instruction stringlengths 12 30k |
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Why can you turn clothing right-side-out? |
Everything I'm wearing is a topological sphere with holes (t-shirts have 4, pants 3, shoes and socks 1) in which cases any hole works, but imagine if you sewed the legs of your pants together so that they form a topological torus with a hole. What happens when you turn them inside out now? |
Everything I'm wearing is a topological sphere with holes (t-shirts have 4, pants 3, shoes and socks 1) in which cases any hole works, but imagine if you sewed the legs of your pants together so that they form a topological torus with a hole (so if you were to wear them your feet would be touching and it would be impos... |
Standard deck has 52 cards, 26 Red and 26 Black. A run is a maximum contiguous block of cards, which has the same color.
Eg.
* (R,B,R,B,...,R,B) has 52 runs.
* (R,R,R,...,R,B,B,B,...,B) has 2 runs.
What is the expected number of runs in a shuffled deck of cards? |
What is the expected number of runs of same color in a standard deck of cards? |
The first question that should be asked of any person presenting you with a large-ish matrix: "is the matrix dense or sparse?" In the latter case, one definitely does not need to allocate space for the zero entries, and should thus look into special techniques to storing them(which as I understand nowadays rely on a l... |
First, a warning. I suspect this response is likely not going to be immediately comprehensibe. There is a formal set-up for your question, there are tools available to understand what's going on. They're not particularly light tools, but they exist and they're worthy of being mentioned. Before I write down the main... |
First, a warning. I suspect this response is likely not going to be immediately comprehensibe. There is a formal set-up for your question, there are tools available to understand what's going on. They're not particularly light tools, but they exist and they're worthy of being mentioned. Before I write down the main... |
The first question that should be asked of any person presenting you with a large-ish matrix: "is the matrix dense or sparse?" In the latter case, one definitely does not need to allocate space for the zero entries, and should thus look into special techniques to storing them(which as I understand nowadays rely on a l... |
First, a warning. I suspect this response is likely not going to be immediately comprehensibe. There is a formal set-up for your question, there are tools available to understand what's going on. They're not particularly light tools, but they exist and they're worthy of being mentioned. Before I write down the main... |
First, a warning. I suspect this response is likely not going to be immediately comprehensibe. There is a formal set-up for your question, there are tools available to understand what's going on. They're not particularly light tools, but they exist and they're worthy of being mentioned. Before I write down the main... |
First, a warning. I suspect this response is likely not going to be immediately comprehensibe. There is a formal set-up for your question, there are tools available to understand what's going on. They're not particularly light tools, but they exist and they're worthy of being mentioned. Before I write down the main... |
Let $\theta_{kl}$ be an angle such that $\cos\theta_{kl}=\frac{1}{2}(\cos(\frac{2\pi k}{n})+\cos(\frac{2\pi l}{n}))$.
Given that definition, if I introduce a new variable $t$ is the following a correct?
$\cos(t\theta_{kl})\approx\frac{1}{2}(\cos(\frac{2\pi kt}{n})+\cos(\frac{2\pi lt}{n}))$
|
When I was studying topology I remember being able to demonstrate that the set of topological surfaces with any number of punctures (including the projective plane, Klein bottle, Moebius strip, double torus, etc.), together with their associated inverses, was _not_ a group with respect to the operations of connection a... |
Let $\theta_{kl}$ be an angle such that $\cos\theta_{kl}=\frac{1}{2}(\cos(\frac{2\pi k}{n})+\cos(\frac{2\pi l}{n}))$.
Given that definition, if I introduce a new variable $t$ is the following a correct?
$\cos(t\theta_{kl})\approx\frac{1}{2}(\cos(\frac{2\pi kt}{n})+\cos(\frac{2\pi lt}{n}))$
**Update:** I'm actu... |
First, a warning. I suspect this response is likely not going to be immediately comprehensibe. There is a formal set-up for your question, there are tools available to understand what's going on. They're not particularly light tools, but they exist and they're worthy of being mentioned. Before I write down the main... |
Let $\theta_{kl}$ be an angle such that $\cos\theta_{kl}=\frac{1}{2}(\cos(\frac{2\pi k}{n})+\cos(\frac{2\pi l}{n}))$.
Given that definition, if I introduce a new variable $t$ is the following a correct?
$\cos(t\theta_{kl})\approx\frac{1}{2}(\cos(\frac{2\pi kt}{n})+\cos(\frac{2\pi lt}{n}))$
**Update:** I'm actu... |
When I was studying topology I remember being able to demonstrate that the set of topological surfaces with any number of punctures (including the projective plane, Klein bottle, Moebius strip, double torus, etc.), together with their associated inverses, was _not_ a group with respect to the operations of connection a... |
First, a warning. I suspect this response is likely not going to be immediately comprehensibe. There is a formal set-up for your question, there are tools available to understand what's going on. They're not particularly light tools, but they exist and they're worthy of being mentioned. Before I write down the main... |
When I was studying topology I remember being able to demonstrate that the set of topological surfaces with any number of punctures (including the projective plane, Klein bottle, Moebius strip, double torus, etc.), together with their associated inverses, was _not_ a group with respect to the operations of connection a... |
I'm starting to read Baby Rudin (Principles of mathematical analysis) now and I wonder whether you know of any companions to it. Another supplementary book would do too. I tried Silvia's notes, but I found them a bit too "logical" so to say. Are they good? What else do you recommend? |
Companions to Rudin? |
In section 4 of the Afred van der Poorten's [A Proof That Euler Missed ...][1] article the following inequality is used:
$v_{p}\displaystyle\binom{n}{m}$ $\leq\left\lfloor\dfrac{\ln n}{\ln p}\right\rfloor-v_{p}(m)\qquad(\ast)$ (*In the original* denoted $\text{ord}$ $_{p}(\cdot)$ instead of $v_p(\cdot)$).
where ... |
In section 4 of the article by Afred van der Poorten's [A Proof That Euler Missed ...][1] the following inequality is used:
$v_{p}\displaystyle\binom{n}{m}$ $\leq\left\lfloor\dfrac{\ln n}{\ln p}\right\rfloor-v_{p}(m)\qquad(\ast)$ (*In the original* denoted $\text{ord}$ $_{p}(\cdot)$ instead of $v_p(\cdot)$).
wh... |
If you want to avoid too much category theory, you can first read this link to understand the definition of monads in [Haskell][1]. Then look at [Wikibooks][2] for a more mathematical look (thanks Jonathan Fischoff).
There are two descriptions that I know of. The first can easily be found by looking at wiki under Mo... |
I'm starting a chapter on Functions and they had the steps shown to reach the p-q equation.
$$ x_1,_2 = -\frac{p}{2} \pm\sqrt{\left(\frac{p}{2}\right)^2 - q}$$
So I wanted to do the same with the Quadratic Equation. I'm using the base linear equation
$$ax+by+c = 0.$$
The solution I have so far is as follow... |
Sorry I don't know how to do tex on websites, but I'm trying to learn.
You just made a small mistake on the final step. In the second to last step, we actually have our full equation as:
(-b/2a) +/- Sqrt((b^2-4ac)/(2a)^2) = (-b/2a) +/- Sqrt(b^2-4ac)/2a
Now we can collect the common factor of 1/2a and get:
(-b... |
I'm starting a chapter on Functions and they had the steps shown to reach the p-q equation.
$$ x_{1,2} = -\frac{p}{2} \pm\sqrt{\left(\frac{p}{2}\right)^2 - q}$$
So I wanted to do the same with the Quadratic Equation. I'm using the base linear equation
$$ax+by+c = 0.$$
The solution I have so far is as follo... |
Sorry I don't know how to do tex on websites, but I'm trying to learn.
You just made a small mistake on the final step. In the second to last step, we actually have our full equation as:
$\frac{-b}{2a}\pm \sqrt\frac{{b^{2}-4ac}}{4a^{2}}=\frac{-b}{2a}\pm \frac{\sqrt{b^{2}-4ac}}{2a}$
Now we can collect the com... |
Sorry I don't know how to do tex on websites, but I'm trying to learn.
You just made a small mistake on the final step. In the second to last step, we actually have our full equation as:
$$\frac{-b}{2a}\pm \sqrt{\frac{{b^{2}-4ac}}{4a^{2}}}=\frac{-b}{2a}\pm \frac{\sqrt{b^{2}-4ac}}{2a}$$
Now we can collect the... |
Sorry I don't know how to do tex on websites, but I'm trying to learn.
You just made a small mistake on the final step. In the second to last step, we actually have our full equation as:
$\frac{-b}{2a}\pm \sqrt{\frac{b^{2}-4ac}{4a^{2}}}=\frac{-b}{2a}\pm \frac{\sqrt{b^{2}-4ac}}{2a}$
Now we can collect the com... |
I'm starting a chapter on Functions and they had the steps shown to reach the p-q equation.
$$ x_{1,2} = -\frac{p}{2} \pm\sqrt{\left(\frac{p}{2}\right)^2 - q}$$
So I wanted to do the same with the Quadratic Equation. I'm using the base linear equation
$$ax+by+c = 0.$$
The solution I have so far is as follo... |
In 1D case, R1 as a typical case of 1D Euclidean space is totally ordered. I was wondering if any 1D Euclidean space E1 is ordered as well? What is it
For higher dimensional case, are both Rn and En ordered? What are the orders?
Thanks and regards! |
Representation theory plays a big role in the group-theoretic approach to special functions.
For example, Willard Miller showed that the powerful Infeld-Hill factorization / ladder method - widely exploited by physicists - is equivalent to the representation theory of four local Lie groups. This lie-theoretic approach... |
Representation theory plays a big role in the group-theoretic approach to special functions.
For example, Willard Miller showed that the powerful Infeld-Hill factorization / ladder method - widely exploited by physicists - is equivalent to the representation theory of four local Lie groups. This lie-theoretic approach... |
Representation theory plays a big role in the group-theoretic approach to special functions.
For example, Willard Miller showed that the powerful Infeld-Hill factorization / ladder method - widely exploited by physicists - is equivalent to the representation theory of four local Lie groups. This lie-theoretic approach... |
This is too long for a comment, so I'll throw in this choice piece by Forman Acton in his "Numerical Methods That Work" on the subject of dealing with large matrices:
> Whenever a person eagerly inquires if
> my computer can solve a set of 300
> equations in 300 unknowns, I must
> quickly suppress the temptation ... |
This is a final exam question in my algorithms class:
$k$ is a taxicab number if $k = a^3+b^3=c^3+d^3$, $a,b,c,d$ are distinct positive integers. Find all taxicab number $k$ such that $a,b,c,d < n$ in $O(n)$ time.
I don't know if the problem had a typo or not, because $O(n^3)$ seems more reasonable. The best I ca... |
In a densely and totally ordered set, induce a order topology from the order.
Are the dense in terms of the order and the dense in terms of the order topology equivalent?
Thanks and regards! |
This is a final exam question in my algorithms class:
$k$ is a taxicab number if $k = a^3+b^3=c^3+d^3$, $a,b,c,d$ are distinct positive integers. Find all taxicab number $k$ such that $a,b,c,d < n$ in $O(n)$ time.
I don't know if the problem had a typo or not, because $O(n^3)$ seems more reasonable. The best I ca... |
In 1D case, R1 as a typical case of 1D Euclidean space is totally ordered. I was wondering if any 1D Euclidean space E1 is ordered as well? What is it
For higher dimensional case, are both Rn and En ordered? What are the orders?
Thanks and regards!
**************
Update:
I meant for an order for an Eucli... |
In 1D case, R1 as a typical case of 1D Euclidean space is totally ordered. I was wondering if any 1D Euclidean space E1 is ordered as well? What is it
For higher dimensional case, are both Rn and En ordered? What are the orders?
Thanks and regards!
**************
Update:
I meant for an order for an Eucli... |
It only seems odd to you because you are used to representing numbers in base 10. What if you used base 7? |
> "Why is "the set of all sets" a
> paradox? It seems like it would be
> fine, to me."
The lesson of this and other set theory paradoxes is that when you define a set in a proof, you are obliged to demonstrate that the set exists and is well-defined. To me, a paradox of Russell's Paradox is that he fell into this ... |
Problem 63 of the 2001 St. Petersburg Mathematical Olympiad, Second Round, 11th grade. Are there three different numbers x, y, z in [0,pi/2] such that the numbers sin x, sin y, sin z, cos x, cos y, cos z can be divided into three pairs with equal sums? |
What is a good PDE book suitable for self study? I'm looking for a book that doesn't require much prerequisite knowledge beyond undergraduate-level analysis. My goal is to understand basic solutions techniques as well as some basic theory. |
Suppose p and q are discrete distributions over k outcomes, q is given, for which p does the following hold?
$p_1 \log q_1 + \ldots + p_k \log q_k \ge q_1 \log q_1 + \ldots + q_k \log q_k$ |
When does entropy bound cross-entropy? |
In section 4 of the article by Afred van der Poorten's [A Proof That Euler Missed ...][1] the following inequality is used:
$\nu_{p}\displaystyle\binom{n}{m}$ $\leq\left\lfloor\dfrac{\ln n}{\ln p}\right\rfloor-\nu_{p}(m)\qquad(\ast)$ (*In the original denoted* $\text{ord}$ $_{p}(\cdot)$ *instead of* $\nu_p(\cdot)$)... |
I would recommend:
Fritz John, [Partial Differential Equations][1] (Applied Mathematical Sciences) ISBN: 0387906096. It is a classical Springer book that contains what you ask for.
Google Books might be a good start before you make your final decision.
[1]: http://books.google.com/books?id=cBib_bsGGLYC&... |
I haven't read a lot of PDE material, but I enjoyed Taylor's [book][1]. It's quite well-written, and also contains introductory material (like Lie derivatives), since it does things on manifolds.
[1]: http://books.google.com/books?id=uphqFYqYm0oC&printsec=frontcover&dq=taylor+partial+differential+equations&hl=en... |
The book by [Strauss](http://www.amazon.com/Partial-Differential-Equations-Walter-Strauss/dp/0470054565/ref=sr_1_1?ie=UTF8&s=books&qid=1282257031&sr=8-1) is pretty good for a first course. For a second one the book by [Evans](http://www.amazon.com/Partial-Differential-Equations-Graduate-Mathematics/dp/0821849743/ref=sr... |
The meaning is, of course, context-dependent...
In the context of differential equations, that a solution to an equation with a "time" variable blows up usually means that the maximal domain for which it is defined is finite, so that at the endpoint of that interval something `bad' happens: either the solution goes ... |
Given a line segment, denoted by it's 2 endpoints (X1, Y1) and (X2, Y2), and a circle, denoted by it's center point (Xc, Yc) and a radius R, how can I tell if the line is a tangent of or runs through this circle? I don't need to be able to discern between tangent or running through a circle, I just need to be able to d... |
How to tell if a line intersects with a circle? |
Given a line segment, denoted by it's 2 endpoints (X1, Y1) and (X2, Y2), and a circle, denoted by it's center point (Xc, Yc) and a radius R, how can I tell if the line segment is a tangent of or runs through this circle? I don't need to be able to discern between tangent or running through a circle, I just need to be a... |
How to tell if a line segment intersects with a circle? |
Find the intersections of the line containing the segment and the circle. This amounts to solving a quadratic equation. If there are no intersections (*i.e.* the solutions of the corresponding equation are non-real), then your segment does not intersect the circle. Now, if there *are* intersections, see whether they ar... |
Find the intersections of the line containing the segment and the circle. This amounts to solving a quadratic equation. If there are no intersections (*i.e.* the solutions of the corresponding equation are non-real), then your segment does not intersect the circle. Now, if there *are* intersections, see whether they ar... |
Maybe this is what you are getting at... Given a line through points $(x_1,y_1)$ and $(x_2,y_2)$, and given a single point $p$ *not necessarily on the line*, we want to find out if we draw a circle of radius $r$ around $p$; Does that circle intersect the line?
If this is what you want then you are looking for the <... |
Maybe this is what you are getting at... Given a line through points $(x_1,y_1)$ and $(x_2,y_2)$, and given a single point $p$ *not necessarily on the line*, we want to find out: *If we draw a circle of radius $r$ around $p$, does that circle intersect the line?*
If this is what you want then you are looking for th... |
Maybe this is what you are getting at... Given a line through points $(x_1,y_1)$ and $(x_2,y_2)$, and given a single point $p$ *not necessarily on the line*, we want to find out: *If we draw a circle of radius $r$ around $p$, does that circle intersect the line?*
If this is what you want then you are looking for th... |
Maybe this is what you are getting at... Given a line through points $(x_1,y_1)$ and $(x_2,y_2)$, and given a single point $p$ *not necessarily on the line*, we want to find out: *If we draw a circle of radius $r$ around $p$, does that circle intersect the line?*
If this is what you want then you are looking for th... |
My nephew was folding laundry, and turning the occasional shirt right-side-out. I showed him a "trick" where I turned it right-side-out by pulling the whole thing through a sleeve instead of the bottom or collar of the shirt. He thought it was really cool (kids are easily amused, and so am I).
So he learned that yo... |
I am interested in image analysis and am looking for an approachable tutorial to the Wiener filter. At some point I am interested in implementing such a filter but I would like to have a deeper understanding of the algorithms I'll be writing. Does anyone have recommendations for papers, web sites, etc. that present goo... |
Suppose I have a matrix $S$ having a one-dimensional nullspace $\lbrace e \rbrace$ such that $S + ee^\top$ is a positive definite symmetric matrix.
Now let $b \in Range(S)$ and suppose I solve the equation $(S + ee^\top)x = b$ is there anyway I can derive the solution $x'$ of the equation $Sx' = b$? I was trying a... |
I am crossposting this question from Stack Overflow since it is more math then programming related.
In Reference to [my question][1] about projecting a planar Polygon to a plane, i came to another problem.
I want to project a base contour which is aligned to the x-y-plane along a projection vector from from a poi... |
Find the intersections of the line containing the segment and the circle. This amounts to solving a quadratic equation. If there are no intersections (*i.e.* the solutions of the corresponding equation are non-real), then your segment does not intersect the circle. Now, if there *are* intersections, see whether they ar... |
In section 4 of the article by Afred van der Poorten's [*A Proof That Euler Missed ...*][1] the following inequality is used:
$\nu_{p}\displaystyle\binom{n}{m}$ $\leq\left\lfloor\dfrac{\ln n}{\ln p}\right\rfloor-\nu_{p}(m)\qquad(\ast)$ (*In the original denoted* $\text{ord}$ $_{p}(\cdot)$ *instead of* $\nu_p(\cdot)... |
The first question that should be asked of any person presenting you with a large-ish matrix: "is the matrix dense or sparse?" In the latter case, one definitely does not need to allocate space for the zero entries, and should thus look into special techniques to storing them(which as I understand nowadays rely on a l... |
First let's make sure I understood your question correctly:
- You have a probability function that is expressed as a sum/product of N parametrized one-dimensional gaussians, each with different mean and standard deviation.
- You want to generate stochastic variables according to this distribution.
Is this corr... |
Maybe this is what you are getting at... Given a line through points $(x_1,y_1)$ and $(x_2,y_2)$, and given a single point $p$ *not necessarily on the line*, we want to find out: *If we draw a circle of radius $r$ around $p$, does that circle intersect the line?*
If this is what you want then you are looking for th... |
Say you have a very dense matrix that is 30000x30000 elements. The very dense matrix comes from the radiosity equation, which I discussed [here](http://math.stackexchange.com/questions/2180/how-can-i-characterize-the-type-of-solution-vector-that-comes-out-of-a-matrix).
Say you have Ax = B. You have B, and A is 3000... |
Find the intersections of the line containing the segment and the circle. This amounts to solving a quadratic equation. If there are no intersections (*i.e.* the solutions of the corresponding equation are non-real), then your segment does not intersect the circle. Now, if there *are* intersections, see whether they ar... |
Could someone verify this proof?
Given the following definition of divisibility $(\forall{a,b \in \mathbb{N}})[a | b \equiv (\exists{c \in \mathbb{N}})(ac=b)]$.
Statement: $(\forall{a,b,c \in \mathbb{N}})[(a|b) \rightarrow (ac|bc)] $ .
Proof:
Let $a,b \in \mathbb{N}$ be arbitrary. Assuming that $a|b$, we c... |
I'm am trying to define a data structure to represent road networks. The immediately obvious structure is that of a graph - a set of nodes and edges that connect pairs of nodes. The nodes would represent things like intersections, and the edges would represent lanes.
However, the basic concept of a graph is insu... |
Maybe this is what you are getting at... Given a line through points $(x_1,y_1)$ and $(x_2,y_2)$, and given a single point $p$ *not necessarily on the line*, we want to find out: *If we draw a circle of radius $r$ around $p$, does that circle intersect the line?*
If this is what you want then you are looking for th... |
Well, what the heck... there has been so much virtual ink spilled at this point that what's another gallon, really?
The principle behind *this* approach is that we can find out algebraically whether the set of coordinates defining the perimeter of a circle with center $(h,k)$ and radius $r$ contains any point of the... |
Maybe this is what you are getting at... Given a line through points $(x_1,y_1)$ and $(x_2,y_2)$, and given a single point $p$ *not necessarily on the line*, we want to find out: *If we draw a circle of radius $r$ around $p$, does that circle intersect the line?*
If this is what you want then you are looking for th... |
A [hyperbolic geometry](http://en.wikipedia.org/wiki/Hyperbolic_geometry) is a non-Euclidian geometry with constant negative curvature. It has the property that given a line and a point, many lines can be drawn containing the point that never meet the given line. The picture below (Circle Limit by M.C.Escher) is a conf... |
If $(G_i,f_{ij})$ is a direct system of topological groups, is it always the case that the topological\group-theoretical direct limit $G:=\varinjlim_iG_i$ is a topological group? (The topology on $G$ is the final topology with respect to the canonical homomorphisms $\psi_i:G_i\rightarrow G$). It is immediate from the ... |
Is a direct limit of topological groups always a topological group? |
Well, what the heck... there has been so much virtual ink spilled at this point that what's another gallon, really?
(**update**: This is precisely the solution [Mariano proposed](http://math.stackexchange.com/users/274/mariano-suarez-alvarez) long before I did, it just happens that I couched this in a different nota... |
A [hyperbolic geometry](http://en.wikipedia.org/wiki/Hyperbolic_geometry) is a non-Euclidian geometry with constant negative curvature. It has the property that given a line and a point, many lines can be drawn containing the point that never meet the given line. The picture below (Circle Limit by M.C.Escher) is a conf... |
Is there an analytic solution for the density function of this complex random variable? |
I direct you to chapter 1 of *Fuchsian Groups* by Svetlana Katok. It has the following proof of the theorem (at least where I learned it) with illustrations:
Let us define hyperbolic area to be: $$ \mu(A)=\int_{A}\frac{dxdy}{y^2}. $$
It can be verified that for all $T \in$ $\text{PSL}(2,\mathbb{R})$ that $\mu(A)=... |
How can I use the chain rule $\frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx}$ to differentiate an expression like $\frac{d}{dx}(y^3)$ ? |
How can I use the chain rule $\frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx}$ to differentiate an expression like $\frac{d}{dx}(y^3)$ ? The solution in the book is $\frac{d}{dx}(y^3) = 3y^2 \frac{dy}{dx}$. I'm not sure how they got it though. |
In fact the result holds a bit more generally, namely:
**LEMMA** $\quad 24 | M^2 - N^2 \;$ if $\; M,N \perp 6, \;$ i.e. coprime to $6.\;$ The proof is easy:
$\quad\quad\quad N\perp 2 \;\Rightarrow\; N = \pm 1, \pm 3 \pmod 8 \;\Rightarrow\; N^2 = 1 \pmod 8$
$\quad\quad\quad N\perp 3 \;\Rightarrow\; N \;\;= \... |
Take any natural number $n$, and factor it as $n=2^{e_1} 3^{e_2} 5^{e_3} ... p^{e_i}$,
where $i$ is the $i$-th prime.
Now map $n$ to the point $n \mapsto (e_1,e_2,\ldots,e_i,0,\ldots)$, where $i$ is the last prime in the factorization
of $n$.
For example,
$$n=123456789
\mapsto
(0,2,\ldots,1,\ldots,1,\ldots)$$ ... |
Take any natural number $n$, and factor it as $n=2^{e_1} 3^{e_2} 5^{e_3} ... p^{e_i}$,
where $i$ is the $i$-th prime.
Now map $n$ to the point $n \mapsto (e_1,e_2,\ldots,e_i,0,\ldots)$, where $i$ is the last prime in the factorization
of $n$.
For example,
$$n=123456789
\mapsto
(0,2,\ldots,1,\ldots,1,\ldots)$$ ... |
Take any natural number $n$, and factor it as $n=2^{e_1} 3^{e_2} 5^{e_3} ... p^{e_i}$,
where $i$ is the $i$-th prime.
Now map $n$ to the point $n \mapsto (e_1,e_2,\ldots,e_i,0,\ldots)$, where $i$ is the last prime in the factorization
of $n$.
For example,
$$n=123456789
\mapsto
(0,2,\ldots,1,\ldots,1,\ldots)$$ ... |
We can compute the total number of k-ary necklaces/bracelets of length n using the Euler totient function:
http://en.wikipedia.org/wiki/Necklace_%28combinatorics%29
Apparently there is a formula for the number of k-ary necklaces/bracelets when the number of beads for each color is fixed.
For example, the numbe... |
I am an undergrad student, and I just finished taking Application of Linear Algebra. Here is what I want from a linear algebra text:
EXAMPLES WHICH ACTUALLY DEMONSTRATE THE POINT.
An example of what I don't ever want to see: the first example in my textbook was to show how to do matrix multiplication. Simple, rig... |
Moron's answer certainly covers your question, but as someone who's not your instructor I'd like to see a few more details in your 'proof' of the first half - can you be more specific about how $q^2/3 \in \mathbb{Z} \Rightarrow 3|q$? While that's easy, it's not necessarily trivial, and you've elided some details there... |
Think about how many times each prime factor must appear on each side of the equation, if you were to break p and q into their prime factorizations. The left side has a 3 in it, how many must the right side have, at least? |
> **Possible Duplicate:**
> [Circular permutations with indistinguishable objects](http://math.stackexchange.com/questions/600/circular-permutations-with-indistinguishable-objects)
<!-- End of automatically inserted text -->
We can compute the total number of k-ary necklaces/bracelets of length n using the Euler ... |
Recently I stumbled upon someone who said he wanted to understand why
$arctanx = \int \displaystyle \frac{1}{1-x^2} $
At first I was confused. This is an easy result in any integral calculus course. But then he explained that although he understood the proof, he wanted to understand it "intuitively". He wanted to... |
Recently I stumbled upon someone who said he wanted to understand why
$\arctan x = \int\dfrac{dx}{1+x^2}$
At first I was confused. This is an easy result in any integral calculus course. But then he explained that although he understood the proof, he wanted to understand it "intuitively". He wanted to see why it ... |
Let $f(x)$ be a continuous function from $\mathbb{R}\rightarrow\mathbb{R}$.
Let's denote $k$-times repeated application of the function, $f(f(f(...f(x)...)))$ as $f^{(k)}(x)$.
Let's call any $x$ a **periodic point** with period $n$ if $f^{(n)}(x)=x$.
Is it true that if a point with period 3 exists, then points... |
Subsets and Splits
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