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The standard proof is that this follows from the <a href="http://en.wikipedia.org/wiki/Taylor_series">Taylor series</a>
$$\arctan x = x - \frac{x^3}{3} + \frac{x^5}{5} \mp ...$$
for the arctangent. This Taylor series is closely related to the Taylor series of the logarithm
$$\log (1 + x) = x - \frac{x^2}{2} ... |
I was looking at movie times today and was struck by the oddly-spaced showing times. For example, at the local Loew's Theater "Tron: Legacy 3D" (127 min.) is playing on two screens at the following interlaced times: 1:00 pm, 1:45 pm, 4:00 pm, 4:45 pm, 7:00 pm, 7:45 pm, 10:00 pm and 10:45 pm. Why not space the times equ... |
> Is there a commutative integral domain R in which:
>
> * every nonzero prime ideal Q is maximal, and
> * there are maximal ideals Q with R/Q of sizes 3, 11, and 27?
This doesn't happen with number rings of Galois extensions of <b>Q</b>, as far as I know, since what happens to 3 seriously happens to 3. You cann... |
According to a paper "Optimal Irreducible Polynomials for GF(2m) Arithmetic" by M. Scott,
"it is in practise always possible to chooose as an irreducible polynomial either a trinomial... or a pentanomial." <a href="http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.75.455"> [PDF link]</a>
In random number gen... |
I was given the question:what is 9+99+999+9999+...+999..99(30 digits) After noticing a trend, I came with the conclusion that the answer would be 28 1's 080. Can anyone confirm my answer and give a reason as to why?
|
I have an old Spanish CG book that calls Perlin Noise a "fractal structure". After reading [this][1] I couldn't deny it or confirm it. Is it a fractal structure? What would it Hausdorff dimension be?
[1]: http://webstaff.itn.liu.se/~stegu/TNM022-2005/perlinnoiselinks/perlin-noise-math-faq.html |
Is Perlin Noise a "fractal"? |
> * G = { x,y : x<sup>2<sup>n+1</sup></sup> = 1, y<sup>4</sup> = 1, xy = yx<sup>−1</sup> }
> * H = { x,y : x<sup>2<sup>n+1</sup></sup> = 1, y<sup>4</sup> = 1, xy = yx<sup>2<sup>n</sup>−1</sup> }
>
> Can these groups of order 2<sup>n+3</sup> and nilpotency class n+1 be distinguished by any reasonable (computable) set... |
Can invariants distinguish these two 2-groups? |
Let $X$ be a CW complex. The (isomorphism classes of) complex line bundles on $X$ are classified by the homotopy classes of maps $X \to \mathbb{CP}^\infty$, that is by the elements of $H^2(X, \mathbb{Z})$.
It is also true that the tensor product of line bundles corresponds to adding cohomology classes. It follows tha... |
I am reading the book by Goldrei on Classical Set Theory. My question is more of a clarification. It is on if we are overloading symbols in some cases. For instance, when we define $2$ as a natural number, we define $$2_{\mathbb{N}} = \{\emptyset,\{\emptyset\} \}$$ When we define $2$ as an integer, $2_{\mathbb{Z}}$ is ... |
I am reading the book by Goldrei on Classical Set Theory. My question is more of a clarification. It is on if we are overloading symbols in some cases. For instance, when we define $2$ as a natural number, we define $$2_{\mathbb{N}} = \{\emptyset,\{\emptyset\} \}$$ When we define $2$ as an integer, $2_{\mathbb{Z}}$ is ... |
This question came up when trying to trying to find a $3\times3$ real matrix $A$ such that
1. $Ax$ is nonzero for nonzero $x$
2. $Ax$ is orthogonal to $x$ for any $x$ in $\mathbb{R}^3$
We know such a matrix cannot exist because $A$ must have an eigenvalue (thus there is some $x$ such that either $Ax = 0$ or $... |
Geometric argument that operators on $\mathbb{R}^3$ have an eigenvalue? |
I am reading the book by Goldrei on Classic Set Theory. My question is more of a clarification. It is on if we are overloading symbols in some cases. For instance, when we define $2$ as a natural number, we define $$2_{\mathbb{N}} = \{\emptyset,\{\emptyset\} \}$$ When we define $2$ as an integer, $2_{\mathbb{Z}}$ is an... |
Imagine the varying frequency generator, which feeds this frequency to the oscilloscope.
Oscilloscope is turned to show signal spectra with FFT.
The image on the screen of the oscilloscope is as expected: single peak, traveling from 1 kHz to 50 kHz. Of course, my generator is not ideal, it gives a number of harmoni... |
If *G* is a free group generated by *n* elements, is it possible to find an isomorphism of *G* with a free group generated by *n-1* (or any fewer number) of elements? |
In tensor notation in Spivak's Calculus on Manifolds, what is that character that looks like a 3? |
A completely different method of recognition is just a random constructive recognition:
1. Choose a random element z
2. If z has order |G|/4, call it x [ probability 25% ]
3. If z has order 4 and if it has more than 2 conjugates, call it y [ probability 50% ]
4. Otherwise, go to 1 [ probability 25% ]
5. If we ar... |
The function $\sec(x)=\frac{1}{\cos(x)}$ also has a series expansion that will give you this result, but it is not its Taylor series. Namely, its [partial fraction expansion][1] is $$\sec(x)=\sum_{n=1}^\infty \frac{(-1)^n(2n-1)\pi }{x^2-\left (n-\frac{1}{2}\right )^2\pi^2},$$
as derived in Example 1 at [this link][2... |
What is the geometric, physical or other meaning of the third derivative of a function at a point?
(Originally [asked on MO][1] by AJAY)
[1]: http://mathoverflow.net/questions/49820/whats-meaning-of-geometrical-meaning-of-third-derivative-of-a-function-at-a-poin
|
cross-post from <a href="http://stackoverflow.com/questions/4481973/permutations-distinct-under-given-symmetry-mathematica-8-group-theory">stack overflow</a>
I have a list of numbers like {2,1,1,0} and I'd like to list all permutations of that list that are not equivalent under given symmetry group. So for instance ... |
What is the geometric, physical or other meaning of the third derivative of a function at a point?
(Originally [asked on MO][1] by AJAY)
If you have interesting things to say about the meaning of the first, and second derivatives, please do.
[1]: http://mathoverflow.net/questions/49820/whats-meaning-of-geome... |
The standard proof is that this follows from the <a href="http://en.wikipedia.org/wiki/Taylor_series">Taylor series</a>
$$\arctan x = x - \frac{x^3}{3} + \frac{x^5}{5} \mp ...$$
for the arctangent. This Taylor series is closely related to the Taylor series of the logarithm
$$\log (1 + x) = x - \frac{x^2}{2} ... |
Fiction "Division by Zero" By Ted Chiang
I read the fiction story <a href="http://www.fantasticmetropolis.com/i/division/full/">"Division by Zero" By Ted Chiang</a>
My interpretation is the character finds a proof that arithmetic is inconsistent.
Is there a formal proof the fiction can't come true? (I don'... |
Several papers give poly-time algorithms for constrained paths on labelled graphs, e.g. [1]
Quote:
----------
Given an alphabet Σ, a (directed) graph G whose edges are weighted and Σ-labeled,
and a formal language L ⊆ Σ∗ , the formal-language-constrained shortest/simple path problem con-
sists of finding a short... |
The recent post [here][1] has led to the following question (consider $\cos(sx) = \sum\nolimits_{k = 1}^\infty {\frac{{( - 1)^k (sx)^{2k} }}{{(2k)!}}}$). I find it instructive, whether it will turn out to be easy/well-known or not.
So, suppose that $(f_k)_{k \geq 1}$ is a sequence of continuous functions on a non-c... |
***How many ways of giving out 6 pieces of candy to 3 children if each child must receive at-least one piece ?***
I think, $ 5 \choose 2$ is the answer,consistent to $ n-1 \choose r $ as this is similar to finding the number of non-zero positive integer solution of the equation $X_1 + X_2 + X_3 = 6$ ,but the corre... |
I have a question about abelian X-group:
Suppose A is abelian X-group and it's artinian and noetherian. f is an X-endomorphism on A, i.e. $f\in \mathrm{End}_X(A)$. Also know that f(f(A))=f(A).
Prove $A=ker(f)\dot{+}f(A)$ i.e. A is direct sum of those two subgroup.
My idea is
firstly show $A=ker(f)+f(A)$ which I... |
You have assumed that $6$ candies are identical but this is not the case here.Here actually you have to compute number of onto functions.
This problem can be modeled be letting A represent the set of candies and B represent the set of children.Then a function $f \colon A \to B $ can be interpreted as giving candy $a... |
You have assumed that $6$ candies are identical but this is not the case here.Here actually you have to compute number of onto functions.
This problem can be modeled be letting A represent the set of candies and B represent the set of children.Then a function $f \colon A \to B $ can be interpreted as giving candy $a... |
I have a question about abelian group:
Suppose A is abelian group and it's artinian and noetherian. f is an endomorphism on A, i.e. $f\in \mathrm{End}(A)$. Also know that f(f(A))=f(A).
Prove $A=ker(f)\dot{+}f(A)$ i.e. A is direct sum of those two subgroup.
My idea is
firstly show $A=ker(f)+f(A)$ which I have pr... |
Actually, such an almost-everywhere divergent Fourier series was constructed by Kolmogorov.
For an explicit example, you can consider a [Riesz product][1] of the form:
$$ \prod_{k=1}^\infty \left( 1+ i \frac{cos 10^k x}{k}\right)$$
which is divergent. For more examples, see [here](http://projecteuclid.org/euc... |
Actually, such an almost-everywhere divergent Fourier series was constructed by Kolmogorov.
For an explicit example, you can consider a [Riesz product][1] of the form:
$$ \prod_{k=1}^\infty \left( 1+ i \frac{cos 10^k x}{k}\right)$$
which is divergent. For more examples, see [here][2] and [here][3].
-------... |
Actually, such an almost-everywhere divergent Fourier series was constructed by Kolmogorov.
For an explicit example, you can consider a [Riesz product][1] of the form:
$$ \prod_{k=1}^\infty \left( 1+ i \frac{cos 10^k x}{k}\right)$$
which is divergent. For more examples, see [here][2] and [here][3].
-------... |
For a historical perspective (i.e. if you want to see how geniuses struggled through things that would be natural today, mostly because of things they later discovered), the nice article [*The Discovery of the Series Formula for π by Leibniz, Gregory and Nilakantha*](http://mathdl.maa.org/mathDL/46/?pa=content&sa=viewD... |
Actually, such an almost-everywhere divergent Fourier series was constructed by Kolmogorov.
For an explicit example, you can consider a [Riesz product][1] of the form:
$$ \prod_{k=1}^\infty \left( 1+ i \frac{\cos 10^k x}{k}\right)$$
which is divergent. For more examples, see [here][2] and [here][3].
------... |
I'm investigating an idea in cryptography that requires irreducible polynomials with coefficients of either 0 or 1 (e.g. over GF[2]). Essentially I am mapping bytes to polynomials. For this reason, the degree of the polynomial will always be an integer multiple of 8.
I would like to build a table of such irreducible... |
I have a question about abelian group (or rather X-group):
Suppose A is abelian group (or rather X-group) and it's artinian and noetherian. f is an endomorphism (or rather X-endomorphism) on A, i.e. $f\in \mathrm{End}(A)$. Also know that f(f(A))=f(A).
Prove $A=ker(f)\dot{+}f(A)$ i.e. A is direct sum of those two su... |
I had this problem on my last exam in calculus, I still don't know, how to solve this, so I would appreciate your help.
Let $a_{ij} \in \mathbb{R}$ for $i,j \in \mathbb{N}$. Suppose, that for every $j$ series $S_j = \sum_{i=1}^{\infty}{a_{ij}}$ is absolutely convergent and, for every $i$ exists finite limit (limit $... |
I disagree with observation 2. It gives a sufficient condition that is not necessary unless you only consider projections onto a 1-dimensional subspace.
A weaker sufficient condition, where I am not sure whether its also necessary
is the following: if there is a decomposition $P_A G = BC$, where $B$ is
any regular... |
As I responded on stackoverflow...
You can take the [infinity/uniform norm][1] of the corresponding tuple, which is defined as
$$\lim_{n \rightarrow \infty} (|x_1|^n + |x_2|^n + |x_3|^n + ...)^{\frac{1}{n}}$$
Or you can just have a maximum/infimum function... what exactly is your problem with using that?
... |
As I responded on stackoverflow...
You can take the [infinity/uniform norm][1] of the corresponding tuple, which is defined as
$$\lim_{n \rightarrow \infty} (|x_1|^n + |x_2|^n + |x_3|^n + ...)^{\frac{1}{n}}$$
Or you can just have a maximum/infimum function... what exactly is your problem with using that?
Al... |
As I responded on stackoverflow...
You can take the [infinity/uniform norm][1] of the corresponding tuple, which is defined as
$$\lim_{n \rightarrow \infty} (|x_1|^n + |x_2|^n + |x_3|^n + ...)^{\frac{1}{n}}$$
Or you can just have a maximum/infimum function... what exactly is your problem with using that?
Al... |
As I responded on stackoverflow...
You can take the [infinity/uniform norm][1] of the corresponding tuple, which is defined as
$$\lim_{n \rightarrow \infty} (|x_1|^n + |x_2|^n + |x_3|^n + ...)^{\frac{1}{n}}$$
Or you can just have a maximum/infimum function... what exactly is your problem with using that?
Al... |
Deat Akhil, I have nothing to add to Matt E's masterful survey on the algebraic/analytic comparison. However since you ask about *topological* line bundles, you have to modify his answer in the following way.
You must replace everywhere $\mathcal O$ by $\mathcal C$, the sheaf of continuous functions. Now things are... |
Given a And\Or-graph induced by the transition function, each node of G corresponds to a state q belonging to set Q of the state of the Automaton, for q with $\delta(q,a)=q1*q2$, the node is a $*-node$ with two successor q1 and q2. For q={true,false}, the node q is a sink-node. Hence, if * is $\vee$ then q is $\vee-nod... |
This is a follow-up to the comment about Abel's theorem below Willie Wong's post.
You don't need Abel's theorem if you proceed as follows (taken from "What is Mathematics?" by Courant and Robbins):
$\sum_{k=0}^{n} (-1)^n x^2 = \frac{1+(-1)^nx^{2n+2}}{1+x^2} = \frac{1}{1+x^2} + \frac{(-1)^nx^{2n+2}}{1+x^2}$
... |
In a And\Or-graph induced by the transition function, each node of G corresponds to a state q belonging to set Q of the state of the Automaton, for q with $\delta(q,a)=q1*q2$, the node is a $*-node$ with two successor q1 and q2. For q={true,false}, the node q is a sink-node. Hence, if * is $\vee$ then q is $\vee-node$,... |
According to a paper "Optimal Irreducible Polynomials for GF(2m) Arithmetic" by M. Scott,
"it is in practise always possible to chooose as an irreducible polynomial either a trinomial... or a pentanomial." <a href="http://www.hyperelliptic.org/SPEED/slides/Scott_optimal_polynomials.pdf"> [PDF link]</a>
In random nu... |
According to a paper "Optimal Irreducible Polynomials for GF(2m) Arithmetic" by M. Scott,
"it is in practise always possible to chooose as an irreducible polynomial either a trinomial... or a pentanomial." <a href="http://www.hyperelliptic.org/SPEED/slides/Scott_optimal_polynomials.pdf"> [talk slides]</a> <a href="htt... |
According to a paper "Optimal Irreducible Polynomials for GF(2m) Arithmetic" by M. Scott,
"it is in practise always possible to chooose as an irreducible polynomial either a trinomial... or a pentanomial." <a href="http://www.hyperelliptic.org/SPEED/slides/Scott_optimal_polynomials.pdf"> [talk slides]</a> <a href="htt... |
I was looking at movie times today and was struck by the oddly-spaced showing times. For example, at the local Loew's Theater "Tron: Legacy 3D" (127 min.) is playing on two screens at the following interlaced times: 1:00 pm, 1:45 pm, 4:00 pm, 4:45 pm, 7:00 pm, 7:45 pm, 10:00 pm and 10:45 pm. Why not space the times equ... |
I was looking at movie times today and was struck by the oddly-spaced showing times. For example, at the local Loew's Theater "Tron: Legacy 3D" (127 min.) is playing on two screens at the following interlaced times: 1:00 pm, 1:45 pm, 4:00 pm, 4:45 pm, 7:00 pm, 7:45 pm, 10:00 pm and 10:45 pm. Why not space the times equ... |
Deat Akhil, I have nothing to add to Matt E's masterful survey on the algebraic/analytic comparison. However since you ask about *topological* line bundles, you have to modify his answer in the following way.
You must replace everywhere $\mathcal O$ by $\mathcal C$, the sheaf of continuous functions. Now things are... |
I seem to have a misunderstanding of how to work with a Taylor series.
Suppose I want to write $f(x)=x e^x$ Taylors expansion of $n$ degree around $0$.
I see two ways:
1) Find the n'th derivative of $f(x)$, this is quite easy: $f^{(n)} (x)=n e^x+x e^x$.
And from the formula I get:
$\displaystyle f(x)=... |
1) Sean Carroll claims that the essentials of the vector concept emerged with Galileo Galilei. Galileo observed that if you kick a ball forwards off a building, or drop a ball off a building, that the balls land at the same time.
Therefore horizontal motion (**kick**) is **separable** from vertical motion (**fall**)... |
I seem to have a misunderstanding of how to work with a Taylor series.
Suppose I want to write $f(x)=x e^x$'s Taylor expansion of $n$ degree around $0$.
I see two ways:
1) Find the $n$th derivative of $f(x)$, this is quite easy: $f^{(n)} (x)=n e^x+x e^x$.
And from the formula I get:
$$f(x)=\sum_{k=1}^... |
Does anyone have a good explanation for why Knot invariants tend to be well organized as polynomials? What exactly is going on and why don't we often see polynomial invariants for classifying other geometric objects? For instance, why don't we just say that the Alexander invariants are a finite set of numbers? Pres... |
Why are knot invariants best organized as polynomials? |
I have a question about Hatcher's proof that the fundamental group of a circle is Z. Specifically, halfway through, ( http://www.math.cornell.edu/~hatcher/AT/ATch1.pdf , page 30), he proves an important a lemma stated rather generally:
Given a map $Y\times I\rightarrow S^1$ (Y is any space, and I is an interval), th... |
Here is my set up.
I am running a simulation of a gambler betting over time
Here are the givens I am using.
Probability of win = 35% which is in B1 table
Odds = 2 which is in B3 table
Kelly # (amount of current bankroll to bet each time) = 2.5% which is in D1 table
Bankroll = $10,000 which is in ... |
I was looking at movie times today and was struck by the oddly-spaced showing times. For example, at the local Loew's Theater "Tron: Legacy 3D" (127 min.) is playing on two screens at the following interlaced times: 1:00 pm, 1:45 pm, 4:00 pm, 4:45 pm, 7:00 pm, 7:45 pm, 10:00 pm and 10:45 pm. Why not space the times equ... |
In Wikipedia and MathPlanet an equivalent definition of a symplectic matrix is given:
$$\left( \begin{array}{ccc}
A & B \\
C & D \end{array} \right)$$
is symplectic if and only if:
$$A^TD-C^TB=I, A^TC=C^TA, D^TB=B^TD$$
but it seems wrong, since, for example:
$$\left( \begin{array}{ccc}
0 & 0 & -1 & 1 ... |
Wikipedia definition of Symplectic Matrix is wrong? |
In Wikipedia and MathPlanet an equivalent definition of a symplectic matrix is given:
$$\left( \begin{array}{ccc}
A & B \\
C & D \end{array} \right)$$
is symplectic if and only if:
$$A^TD-C^TB=I, A^TC=C^TA, D^TB=B^TD$$
but it seems wrong, since, for example:
$$\left( \begin{array}{ccc}
0 & 0 & -1 & 1 ... |
What are some good ways to motivate the material on infinite series that appears at the end of a typical American Calculus II course?
My students in this course are generally from biochemistry, computer science, economics, business, and physics (with a few humanities folks taking the course for fun) - not just math ... |
As I mentioned in comments above, Kolmogorov's example is for a discontinuous function in $L^1$.
For a continuous function whose Fourier series diverges at all rational multiples of $2\pi$ (and hence on a dense set) see [Katznelson's book: An Introduction to Harmonic Analysis][1] Chapter 2, Remark after proof of Th... |
As I mentioned in comments below, Kolmogorov's example is for a discontinuous function in $L^1$.
For a continuous function whose Fourier series diverges at all rational multiples of $2\pi$ (and hence on a dense set) see [Katznelson's book: An Introduction to Harmonic Analysis][1] Chapter 2, Remark after proof of Th... |
The vector force field F=(yi,-xj) has a curl of -2. The acceleration of a particle in space is given by:
ax=y/m
ay=-x/m
This vector field has a divergence of 0. Will particles in this vector FORCE field move in circles or will they spiral outwards in a crazy fashion? My conundrum is this, the acceleration of the par... |
The Rule of Three is just doing the same thing you are doing: if $\$20,000$ is $83$%, then $\frac{20000}{83}$ is $1$% of the original. Since $X$ is the original, then $\frac{X}{100}$ is **also** $1$% of the original. That means that the two quantities are equal, so we have:
$$\frac{20,000}{83} = \frac{X}{100}.$$
From... |
Here is my set up.
I am running a simulation of a gambler betting over time
Here are the givens I am using.
Probability of win = 35% which is in B1 table
Odds = 2 which is in B3 table
Kelly # (amount of current bankroll to bet each time) = 2.5% which is in D1 table
Bankroll = $10,000 which is in ... |
Hi i am trying to understand a proof, I dont understand one of the steps.
could someone shed some light why the following is true.
if $f(x)$ is a non-increasing and continuous function and $a>b$ then $\{x\colon f(x) > f(b) \} \subset \{x\colon f(x) > a \} $
i dont see why this is true. any insights ?
thanks |
The Rule of Three is just doing the same thing you are doing: if $\$20,000$ is $83$%, then $\frac{20000}{83}$ is $1$% of the original. Since $X$ is the original, then $\frac{X}{100}$ is **also** $1$% of the original. That means that the two quantities are equal, so we have:
$$\frac{20,000}{83} = \frac{X}{100}.$$
From... |
i was given the following question:
calculate 1-1/2+1/4-1/8+1/16-1/32+1/64-...-1/512+...;express the answer in a simple fraction.
My approach was to use the following formula-1/1-r where r is the common ratio. In the end i got 2/3. Am i correct? |
Is the convergence of the following series 2/3? |
I need to find out $\log_g {-1}$ in $\mathbb{Z}_n$ where $n$ is an odd prime and $g$ is a primitive root mod $n$. How do I do that? Thanks. |
1) Sean Carroll claims that the essentials of the vector concept emerged with **Galileo Galilei**. Galileo observed that if you kick a ball forwards off a building, or drop a ball off a building, that the balls land at the same time.
Therefore horizontal motion (**kick**) is **separable** from vertical motion (**fal... |
Yes, and your reason is correct. If you want a simple way to check your work on such things, you could use a computer algebra system like Mathematica, or simply Wolfram Alpha:
http://www.wolframalpha.com/input/?i=sum+%28-1%2F2%29^n |
Yes, and your reason is correct. If you want a simple way to check your work on such things, you could use a computer algebra system like Mathematica, or simply Wolfram Alpha:
http://www.wolframalpha.com/input/?i=sum+%28-1%2F2%29%5En |
**Question**:
We say that a convex lattice polytope $P\subset \mathbb{R}^d$ is a _smooth Fano polytope_ if:
1. The origin is contained in the interior of $P$
2. The vertices of every facet of $P$ are a $\mathbb{Z}$-basis of $\mathbb{Z}^d$
Now, suppose we have a set $V=\{v_1,\ldots,v_n\}\subset \mathbb{Z}^d$... |
How to check if a polytope is a smooth Fano polytope? |
I was given the following question:
>Calculate
$$1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\frac{1}{16}- \frac{}{32}+\frac{1}{64}-\cdots -\frac{1}{512}+\cdots;$$
express the answer in a simple fraction.
My approach was to use the following formula: $\frac{1}{1-r}$ where $r$ is the common ratio. In the end I got $2/... |
Does the following series converge to 2/3? |
I was given the following question:
>Calculate
$$1-\frac12+\frac14-\frac18+\frac1{16}- \frac1{32}+\frac1{64}-\cdots -\frac1{512}+\cdots;$$
and express the answer as a simple fraction.
My approach was to use the following formula: $\frac1{1-r}$ where $r$ is the common ratio. In the end I got $2/3$. Am I correct... |
Does the following series converge to 2/3? |
The vector force field F=(yi,-xj) has a curl of -2. The acceleration of a particle in space is given by:
ax=y/m
ay=-x/m
This vector field has a divergence of 0. Will particles in this vector FORCE field move in circles or will they spiral outwards in a crazy fashion? My conundrum is this, the acceleration of the par... |
I was given the following question:
>Calculate
$$1-\frac12+\frac14-\frac18+\frac1{16}- \frac1{32}+\frac1{64}-\cdots -\frac1{512}+\cdots;$$
and express the answer as a simple fraction.
My approach was to use the following formula: $\frac1{1-r}$ where $r$ is the common ratio. In the end I got $2/3$. Am I correct... |
I was reading up on class field theory, and I have a question. On wiki (http://en.wikipedia.org/wiki/Artin_reciprocity), one formulation is that there's *some* modulus for which $I^c_K/i(K_{c,1})Nm_{L/K}(I^c_L)$ is isomorphic to $Gal(L/K)$.
Another formulation on the same page is:
$C_K/N_{L/K}(C_L)$ is isomorphic t... |
I was reading up on class field theory, and I have a question. On wiki (http://en.wikipedia.org/wiki/Artin_reciprocity), one formulation is that there's *some* modulus for which $I^c_K/i(K_{c,1})Nm_{L/K}(I^c_L)$ is isomorphic to $Gal(L/K)$.
Another formulation on the same page is:
$C_K/N_{L/K}(C_L)$ is isomorphic t... |
One of the perks of my research topic (Latin squares) is that it's somewhat possible to explain what I do to those with a fairly minimal mathematics background. I'll pose a typical question that comes up in one of my main research topics as a problem here -- I'm hoping that this will whet the appetite of someone cons... |
Does there exist a Latin square of order 16 that admits a specific automorphism? |
<blockquote>Is there a formal proof the fiction can't come true?</blockquote>
No, by <a href="http://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems#Second_incompleteness_theorem">Gödel's second incompleteness theorem</a>, formal systems can prove their own consistency if and only if they are inconsiste... |
> How many stripes can you paint on a 2-group of fixed size?
A group of order 2<sup>a</sup> p<sup>b</sup> is solvable, by Burnside's theorem, so its chief factors are either abelian 2-groups or abelian p-groups. Such a group is a **zebra group** if its chief factors in any chief series alternate between the 2 and t... |
I believe there is also a proof by associating the tiling with an electric network!
This is a section in the book "Modern Graph Theory" by Béla Bollobás.
For instance see this: http://books.google.com/books?id=SbZKSZ-1qrwC&pg=PA46
A snapshot of the page which talks about Dehn's theorem and it's p... |
It is well-known that there is only one "kind" of line, and that there are three "kinds" of quadratic curves (the nature of which depends on the sign of a so-called "discriminant").
It is noteworthy that many of the named cubic curves look rather similar: the [folium of Descartes][1], the [trisectrix of Maclaurin][2... |
How many cubic curves are there? |
I believe there is also a proof by associating the tiling with an electric network!
This is a section in the book "Modern Graph Theory" by Béla Bollobás.
It is titled "Squaring the Square" and is numbered section II.2. See this: http://books.google.com/books?id=SbZKSZ-1qrwC&pg=PA46.
A snapshot of... |
I'm investigating an idea in cryptography that requires irreducible polynomials with coefficients of either 0 or 1 (e.g. over GF[2]). Essentially I am mapping bytes to polynomials. For this reason, the degree of the polynomial will always be an integer multiple of 8.
I would like to build a table of such irreducible... |
Is it true that $\frac1{n!} \int_0^\infty x^{n-k} (1-x)^k e^{-x}\,dx \approx e^{-k/n}$ when $k$ and $n$ are large integers with $k \le n$?
This quantity is the probability that a random permutation of $n$ elements does not fix any of the first $k$ elements.
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