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I'm not sure if I get the question but I'll wager a guess.
I'll do 1D.
1D walks are building binary strings, 010101, etc.
Say take six steps. Then 111111 is just as likely as 101010.
However, how many of the possible sequences have six ones? 1. How many of the possibly sequences have three ones and three ... |
There is a little triviality that has been referred to as the <a href="http://www.johndcook.com/blog/2010/01/13/soft-maximum">"soft maximum"</a> over on <a href="http://www.johndcook.com/blog/">John Cook's Blog</a> that I find to be fun, at the very least.
The idea is this: given a list of values, say $x_1,x_2,\l... |
I'll do 1D.
1D walks are building binary strings, 010101, etc.
Say take six steps. Then 111111 is just as likely as 101010.
However, how many of the possible sequences have six ones? 1. How many of the possibly sequences have three ones and three zeros? Much more.
That number is called multiplicity, and ... |
There is a little triviality that has been referred to as the <a href="http://www.johndcook.com/blog/2010/01/13/soft-maximum">"soft maximum"</a> over on <a href="http://www.johndcook.com/blog/">John Cook's Blog</a> that I find to be fun, at the very least.
The idea is this: given a list of values, say $x_1,x_2,\l... |
How does one prove that a simple (steps of length 1 in directions parallel to the axes) symmetric (each possible direction is equally likely) random walk in 1 or 2 dimensions returns to the origin with probability 1?
***edit***: note that while returning to the origin is guaranteed (p = 1) in 1 and 2 dimensions, it ... |
**EDIT**: Given the update, this proof must be wrong. I wonder where the error is though?
I can prove the 1 dimensional case a bit more formally than [Jonathan][1]. First we only look at the absolute values. Let us try to calculate the probability of this never exceeding x. The probability of 2x+1 consecutive moves ... |
Polygons are, in this question, defined as non-unique if they similar to another (by rotation, reflection, translation, or scaling).
Would this answer be any different if similar but non-identical polygons were allowed? And if only if rotated/translated by rational coefficients?
Would this answer be any differen... |
Is the set of all unique (convex) polygons countable? If so, by what bijection to the natural numbers? |
In laymen's terms, as much as possible: What is the Riemann-Zeta function, and why does it come up so often with relation to prime numbers? |
What is the Riemann-Zeta function? |
Which books would you recommend about Recreational Mathematics? |
What is a intuitive explanation of a Markov Chain, and how they work? Please provide at least one practical example. |
What is a Markov Chain? |
Ignoring Limits, I would like to know if this is a valid explanation for why 0/0 is undefined.
>x = 0/0
x * 0 = 0
>Hence There are an infinite number of values for x as anything multiplied by 0 is 0.
However, it seems to have got comments, with two general themes.
Once is that you lose the value... |
(This was asked due to the comments and downvotes on this [Stackoverflow](http://stackoverflow.com/questions/3236489/why-is-0-divided-by-0-an-error/3236541#3236541) answer)
Ignoring Limits, I would like to know if this is a valid explanation for why 0/0 is undefined.
>x = 0/0
x * 0 = 0
>Hence There... |
Here there is another attempt at an explanation.
We know that the sum of the inverse of the positive numbers, 1 + 1/2 + 1/3 + ..., diverges. Euler shown that the sum of the inverse of the squares, 1/(1<sup>2</sup>) + 1/(2<sup>2</sup>) + 1/(3<sup>2</sup>) + ..., has a finite sum, namely π<sup>2</sup>/6. Mathemati... |
Here there is another attempt at an explanation.
We know that the sum of the inverse of the positive numbers, 1 + 1/2 + 1/3 + ..., diverges. Euler shown that the sum of the inverse of the squares, 1/(1<sup>2</sup>) + 1/(2<sup>2</sup>) + 1/(3<sup>2</sup>) + ..., has a finite sum, namely π<sup>2</sup>/6. Mathemati... |
(This was asked due to the comments and downvotes on this [Stackoverflow](http://stackoverflow.com/questions/3236489/why-is-0-divided-by-0-an-error/3236541#3236541) answer. I am not that good at maths, so was wondering if I had made any basic mistakes)
Ignoring limits, I would like to know if this is a valid explana... |
(This was asked due to the comments and downvotes on this [Stackoverflow](http://stackoverflow.com/questions/3236489/why-is-0-divided-by-0-an-error/3236541#3236541) answer. I am not that good at maths, so was wondering if I had made any basic mistakes)
Ignoring limits, I would like to know if this is a valid explana... |
Quasi-coherent sheaves on affine schemes (say $Spec(A)$) are obtained by taking an $A$-module $M$ and the associated sheaf (by localizing $M$). This gives an equivalence of categories between $A$-modules and q-c sheaves on $Spec(A)$.
Let $R$ be a graded ring, $R = R_0 + R_1 + \dots$ (direct sum). Then we can, give... |
Here there is another attempt at an explanation.
We know that the sum of the inverse of the positive numbers, 1 + 1/2 + 1/3 + ..., diverges. Euler shown that the sum of the inverse of the squares, 1/(1<sup>2</sup>) + 1/(2<sup>2</sup>) + 1/(3<sup>2</sup>) + ..., has a finite sum, namely π<sup>2</sup>/6. Mathemati... |
Okay, so this question was bound to come up sooner or later- the hope was to ask it well before someone asked it badly...
###We all love a good puzzle
To a certain extent, any piece of mathematics is a puzzle in some sense: whether we are classifying the homological intersection forms of four manifolds or calculati... |
Ultrafinitism is basically resource-bounded constructivism: proofs have constructive content, and what you get out of these constructions isn't much more than you put in.
Looking at the universal and existential quantifier should help clarify things. Constructively, a universally quantified sentence means that if ... |
unknown source:
Could the plane be colored with two different colors (say, red and blue) so that there is no equilateral triangle whose vertices are all of the same color? |
You are standing on a cliff at a height *h* above the sea. You are capable of throwing a stone with velocity *v* at any angle *a* between horizontal and vertical. What is the value of *a* when the horizontal distance travelled *d* is at a maximum?
On level ground, when *h* is zero, it's easy to show that *a* needs t... |
What is the optimum angle of projection when throwing a stone off a cliff? |
What does it mean (in a mathematically rigorous way) to claim something is "generic?" How does this coincide with the Zariski topology? |
Given a curve how do you intuitively construct the picture of its projective dual? I know points --> lines, lines--> points but for something like the swallowtail this is not really obvious. |
According to Wikipedia, Godel's incompleteness theorem states:
> No consistent system of axioms whose
> theorems can be listed by an
> "effective procedure" (essentially, a
> computer program) is capable of
> proving all facts about the natural
> numbers.
This obviously includes our current system. So has an... |
Has anyone ever proposed additional axioms? |
You are standing on a cliff at a height *h* above the sea. You are capable of throwing a stone with velocity *v* at any angle *a* between horizontal and vertical. What is the value of *a* when the horizontal distance travelled *d* is at a maximum?
On level ground, when *h* is zero, it's easy to show that *a* needs t... |
Most of us know that, being deterministic, computers cannot generate true [random numbers](http://en.wikipedia.org/wiki/Pseudorandom_number_generator).
However, let's say you have a box which generates truly random binary numbers, but is biased: it's more likely to generate either a `1` or a `0`, but you don't know... |
Assuming you have unlimited time and cash, is there a strategy that's guaranteed to win at roulette? |
**Fork in the road 1**
You're on a path on an island, come to a fork in the road. Both paths lead to villages of natives; the entire village either always tells the truth or always lies *(both villages could be truth-telling or lying villages, or one of each)*. There are two natives at the fork - they could both b... |
**Fork in the road 2**
You're once again at a fork in the road, and again, one path leads to safety, the other to doom.
There are three natives at the fork. One is from a village of truth-tellers, one from a village of liars, one from a village of random answerers. Of course you don't know which is which.
Mo... |
Adding additional axioms would make more truths provable. But it wouldn't make all truths provable (unless the axiom was inconsistent with the already given ones, in which case all falsehoods would also be provable too).
So adding additional axioms isn't going to help make all the truths provable. I guess you could ... |
How do we know if a particular function can be represented as a power series? And once we have come up with a power series representation, how does one figure out its radius of convergence ? |
What functions can be represented as power series? |
How do we know if a particular function can be represented as a power series? And once we have come up with a power series representation, how does one figure out its radius of convergence ?
Edit: In particular, I was thinking of Taylor series representations here. |
How do we know if a particular function can be represented as a power series? And once we have come up with a power series representation, how does one figure out its radius of convergence ? |
When I look at the Taylor series for e^x and the volume formula for oriented simplexes, it makes e^x look like it is, at least almost, the sum of simplexes volumes from n to infinity. Does anyone know of a stronger relationship beyond, "they sort of look similar"?
Here are some links:
Volume formula
http://e... |
Is there a relationship between e and the sum of n-simplexes volumes? |
It is a theorem in elementary number theory that if $p$ is a prime and congruent to 1 mod 4, then it is the sum of two squares. Apparently there is a trick involving arithmetic in the gaussian integers that lets you prove this quickly. Can anyone explain it? |
How do you prove that a prime is the sum of two squares iff it is congruent to 1 mod 4? |
**Fork in the road 1**
Y'r n pth n n slnd, cme t frk n th rd. Bth pths ld t vllgs f ntvs; th ntr vllg thr lwys tlls th trth r lwys ls *(bth villgs cld b trth-tllng r lyng vllgs, r n f ch)*. Thr r tw ntvs t th frk - thy cld bth b frm th sm vllg, r frm dffrnt vllgs *(s bth cld b trth-tllrs, both lrs, r ne f ch)*.
... |
I'm learning Linear Algebra using MIT's Open Courseware [Course 18.06][1]
Quite often, the professor says "... assuming that the matrix is invertible ...".
Somewhere in lecture he says that using a determinant on an $n \times n$ matrix is on the order of $O(n!)$ operations, where an operation is a multiplication ... |
What is the most efficient way to determine if a matrix is invertible? |
**Frk n th rd 1**
Y'r n pth n n slnd, cme t frk n th rd. Bth pths ld t vllgs f ntvs; th ntr vllg thr lwys tlls th trth r lwys ls *(bth villgs cld b trth-tllng r lyng vllgs, r n f ch)*. Thr r tw ntvs t th frk - thy cld bth b frm th sm vllg, r frm dffrnt vllgs *(s bth cld b trth-tllrs, both lrs, r ne f ch)*.
n p... |
I'm learning Linear Algebra using MIT's Open Courseware [Course 18.06][1]
Quite often, the professor says "... assuming that the matrix is invertible ...".
Somewhere in the lecture he says that using a determinant on an $n \times n$ matrix is on the order of $O(n!)$ operations, where an operation is a multiplicat... |
Given n distinct objects, there are n! permutations of the objects and n!/n "circular permutations" of the objects (orientation of the circle matters, but there is no starting point, so 1234 and 2341 are the same, but 4321 is different).
Given n objects of k types (where the objects within each type are indistinguis... |
From a standard 52-card deck, how many ways are there to pick a hand of $k$ cards that includes one card from all four suits?
I know that for any specific $k$, it's possible to break it up into cases based on the partitions of $k$ into four parts. For example, if I want to choose a hand of six cards, I can break it ... |
Congruence transformations (isometries) and similarity transformations (isometries + dilations) should be constructable. What about other affine transformations? Other conformal mappings? |
What transformations of the plane are geometrically constructable (compass & straight edge)? |
For a long time I've eschewed bulky and inelegant calculators for the use of my trusty trig/log-log slide rule. For those unfamiliar, here is a simple [slide rule simulator using Javascript](http://www.antiquark.com/sliderule/sim/virtual-slide-rule.html).
To demonstrate, find the LL_3 scale, which is on the back of... |
Why does the log-log scale on my Slide Rule work? |
Grigory has already answered your particular question. However, I wanted to point out that your question "How do you prove that a group specified by a presentation is infinite?" has no good answer *in general*. Indeed, in general the question of whether a group presentation defines *the trivial group* is undecidable. |
Congruence transformations (isometries) and similarity transformations (isometries + dilations) should be constructable. What about other affine transformations? Other conformal mappings?
***edit***: by constructable, I mean given the defining information for the transformation in a geometric way (e.g. a dilation ... |
Assuming I can play forever, what are my chances of coming out ahead in a coin flipping series?
Let's say I want "heads"...then if I flip once, and get heads, then I win, because I've reached a point where I have more heads than tails (1-0). If it was tails, I can flip again. If I'm lucky, and I get two heads in a... |
Given enough time, what are the chances I can come out ahead in a coin toss contest? |
If you're after Olmpiad-level books, get [The IMO Compendium][1] which is a collection of problems from the International Math Olympiad, 1959-2004. You can find similar books with national Olympiad problems by going to Amazon and searching for "mathematical Olympiad".
Two books that offer collections of techniques u... |
Are there any results from algebraic geometry that have led to an interesting "real world" application? |
What are some applications outside of mathematics for algebraic geometry? |
> A 10-year loan of $500 is repaid with
> payments at the end of each year. The lender charges interest at an
> annual effective rate of 10%.
> Each of the first ten payments is
> 150% of the amount of interest due.
> Each of the last ten payments is X.
> Calculate X.
I came across this practice question ... |
How to determine annual payments on a partially repaid loan? |
> A 10-year loan of $500 is repaid with
> payments at the end of each year. The lender charges interest at an
> annual effective rate of 10%.
> Each of the first ten payments is
> 150% of the amount of interest due.
> Each of the last ten payments is X.
> Calculate X.
I came across this practice question wh... |
Broadly speaking, algebraic geometry is used a lot in some areas of robotics and mechanical engineering. Real algebraic geometry, for example, is important to the development of CAD systems (think NURBS, computing intersections of primitives, etc.) And AG comes up in robotics when it is important to figure out, say, wh... |
Suppose I have a loan of M dollars. At the end of each year, I am charged interest at rate R and make a repayment of P. The loan is repaid after n years.
1. How long (n) does it take to repay the loan if I am given the other variables?
2. How much are the repayments of P if I am given the other variable?
3. Sup... |
*The Problem:*
Two trains travel on the same track towards each other, each going at a speed of 50 kph. They start out 50km apart. A fly starts at the front of one train and flies at 75 kph to the front of the other; when it gets there, it turns around and flies back towards the first. It continues flying back and f... |
*The Problem:*
Two trains travel on the same track towards each other, each going at a speed of 50 kph. They start out 50km apart. A fly starts at the front of one train and flies at 75 kph to the front of the other; when it gets there, it turns around and flies back towards the first. It continues flying back and f... |
Representation theory is a subject I want to like (it can be fun finding the representations of a group), but it's hard for me to see it as a subject that arises naturally or why it is important. I can think of two mathematical reasons for studying it:
1) The character table of a group is packs a lot of information... |
Why is the volume of a cone one third of the volume of a cylinder? |
In my AP chem class, I often have to balance chemical equations like the following:
Al + O<sub>2</sub> ---> Al<sub>2</sub>O<sub>3</sub>
The goal is to make both side of the arrow have the same amount of atoms by adding compounds in the equation to each side.
An solution
2Al + 3O<sub>2</sub> ---> 2Al<sub>2</... |
Balance chemical equations without trial and error? |
When differentiated with respect to $r$, the derivative of $\pi r^2$ is $2 \pi r$, which is the circumference of a circle.
Similarly, when the formula for a sphere's volume $\frac{4}{3} \pi r^3$ is differentiated with respect to $r$, we get $4 \pi r^2$.
Is this just a coincidence, or is there some deep explanatio... |
Why is the derivative of a circle's area its perimeter (and similarly for spheres)? |
> A 10-year loan of $500 is repaid with
> payments at the end of each year. The lender charges interest at an
> annual effective rate of 10%.
> Each of the first ten payments is
> 150% of the amount of interest due.
> Each of the last ten payments is X.
> Calculate X.
I came across this practice question wh... |
1. The representation theory of finite groups can be used to prove results about finite groups themselves that are otherwise much harder to prove by "elementary" means. For instance, the proof of [Burnside's theorem][1] (that a group of order $p^a q^b$ is solvable). A lot of the classification proof of finite simple g... |
> A 10-year loan of $500 is repaid with
> payments at the end of each year. The lender charges interest at an
> annual effective rate of 10%.
> Each of the first ten payments is
> 150% of the amount of interest due.
> Each of the last ten payments is X.
> Calculate X.
I came across this practice question wh... |
1. The representation theory of finite groups can be used to prove results about finite groups themselves that are otherwise much harder to prove by "elementary" means. For instance, the proof of [Burnside's theorem][1] (that a group of order $p^a q^b$ is solvable). A lot of the classification proof of finite simple g... |
> A 10-year loan of $500 is repaid with
> payments at the end of each year. The lender charges interest at an
> annual effective rate of 10%.
> Each of the first ten payments is
> 150% of the amount of interest due.
> Each of the last ten payments is X.
> Calculate X.
I came across this practice question wh... |
The 24 game is as follows. Four numbers are drawn; the player's objective is to make 24 from the four numbers using the four basic arithmetic operations (in any order) and parentheses however one pleases.
Consider the following generalization. Given $n+1$ numbers, determine whether the last one can be obtained from... |
Is the 24 game NP-complete? |
I am working on computing phase diagrams for alloys. These are
blueprints for a material that show what phase, or combination of
phases, a material will exist in for a range of concentrations and
temperatures (see <a
href="http://web.cos.gmu.edu/~tstephe3/talks/SIAMMaterialsScience2010.pdf">this
pdf presentation<... |
I am working on computing phase diagrams for alloys. These are
blueprints for a material that show what phase, or combination of
phases, a material will exist in for a range of concentrations and
temperatures (see <a
href="http://web.cos.gmu.edu/~tstephe3/talks/SIAMMaterialsScience2010.pdf">this
pdf presentation<... |
I am working on computing phase diagrams for alloys. These are
blueprints for a material that show what phase, or combination of
phases, a material will exist in for a range of concentrations and
temperatures (see <a
href="http://web.cos.gmu.edu/~tstephe3/talks/SIAMMaterialsScience2010.pdf">this
pdf presentation<... |
I am working on computing phase diagrams for alloys. These are
blueprints for a material that show what phase, or combination of
phases, a material will exist in for a range of concentrations and
temperatures (see <a
href="http://web.cos.gmu.edu/~tstephe3/talks/SIAMMaterialsScience2010.pdf">this
pdf presentation<... |
The knight's tour is a sequence of 64 squares on a chess board, where each square is visted once, and each subsequent square can be reached from the previous by a knight's move. Tours can be cyclic, if the last square is a knight's move away from the first, and acyclic otherwise.
There are several symmetries among ... |
How many knight's tours are there? |
If we replace the axiom that 'there exists an infinite set' with 'all sets are finite', how would mathematics be like? My guess is that, all the theory that has practical importance would still show up, but everything would be very very unreadable for humans. Is that true?
We would have the natural numbers, athoug... |
I just got out from my Math and Logic class with my friend. During the lecture, a well-known math/logic puzzle was presented:
>The King has $1000$ wines, $1$ of which is poisoned. He needs to identify the poisoned wine immediately, so he hires the protagonist, a Mathematician. The king offers you his expendable s... |
Can someone help me with a math/logic problem my friend and I are struggling with? |
When I studied complex analysis, I could never understand how once-differentiable complex functions could be possibly be infinitely differentiable. After all, this doesn't hold for functions from R2 to R2. Can anyone explain what is different about complex numbers? |
Why are differentiable complex functions infinitely differentiable? |
If we replace the axiom that 'there exists an infinite set' with 'all sets are finite', how would mathematics be like? My guess is that, all the theory that has practical importance would still show up, but everything would be very very unreadable for humans. Is that true?
We would have the natural numbers, athoug... |
An extreme form of constructivism is called *finitisim*. In this form, unlike the standard axiom system, infinite sets are not allowed. There are important mathematicians, such as Kronecker, who supported such a system. I can see that the natural numbers and rational numbers can easily defined in a finitist system, ... |
If all sets were finite, how could the real numbers be defined? |
The 24 game is as follows. Four numbers are drawn; the player's objective is to make 24 from the four numbers using the four basic arithmetic operations (in any order) and parentheses however one pleases.
Consider the following generalization. Given n+1 numbers, determine whether the last one can be obtained from t... |
An extreme form of constructivism is called *finitisim*. In this form, unlike the standard axiom system, infinite sets are not allowed. There are important mathematicians, such as Kronecker, who supported such a system. I can see that the natural numbers and rational numbers can easily defined in a finitist system, ... |
An extreme form of constructivism is called *finitisim*. In this form, unlike the standard axiom system, infinite sets are not allowed. There are important mathematicians, such as Kronecker, who supported such a system. I can see that the natural numbers and rational numbers can easily defined in a finitist system, ... |
Chicken is a famous game where two people drive on a collision course straight towards each other. Whoever swerves is considered a 'chicken' and loses, but if nobody swerves, they will both crash. So the payoff matrix looks something like this:
B swerves B straight
A swerves ... |
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