instruction stringlengths 12 30k |
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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... |
Suppose {x_t} is a random walk with mean zero. (either discrete or continuous time)
Fix a time T.
What is: Probability[x_t < 0, for all t <= T]?
in words, what's the probability the random walk from 0 to T has never been above 0?
Thank you. |
If you are interested in computer science, a very useful thing to remember is that
$$ \log n! = \theta(n \log n) $$
Now given this, can you tell what $f(n)$ is, if $ \log (n!^{n}) = \theta(f(n))$?
How would that compare to the right hand side?
|
The two element group has a centre (of course it does, it's abelian) but no outer automorphisms. I can come up with a (convoluted) countably infinite non-abelian example. Is there a finite one? If so what is the smallest?
Is there a name for this sort of group? ("Outerless?" Since a group without a centre is "centr... |
Say we have a partially ordered set $(S,\preceq)$, and some subset $E\subseteq S$ such that $E$ is bounded below and $\inf E$ exists. My question is, since $S$ is not totally ordered is it possible to have two lower bounds that are not comparable, and hence be unable to say whether one lower bound is really the greates... |
Does the existence of an infimum imply that the set of lower bounds of a set is totally ordered? |
I was preparing for an area exam in analysis and came across a problem in the book *Real Analysis* by Haaser & Sullivan. From p.34 Q 2.4.3, If the field *F* is isomorphic to the subset *S'* of *F'*, show that *S'* is a subfield of *F'*. I would appreciate any hints on how to solve this problem as I'm stuck, but that's ... |
Examples of fields of characteristic 0? |
Another name (that I've only ever seen when someone else asked this [question][1]) is "logarithmand".
[1]: http://www.google.com/url?sa=t&source=web&cd=1&ved=0CBQQFjAA&url=http%3A%2F%2Fen.wikipedia.org%2Fwiki%2FWikipedia%3AReference_desk%2FArchives%2FMathematics%2F2010_February_17&rct=j&q=%22logarithmand%22&ei=X... |
This is a follow up question to one I posted earlier. I'm trying to decide that if for $(S,\preceq)$ a partially ordered set and $E\subseteq S$, one has $L(E)=\langle s]$ for some $s\in S$ iff $\inf E$ exists, and in particular, $L(E)=\langle\inf E]$.
I'm simply curious about showing that $L(E)\subseteq\langle\inf ... |
Can one show that $L(E)=\langle s]$ on a partially ordered set? |
This is a follow up question to one I posted earlier. I'm trying to decide that if for $(S,\preceq)$ a partially ordered set and $E\subseteq S$, one has $L(E)=\langle s]$ for some $s\in S$ iff $\inf E$ exists, and in particular, $L(E)=\langle\inf E]$.
I'm simply curious about showing that $L(E)\subseteq\langle\inf ... |
If I correctly understand your question, it boils down to whether $\inf E$ is "larger" than or equal to every lower bound of $E$. Isn't this the definition? A greatest lower bound of a set must be, by definition, comparable to and "larger" than (or equal to) every lower bound.
I think the confusion arises because,... |
The statement is the following: If you take a variety with the *analytic* topology (or a complex manifold), then yes, sheaf cohomology of the constant sheaf agrees with singular cohomology. More generally, sheaf cohomology of a constant sheaf on a *locally contractible space* agrees with singular cohomology of that spa... |
Can you give me some tips?
Let e1,e2,... be iid normal mean 0 variance 1. Let Xt := e1+...+et, for t=1,2.. and X0 := 0. (So we have a discrete-time random walk whose steps are iid N(0,1))
Define first-positive time T0 := inf{t>0 : Xt >0}.
What is the CDF of T0, i.e. what is Prob(T0 <= t) ?
And second (pro... |
Could someone give me some tips?
Let e1,e2,... be iid normal mean 0 variance 1. Let Xt := e1+...+et, for t=1,2.. and X0 := 0. (So we have a discrete-time random walk whose steps are iid N(0,1))
Define first-hitting time T0 := inf{t>0 : Xt <0}.
What is the CDF of T0, i.e. what is Prob(T0 <= t) ?
And second... |
Mathematics is generally seen as a precise and specific language. Surely if there is ambiguity in the phrase "in the sequel" it should be avoided. There are far more accurate ways of expressing both the meanings that phrase could carry so why create needless confusion? |
Okay, I can't resist: here is a quick answer.
I am construing the question in the following way: "Is there some criterion for a subset of $[0,\infty)$ to be all of $[0,\infty)$ which is (a) analogous to the principle of mathematical induction on $\mathbb{N}$ and (b) useful for something?"
The answer is yes, at leas... |
On going through this post http://math.stackexchange.com/questions/2120/does-there-exist-an-f-mathbbn-to-mathbbn-such-that-sum-fn-n2-conve i happened to get the following 2 problems into my mind:
Let $f: \mathbb{N} \to \mathbb{N}$ be a bijection. Then does the series $$\sum\limits_{n=1}^{\infty} \frac{1}{nf(n)}$$ co... |
A [couple of proofs][1] in italian.
[1]: http://www.scienzematematiche.it/forum/viewtopic.php?f=7&t=2322 |
> Is there an analogous classification for fields of characteristic zero?
Yes, but it is somewhat useless and nobody would call it a classification.
Every field of characteristic zero has the form $Quot(\mathbb{Q}[X]/S)$, where $X$ is a set of variables and $S$ is a set of polynomials in $\mathbb{Q}[X]$ (which ... |
Hints. For the first series, prove that
$$\sum_{n=1}^N\frac1{nf(n)}\le\sum_{n=1}^N\frac1{n^2}.$$
For the second, what would happen if $f$ were an involution,
and in each pair $(n,f(n))$ one term was much bigger than the other? |
I can't offer specific help, but I'd recommend thumbing through [Concrete Mathematics][1] looking for techniques. It has many sums that look similar, though of course the difficulty is in the details.
There's also the book [A=B][2], but Concrete Mathematics gives an introduction to the content of A=B and in my opin... |
In my previous question, [Weakly-compact cardinals](http://math.stackexchange.com/q/4198/622), I was asking about weakly-compact cardinals and equivalent definitions to the basic one, which is $\kappa \to (\kappa)^2$.
One of which was that $\kappa$ has the *tree property* if any tree of cardinality $\kappa$ for whic... |
I was wondering if a field is an algebra over itself?
Also is a ring an algebra over itself? If not, does the ring require to be commutative?
Thanks and regards! |
Is a field (ring) an algebra over itself? |
Consider a situation where a person has partial knowledge, but we have a more complete picture. For example, the person might know that 1/3 of fish are red, but we know that the particular species is actually red 2/3 of the time. Do these two separate probabilities have special names? If they don't have any standard na... |
I was wondering if a field is an algebra over itself (http://en.wikipedia.org/wiki/Algebra_over_a_field)?
Also is a ring an algebra over itself (http://en.wikipedia.org/wiki/Algebra_(ring_theory)? If not, does the ring require to be commutative?
Thanks and regards! |
What is the area of the region bounded by $y = f_n(x)$ and the $x$-axis as $n$ gets large?
$f_0=1-|x|$
$f_n=1-|1-2f_{n-1}(x)|$
By trying various values for $n$ and $x$ I got that $f_n(x) = 2^{n-1}\cdot f_0(x)$
By integrating this I got that the area is given by $2^{n-1}\cdot x\cdot (2-|x|)$.
However, ... |
We can divide 7^17 - 7^15 by?
The answer is six ,but the solution?
Thanks in advance. |
We can divide 7^17 - 7^15 by? |
I'm currently getting confused about indices in some spectral sequences. Assume we work in the category of modules for simplicity. Let $A^\cdot$ be a (bounded on the right) complex and let $B^\cdot$ (I don't think we have to assume anything about the boundedness of $B$). I want to compute $Ext^n(A^\cdot,B^\cdot)$, whic... |
Say P is a convex Euclidean polytope, where the origin is not contained in any bounding hyperplane containing a facet of P, with n facets given by f ◦ x = 1 and m vertices v. (That is, each facet equation has been multiplied through as needed to obtain 1 on the right side.)
Is it true that the polytope P’ with m facet... |
What is the area of the region bounded by $y = f_n(x)$ and the $x$-axis as $n$ gets large?
$f_0=1-|x|$
$f_n=1-|1-2f_{n-1}(x)|$
Assume the domain of $f$ to be the real numbers.
By trying various values for $n$ and $x$ I got that $f_n(x) = 2^{n-1}\cdot f_0(x)$
By integrating this I got that the area is ... |
What is the area of the region bounded by $y = f_n(x)$ and the $x$-axis as $n$ gets large?
$f_0=1-|x|$
$f_n=1-|1-2f_{n-1}(x)|$
Assume the domain of $f$ to be the real numbers.
By trying various values for $n$ and $x$ I got that $f_n(x) = 2^{n}\cdot f_0(x)$
By integrating this I got that the area is gi... |
Suppose I have a 4-index, real-valued tensor $\lambda_{ij}^{ab}$ where $i$ and $j$ are from some subspace $O$ of the whole numbers, and $a$ and $b$ are from some other subspace $V,$ which is disjunct from $O.$ Suppose also that we know that $\lambda$ is antisymmetric. If we impose the constraint that $\lambda$ is Her... |
In my previous question, [Weakly-compact cardinals](http://math.stackexchange.com/q/4198/622), I was asking about weakly-compact cardinals and equivalent definitions to the basic one, which is $\kappa \to (\kappa)^2$.
One of which was that $\kappa$ is inaccessible and has the *tree property* if any tree of cardinali... |
What exactly do you want?
If you want step-by-step software algorithm to draw a Bezier curve,
and the ["Don Lancaster's Guru's Lair Cubic Spline Library"][1] doesn't have the answer,
then the StackOverflow.com website is a better place to ask such questions.
[1]: http://www.tinaja.com/cubic01.asp |
In my previous question, [Weakly-compact cardinals](http://math.stackexchange.com/q/4198/622), I was asking about weakly-compact cardinals and equivalent definitions to the basic one, which is $\kappa \to (\kappa)^2$.
One of which was that $\kappa$ is inaccessible and has the *tree property* (that is if any tree of ... |
There are a powerful **algorithms generalizing telescopy** (Gosper, Zeilberger et al.) that easily tackle this case of the **Chu-Vandermonde identity** and much more complicated sums. For example, see [this paper][1] which gives as an application a very interesting q-analogy - namely that L. J. Rogers' classical finite... |
Suppose $K$ is the splitting field of $x^3-2$ over $\mathbb{Q}$, and let $\mathcal{O}_K$ denote its ring of integers. I want to show that for any prime number $p$, $p\mathcal{O}_K$ is not a prime ideal of $\mathcal{O}_K$. The hint I was given was to consider the factorization of $x^3-2$ over $F_p$.
**My question(s... |
It really should be easy to see that $2$ ramifies (there an ideal $\mathfrak{p}$
with $\mathfrak{p}^3=2\mathfrak{p}$). Proving that $3$ ramifies is a little trickier.
If we could find an element of $\mathfrak{O}$ whose minimal polynomial is Eisenstein
at $3$, that would do. An Eisenstein polynomial at $3$ would be
... |
It really should be easy to see that $2$ ramifies (there an ideal $\mathfrak{p}$
with $\mathfrak{p}^3=2\mathfrak{O}$). Proving that $3$ ramifies is a little trickier.
If we could find an element of $\mathfrak{O}$ whose minimal polynomial is Eisenstein
at $3$, that would do. An Eisenstein polynomial at $3$ would be
... |
*This problem is taken from Vojtěch Jarník International Mathematical Competition 2010, [Category I, Problem 1](http://vjimc.osu.cz/j20/j20problems1.pdf). <sub><sup>— edit by KennyTM</sup></sub>*
---
On going through this post http://math.stackexchange.com/questions/2120/does-there-exist-an-f-mathbbn-to-mathbbn-s... |
Let's work inside $L=\mathbb{Q}(\sqrt[3]2)$.
It really should be easy to see that $2$ ramifies (there an ideal $\mathfrak{p}$
with $\mathfrak{p}^3=2\mathfrak{O}_L$). Proving that $3$ ramifies is a little trickier.
If we could find an element of $\mathfrak{O}_L$ whose minimal polynomial is Eisenstein
at $3$, that wo... |
We can divide $7^{17} - 7^{15}$ by?
The answer is $6$, but how?
Thanks in advance. |
We can divide $7^{17} - 7^{15}$ by? |
I am trying to understand [**pairwise testing**][1].
How many combinations of tests would be there for example, if
a: 1-m
b: 1-n
c: 1-p
a, b and c can take m, n and p distinct values respectively. **What are the total number of pairwise combinations possible?**
[1]: http://en.wikipedia.org... |
How to derive the number of pairwise combinations of a set of factors? |
I don't see what you're doing. Are you trying to show that:
$$\int_{\mathbb R^2} f(x) g(x) \, dx = \int_{\mathbb R^2} \hat{f(\xi)} \overline{\hat{g(\xi)}} \, d\xi$$?
This is usually called Parseval's identity.
You can prove it as follows (even in $\mathbb R^n$), first prove by Fubini (very simple) that:
$$\in... |
I am trying to understand [**pairwise testing**][1].
How many combinations of tests would be there for example, if
>`a` can take values from 1 to m
>`b` can take values from 1 to n
>`c` can take values from 1 to p
`a`, `b` and `c` can take m, n and p distinct values respectively. **What are the total num... |
Let's work inside $L=\mathbb{Q}(\sqrt[3]2)$.
It really should be easy to see that $2$ ramifies (there an ideal $\mathfrak{p}$
with $\mathfrak{p}^3=2\mathfrak{O}_L$). Proving that $3$ ramifies is a little trickier.
If we could find an element of $\mathfrak{O}_L$ whose minimal polynomial is Eisenstein
at $3$, that wo... |
I am trying to understand [**pairwise testing**][1].
How many combinations of tests would be there for example, if
>`a` can take values from 1 to m
>`b` can take values from 1 to n
>`c` can take values from 1 to p
`a`, `b` and `c` can take m, n and p distinct values respectively. **What are the total num... |
If each parameter had $10$ choices you'd be testing $300$ vs $1000$ combinations, namely hold $\rm a$ constant and vary $\rm b,c$ through $10\cdot 10 = 100$ values. Similarly hold, $\rm b$ constant; then $\rm c$. As the number of variables $\rm k$ increases you get better savings, roughly $\rm (k N)^2$ vs. $\rm N^k$, w... |
Given a sequence of row sums $r_1, \ldots, r_m$ and column sums $c_1, \ldots, c_n$, all positive, find a matrix $A_{m\times n}$ consistent with the given row and column sums that has the fewest nonzero entries. That is, we want to minimize $|\\{a_{ij} \ne 0\\}|$ under the constraint that $\sum_j a_{ij} = r_i$ and $\sum... |
Some years ago I was interested in the following Markov chain
whose state space is the positive integers. The chain begins at state "1",
and from state "n" the chain next jumps to a state uniformly
selected from {n+1,n+2,...,2n}.
As time goes on, this chain goes to infinity, with occasional
large jumps. In any ... |
**HINT** $\rm\quad X^{n+2} - X^n \;=\; (X^2 - 1)\: X^n \;=\; (X-1)\: (X+1)\: X^n$
Often number identities are more perceptively viewed as special cases of function or polynomial identities. For example, [Aurifeuille, Le Lasseur and Lucas][1] discovered so-called [**Aurifeuillian factorizations**][2] of cyclotomic ... |
For computing the present worth of an infinite sequence of equally spaced payments $(n^{2})$ I had the need to evaluate
$\displaystyle\sum_{n=1}^{\infty}\frac{n^{2}}{x^{n}}=\dfrac{x(x+1)}{(x-1)^{3}}$.$\qquad x>1$.
The method I used was based on the geometric series $\displaystyle\sum_{n=1}^{\infty}x^{n}=\dfrac{x}... |
If each parameter had $10$ choices you'd be testing $300$ vs $1000$ combinations, namely hold $\rm a$ constant and vary $\rm b,c$ through $10\cdot 10 = 100$ values. Similarly hold, $\rm b$ constant; then $\rm c$. As the number of variables $\rm k$ increases you get better savings, roughly $\rm (k N)^2$ vs. $\rm N^k$, w... |
I asked this question on mathoverflow, but it was deemed too simple, so I'm posting here instead --
Is there a nice way to characterize an orthonormal basis of eigenvectors of the following $d\times d$ matrix?
$$\mathbf{I}-\frac{1}{d} \mathbf{v}\mathbf{v}'$$
Where $\mathbf{v}$ is a $d\times 1$ vector of 1's. ... |
I have a 3D plane defined by three points: $P_0$ , $P_1$ and $P_2$. How to check whether a point $P$ is located right on the 3D plane? |
If we have an algebraic number α with (complex) absolute value 1, it does not follow that α is a root of unity (i.e., that α<sup>*n*</sup>=1 for some *n*). For example, (3/5 + 4/5 *i*) is not a root of unity.
But if we assume that α is an algebraic integer with absolute value 1, does it follow t... |
Are all algebraic integers with absolute value 1 roots of unity? |
I have a 3D plane defined by three points: $P_0$ , $P_1$ and $P_2$. How to check whether a point $P$ is located right on the 3D plane?
So, for example, if I have a plane defined by $({0,0,0})$, $({10,0,0})$ and $({0,10,0})$, then the point $({50,0,0})$ is considered ***not*** located on the plane, whereas the point ... |
I have a 3D plane defined by three points: $P_0$ , $P_1$ and $P_2$. How to check whether a point $P$ is located right on and ***inside*** the 3D plane?
So, for example, if I have a plane defined by $({0,0,0})$, $({10,0,0})$ and $({0,10,0})$, then the point $({50,0,0})$ is considered ***not*** located on the plane, w... |
I have a 3D plane defined by three points: $P_0$ , $P_1$ and $P_2$. How to check whether a point $P$ is located right on and ***inside*** the 3D triangle?
So, for example, if I have a plane defined by $({0,0,0})$, $({10,0,0})$ and $({0,10,0})$, then the point $({50,0,0})$ is considered ***not*** located on the plane... |
If you're willing to accept the [fundamental theorem of arithmetic][1] (which can be proved by elementary -i.e., "ideal free"- methods, see the link) that states that any integer different from $\pm 1$ can be written as a unique product (up to ordering of the factors) of prime numbers, then the equivalence of both defi... |
Usually, applying the conversion formulae
$r^2=x^2+y^2$
$\cos\;\theta=\frac{x}{r}$
$\sin\;\theta=\frac{y}{r}$
to transform an equation in polar coordinates to an implicit Cartesian equation is quite straightforward for an equation of the form
$r=f(\cos(n\theta),\sin(n\theta))$
with $n$ an integer, tha... |
Contrast these two methods for calculating n!:
- Counting permutations of a n-element set one-by-one (this is what n! counts), vs.
- Multiplying together the numbers 1,2,...,n.
Therefore n! represents an amazingly efficient method for counting permutations! So I think (and I don't think I'm the only one) that... |
We could do something simple, since we have:
\begin{equation}
\cos\left(\frac{p}{q}\theta\right)=\frac{1}{2}\left(e^{i\frac{p}{q}\theta}+e^{-i\frac{p}{q}\theta}\right)=\frac{1}{2}\left(\left(\frac{x}{r}+i\frac{y}{r}\right)^{\frac{p}{q}}+\left(\frac{x}{r}-i\frac{y}{r}\right)^{\frac{p}{q}}\right)
\end{equation}
That ... |
I can't offer specific help, but I'd recommend thumbing through [Concrete Mathematics][1] looking for techniques. It has many sums that look similar, though of course the difficulty is in the details.
There's also the book [A=B][2], but Concrete Mathematics gives an introduction to the content of A=B and in my opin... |
I'm starting a very long quest to learn about math so I can program games. I'm mostly a coorporate developer and it's somewhat boring and non exiting. When I began my career I chose it because I wanted to create games.
I'm told Linear Algebra is the best place to start - where should I go? |
Where to start learning Linear Algebra? |
If the matrix is positive definite, then all its eigenvalues are strictly positive. Is the inverse also true? That is, if the eigenvalues are strictly positive, then matrix is positive definite? Can you give example of 2x2 matrix with 2 positive eigenvalues but is not positive definite? |
if eigenvalues are positive, is the matrix positive definite? |
If the matrix is positive definite, then all its eigenvalues are strictly positive. Is the converse also true? That is, if the eigenvalues are strictly positive, then matrix is positive definite? Can you give example of 2x2 matrix with 2 positive eigenvalues but is not positive definite? |
This is a riff on some of the comments about what might constitute an "answer" and what "closed form" might mean: although it's somewhat facetious, it's intended to prompt thoughts about these issues.
Our base-10 number system interprets a string ${a_n}{a_{n-1}} \cdots {a_0}$ as the sum $\sum_{i=0}^{n} a_i 10^i $ (w... |
You are right: Linear Algebra is not just the "best" place to start. It's THE place to start.
Among all the books cited in [Wikipedia - Linear Algebra][1], I would recommend:
* Strang, Gilbert, Linear Algebra and Its Applications (4th ed.)
Strang's book has at least two reasons for being recomended. First... |
You are right: Linear Algebra is not just the "best" place to start. It's THE place to start.
Among all the books cited in [Wikipedia - Linear Algebra][1], I would recommend:
* Strang, Gilbert, Linear Algebra and Its Applications (4th ed.)
Strang's book has at least two reasons for being recomended. First... |
We could do something simple, since we have:
\begin{equation}
\cos\left(\frac{p}{q}\theta\right)=\frac{1}{2}\left(e^{i\frac{p}{q}\theta}+e^{-i\frac{p}{q}\theta}\right)=\frac{1}{2}\left(\left(\frac{x}{r}+i\frac{y}{r}\right)^{\frac{p}{q}}+\left(\frac{x}{r}-i\frac{y}{r}\right)^{\frac{p}{q}}\right)
\end{equation}
That ... |
In [this][1] thread, the question was to find a $f: \mathbb{R} \to \mathbb{R}$ such that
$$f(f(x)) = f(x) + x$$
(which was revealed in the comments to be solved by $f(x) = \varphi x$ where $\varphi$ is the [golden ratio][2] $\frac{1+\sqrt{5}}{2}$)
<hr>
Having read about [iterated functions][3] shortly befor... |
Suppose I have a non-negative real-valued function over $d$-dimensional real vectors as follows
$$f(\mathbf{x})=\exp(-\mathbf{x}' A \mathbf{x})$$
Where $A$ is some symmetric positive definite $d\times d$ matrix. What is the normalization factor to turn this into a valid density over the following set?
$$S_d=\\... |
Suppose I have a non-negative real-valued function over $d$-dimensional real vectors as follows
$$f(\mathbf{x})=\exp(-\mathbf{x}' A \mathbf{x})$$
Where $A$ is some symmetric positive definite $d\times d$ matrix. What is the normalization factor to turn this into a valid density over the following set?
$$S_d=\\... |
I don't see what you're doing. Are you trying to show that:
$$\int_{\mathbb R^2} f(x) g(x) \, dx = \int_{\mathbb R^2} \widehat{f(\xi)} \overline{\widehat{g(\xi)}} \, d\xi$$?
This is usually called Parseval's identity.
You can prove it as follows (even in $\mathbb R^n$), first prove by Fubini (very simple) that:
... |
I'm not sure if this a proper recurance relation *per se* but I'd be interested in the methodology in solving a recurrence relation of the following form:
$Z_0 = 1$
$Z_1 = x_1$
$Z_2 = x_1Z_1 + x_2 = x_1^2 + x_2$
$Z_3 = x_1Z_2 + x_2Z_1 + x_3 = x_1^3 + x_1x_2 + x_1x_2 + x_3$
$Z_n = x_1 Z_{n-1} + x_2 Z_{n-2... |
I'm having a hard time solving this problem. Would appreciate any hint! (and special thanks to Byron Schumuland for answering my previous 2 questions. This 3rd Q is different.)
Let et e1,e2,... be iid normal mean 0 variance 1. Let Xt := e1+...+et, for t=1,2.. and X0 := 0. (So we have a discrete-time random walk who... |
I have a constant area $A$ , and I can mold that area into a $n$ -side polygon, where $n>=3$. The issue now is how to find the $n$ such that it can contain the most number of circles, each with a constant radius $R$ ?
|
I've been reading a bit about how the set of bounds changes for a set depending on what superset one works with. I considered the sets $S\subseteq T\subseteq\mathbb{Q}$ and worked out a few contrived examples:
If $S=T=$ {$x\in\mathbb{Q}\ | \ x^2\lt 2$}, so here $S$ is not bounded above in $T$, but it is bounded abou... |
For $S\subseteq T\subseteq\mathbb{Q}$, it is possible for $\sup_T S$ to exist, but $\sup_\mathbb{Q} S$ does not? |
Consider the following sequence $a_n$:
$a_1 = 0$
$a_n = 1 + \frac{1}{2^n-2} \sum_{i=1}^{n-1} \binom{n}{i} a_i$
The first few terms are $0,1,\frac{3}{2},\frac{13}{7},\frac{15}{7}$.
The sequence comes out of the analysis of a certain process, whose asymptotics we expect to be about $\log_2 n$. Numerical compu... |
A recurrence that wiggles? |
I have a constant area $A$ , and I can mold that area into a [**regular**][1] $n$ -side polygon, where $n>=3$. The issue now is how to find the $n$ such that it can contain the most number of circles, each with a constant radius $R$ ?
[1]: http://en.wikipedia.org/wiki/Regular_polygon |
If $\varphi:U\subset \mathbb{R}^n \to \mathbb{R}^m$ is $C^1$, let $\mathrm{T}\varphi:\mathrm{T}U \to \mathrm{T}R^m$ be its tangent map. The inverse function theorem tells us that $\ker(\mathrm{T}\varphi(x))$ is zero iff $\varphi$ is injective in some neighborhood of $x$. If the kernel is non-zero, what can we say about... |
I have a constant area $A$ , and I can mold that area into a [**regular**][1] $n$ -side polygon, where $n>=3$. The issue now is how to find the $n$ such that it can contain the most number of circles, each with a constant radius $R$ ?
**Edit to clarify the question: the area must be a regular $n$ -side polygo... |
This is the Erdos-Renyi "20 Questions" problem. You want to guess an integer in a set of N elements by selecting subsets and asking questions of the form "and is it in *this* subset?". For the question to provide information the subset cannot be empty or the entire set of uneliminated possibilities, and if an informa... |
A number $n$ is perfect if it's equal to the sum of its divisors (smaller than itself). A well known theorem by Euler states that every even perfect number is of the form $2^{p-1}(2^p-1)$ where $2^p-1$ is prime (this is what is called a **Mersenne** prime).
How is this theorem proved? The converse (showing that ever... |
How to show that all even perfect numbers are obtained via Mersenne primes? |
This is the Erdos-Renyi "20 Questions" problem. You want to guess an integer in a set of N elements by selecting subsets and asking questions of the form "and is it in *this* subset?". For the question to provide information the subset cannot be empty or the entire set of uneliminated possibilities, and if an informa... |
I am trying to compute the asymptotic growth-rate in a specific combinatorial problem depending on a parameter w, using the Transfer-Matrix method. This amounts to computing the largest eigenvalue of the corresponding matrix.
For small values of w, the corresponding matrix is small and I can use the so-called power ... |
Computing the largest Eigenvalue of a very large sparse matrix? |
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