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As part of my solution to a problem, I come to a point where I need to find the solutions to
$-2\partial_{T}B\left(T\right)+\frac{3}{4}B\left(T\right)\left(A\left(T\right)^{2}+1\right)=0$
$2\partial_{T}A\left(T\right)+\frac{3}{4}A\left(T\right)\left(B\left(T\right)^{2}-A\left(T\right)^{2}\right)=0$
where $\parti... |
What are the conditions for torsion to be zero other than having a plane curve?
The only thing I can thing of is an equation that have the torsion that cancels out each other. |
$$S = \sum_{k=1}^{\infty} \frac{cos(\theta\log(k))}{k^a}$$
How do I go about finding the value of S, given that $\theta \to \infty$ and $0 < a < 1$.
> Any special techniques that might be
> helpful in calculating this sum? |
Write your series as
$$\sum_{n=1}^\infty \sum_{m=1}^\infty z^{mn},$$
and consider $mn=k,$ say. Does this help you? |
Write your series as
$$\sum_{n=1}^\infty \sum_{m=1}^\infty z^{mn},$$
and consider $mn=k,$ say. Does this help you?
If you get stuck look [here.][lambert]
[lambert]: http://en.wikipedia.org/wiki/Lambert_series |
There isn't any simple approximation of the sort you seek. Simply defining an angle $\theta_{k\ell}$ such that $\cos(\theta_{k\ell}) = \tfrac12 \bigl( \cos(\tfrac{2\pi k}n) + \cos(\tfrac{2\pi \ell}n) \bigr)$ is not sufficient. There are infinitely many different angles $\theta_{k\ell}$ which satisfy this equation, afte... |
Or am I completely off course here?
|
Is the difference between an integral and a function that an integral describes an area, while a function describes a line? |
There isn't any simple approximation of the sort you seek. Simply defining an angle $\theta_{k\ell}$ such that $\cos(\theta_{k\ell}) = \tfrac12 \bigl( \cos(\tfrac{2\pi k}n) + \cos(\tfrac{2\pi \ell}n) \bigr)$ is not sufficient. There are infinitely many different angles $\theta_{k\ell}$ which satisfy this equation, afte... |
Am I correct to say:
A difference between an integral and a function is: that an integral can describe an area on a graph, while a function can't?
Or am I completely off course here?
|
Can a function describe an area on a graph like an integral? |
Is there a more efficient algorithm besides [Gram-Schmidt][1] that would produce an orthonormal matrix of rank N, with first row equal to [1 1 1 1 1 ... 1] / sqrt(N)?
[1]: http://en.wikipedia.org/wiki/Gram-Schmidt_process#Algorithm |
Is there a more efficient algorithm besides [Gram-Schmidt][1] that would produce an orthonormal matrix of rank N, with first row equal to [1 1 1 1 1 ... 1] / sqrt(N)?
e.g. for N = 3, the matrix $\begin{align}
\mathsf A_3 &= \begin{bmatrix}
1/\sqrt 3 & 1/\sqrt 3 & 1/\sqrt 3 \\
2/\sqrt 6 & -1/\sqrt 6 & -1/\sqrt... |
Is there a more efficient algorithm besides [Gram-Schmidt][1] that would produce an orthonormal matrix of rank N, with first row equal to [1 1 1 1 1 ... 1] / sqrt(N)?
e.g. for N = 3, the matrix $\begin{align}
\mathsf A_3 &= \begin{bmatrix}
1/\sqrt 3 & 1/\sqrt 3 & 1/\sqrt 3 \\\\
2/\sqrt 6 & -1/\sqrt 6 & -1/\sq... |
I don't understand what is *polynomial fitting*.
Can anyone explain me how to fit a polynomial?
Like $x^3 - 3x^4 - 4x + 1$.
Thanks in advance! |
As part of my solution to a problem, I come to a point where I need to find the solutions to
$-2\partial_{T}B\left(T\right)+\frac{3}{4}B\left(T\right)\left(A\left(T\right)^{2}+B\left(T\right)^{2}\right)=0$
$2\partial_{T}A\left(T\right)+\frac{3}{4}A\left(T\right)\left(B\left(T\right)^{2}+A\left(T\right)^{2}\right)=0... |
$$S = \sum_{k=1}^{\infty} \frac{cos(\theta\log(k))}{k^a}$$
How do I go about finding the value of S, given that $\theta \to \infty$ and $0 < a < 1$.
> Any special techniques that might be
> helpful in calculating this sum?
EDIT:
Just to give some background,
I was actually trying to figure out
$$\sum_{k... |
I'm looking for an explanation on how reducing the Hamiltonian cycle problem to the Hamiltonian path's one (to proof that also the latter is NP-complete). I couldn't find any on the web, can someone help me here? (linking a source is also good).
Thank you. |
I don't understand what is *polynomial fitting*.
Can anyone explain me how to fit a curve?
Thanks in advance!
EDITED |
I don't understand what is *polynomial fitting*.
Can anyone explain me how to fit a curve to given points?
Thanks in advance!
EDITED |
I have a problem of the following form:
minimize $\|Dx\|_2$
subject to $\|x*x\|_2 = 1$
where $x\in\mathbb R^n$, $D$ is a given diagonal matrix of positive entries, and $*$ represents convolution, i.e. $(x*x)\_n = \sum_{i+j=n}x_ix_j$ and $x*x\in\mathbb R^{2n-1}$
What approach could be used in dealing with t... |
The scatter graph at the [Wikipedia article][1] seems to suggest so. Has anyone attacked this before? Is there a known proof?
[1]: http://en.wikipedia.org/wiki/Pythagorean_triple#Distribution_of_triples |
Is it possible to get arbitrarily near any acute angle with Pythagorean triangles? |
Let G={x,y|x^2=y^3=(xy)^3=1}
I would like to show that G is isomorphic to A_4.
Let f:F_2->G be a surjective homomorphism from the free group on two elements to G. Let f map x to (12)(34) and y to (123). I'm not sure how to show that these elements generate the kernel of f. If they do generate the kernel, how do I c... |
Let $G=\{x,y|x^2=y^3=(xy)^3=1\}$
I would like to show that G is isomorphic to A_4.
Let f:F_2->G be a surjective homomorphism from the free group on two elements to G. Let f map x to (12)(34) and y to (123). I'm not sure how to show that these elements generate the kernel of f. If they do generate the kernel, how do... |
Let $G=\\{x,y|x^2=y^3=(xy)^3=1\\}$
I would like to show that G is isomorphic to A_4.
Let f:F_2->G be a surjective homomorphism from the free group on two elements to G. Let f map x to (12)(34) and y to (123). I'm not sure how to show that these elements generate the kernel of f. If they do generate the kernel, how ... |
Let $G=\\{x,y|x^2=y^3=(xy)^3=1\\}$
I would like to show that G is isomorphic to A_4.
Let $f:\mathbb{F}_{2} \to G$ be a surjective homomorphism from the free group on two elements to $G$. Let $f$ map $x \to (12)(34)$ and $y \mapsto (123)$. I'm not sure how to show that these elements generate the kernel of $f$. If t... |
Let $G=\\{x,y|x^2=y^3=(xy)^3=1\\}$
I would like to show that G is isomorphic to A_4.
Let $f:\mathbf{F}_{2} \to G$ be a surjective homomorphism from the free group on two elements to $G$. Let $f$ map $x \to (12)(34)$ and $y \mapsto (123)$. I'm not sure how to show that these elements generate the kernel of $f$. If t... |
Abelsh, your definition of $f$ does not make sense: you cannot define the image of $x$ to **be** $(12)(34)$ unless you already know that $G$ **is** a group of permutations.
Instead, what you want to do is define $f$ from $\mathbf{F}_2$ to $A_4$ by defining it as you want: $f(x) = (12)(34)$ and $f(y)=(123)$. If you d... |
What is the least number of comparisons we need to find the median of 6 distinct numbers?
I am able to find the answer to the median of 5 distinct numbers to be 6 comparisons, and it makes sense, however in the case of 6 distinct numbers I can't find an answer.
The best I was able to do it in by hand was 9 compar... |
What is the least number of comparisons we need to find the median of 6 distinct numbers?
I am able to find the answer to the median of 5 distinct numbers to be 6 comparisons, and it makes sense, however in the case of 6 distinct numbers I can't find an answer.
The best I was able to do it in by hand was 9 compar... |
I'm familiar with [Householder reflections][1]; they are a simple transformation that, given a normal vector, describes reflection in the hyperplane perpendicular to that vector.
But do Householder reflections describe the complete space of matrices describing reflection about the origin, or are there others?
... |
do Householder reflections describe all reflections? |
I'm familiar with [Householder reflections][1]; they are a simple transformation that, given a normal vector, describes reflection in the hyperplane perpendicular to that vector.
But do Householder reflections describe the complete space of matrices describing reflection about the origin, or are there other reflecti... |
What is the least number of comparisons we need to find the median of 6 distinct numbers?
I am able to find the answer to the median of 5 distinct numbers to be 6 comparisons, and it makes sense, however in the case of 6 distinct numbers I can't find an answer.
The best I was able to do it in by hand was 9 compar... |
It takes at least 8.
A lower bound to find all the $k$ smallest elements is given here: [Lower bounds for Selection][1].
Which says we need at least
$$ n - k + \sum_{n+1-k < j \le n} \lceil \log_{2} j \rceil $$
For $n=6, k=3$, this gives $9$.
$$ 6 - 3 + \lceil \log_{2} 5 \rceil + \lceil \log_{2} 6 \rceil... |
I'm trying to come up with some math problems (word or otherwise) that get to the meaning of adding zero, but I'm getting stuck because it seems just too simple to me.
I have come up with questions like "John is having a birthday party and he invites five friends over but nobody shows up. How many people are at the ... |
What are some deep questions that are applicable to first graders in regards to adding zero? |
In fact, more is true -- consider the triples $(3,4,5), (-7,24,25), (-117,44,125), (-527,-336,625), \ldots$, where the $n$th triple is
$$ (a_n, b_n, c_n) := (Re((3+4i)^n), Im((3+4i)^n), 5^n). $$
Since $\tan^{-1} 4/3$ is irrational, the sequence of pairs $(a_n/c_n,b_n/c_n)$
$$ (3/5, 4/5), (-7/25, 24/25), (-117... |
**HINT** $\ \ \ \rm bd = ac\ \:\Rightarrow\:\ (a,b)\:(b,c)\: =\ b\: (a,b,c,d)\ $ via GCD distributive, associative laws.
Alternatively, if you know $\rm\ [a,b]\: (a,b)\: =\: ab\ $ then using this LCM*GCD law
we have $\rm\ \ \: a,b\:|\:ac\ \:\Rightarrow\ \: [a,b]\:|\:ac\ \ \Rightarrow\:\ ab\:|(a,b)\:ac\ \Rightarro... |
Is it possible to have 2 different points (non-degenerate) in n-dimensions for any value of n where n>1, share the same cost and both be visited by the simplex method? |
Simplex - 2 different points with the same cost? |
How can I compute the pmf of a random sum(1 to N) of iid bernoulli random variables, where N is distributed geometrically? |
How can I compute the pmf of a random sum? |
It takes at least 8 and can be done in exactly 8.
**Lower Bound**
A lower bound to find all the $k$ smallest elements is given here: [Lower bounds for Selection][1].
Which says we need at least
$$ n - k + \sum_{n+1-k < j \le n} \lceil \log_{2} j \rceil $$
For $n=6, k=3$, this gives $9$.
$$ 6 - 3... |
Perhaps this answer will use too much technology. Still, I think it's pretty.
Consider $A_4$ as the group of orientation-preserving symmetries of a tetrahedron $S$. The quotient $X=S/A_4$ is a 2-dimensional [orbifold][1]. Let's try to analyse it.
Two-dimensional orbifolds have three different sorts of singular... |
I was asked to find a solution to
$$\frac{\sin^2(nx)}{n^2\sin^2(x)}=2^{-1/2}$$
where $n$ is a fixed integer greater than 1.
Numerically, there's a solution just above 1/n so I decided to find this one (rather than one of the other roots). I hoped to find an obvious pattern that I could use to find some sort of cl... |
A function $f: \mathbb{R} \to \mathbb{R}$ is said to have the intermediate-value property if for any $a$,$b$ and $\lambda \in [f(a),f(b)]$ there is a $x \in [a,b]$ such that $f(x)=\lambda$
A function $f$ is injective if $f(x)=f(y) \Rightarrow x=y$
Now it is the case that every injective function with the intermed... |
Injective function with intermediate value property is continuous. Better proof? |
I had already posted this on mathoverflow and was advised to post the same here. So here it goes:
X={x,y,z|$x^2+y^2+z^2≤1$ and $ z≥0$} i.e. X is the top half of a 3-Disk.
Z=X/E, where E is the equivalence relation on the the plane z = 0 which is as follows:
(x,y,0)∼(−x,−y,0).
I was told that this space is e... |
First consider just the hemisphere $X'\subseteq X$, and define $Z'= X'/E$. This is the real projective plane, in one of its many incarnations. You may have seen $\mathbb{RP}^2$ presented as the space of lines through the origin in $\mathbb{R}^3$. To get from that presentation to this one, note that each such line ... |
What is the least number of comparisons we need to find the median of 6 distinct numbers?
I am able to find the answer to the median of 5 distinct numbers to be 6 comparisons, and it makes sense, however in the case of 6 numbers I can't find an answer.
The best I was able to do it in by hand was 9 comparisons. C... |
s and t are two nodes in a graph.
How to prove that the maximum number of edge-disjoint cut sets which divide s and t equals the length of the shortest path from s to t? |
Number of cut sets and the length of the shortest path? |
A very simple solution to the [diffusion equation][1] is $u(x,t)=x^2+2 t$
**My question:**<br>
How can this be a solution to the diffusion equation when nothing really diffuses, but just stays the same - see plot: [Here][2]
I can see that it is a solution but can't believe my eyes... Is there some kind of intuit... |
I'm currently reading Hilton & Stammbach's A First Course in Homological Algebra, and the following point has stumped me:
In section 1.8, they construct co-free modules ("left moodule" over some ring) as essentially coming from the right adjoint to the forgetful functor from $\Lambda$-Modules to Abelian Groups. On t... |
Why is the co-free module defined as the right adjoint to the forgetful functor to Ab rather than Set? |
Quaternions came up while I was interning not too long ago and it seemed like no one really know how they worked. While eventually certain people were tracked down and were able to help with the issue, it peaked my interest in quaternions.
After reading many articles and a couple books on them, I began to know the f... |
How can one intuitively think about quaternions? |
Here are a few pointers, from the more obtuse to the more direct.
1. "Meeting a friend in a corridor, Wittgenstein said: 'Tell me, why do people always say it was *natural* for men to assume that the sun went round the earth, rather than that the earth was rotating?' His friend said, 'Well, obviously, because it *l... |
Can any of you guys think of a topological space $(X,\tau)$ and a family of subsets $\{A_{s}}_{s \in S\}$ of $X$ such that for a certain $x \in X$ you can find a subset $V$ such that $x \in V$ and $\{s \in S: V \cap A_{s} \neq \emptyset\}$ is finite whereas for every $W \subseteq X$, with $x \in W$, we have that $\{s \... |
Okay, I have this not so pretty 2nd order non-linear ODE I should be able to solve numerically.
$$f''(R) + \frac{2}{R} f'(R)=\frac{0.7}{R} \left( \frac{1}{\sqrt{f(R)}} - \frac{0.3}{\sqrt{1-f(R)}} \right),$$
$$f(1)=1.$$
The function around the origin is behaving very wildly.
I was thinking of breaking this ... |
Can any of you guys think of a topological space $(X,\tau)$ and a family of subsets $A_{s}$_{s \in S}$ of $X$ such that for a certain $x \in X$ you can find a subset $V$ such that $x \in V$ and $\{s \in S: V \cap A_{s} \neq \emptyset\}$ is finite whereas for every $W \subseteq X$, with $x \in W$, we have that $\{s \in ... |
Can any of you guys think of a topological space $(X,\tau)$ and a family of subsets {A<sub>s</sub>}<sub>${s \in S}$</sub> of $X$ such that for a certain $x \in X$ you can find a subset $V$ such that $x \in V$ and {$s \in S: V \cap A_{s} \neq \emptyset$} is finite, whereas for every $W \subseteq X$, with $x \in W$, we h... |
Can any of you guys think of a topological space $(X,\tau)$ and a family of subsets {A<sub>s</sub>}<sub>${s \in S}$</sub> of $X$ such that for a certain $x \in X$ you can find a subset $V$ such that $x \in V$ and {$s \in S: V \cap A_{s} \neq \emptyset$} is finite, whereas for every $W \subseteq X$, with $x \in W$, we h... |
Based on a [previous question][1], the answer I got for the "sum of products" method was:
RC' + RCG' + RCGM'
Just for completeness, can someone help walk me through to how this was arrived at? Thanks again in advance!
[1]: http://math.stackexchange.com/questions/7126/boolean-algebra-simplification-dont-... |
I'm wondering if a trick I have will work.
I'll give the basic idea first, and then I'll give some additional complications and/or restrictions.
I think I might be looking for some kind of residue in a ring of polynomials.
I have a series of $n$ numbers, where $n$ is odd. I'd like to get the first half of thi... |
Can someone help direct me towards information on calculating the <a href="http://mathworld.wolfram.com/PolynomialRemainder.html">polynomial remainder</a>?
I know how to do the basics, but what if I don't have access to the polynomial in expanded form?
For example, $\frac{1}{1-x}$ represents the polynomial $\sum_... |
Given two values $a$ and $b$, how should one go about solving the following inequality for $r$:
$a \le b^{r}-r$
Applying $log_{b}$ on both sides of the inequality doesn't help me much since that yields the following:
$log_{b}{a} \le log_{b}{\(b^{r}-r\)}$
I know that
$\log_{z}{x} - \log_{z}{y} = \frac{\l... |
Can someone help direct me towards information on calculating the <a href="http://mathworld.wolfram.com/PolynomialRemainder.html">polynomial remainder</a>?
I know how to do the basics, but what if I don't have access to the polynomial in expanded form?
For example, $\frac{1}{1-x}$ represents the polynomial $\sum_... |
How do we solve $a \le b^{r}-r$ for $r$? |
A metric space $X$ is totally bounded if for any $r>0$, $X$ can be covered by a finite number of balls of radius $r$.
For example, $\mathbb{R}^n$ are totally bounded.
Are there any interesting necessary and/or sufficient conditions for a metric space to be totally bounded?
[Background: I was trying to generali... |
Which metric spaces are totally bounded? |
A subset $S$ of a metric space $X$ is totally bounded if for any $r>0$, $S$ can be covered by a finite number of $X$-balls of radius $r$.
A metric space $X$ is totally bounded if it is a totally bounded subset of itself.
For example, bounded subsets of $\mathbb{R}^n$ are totally bounded.
Are there any interest... |
A metric space is totally bounded if and only if its completion is compact. (However, a little googling reveals that totally bounded doesn't necessarily imply compact completion without the axiom of choice.) This gives one explanation of the result in your background, because uniformly continuous functions extend to ... |
Can someone help direct me towards information on calculating the polynomial remainder?
I know how to do the basics, but what if I don't have access to the polynomial in expanded form?
For example, $\frac{1}{1-x}$ represents the polynomial $\sum_{i=0}^\infty{x^i}$. I want to be able to find the remainder using a... |
Everything that follows takes place in the Borel $\sigma$-algebra with Lebesgue measure.
The Lebesgue mean of $f$ at $x$ is defined as $\displaystyle \lim_{\epsilon\to 0} \int_{x-\epsilon}^{x+\epsilon} \frac{f}{2\epsilon}$. The function defined by $\[f\](x) =$ "Lebesgue mean of $f$ at $x$" is equal to $f$ almost eve... |
What is the Lebesgue mean of the fat Cantor set? |
Suppose I have a variable $s$ that has a geometric distribution with success parameter x. So the probability of success on trial $s$ is
$p_s = (1 - x)^{s - 1} x$,
Consider the following function of $s$
$V(s) = \delta^s$,
where $\delta < 1 $. Then the expected value of this function is given by
$\su... |
This is the special case $\rm\ x,\:n,\:a = 2,\:1,\:0\ $ in Ramanujan's second notebook, chapter XII, entry 4:
$$\rm x + n + a\ =\ \sqrt{ax + (n+a)^2 + x\sqrt{a(x+n) + (n+a)^2 + (x+n) \sqrt{\cdots}}} $$
Below is Ramanujan's solution of the given special case - which was submitted to a journal in April 1911. Note t... |
Let $\Sigma$ be a $n-1$ dimensional space-like submanifold of a $n+1$ dimensional space-time $(V,g)$ and let $x \in \Sigma$. Then $(T_x \Sigma)^\perp$ is of dimension $2$ and is time-like. Such a $2$-plane as $(T_x \Sigma)^\perp$ cuts the future lightcone at $x$ along two linearly independent future null directions. Le... |
Let $\Sigma$ be a $n-1$ dimensional space-like submanifold of a $n+1$ dimensional space-time $(V,g)$ and let $x \in \Sigma$. Then $(T_x \Sigma)^\perp$ is of dimension $2$ and is time-like. Such a $2$-plane as $(T_x \Sigma)^\perp$ cuts the future lightcone at $x$ along two linearly independent future null directions. Le... |
If a space has curvature, then the curvature can be seen intrinsically by finding sums of angles in triangles made of geodesics. Under general relativity, space-time is curved on local scales. On global scales, experiments have determined that space-time is almost totally flat.
If one were to, for each of three points... |
Quaternions came up while I was interning not too long ago and it seemed like no one really know how they worked. While eventually certain people were tracked down and were able to help with the issue, it piqued my interest in quaternions.
After reading many articles and a couple books on them, I began to know the f... |
Let me note that http://en.wikipedia.org/wiki/Vector-valued_form seems to be highly relevant. They essentially define the bracket. Note that in the case of a 1-form A valued in $\mathfrak{g}$ you get $[A,A]=[A\wedge A (v \otimes w)]= A(v)\otimes A(w)- A(w)\otimes A(v)$, which has value in $\mathfrak{g} \otimes \mathfr... |
More specifically, assume d taken from [1, 2^n] and m is taken from [1, n]. What is an upper bound on the probability that d is a multiple of m? |
For an arbitrary positive integer d and random modulus m, what is the probability that d mod m = 0? |
Let me note that http://en.wikipedia.org/wiki/Vector-valued_form seems to be highly relevant. They essentially define the bracket. Note that in the case of a 1-form A valued in $\mathfrak{g}$ you get $[A,A]=[A\wedge A (v \otimes w)]= A(v)\otimes A(w)- A(w)\otimes A(v)$, which has value in $\mathfrak{g} \otimes \mathfr... |
The Complex plane, set of all z=x+iy where x and y are real, surface area equals cross product of x and y equals aleph-something (that's not the question). Projecting the plane onto a sphere, and adding the complex infinity to the set (without a value for arg(z)), gives the riemann sphere, which, being a sphere, is com... |
Is the riemann sphere compact even though the complex plane isn't? |
True or False? If $\sum n=1$ to $\infty ((a_{n})^2)$ is convergent, and$ \sum n=1$ to $\infty (\pi / 2 - b_{n})$ is divergent, then $\{a_{n}(sin(b_{n})-1)\}$ is a convergent sequence.
Sorry if you are reading this right now, I am editing my work in. |
True or False? If $\sum n=1$ to $\infty ((a_{n})^2)$ is convergent, and$ \sum n=1$ to $\infty (\pi / 2 - b_{n})$ is divergent, then $\{a_{n}(sin(b_{n})-1)\}$ is a convergent series
So we know that $(a_{n})^2$ is convergent.
Therefore, $lim n-> \infty (a_{n})^2 = 0$
I'm not sure what this tells us about or how to r... |
True or False?
If $\displaystyle \sum_{n=1}^{\infty}a_{n}^2$ is convergent, and$\displaystyle \sum_{n=1}^{\infty} (\frac{\pi}{2} - b_{n})$ is divergent, then $\displaystyle \{a_{n}(\sin(b_{n})-1)\}$ is a convergent series
So we know that $(a_{n})^2$ is convergent.
Therefore, $\lim_{n \to \infty} a_{n}^2 = 0$
I... |
True or False?
If $\displaystyle \sum_{n=1}^{\infty}a_{n}^2$ is convergent, and$\displaystyle \sum_{n=1}^{\infty} (\frac{\pi}{2} - b_{n})$ is divergent, then $\displaystyle \{a_{n}(\sin(b_{n})-1)\}$ is a convergent sequence.
So we know that $(a_{n})^2$ is convergent.
Therefore, $\lim_{n \to \infty} a_{n}^2 = 0$... |
I have a text book which gives two diagrams of beads suspended from strings with different angles, and worked examples to calculate the tension in the strings.
The first bead is suspended by 2 strings attached to a rod, both strings at angles to the vertical (the bead is at the bottom, the rod is across the top). Th... |
If a bead is suspended by 2 strings, one vertical, the other diagonally, why are the tensions equal? |
Suppose we look at $k$-digit strings with digits between $1$ and $n$. How many distinct classes of strings are there when digit identity and order doesn't matter?
More formally, what is the number of equivalence classes over strings $d_1,\ldots,d_k$ with $d_i\in \\{1,\ldots, n\\}$ where strings $d,\hat{d}$ are consi... |
What does a compact set look like? |
Is there a simple explanation of what the Laplace transformation do exactly and how they work? Reading my math book has left me in a foggy haze of proofs that I don't completely understand. I'm looking for an explanation on lamans terms so that I understand what it is doing as I make these seemingly "magical" transfor... |
For example:
the limit of e^t as t approaches infinity is simply infinity.
but the limit of e^(t^2) cannot be found because it explodes? Is this due to the fact that there are differing types of infinity and the second limit produces a non-"countable" infinity?
[see referenced question](http://math.stackexcha... |
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