instruction stringlengths 12 30k |
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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 bou... |
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... |
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 $e^{-\ga... |
If W is randomly chosen with the PDF P(x), then the expectation value should be
>  |
If W is randomly chosen with the PDF P(x), then the expectation value should be
> ![$E[e^{-\gamma W}]=\int_{-\infty}^\infty P(x) e^{-\gamma x} dx$](http://mathcache.appspot.com/?tex=%5cpng%5c%5bE%5Be%5E%7B-%5Cgamma%20W%7D%5D%3D%5Cint_%7B-%5Cinfty%7D%5E%5Cinfty%20P%28x%29%20e%5E%7B-%5Cgamma%20x%7D%20dx%5c%5d)
And... |
If W is randomly chosen with the PDF P(x), then the expectation value should be
> ![$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... |
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) ... |
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 $m$... |
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 hi... |
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 lengt... |
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 growi... |
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 equa... |
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 lengt... |
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... |
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 $1$?
|
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 ... |
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 usual... |
There is a theorem of Riemann to that effect. How to prove it?
Note: This was asked by Kenny in the beta for "calculus".
|
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) mention... |
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?
|
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?
|
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 equivale... |
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.or... |
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, p... |
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/... |
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... |
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 equa... |
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... |
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, ... |
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... |
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 the... |
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... |
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.
G... |
<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. ... |
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... |
(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 s... |
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... |
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... |
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 ... |
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... |
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^{-\ga... |
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^{-\ga... |
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^{-\ga... |
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... |
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,..... |
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,..... |
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.
Ca... |
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"; Z... |
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... |
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 ... |
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 s... |
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... |
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, ... |
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 ungrammati... |
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/e7p7... |
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. Als... |
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... |
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 o... |
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 requi... |
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 usual... |
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 requi... |
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 ... |
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 f... |
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?
Ο β 3.14..
Ο β 3.14..
|
Approximation symbol: Is Ο β 3.14.. equal to Ο β 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 β equal to β?
Ο β 3.14..
Ο β 3.14..
|
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