By analogy to the case of $U(p)$, one might expect the squaring map to be two-to-one onto its image so that exactly half of the elements are squares. But that is not what is happening here: indeed
$1^2 \equiv 3^2 \equiv 5^2 \equiv 7^2 \equiv 1 \pmod 8$,
so the subgroup of squares is again trivial. What's different? Since $\mathbb{Z}/8\mathbb{Z}$ is not a field, it is legal for a given element to have more than two square roots, but a more insightful answer comes from the structure of the groups $U(n)$. For any odd prime $p$, the group $U(p)$ is *cyclic* of order $p-1$ ("existence of primitive roots"). It is easy to see that in any cyclic group of even order, exactly half of the elements are squares. So $U(8)$ must not be cyclic, so it must be the other abelian group of order $4$, i.e., isomorphic to the Klein $4$-group $C_2 \times C_2$.
More generally, if $p$ is an odd prime number and $a$ is a positive integer, then
$U(p^a)$ is cyclic of order $p^{a-1}(p-1)$ hence isomorphic to $C_{p^{a-1}} \times C_{p-1}$, whereas for any $a geq 2$, the group $U(2^a)$ is isomorphic to $C_{2^{a-2}} \times C_2$. This is one of the first signs in number theory "there is something odd about the prime $2$".
|
I am looking for functions and/or constants that when being integrated from minus infinity to infinity produce 1. I think the Dirac delta function is one example but perhaps there are some more? References on useful material is also greatly appreciated. |
Let $pi(x)$ be the number of primes not greater than $x$.
<a href="http://en.wikipedia.org/w/index.php?title=Prime-counting_function">Wikipedia article</a> says that $pi(10^23) = 1,925,320,391,606,803,968,923$.
The question is how to calculate $pi(x)$ for large $x$ in a reasonable time? What algorithms do exist for that? |
This could be a trivial question, but what is exactly the difference of between these two expressions? Am I correct to state the both interchangeably whenever I need to express the approximation of pi?
π ≈ 3.14..
π ≒ 3.14..
|
Approximation symbol: Is π ≈ 3.14.. equal to π ≒ 3.14.. ? |
This could be a trivial question, but what is exactly the difference of between these two expressions? Am I correct to state the both interchangeably whenever I need to express the approximation of pi? ie, is ≈ equal to ≒?
π ≈ 3.14..
π ≒ 3.14..
|
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