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Let $G$ be a group of permutations of $n$ objects. Let subsets $A$,$B$ of the objects be equivalent if $A=g(B)$ for some $g \in G$.
Is there a way to show that a bijection exists between the equivalence classes and the $2$-colorings of the objects that are invariant under those permutations in $G$?
So Far, I hav... |
I am a calculus TA a bit stumped at a question a student has posed me.
What is the method for finding the sum of the series $$\sum_{n=0}^\infty\frac{1+n}{3^n}$$ If you split the numerator up, one becomes a geometric series summing to 3/2, and the other sums to 3/4 by <a href="http://www.wolframalpha.com/input/?i=sum... |
Find Fourier cosine transform of $e^{-a^2 x^2}$ and hense evaluate Fourier sine transform of $x\cdot e^{-a^2x^2}$.
I can solve this question only if there is $x$ instead of $x^2$ in the exponential function $e^{-a^2x^2}$. Because in this situation i can use integral formula :-
$$\int e^{-ax} \cos sx\,dx = \frac{e^... |
I am a calculus TA a bit stumped at a question a student has posed me.
What is the method for finding the sum of the series $$\sum_{n=0}^\infty\frac{1+n}{3^n}$$ If you split the numerator up, one becomes a geometric series summing to 3/2, and the other sums to 3/4 by <a href="http://www.wolframalpha.com/input/?i=sum... |
Please explain how Conditionally Convergent can be valid? |
Let $G$ be a group of permutations of $n$ objects. Let subsets $A$,$B$ of the objects be equivalent if $A=g(B)$ for some $g \in G$.
Is there a way to show that a bijection exists between the equivalence classes and the $2$-colorings of the objects that are invariant under those permutations in $G$?
So Far, I hav... |
Let $G$ be a group of permutations of $n$ objects. Let subsets $A$,$B$ of the objects be equivalent if $A=g(B)$ for some $g \in G$.
Is there a way to show that a bijection exists between the equivalence classes and the $2$-colorings of the objects that are invariant under those permutations in $G$?
So Far, I hav... |
I am a calculus TA a bit stumped at a question a student has posed me.
What is the method for finding the sum of the series $$\sum_{n=0}^\infty\frac{1+n}{3^n}$$ If you split the numerator up, one becomes a geometric series summing to 3/2, and the other sums to 3/4 by <a href="http://www.wolframalpha.com/input/?i=sum... |
Let $G$ be a group of permutations of $n$ objects. Let subsets $A$,$B$ of the objects be equivalent if $A=g(B)$ for some $g \in G$.
Is there a way to show that a bijection exists between the equivalence classes and the $2$-colorings of the objects that are invariant under those permutations in $G$?
So Far, I hav... |
**Hint:** Suppose $\lambda$ is an eigenvalue. Then there exists an eigenvector $\mathbf{v}$ corresponding to $\lambda$. Write $\mathbf{v}=\mathbf{w}_1+\mathbf{w}_2$; since $\mathbf{v}$ is nonzero, at least one of $\mathbf{w}_1$ and $\mathbf{w}_2$ is nonzero.
Now, evaluate $T(\mathbf{v})$; since $\mathbf{V}$ is a *d... |
Let's say that I have a taxe of 17% over my services and I want to receive $ 20.000,00 regardless the taxe.
So I want something like:
X - 17% = $ 20.0000
Where X is the final value that I need to charge to receive $ 20.000
How do I calculate this? |
HINT 1:
$$
\int_{ - \infty }^\infty {\cos (sx)e^{ - a^2 x^2 } \,{\rm d}x} = \int_{ - \infty }^\infty {[\cos (sx) + {\rm i}\sin (sx)]e^{ - a^2 x^2 } \,{\rm d}x}.
$$
HINT 2:
Characteristic function of the (centered) normal distribution: see [this][1].
**EDIT**: The first hints didn't help. So,
$$
\in... |
Let's say that I have a taxe of `17%` over my services and I want to receive 20.000,00 regardless the taxe.
So I want something like:
X - 17% = $ 20.0000
Where X is the final value that I need to charge to receive 20.000
How do I calculate this? |
Let's say that I have a tax of 17% over my services and I want to receive \$20,000.00 regardless (after paying) the tax.
So I want something like:
$$X - 17\% = \$20,000.00$$
Where $X$ is the final value that I need to charge to receive \$20,000
How do I calculate this? |
HINT 1:
$$
\int_{ - \infty }^\infty {\cos (sx)e^{ - a^2 x^2 } \,{\rm d}x} = \int_{ - \infty }^\infty {[\cos (sx) + {\rm i}\sin (sx)]e^{ - a^2 x^2 } \,{\rm d}x}.
$$
HINT 2:
Characteristic function of the (centered) normal distribution: see [this][1].
**EDIT**: The first hints didn't help. So,
$$
\in... |
HINT 1:
$$
\int_{ - \infty }^\infty {\cos (sx)e^{ - a^2 x^2 } \,{\rm d}x} = \int_{ - \infty }^\infty {[\cos (sx) + {\rm i}\sin (sx)]e^{ - a^2 x^2 } \,{\rm d}x}.
$$
HINT 2:
Characteristic function of the (centered) normal distribution: see [this][1].
**EDIT**: The first hints didn't help. So,
$$
\in... |
Draw two balls centered at the origin in R^3, where $r_1 < r_2$. What is the topological name for the boundary created? Does it even make sense to talk about a figure with an inside boundary and an outside one? |
What is topological name a sphere with a hole in it? |
Let's say that I have a tax of 17% over my services and I want to receive \$20,000.00 regardless (after paying) the tax.
So I want something like:
$$X - 17\% = \$20,000.00$$
Where $X$ is the final value that I need to charge to receive $\$20,000$.
How do I calculate this? |
Ok, so I just finished a Discrete Math class where we learned about all types of proofs and counting and graphs, etc. So I know the basics, but never thought I'd need to use it in the real world. However, a friend of mine showed me a puzzle game, and he said that it's *always* possible to win this game, even though it ... |
Consider the [Chernoff bound](http://en.wikipedia.org/wiki/Chernoff_bound): It assumes $X_1, \ldots, X_n$ are independent Bernoulli random variables, each having probability $p > 1/2$, and seeks the probability of simultaneous occurrence of more than $n/2$ of the events.
I once heard that, in the proof of the Cherno... |
If we have a two lattices (totally ordered) - one for subgroups, one for factor groups, and we know order of some set - is group on this set (supplying thas two lattices) uniquely determined by that lattices - up to isomorphism (if exists)? Or is there any counterexample?
If that's true, are sufficient conditions on... |
If we have a two lattices (totally ordered) - one for subgroups, one for factor groups, and we know order of the group we want to have these subgroup and factor group lattices, is such a group unique up to isomorphism (if exists)? Or is there a counterexample?
If that's true, are sufficient conditions on the order a... |
Ok, so I just finished a Discrete Math class where we learned about all types of proofs and counting and graphs, etc. So I know the basics, but never thought I'd need to use it in the real world (not anytime soon at least). However, a friend of mine showed me a puzzle game, and he said that it's *always* possible to wi... |
What is topological name of a sphere with a hole in it? |
If we have a two lattices (partially ordered) - one for subgroups, one for factor groups, and we know order of the group we want to have these subgroup and factor group lattices, is such a group unique up to isomorphism (if exists)? Or is there a counterexample?
If that's true, are sufficient conditions on the order... |
Is it possible to construct a non-measurable set of a given outer measure $x \in [0,1]$? This will probably require the axiom of choice. Does anyone have a suggestion? |
Anyone know how to convert from lat/lng to EPSG:27700? |
Formula to convert from lat/lng to EPSG:27700? |
For a way to find the limit without using more advanced concepts like McLaurin series etc...
Consider
$$\dfrac{\tan(2x) - 2x}{(2x)^2} = \dfrac{ \frac{2\tan x}{1-\tan^2 x} - 2x}{4x^2}$$
$$ = \dfrac{(2\tan x - 2x) + 2x \tan^2 x}{4x^2(1 - \tan^2 x)} = (\dfrac{\tan - x}{2x^2} + \dfrac{x\tan^2 x}{2x^2}) \dfrac{1}{... |
For a way to find the limit without using more advanced concepts like McLaurin series etc...
Consider
$$\dfrac{\tan(2x) - 2x}{(2x)^2} = \dfrac{ \frac{2\tan x}{1-\tan^2 x} - 2x}{4x^2}$$
$$ = \dfrac{(2\tan x - 2x) + 2x \tan^2 x}{4x^2(1 - \tan^2 x)} = (\dfrac{\tan - x}{2x^2} + \dfrac{x\tan^2 x}{2x^2}) \dfrac{1}{... |
In multivariable calculus, I was taught to compute the chain rule by drawing a "tree diagram" (a directed acyclic graph) representing the dependence of one variable on the others. I now want to understand the theory behind it.
Examples:
Let $y$ and $x$ both be functions of $t$.
Let $z$ be a function of both $x$ an... |
chain rule using tree diagram, why does it work? |
For a way to find the limit without using more advanced concepts like McLaurin series etc...
Consider
$$\dfrac{\tan(2x) - 2x}{(2x)^2} = \dfrac{ \frac{2\tan x}{1-\tan^2 x} - 2x}{4x^2}$$
$$ = \dfrac{(2\tan x - 2x) + 2x \tan^2 x}{4x^2(1 - \tan^2 x)} = (\dfrac{\tan x - x}{2x^2} + \dfrac{x\tan^2 x}{2x^2}) \dfrac{1... |
Is it possible to construct a non-measurable set in $[0,1]$ of a given outer measure $x \in [0,1]$? This will probably require the axiom of choice. Does anyone have a suggestion?
**Edit:** I forgot to mention that the set has to be in $[0,1]$ which seemed "clear" in my head, but which obviously isn't. |
The tensor product $M\otimes N$ of abelian groups cannot be non-zero and free unless both $M$ and $N$ are free. This is implied by the following facts about abelian groups.
1. If $m\in M$ is nonzero and $n\in N$ is not a torsion element, then the product $m\otimes n\in M\otimes N$ is nonzero. Equivalently, for any ... |
Let's say that I have a tax of 17% over my services and I want to receive $20,000.00 regardless (after paying) the tax.
So I want something like:
$$X - 17\% = \$20,000.00$$
Where $X$ is the final value that I need to charge to receive $\$20,000$.
How do I calculate this? |
This was my answer to the question as originally phrased: Once you have one nonmeasurable set with finite outer measure, you can get all positive values by rescaling.
With the additional requirement that the set is contained in $[0,1]$, this argument will work as long as you have a nonmeasurable subset of outer me... |
Here is an attempt at a geometric proof.
![alt text][1]
(Please pardon the shoddy paint job).
Consider $\triangle ABC$ such that $\angle{BCA} = x$. Let $BC=1$ and so $AB = \tan x$.
Let $BE$ be the arc of radius 1 and angle $x$ drawn with $C$ as the center (note that $E$ is on AC, between $A$ and $D$ and i... |
Is the projective closure of an infinite affine variety (over an algebraically closed field, I only care about the classical case right now) always strictly larger than the affine variety? I know it is an open dense subset of its projective closure, but I don't think it can actually be its own projective closure unless... |
Here is an attempt at a geometric proof.
![alt text][1]
(Please pardon the shoddy paint job).
Consider $\triangle ABC$ such that $\angle{BCA} = x$. Let $BC=1$ and so $AB = \tan x$.
Let $BE$ be the arc of radius 1 and angle $x$ drawn with $C$ as the center (note that $E$ is on AC, between $A$ and $D$ and i... |
Here is an attempt at a geometric proof.
![alt text][1]
(Please pardon the shoddy paint job).
Consider $\triangle ABC$ such that $\angle{BCA} = x$. Let $BC=1$ and so $AB = \tan x$.
Let $BE$ be the arc of radius 1 and angle $x$ drawn with $C$ as the center (note that $E$ is on AC, between $A$ and $D$ and i... |
(i) $ \forall x < p: f(x) < f(p-)$, likewise $ \forall x > p: f(x) > f(p+)$ (because $f$ is strictly increasing).
$f(p-) \leq f(p) \leq f(p+)$ (because $f$ is strictly increasing).
If $f(p-) \neq f(p) \neq f(p+)$, let $a_p = f(p-)$ and $b_p = f(p+)$.
It follows that $\forall x < p: f(x) < a_p$, likewise $\fora... |
Using [Gauss's algorithm][1] we have $\rm\displaystyle\ \frac{1}8\ \equiv\ \frac{4}{32}\ \equiv\ \frac{4}{3}\ \equiv\ \frac{40}{30}\ \equiv\ \frac{11}{1}\ \ (mod\ 29)$
I.e. scale $\rm A/B\ \to AN/BN\ $ by the least $\rm\:N\:$ so that $\rm\ BN > 29\:.\ $ Then reduce the numerator and denominator $\rm\ mod\ 29,\:$ ... |
Is it possible to construct a non-measurable set in $[0,1]$ of a given outer measure $x \in [0,1]$? This will probably require the axiom of choice. Does anyone have a suggestion?
**Edit:** I forgot to mention that the set has to be in $[0,1]$ which seemed "clear" in my head, but which obviously isn't.
**Edit 2:**... |
**Edit** this is just wrong. Maybe the argument can be salvaged, but my first "fact" below is plain wrong. I'll come back to this...
The tensor product $M\otimes N$ of abelian groups cannot be non-zero and free unless both $M$ and $N$ are free. This is implied by the following facts about abelian groups.
1. If $... |
Using [Gauss's algorithm][1] we have $\rm\displaystyle\ \frac{1}8\ \to\ \frac{4}{32}\ \equiv\ \frac{4}{3}\ \to\ \frac{40}{30}\ \equiv\ \frac{11}{1}\ \ (mod\ 29)$
I.e. scale $\rm A/B\ \to AN/BN\ $ by the least $\rm\:N\:$ so that $\rm\ BN > 29\:.\ $ Then reduce the numerator and denominator $\rm\ mod\ 29,\:$ and it... |
This was my answer to the question as originally phrased: Once you have one nonmeasurable set with finite outer measure, you can get all positive values by rescaling.
With the additional requirement that the set is contained in $[0,1]$, this argument will work as long as you have a nonmeasurable subset of outer me... |
This was my answer to the question as originally phrased: Once you have one nonmeasurable set with finite outer measure, you can get all positive values by rescaling.
With the additional requirement that the set is contained in $[0,1]$, this argument will work as long as you have a nonmeasurable subset of outer me... |
If $E$ is Lebesgue measurable in $\mathbb{R}^n$ and $I=[a,b]$ how do I show that $E\times I$ is measurable in $\mathbb{R}^{n+1}$?
Jonas:
I'm using $\mu^*(E)=\inf \{ \sum \mathrm{Vol}(I_k) \mid E\subseteq \cup I_k\}$ and for every $\epsilon \gt 0$ there exists an open set $G$ containing $E$ such that $\mu^*(G-E)\l... |
The tensor product $M\otimes N$ of abelian groups cannot be non-zero and free unless both $M$ and $N$ are free.
The argument is is follows (with apologies for my original flawed answer).
Let $T$ be the torsion sub-module of $N$, so that $N/T$ is torsion free and $0\to T\to N\to N/T\to 0$ is an exact sequence. The... |
Let's say I have a list of points( in 2D coordinates!) with the vertex $V$, how to find all the triangular faces $F$ ( which has three edges $E$) that covers **all** of the points? In other words, I'm looking for a mesh that connects all the points together with well behaved triangles.
The conditions are that the fa... |
Let's say I have a list of points( in 2D coordinates!) with the vertex $V$, how to find all the triangular faces $F$ ( which has three edges $E$) that covers **all** of the points? In other words, I'm looking for a mesh that connects all the points together with well behaved triangles.
The conditions are that the fa... |
The tensor product $M\otimes N$ of abelian groups cannot be non-zero and free unless both $M$ and $N$ are free. This follows from the following facts about tensor products.
1. Any subgroup of a free abelian group is free. This holds even in the infinitely generated case. Wikipedia has a proof in its article on [fre... |
The tensor product $M\otimes N$ of abelian groups cannot be non-zero and free unless both $M$ and $N$ are free. This follows from the following facts.
1. Any subgroup of a free abelian group is free. This holds even in the infinitely generated case. Wikipedia has a proof in its article on [free abelian groups][1].
... |
if I have 2 lists of time intervals :
List1 :
1. 2010-06-06 to 2010-12-12
2. 2010-05-04 to 2010-11-02
3. 2010-02-04 to 2010-10-08
4. 2010-04-01 to 2010-08-02
5. 2010-01-03 to 2010-02-02
and List2 :
1. 2010-06-08 to 2010-12-14
2. 2010-04-04 to 2010-10-10
3. 2010-02-02 to 2010-12-16
What ... |
In multivariable calculus, I was taught to compute the chain rule by drawing a "tree diagram" (a directed acyclic graph) representing the dependence of one variable on the others. I now want to understand the theory behind it.
Examples:
Let $y$ and $x$ both be functions of $t$.
Let $z$ be a function of both $x$ an... |
The tensor product $M\otimes N$ of abelian groups cannot be non-zero and free unless both $M$ and $N$ are free. This follows from the following facts.
1. Any subgroup of a free abelian group is free. This holds even in the infinitely generated case. Wikipedia has a proof in its article on [free abelian groups][1].
... |
What would be the best way to calculate some sort of correlation or similarity factor between two lists of time intervals :
List1 :
1. 2010-06-06 to 2010-12-12
2. 2010-05-04 to 2010-11-02
3. 2010-02-04 to 2010-10-08
4. 2010-04-01 to 2010-08-02
5. 2010-01-03 to 2010-02-02
and List2 :
1. 2010-06-0... |
There's a question in Spivak's *Calculus* (I don't happen to have the question number in front of me) that develops the concept of continued fraction, specifically the continued fraction representation of $\sqrt{2}$. Having only recently become aware of continued fractions, I'm trying to work my way through this probl... |
There's a question in Spivak's *Calculus* (I don't happen to have the question number in front of me) that develops the concept of continued fraction, specifically the continued fraction representation of $\sqrt{2}$. Having only recently become aware of continued fractions, I'm trying to work my way through this probl... |
For a two-sphere embedded in R^4,how can you check whether or not there is an ambient isotopy to the "standard" 2-sphere (the set of points (x,y,z,0) in R^4 distance 1 from the origin)?
Knot theory was discussed in an intro topology course I took and I'm wondering about further generalizations of the concept. It seem... |
For a two-sphere embedded in $\mathbb{R}^4$,how can you check whether or not there is an ambient isotopy to the "standard" 2-sphere (the set of points $(x,y,z,0)$ in $\mathbb{R}^4$ distance 1 from the origin)?
Knot theory was discussed in an intro topology course I took and I'm wondering about further generalization... |
The adelics seem counter-intuitive. I wonder how they came up originally, and what was the immediate reward for introducing them. What was the politics of introducing the adelics into mathematical mainstream? |
Does the order, lattice of subgroups, and lattice of factor groups, uniquely determine a group up to isomorphism? |
Consider the curve $\frac{1}{x}$ where $x \geq 1$. Rotate this curve around the x-axis.
One Dimension - Clearly this structure is infinitely long.
Two Dimensions - Surface Area = $2\pi\int_∞^1\frac{1}{x}dx = 2\pi(\ln ∞ - \ln 1) = ∞$
Three Dimensions - Volume = $\pi\int_∞^1{x}^{-2}dx = \pi(-\frac{1}{∞} + \fr... |
How can a structure have infinte length and infinte surface area, but have finte volume? |
How can a structure have infinte length and infinte surface area, but have finite volume? |
Let $S$ be a ruled surface over a curve of genus $g$. Is it possible to compute the second Chern class of $S$ in terms of $g$? |
Here is an attempt at a geometric proof.
![alt text][1]
(Figure thanks to J.M)
Consider $\triangle ABC$ such that $\angle{BCA} = x$. Let $BC=1$ and so $AB = \tan x$.
Let $BE$ be the arc of radius 1 and angle $x$ drawn with $C$ as the center (note that $E$ is on AC, between $A$ and $D$ and is kind of hidde... |
In multivariable calculus, I was taught to compute the chain rule by drawing a "tree diagram" (a directed acyclic graph) representing the dependence of one variable on the others. I now want to understand the theory behind it.
Examples:
Let $y$ and $x$ both be functions of $t$.
Let $z$ be a function of both $x$ an... |
Here is an attempt at a geometric proof.
![alt text][1]
(Figure thanks to J.M)
Consider $\triangle ABC$ such that $\angle{BCA} = x$. Let $BC=1$ and so $AB = \tan x$.
Let $BE$ be the arc of radius 1 and angle $x$ drawn with $C$ as the center (note that $E$ is on AC, between $A$ and $D$ and is kind of hidde... |
Just wondering:
1. How to solve ultrahyperbolic PDEs? Is there any analytical solution for linear ultrahyperbolic PDEs?
2. If there are only numerical solutions, are the solutions' behavior similar to those of nonlinear eqns? I mean, like an anharmonic oscillator, chaotic but deterministic?
Information of refere... |
I've always had the following silly(?) doubt about convergent sequences.
Given a finite sequence $A_{1:n}=(a_1,a_2\ldots ,a_n)$, define its <i> reverse </i> as $rev(A) = (a_n,a_{n-1},..a_1)$. Further, if $A$ is infinite, define its reverse as the limit of the reverses of its initial subsequences, i.e. as the limit... |
The ideles came first, and were introduced by Chevalley as part of his program of giving a (then) new proof of class field theory which avoided the analysis of $L$-function. The adeles came a little later; they were originally called *valuation vectors*, and were introduced by Artin and Whaples. It took a little time... |
The surface integral of a vector field, $F$ over a surface $S$ (or flux as it is called) is, as stated above, the net flow of fluid through the surface. The purpose of orientation is to simply indicate which side of the surface is the "positive" side. For instance, if I calculate flux to be +10, this means there is a... |
Is there an algorithm to find out all combinations of series, parallel, and series-parallel arrangements given $n$ identical resistors, $R$? All of them must be used.
This might even extend to differently-valued resistors, but I'll just focus on identical resistors. |
The ideles came first, and were introduced by Chevalley as part of his program of giving a (then) new proof of class field theory which avoided the analysis of $L$-function. The adeles came a little later; they were originally called *valuation vectors*, and were introduced by Artin and Whaples. It took a little time... |
Assume that $V$ is an $n$-dimensional Euclidean space. Then, given an orthonormal basis $(e_i)_{i=1}^n$ and an arbitrary n-tuple of vectors $(u_i)_{i=1}^n$, there exists a sequence of $n$ isometries $H_1,...,H_n$ such that
- $H_i$ is either *a hyperplane reflection* or the identity;
- every vector of the form
$... |
Concerning [this](http://math.stackexchange.com/questions/13646/how-to-derive-the-following-identity) question of mine which involves telescoping in the solution, I was wondering if it is possible to express
$\displaystyle \ln(x)=\mathop {\lim }\limits_{n \to \infty } n\left(x^{\frac{1}{n}}-1 \right)$
and
$\di... |
Can $\displaystyle \ln(x)=\mathop {\lim }\limits_{n \to \infty } n\left(x^{\frac{1}{n}}-1 \right)$ be expressed as an infinite teslescoping product? |
Concerning [this](http://math.stackexchange.com/questions/13646/how-to-derive-the-following-identity) question of mine which involves telescoping in the solution, I was wondering if it is possible to express
$\displaystyle \ln(x)=\mathop {\lim }\limits_{n \to \infty } n\left(x^{\frac{1}{n}}-1 \right)$
and
$\di... |
One partition of 8 is 5 + 3, but if we then partition each of the 5 and 3 we could get (3+2) + (2+1), and then partition again to get ((2+1)+(1+1)) + ((1+1)+1) and finally (((1+1)+1)+(1+1)) + ((1+1)+1). 5+3 could also be expanded as (4+1)+(2+1), then ((2+2)+1)+((1+1)+1), then (((1+1)+(1+1))+1)+((1+1)+1).
This questi... |
I am trying to figure out how many permutations exist in a set where none of the numbers equal their own position in the set for example, 3,1,5,2,4 is an acceptable permutation where 3,1,2,4,5 is not because 5 is in position 5. I know that the number of total permutations is n!. is there a formula for how many are ac... |
Recall that complex topological $K$-theory is representable on reasonable spaces by the space $BU \times \mathbb{Z}$ (where $BU$ is a colimit of various infinite Grassmannians), and that the total Chern class provides a natural map $\mathrm{Vect}(B) \to H^*(B)$ for every such space $B$. By the multiplicativity propert... |
What's the map $BU \times \mathbb{Z} \to \prod K(\mathbb{Z},n)$ representing the total Chern class? |
Concerning [this](http://math.stackexchange.com/questions/13646) question of mine which involves telescoping in the solution, I was wondering if it is possible to express
$$\ln(x)=\lim_{n\to\infty}n\left(x^{\frac1{n}}-1\right)$$
and
$$e^x=\lim_{n\to\infty}\left(1+\frac{x}{n}\right)^n$$
as infinite telescopi... |
Can $\ln(x)=\lim_{n\to\infty} n\left(x^{\frac1{n}}-1 \right)$ be expressed as an infinite telescoping product? |
I am trying to figure out how many permutations exist in a set where none of the numbers equal their own position in the set; for example, 3,1,5,2,4 is an acceptable permutation where 3,1,2,4,5 is not because 5 is in position 5. I know that the number of total permutations is n!. Is there a formula for how many are ac... |
I'm trying to compute the variational distance between two negative exponential distributions. If $a$ denotes the value of $x$ where both curves are equal, then:
$$\int_0^{+\infty}\left|\lambda_1e^{-\lambda_1x}-\lambda_2e^{-\lambda_2x}\right|dx$$
$$=-\int_{0}^a(\lambda_1e^{-\lambda_1x}-\lambda_2e^{-\lambda_2x})dx... |
There's a question in Spivak's *Calculus* (<strike>I don't happen to have the question number in front of me</strike> in the 2nd Edition, it's Chapter 21, Problem 7) that develops the concept of continued fraction, specifically the continued fraction representation of $\sqrt{2}$. Having only recently become aware of c... |
Recall that complex topological $K$-theory is representable on reasonable spaces by the space $BU \times \mathbb{Z}$ (where $BU$ is a colimit of various infinite Grassmannians), and that the total Chern class provides a natural map $\mathrm{Vect}(B) \to H^*(B)$ for every such space $B$. By the multiplicativity propert... |
1. Can a CFG over large alphabet describe a language, where each terminal appears even number of times?
2. If yes, would the Chomsky Normal Form be polynomial in |Σ| ?
EDIT: What about a language where each terminal appears 0 or 2 times?
EDIT: Solving this may give interesting extension to <a href="http://math... |
**HINT** $\rm\displaystyle\quad \frac{A}B < \frac{X}Y < \frac{C}D\ \iff\ X = \frac{MA+NC}{BC-AD},\ Y = \frac{MB+ND}{BC-AD}\quad$ for some integers $\rm\:M,N>0$
This has a nice geometric interpretation in terms of plane lattices - which provides a vivid geometric interpretation of good approximations by Farey mediant... |
How can a structure have infinite length and infinite surface area, but have finite volume? |
How do i Find the Area or the volume(Or Formula) of a area that is rectangular but has a cone slope to a drain ? |
Yes. If $F'=f$ and $\gamma:[a,b]\to D$ is a curve, then $\int_\gamma f = F(\gamma(b))-F(\gamma(a))$. For example, see Theorem 1.18 on [page 65][1] of John B. Conway's *Functions of one complex variable*.
[1]: http://books.google.com/books?id=9LtfZr1snG0C&lpg=PP1&ots=OePFw5q3Ex&dq=conway%20complex&pg=PA65#v=one... |
I prefer words, really. What the mean value theorem tells us is that at some point on the interval $[a,b]$ the function $f(x)$ (continuous and differentiable on this interval) has a gradient equal to the slope of the line connecting $a$ and $b$.
Why should this be? Well, the trivial case we can handle, namely that i... |
How do I find the area or the volume (or a formula) for a region that is rectangular but has a cone slope to a drain ? |
Why is integral of $x$ from $-1$ to $1$ $x^2/2$ rather than $x^2$? |
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