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*Notation*: $b_{0}+\underset{n=1}{\overset{\infty }{\mathbb{K}}}\left( a_{n}/b_{n}\right) $ is the [Gauss Notation for generalized continued fractions][1].
----------
**Description of the [Bauer-Muir transformation][2]**
(Based on [pp. 76-77][3] of Lisa Lorentzen and Haakon Waadeland's [book][4] (A)... |
I think Cauchy's integral formula and the Hilbert transform can be used to prove one direction, but is this an equivalence or only an implication?
**edit** for clarification:
Is a function $f : \mathbb C \to \mathbb C, z\mapsto f(z)$ analytical $\Leftrightarrow$ The Fourier-Transform $\mathcal F\\{f\\}(\omega) = N ... |
I presume you want to colour the *faces* of the cube?
If so there are $2^6$ colourings as there are six faces.
But do you want two colourings which are equivalent under
rotations (or rotations and reflections) to count as as the same?
If so then this is a classic application of
[Burnside's lemma][1].
**Added**
... |
Let $H$ be a subgroup of $K$ which is a subgroup of $G$.
If there's a retraction of $G$ onto $H$, it restricts to
a retraction of $K$ onto $H$. So if a group $G$ has this
property (say it's "retractible") then each subgroup $G$
is retractible. Which cyclic groups are retractible? |
Let $H$ be a subgroup of $K$ which is a subgroup of $G$.
If there's a retraction of $G$ onto $H$, it restricts to
a retraction of $K$ onto $H$. So if a group $G$ has this
property (let's say it's "retractible") then each subgroup of $G$
is retractible. Which cyclic groups are retractible? |
If $h(x,t)$ is measurable and the measures involved are $\sigma$-finite, does there exist a sequences of functions
$$h_n(x,t) = \sum_{j=1}^{N_n} f_{j,n}(x) 1_{F_{j,n}}(t)$$
where the sets are pairwise disjoint (in $j$) such that $h_n \to h$ pointwise almost everywhere?
I know that if we fix $x$ we can get a se... |
Connections between K-Theory and PDEs? |
What is an example of a finite model in first order logic having a unique undefinable element? |
This is (a slight paraphrase) of question 1.3.14 in Chang and Keisler's Model Theory book.
"Show that for each natural number $n$, there is a language $L_n$ and finite model $M_n$ of $L$ such that $M_n$ has precisely $n$ undefinable elements."
Here, an element $x\in M$ is definable if there is a (first order) for... |
Let $K$ be a number field and $L$ an infinite algebraic extension of $K$. Fix a non-trivial absolute value $v$ on $K$ (so $v$ is induced either by an embedding into the complex numbers or by a prime ideal in the ring of integers of $K$). If $K_v$ is the corresponding completion and $\overline{K}_v$ a choice of algebrai... |
What is the relationship between non-Archimedean places of infinite extensions of number fields and primes in the ring of integers? |
What are linear combinations of a column space? What equation to they satisfy? Why are they useful in leaner algebra? |
What are linear combinations of a column space? |
What is the homogeneous problem? What is the purpose of null space in this context? |
What is the homogeneous problem? |
What is the homogeneous problem? What is the purpose of null space of a vector in this context? |
Isaac, the auxiliary question that we should ask ourselves before answering your question is: **what is the "right" (i. e., most mathematically relevant/interesting) generalization of the Mandelbrot set to other parametrized families of functions?** Of course, you can take some arbitrary holomorphic family of functions... |
Once someone else has done the hard work and come up with the answer,
as Qiaochu has done, then it's easy to prove it on an *ad hoc*
basis as this interloper (me) will do.
Consider an $n$-cube with faces coloured red and blue. Consider a set
of $n$ balls. Colour the $j$-th ball red, blue or green according to the... |
Suppose $G$ is a compact topological group. We can construct the profinite completion of $G$; let's call this $\Gamma$.
My questions are:
1) Assuming that we know nothing about the (original) topology of $G$ other than that it is compact, is there anything we can say linking the topology of $G$ to the topology o... |
It's worth emphasizing that, besides unique factorization, there are also deeper structural reasons why one doesn't employ the convention that 1 is a prime, or a prime ideal. Below I discuss the motivations for for the convention that 0 is a prime ideal but 1 is not.
The reason one permits zero as a prime ideal is b... |
Can someone shed a light on how to solve something like this?
I've been looking for permutations but so far I found it very confusing.
As I'm not at home atm, I can't reach for my math books either for an answer.
Maybe you can help?
Given:
- a 5x5 grid
- each cell can have 2 possible values: 0 or 1
- the re... |
Can someone shed a light on how to solve something like this?
I've been looking for permutations but so far I found it very confusing.
As I'm not at home atm, I can't reach for my math books either for an answer.
Maybe you can help?
Given:
- a 5x5 grid
- each cell can have 2 possible values: 0 or 1
- each r... |
Hint: What does a<1/a say about a? What does b<1/b say about b? |
Suppose that $a$ were positive. Then, multiplying the string of ineqs. (doesn't change their direction) by $a$ gives $a^2 <1 < ba < a/b$. From here we see that $b>1$ (why?), but then, from the last ineq., $b^2 a < a$ which is impossible (why?). Therefore $a<0$. Now multiply again by $a$ and conclude. |
In the literature it is stated that to each quadratic irrational $\gamma=\frac{P+\sqrt{D}}{Q}$ there is a corresponding ideal $I=[|Q|/\sigma , (P+\sqrt{D})/\sigma]$, where $\sigma=1$, if $\Delta \equiv0$ mod $4$ and $\sigma=2$, otherwise.
Thus, in the case of $\frac{2+\sqrt{13}}{3}$ the associated ideal must be $I=[... |
How do I use second order Taylor method to solve a system of non-linear equations? Is there a good reference that gives details? I found mentions of it in a dozen of numerical analysis books, but no examples
Specifically, $f:\mathbb{R}^n \to \mathbb{R}^m$, solve $f(\mathbf{x})=\mathbf{0}$ using second order Taylor e... |
**EDIT 4 (SEP 30, 2010, LAST EDIT)**
Thank you very much to Martin Brandenburg!
Let $\lambda$ be the topology on $\mathbb Q\times\mathbb R^\infty$ coming from the realization of this set as the inductive limit of the $\mathbb Q\times\mathbb R^i$, $i\ge1$.
I'll try to prove that
> $\lambda$ coincides wit... |
How can you find an imaginary Root of Unity? |
How can you find the complex roots of i? |
I'm going to ask a very silly question, so I'm begging you to be understanding if it is absolutely trivial, or if it's an exercise in some Bourbaki. I'm afraid of asking you, because the question entails this particular case: is the addition of (a finite, but variable quantity of) real numbers a continuous function?
... |
Once someone else has done the hard work and come up with the answer,
as Qiaochu has done, then it's easy to prove it on an *ad hoc*
basis as this interloper (me) will do.
Consider an $n$-cube with faces coloured red and blue. Consider a set
of $n$ balls. Colour the $j$-th ball red, blue or green according to the... |
I'm looking for a convenient upper bound on the integral $\int_y^\infty x^k \exp(-(x-\mu)^2/2) dx$ for (possibly large) positive integer k. This is equivalent to finding higher moments of a truncated normal distribution.
Of course "convenient" is in the eye of the beholder, but I'd like some sort of fairly simple e... |
What is the largest (acyclic, connected) transitively reduced acyclic graph on $n$ vertices, for every $n$?
How many edges $e$ does it have? Does $e$ grow quadratically with $n$? |
How big are transitively reduced graphs? |
What is the largest (acyclic, connected) transitively reduced acyclic graph on $n$ vertices, for every $n$?
How many edges $e$ does it have? How does $e$ grow as a function of $n$? Faster than $n^2/4$? |
If you start with an infinite set, you can have a sequence of nested sets which converge to a single point. (ie Intersection of (-1/n, 1/n) as n->infinity))
However, at no time during the sequence is there a first element with only one point. In fact, for any finite (but unbounded) n, the number of points in the int... |
Suppose $x_{1}, x_{2}, y_{1}, y_{2}$ are known integers which satisfy
$y_{1}= \gamma x_{1}$ mod $c$
and
$y_{2}= \gamma x_{2}$ mod $c$
where $\gamma$ and $c$ are unknown integers. Is there a way to determine the values of $\gamma$ and $c$?
|
Suppose $x_1, x_2, y_1, y_2$ are known integers which satisfy
$y_1 \equiv \gamma x_1 \pmod c$
and
$y_2 \equiv \gamma x_2 \pmod c$
where $\gamma$ and $c$ are unknown integers. Is there a way to determine the values of $\gamma$ and $c$?
|
I had asked this question: http://math.stackexchange.com/questions/4412/characterising-continuous-functions some time back, and this question is more or less related to that question.
Suppose we have a function $f: \mathbb{R} \to \mathbb{R}$ and suppose the set $G = \\{ (x,f(x) : x \in \mathbb{R}\\}$ is connected... |
If you look at [this previous problem][1], I asked how to find the sum of all the areas between two taxicab geometry circles. However, upon learning about [$L^p$ norms][2], I thought it would be pretty interesting to extend the problem to all $L^p$ norm circles, not just $L^1$ (taxicab).
If $p=1$, then the result ha... |
I have recently been working out the full game tree for tic-tac-toe, just for fun. I am using the well known equivalence relation of rotations/reflections to simplify this tree in the standard way (which begins by noting that there are only 3 opening moves: edge, corner, and center). I actually could not find an imag... |
I have recently been working out the full game tree for tic-tac-toe, just for fun. I am using the well known equivalence relation of rotations/reflections to simplify this tree in the standard way (which begins by noting that there are only 3 opening moves: edge, corner, and center). I actually could not find an imag... |
Once someone else has done the hard work and come up with the answer,
as Qiaochu has done, then it's easy to prove it on an *ad hoc*
basis as this interloper (me) will do.
Consider an $n$-cube with faces coloured red and blue. Consider a set
of $n$ balls. Colour the $j$-th ball red, blue or green according to the... |
Given is a sequence $\langle a_1,a_2,\ldots,a_n\rangle$ over the alphabet $\\{1,2,\ldots,m\\}$ chosen uniformly at random among the $m^n$ possibilities. What is the expected size of the *set* $\\{a_1,a_2,\ldots,a_n\\}$?
If $m=n$ it [seems][1] the answer tends to $(1-1/e)n$ as $n\to\infty$, but I don't know why.
I... |
How many bins do random numbers fill? |
Since the transitively reduced graph $G$ is acyclic, there is a corresponding undirected simple graph $G'$, which can be formed by ignoring the direction of the edge. The number of edges in $G$ is same as the number of edges in $G'$.
For $G$ to be transitively reduced, a necessary condition is the $G'$ must not cont... |
This is probably a very simple question to all of you, but I have never taken a physics class, so I am lost for what I need to do for a programming problem. I will need to specify a force by inputting the newtons x, newtons y, and newtons z.
How can I calculate these 3 forces based on the desired angle and desired f... |
What is the center of V, the Klein 4 group? |
I have recently been working out the full game tree for tic-tac-toe, just for fun. I am using the well known equivalence relation of rotations/reflections to simplify this tree in the standard way (which begins by noting that there are only 3 opening moves: edge, corner, and center). I actually could not find an imag... |
I have recently been working out the full game tree for tic-tac-toe, just for fun. I am using the well known equivalence relation of rotations/reflections to simplify this tree in the standard way (which begins by noting that there are only 3 opening moves: edge, corner, and center). I actually could not find an imag... |
it has been years since I have done logs, I remember something like this:
x^log(z)y = y^log(z)x
(where z is the base)
Is that correct? It doesn't seem so, since
3^log(2)4 != 4^log(2)3
Am I right in that assumption?
If I am, does
x^log(z)y
convert to anything easily?
**Also, can someo... |
it has been years since I have done logs, I remember something like this:
$$x^{\log(z)y} = y^{\log(z)x}$$
(where $z$ is the base)
Is that correct? It doesn't seem so, since
$$3^{\log(2)4} \neq 4^{\log(2)3}$$
Am I right in that assumption?
If I am, does
$$x^{\log(z)y}$$
convert to anything easily?
|
it has been years since I have done logs, I remember something like this:
$$x^{\log_z(y)} = y^{\log_z(x)}$$
(where $z$ is the base)
Is that correct? It doesn't seem so, since
$$3^{\log_2(4)} \neq 4^{\log_2(3)}$$
Am I right in that assumption?
If I am, does
$$x^{\log_z(y)}$$
convert to anything easi... |
Here is the dot » https://i.stack.imgur.com/aQBA2.jpg
And here is how I arrived at it » http ://i.imgur.com/4LBJi.jpg
Pretty simple IFS using a 2 x 2 grid as the base for the iterations. Is there a way to describe this point as well as its siblings, all of which lie on the dotted red line(pic 2), using an equatio... |
How do I plot this red dot? |
Here is the dot
And here is how I arrived at it »
Pretty simple IFS using a 2 x 2 grid as the base for the iterations. Is there a way to describe this point as well as its siblings, all of which lie on the dotted red... |
Suppose I have a weakly sequentially continuous linear operator T between two normed linear spaces X and Y (i.e. $x_n \rightharpoonup x$ in $X$ $\Rightarrow$ $T(x_n) \rightharpoonup T(x)$ in $Y$). Does this imply that my operator T must be bounded? |
I have recently been working out the full game tree for tic-tac-toe, just for fun. I am using the well known equivalence relation of rotations/reflections to simplify this tree in the standard way (which begins by noting that there are only 3 opening moves: edge, corner, and center). I actually could not find an imag... |
<p>Consider a canonical eliptical cone C with its vertex at the origin, with height h, and with a base given by:</p>
<pre>(x/a)<sup>2</sup>+(y/b)<sup>2</sup> = 1; z = h</pre>
<p>a, b not equal to 0. Given a point p = (p<sub>x</sub>, p<sub>y</sub>, p<sub>z</sub>) how would you determine if p is interior to C? I real... |
How do I determine if a point is interior to an elliptical cone? |
Suppose I have a weakly sequentially continuous linear operator T between two normed linear spaces X and Y (i.e. $x_n \stackrel {w}{\rightharpoonup} x$ in $X$ $\Rightarrow$ $T(x_n) \stackrel {w}{\rightharpoonup} T(x)$ in $Y$). Does this imply that my operator T must be bounded?
|
This is an old exam problem I found online. For any positive integers $a,b$, one has $a^4|b^3$ implies $a|b$. Clearly if $a^4|b^3$, then $a|b^3$. It seemed simple on first reading, but I can't figure out how to show that $a|b$ follows, since $a$ is not necessarily prime. Is there some observation I'm missing? |
For any positive integers $a,b$, one has $a^4|b^3$ implies $a|b$? |
Why does the translates of a scaling function form a partition of unity? |
Why do the translates of a scaling function form a partition of unity? |
**EDIT 5 (OCT 1, 2010, LAST EDIT)**
I'm realizing with horror that I'm unable to check, in the proof of Lemma 5.5 [page 64](http://www.iecn.u-nancy.fr/~gaillard/milnor_64.pdf) in Milnor and Stasheff's **Characteristic Classes**, that $U$ is open.
**END OF EDIT 5**
----------
**EDIT 4 (SEP 30, 2010)**
... |
I think this is asked as a standard exercise in books about wavelets (e.g. exercise 7.2 in [Mallat's book][1]), but I couldn't find a proof. Let $\phi$ be a scaling function (see definition below). I would like to learn why
$$\sum_{k\in\mathbb Z} \phi(x-k) = 1 $$
almost everywhere.
**Definition.** A sequenc... |
I hope I'm at right place, as I asked this question at stackoverflow.com where the question was closed and sugested I go ask here. I'll try to be more comprehensive also, looking at couple of replies I got there
The question is simple for those who know how algorhitms work, and as I'm physics student I'm not sure ab... |
I have attempted to google for this, but the searching is marred by the close relationship of Jordan Algebras and Quantum Mechanics. I have been passively thinking of this question for ages, tonight I finally post it. :)
<strike>My question is fairly simple. Similar to the universal enveloping algebra for Lie algebr... |
*MATH 4135/5135: Introduction to Category Theory* by Peter Selinger
(17pp). Concise course outline. Only wish it covered more topics. Available in PS or PDF format.
<br> http://www.mscs.dal.ca/~selinger/4135/
*Handbook of Categorical Algebra* (Encyclopedia of Mathematics and its Applications) by Francis Borceux. R... |
Let's say we have some topological space.
Axiom $T_1$ states that for any two points $y \neq x$, there is an open neighborhood $U_y$ of $y$ such that $x \notin U_y$.
Then we say that a topological space is $T_4$ if it is $T_1$ and also satisfies that for any two closed, non-intersecting sets $A,B$, there are open... |
Why is $T_1$ required for a topological space to be $T_4$? |
Let us say that a rubik's cube in a particular configuration is in a particular "state". All other configurations of this cube (other "states"), which can be achieved by rotations of the cube can be thought to be connected to each other... rotations are like walking the edges of the graph where the different states are... |
How many disconnected graphs of the rubiks cube exist? |
Let us say that a Rubik's cube in a particular configuration is in a particular "state". All other configurations of this cube (other "states"), which can be achieved by rotations of the cube can be thought to be connected to each other... rotations are like walking the edges of the graph where the different states are... |
How many disconnected graphs of the Rubik's cube exist? |
How to prove connect sum doesn't depend on the choices of disks(which would be cutted) and different gluing of boundary spheres? Is it still true in differential category? Thanks! |
How to prove connect sum of two manifolds doesn't depend on the choices of balls(which would be cutted) and different gluing of boundary spheres? Is it still true in differential category? Thanks! |
(Apologies for the vague way the question is stated.)
We define the function like this:
(1) $f(x) = x$ for $0\leq x \leq a$ (for some $a \geq 15 \in \mathbb{R}$) $\land$ $f(x) = a$ if $ x > a$ . For $x<0$, it doesn't really matter what $f(x)$ looks like.
Has someone studied this type of function? Does it ha... |
Does the following function have a name and what properties does it have? |
You are looking at the problem of mathematics without *knowing* mathematics. Most of what you write applies to Chinese, provided you do not understand Chinese.
For one thing: A question like «does addition really have any meaning outside of the context of placing two objects in a box and predicting how many are ins... |
I'm trying to learn about the Selberg zeta function, but it seems like introductory texts assume more knowledge of Riemannian geometry than I'm comfortable with.
I have some basic questions that someone might be able to help with:
Is the composition of two closed geodesics itself a closed geodesic?
Is composit... |
**EDIT 4 (SEP 30, 2010)**
Thank you very much to Martin Brandenburg!
Let $\lambda$ be the topology on $\mathbb Q\times\mathbb R^\infty$ coming from the realization of this set as the inductive limit of the $\mathbb Q\times\mathbb R^i$, $i\ge1$.
I'll try to prove that
> $\lambda$ coincides with the produ... |
This analysis is correct, but not complete enough to convince me. For example, why is the fate of the last person settled as soon as the first person's seat chosen? Why will any other seat but the first person's or the last person's be taken by the time the last person boards?
I had to fill in the holes for myself... |
I'm looking for good online resources for 2nd order system dynamics. Any recommendations?
I'm looking for stuff that ideally includes discussion of Q, damping ratio, overshoot, bode plots, for systems with transfer functions of
$$ H(s) = \frac{1}{\tau_2^2 s^2 + \tau_1 s + 1} $$
$$ H(s) = \frac{\tau_1 s + 1}{... |
A student of mine has cooked up a new graphical notation for computing with knots on surfaces. The trouble is, writing up his results is difficult due to his new notation. Is there a good "drawing tool" for mathematics that anyone can suggest? |
A good, free, graphics package for mathematics? |
I'm looking for good online resources for 2nd order system dynamics. Any recommendations?
I'm looking for stuff that ideally includes discussion of Q, damping ratio, overshoot, bode plots, for systems with transfer functions of
$$ H_1(s) = \frac{1}{\tau_2^2 s^2 + \tau_1 s + 1} $$
$$ H_2(s) = \frac{\tau_1 s +... |
Given that I know the point coordinates of A and B on segment AB and the expected length of a perpendicular segment CD crossing the middle of AB, how do I calculate the point coordinates of segment CD? |
I have read a few weeks ago that some mathematical researchers have discover that the maximum number of movements tosolve any initial configuration of a Rubik's cube has been downsized to 20.
How do they calculate such a number ?
Thanks. |
How is done the calculation od the mximum number of movement to solve any configuration of Rubik's Cube ? |
How is done the calculation of the minimum number of movement to solve any configuration of Rubik's Cube ? |
I have read a few weeks ago that some mathematical researchers have discover that the minimum number of movements to solve any initial configuration of a Rubik's cube has been downsized to 20.
How do they calculate such a number ?
Thanks. |
I have a two state dynamical system. The two state variables are $P$ and $Z$ and $a,b,c,d$ are parameters. After manipulating the system I get the first line in the image.
I tried the conserved quantity $E\left(P,Z\right)$ but it doesn't seem to be conserved at all. Is it possible for there to be another conserved q... |
I have a two state dynamical system. The two state variables are $P$ and $Z$ and $a,b,c,d$ are parameters. After manipulating the system I get the first line in the image.
I tried the conserved quantity $E\left(P,Z\right)$ but it doesn't seem to be conserved at all. Is it possible for there to be another conserved q... |
If you are given of a ring, how do you find its ideals? |
If you are given of a ring, how do you find its ideals? |
I have a two state dynamical system. The two state variables are $P$ and $Z$ and $a,b,c,d$ are parameters. The system equations are:
$\frac{dP}{dt}=a\cdot P-b\cdot PZ=P\left(a-bZ\right)$
$\frac{dZ}{dt}=c\cdot PZ-d\cdot Z=Z\left(cP-d\right)$
Multiplying both sides by $\frac{1}{PZ}$ :
$\frac{dP}{dt}\frac{1}{P... |
I have a two state dynamical system. The two state variables are $P$ and $Z$ and $a,b,c,d$ are parameters. The system equations are:
$\frac{dP}{dt}=a\cdot P-b\cdot PZ=P\left(a-bZ\right)$
$\frac{dZ}{dt}=c\cdot PZ-d\cdot Z=Z\left(cP-d\right)$
Multiplying both sides by $\frac{1}{PZ}$ :
$\frac{dP}{dt}\frac{1}{P... |
I have a two state dynamical system. The two state variables are $P$ and $Z$ and $a,b,c,d$ are parameters. The system equations are:
$\frac{dP}{dt}=a\cdot P-b\cdot PZ=P\left(a-bZ\right)$
$\frac{dZ}{dt}=c\cdot PZ-d\cdot Z=Z\left(cP-d\right)$
Multiplying both sides by $\frac{1}{PZ}$ :
$\frac{dP}{dt}\frac{1}{P... |
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