instruction stringlengths 12 30k |
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1.**How many number of three digit even numbers than can be formed out of the digits 0 to 9 ?**
The question seems confusing since there is no mention of whether repetition is allowed or not ?! since if repetition is allowed then the answer would be **450** (`5*9*10`) where as if not then the answer would be **328*... |
I'm curious about the rights of ownership regarding problems presented in math textbooks. Can a math problem in general be considered intellectual property of its "inventor"? I'm assuming textual word problems can be "owned" in the same way other literature is by publishers/authors but it becomes more gray with purely ... |
Can someone copyright/own a math problem? |
This is a follow-up to <a href="http://math.stackexchange.com/questions/6828/proving-sqrt1-x2-ge-texterf-sqrt-log-x">this</a> question. I came across a tighter potential bound and checked it numerically for $0\le x\le 5$. I think it holds for all positive $x$, can anyone see a proof?
$$1-\exp(-4x^2/ \pi) \ge \text{e... |
This is a follow-up to <a href="http://math.stackexchange.com/questions/6828/proving-sqrt1-x2-ge-texterf-sqrt-log-x">this</a> question. I came across a tighter potential bound and checked it numerically for $0\le x\le 5$. I think it holds for all positive $x$, can anyone see a proof?
$$1-\exp(-4x^2/ \pi) \ge \text{e... |
I've always been taught that one way to look at complex numbers is as a cartesian space, where the "real" part is the x component and the "imaginary" part is the y component.
In this sense, these complex numbers are like vectors, and they can be added geometrically like normal vectors can.
However, is there a geo... |
Geometric interpretation of the multiplication of complex numbers? |
What is, in the language of Schemes, a G-galois branched cover? |
$r$ is a primitive root for the prime $p$
$x_1\equiv r^a (mod p)$
$x_2\equiv r^b (mod p)$
If $gcd(b, p-1)=1$, how can I determine $r$ if $p$, $x_2$, and $b$ are known?
|
$r$ is a primitive root for the prime $p$
$x_1\equiv r^a \pmod{p}$
$x_2\equiv r^b \pmod{p}$
If $gcd(b, p-1)=1$, how can I determine $r$ if $p$, $x_2$, and $b$ are known?
|
I understand that the only free groups that are abelian are 1 and Z, hence a difference between 'free abelian groups' and 'abelian free groups'.
Can someone please tell me what are the solvable free groups?
How can I construct 'free solvable groups'?
What is known about free groups in other classes of groups l... |
Free groups in some classes? |
I asked this very question to a copywriter some years ago and he said that you can not copyright protect a formula but he also said that there are copyright protected algorithms.
So, where is the borderline? I do not know, but I guess this has to be tested in each case. Here is a [discussion][1] on the question, an... |
I asked this very question to a copywriter some years ago and he said that you can not copyright protect a formula but he also said that there are copyright protected algorithms.
So, where is the borderline? I do not know, but I guess this has to be tested in each case. Here is a [discussion][1] on the question, an... |
I'm trying to understand the approach to calculating an answer to the following question.
I need to do some processing on a list of n items.
I know that the total processing time takes log2(n) milliseconds.
Problem: What is the maximum number of items that can be processed in 10 seconds (i.e., 10,000 millisecond... |
Okay, I have this not so pretty 2nd order non-linear ODE I should be able to solve numerically.
f''(R)+(2/R)f'(R)=(.7/R)((1/sqrt(f))-((0.3)/sqrt(1-f))), f'(0)=1, f(1)=1
I was thinking of breaking this guy up into a system of two first order ODE's and then solve, but I have no idea how to set this up. What method ... |
I've got a 2D quadratic Bézier curve which, by construction, is a f(x) function : no loops, a single solution for each defined x.
Is there a common mean to convert this curve to a 3rd degree polynomial ? 3 should be enough since there can only be two "bumps".
Thanks! |
The following is a geometry puzzle from a math school book. Even though it has been a long time since I finished school, I remember this puzzle quite well, and I don't have a nice solution to it.
So here is the puzzle:
![alt text][1]
The triangle ABC is known to be isosceles, that is, AC=BC. The labelled angle... |
Okay, I have this not so pretty 2nd order non-linear ODE I should be able to solve numerically.
$$f''(R) + \frac{2}{R} f'(R)=\frac{0.7}{R} \left( \frac{1}{\sqrt{f(R)}} - \frac{0.3}{\sqrt{1-f(R)}} \right),$$
$$f'(0)=1, f(1)=1.$$
I was thinking of breaking this guy up into a system of two first order ODE's and t... |
A smooth surface $S$ embedded in $\mathbb{R}^3$ whose metric is inherited from $\mathbb{R}^3$
(i.e., distance measured by shortest paths on $S$) is a Riemannian 2-manifold: differentiable
because smooth and with the metric just described. Two questions:
<ol>
<li> Are such surfaces a subset of all Riemannian 2-mani... |
Riemannian 2-manifolds not realized by surfaces in $\mathbb{R}^3$? |
My suggestion:
(A) Start with the definition of absolute value $\left\vert z\right\vert $, with $z=xy,x,y\in\mathbb{R}$
$$\left\vert xy\right\vert =xy\qquad \text{if}\ xy\geq 0\iff \left( x\geq 0\wedge y\geq 0\right) \vee \left( x\leq 0\wedge y\leq 0\right) $$
$$\left\vert xy\right\vert =-xy\qquad \text{if}\ x... |
Are there any odd positive numbers that satisfy the equation:
$a^2 - b^3 = 4$
?
I am certain that there are none but can't prove it. How would you prove that? |
I wanted to prove that $\varphi$ is multiplicative (that for $(a,b)=1$, $\varphi(ab)=\varphi(a)\varphi(b)$) using the following idea:
* Define $\varphi'$ by $n = \varphi'(n) + \varphi(n)$.
* Multiply out $\varphi(a)\varphi(b) = (a - \varphi'(a))(b - \varphi'(b)) = ab - \varphi'(a)b - a \varphi'(b) + \varphi'(a)\var... |
How does the factoring of polynomials over Galois fields work? I cannot seem to understand the basic concept.
**For example:** How do I factorize x<sup>6</sup> - 1 over GF(3)? I know that the result is (x+1)<sup>3</sup> (x+2)<sup>3</sup>, but I'm unable to compute it myself.
I've studied the articles on Wikipedia... |
How do I factorize polynomial over Galois field? |
A smooth surface $S$ embedded in $\mathbb{R}^3$ whose metric is inherited from $\mathbb{R}^3$
(i.e., distance measured by shortest paths on $S$) is a Riemannian 2-manifold: differentiable
because smooth and with the metric just described. Two questions:
<ol>
<li> Are such surfaces a subset of all Riemannian 2-mani... |
Let $T_\varphi$ be an analytic Toeplitz operator (meaning $\varphi \in H^\infty$). Further let $K$ be a compact operator that commutes with $T_\varphi$. Now I want to show that the spectrum of $K$ only consists of $0$.
I'm trying the following things: $T_\varphi K$ is compact so we can apply the spectral theorem for... |
u + v = 2
ux + vy = 1
ux^2 + vy^2 = - 1
ux^3 + vy^3 = -5
Can anyone give me a hint for solving this ? I'm kinda stuck. |
$$u + v = 2$$
$$ux + vy = 1$$
$$ux^2 + vy^2 = - 1$$
$$ux^3 + vy^3 = -5$$
Can anyone give me a hint for solving this ? I'm kinda stuck. |
Some of my favorite ways of engaging students from jump street are:
1. the Chinese magic square
2. Erostophenes’ way of calculating the circumference of the Earth
3. the “Why are manhole covers always round?” question
4. the Twin-Sisters paradox
5. Morley’s Miracle
6. Pythagorean Triples
7. Euler’s formula for c... |
Suppose you want to accumulate $12,000$ in a $5 \%$ account by making a level deposit at the beginning of each of the next 9 years. Find the required level payment. So this seems to be an annuity due problem. I know the following:
$ \displaystyle \sum_{k=1}^{n} \frac{A}{(1+i)^{k}} = \frac{A}{1+i} \left[\frac{1- \lef... |
Some of my favorite ways of engaging students from jump street are:
1. the Chinese magic square
2. Erostophenes’ way of calculating the circumference of the Earth
3. the “Why are manhole covers always round?” question
4. the Twin-Sisters paradox
5. Morley’s Miracle
6. Pythagorean Triples
7. Euler’s formula for... |
I'm reading *A Computational Introduction to Number Theory and Algebra*, which can be found [here][1] as a free download. From the book's exercises, I'm stuck with a proof to show that $gcd(a,b)=|a| \iff a | b$.
My current reasoning goes as follows:
Since $gcd(a,b) = |a|$,
we see that
$aZ + bZ = |a|Z$.
Since... |
Okay, I have this not so pretty 2nd order non-linear ODE I should be able to solve numerically.
$$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.$$
as for f(0) the function seems approaches infinity (please see picture),
I was thinking... |
Given two Riemmanian Manifolds of dimension 2, and a point on each. If the scalar curvatures are isomorphic (as functions) in some nbhds of these points, are then the manifolds necessarily locally isometric? |
Is the scalar curvature the only isometric invariant of a Riemannian 2-manifold? |
This is [true if the curvatures are constant][1] (Minding's theorem), but not in general. I actually don't know a counterexample, but Stillwell [points out on page 344][2] of *Mathematics and its history* that Strubecker's *Differentialgeometrie* gives such an example in Volume III on page 124.
[1]: http://en.w... |
Okay, I have this not so pretty 2nd order non-linear ODE I should be able to solve numerically.
$$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 ... |
This is [true if the curvatures are constant][1] (Minding's theorem), but not in general. I actually don't know a counterexample, but (edit: I think that) Stillwell [points out on page 344][2] of *Mathematics and its history* that Strubecker's *Differentialgeometrie* gives such an example in Volume III on page 124. (... |
During the summer, I did an REU where we focused primarily on one-dimensional dynamics and more specifically kneading theory. One thing that I was always confused about is why the Schwarzian derivatives always seem to pop up in discussions of iterated dynamics on the real line. I understand what a Schwarzian derivative... |
Schwarzian Derivative and One-Dimensional Dynamics - how are they connected? |
This is [true if the curvatures are constant][1] (Minding's theorem), but not in general. I actually don't know a counterexample, but (edit: I think that) Stillwell [points out on page 344][2] of *Mathematics and its history* that Strubecker's *Differentialgeometrie* gives such an example in Volume III on page 124. (... |
This is [true if the curvatures are constant][1] (Minding's theorem), but not in general. <strike>I actually don't know a counterexample, but (edit: I think that)</strike> Stillwell [points out on page 344][2] of *Mathematics and its history* that Strubecker's *Differentialgeometrie* gives <strike>such an</strike> a r... |
Krull's Hauptidealsatz (principal ideal theorem) says that for a Noetherian ring $R$ and any $r\in R$ which is not a unit or zero-divisor, all primes minimal over $(r)$ are of height 1. How badly can this fail if $R$ is a non-Noetherian ring? For example, if $R$ is non-Noetherian, is it possible for there to be a minim... |
How badly can Krull's Hauptidealsatz fail for non-Noetherian rings? |
I get $F = A \frac{(1+i)^{2n}-(1+i)^n}{i(1+i)^n}$. But when I solve for $A$, I get the level payment as being $1088.28$. But the correct answer is $1036.46$. Why is there a discrepancy? |
During the summer, I did an REU where we focused primarily on one-dimensional dynamics and more specifically kneading theory. One thing that I was always confused about is why the Schwarzian derivatives always seem to pop up in discussions of iterated dynamics on the real line. I understand what a Schwarzian derivative... |
I'm teaching a group theory course now, and I wanted to give my students a proof that every subgroup of a cyclic group is cyclic. The easiest way I could think to do this is to say that any cyclic group corresponds to a $\mathbb{Z}/n\mathbb{Z}$ and then (implicitly) use the first isomorphism theorem (which they haven't... |
How to show every subgroup of a cyclic group is cyclic? |
See Proposition 5 in
http://www.math.uga.edu/~pete/4400algebra2point5.pdf
Note that part of this argument is left as an exercise for the reader. It *is* an incarnation of the same argument that shows that a Euclidean domain is a PID, but one certainly doesn't need this terminology or any prior results. Namely,... |
Prove the following: If $H$ is a subgroup of finite index in a group $G$, and $K$ is a subgroup of $G$ containing $H$, then $K$ is of finite index in $G$ and $[G:H] = [G:K][K:H]$.
So this is basically a bijective proof? The number of cosets of $H$ in $G$ equals the number of cosets of $K$ in $G$ times the number o... |
See Proposition 5 in
http://www.math.uga.edu/~pete/4400algebra2point5.pdf
Note that part of this argument is left as an exercise for the reader. It *is* an incarnation of the same argument that shows that a Euclidean domain is a PID, but one certainly doesn't need this terminology or any prior results. Namely,... |
Suppose $f(x) = \begin{cases} 0 \ \ \text{if} \ x \in \mathbb{R}- \mathbb{Q} \newline \frac{1}{q} \ \ \text{if} \ x \in \mathbb{Q} \ \text{and} \ x = \frac{p}{q} \ \text{in lowest terms} \end{cases}$
(i) Is $f$ continuous on the irrationals?
(ii) Is $f$ continuous on the rationals?
For (i) you could use the sequ... |
I'm in the process of writing my Master's Thesis on automata theory. The writing must be in English which isn't my mother tongue.
So the question is, given that this is my first time long (hundred pages) writing and since English is a second language for me, what is the book you recommend for academic writing style... |
very sorry for the noobish post, I'm studying math on my own and lot of times im very confused about little things.
Here is my question, the original problem starts out like this |x -2| >= |2x -3| all of I get until this step...
How does: *(x-1)(3x-5) <= 0*
Become solved as : *1 <= x <= 5/3*
if it were sepa... |
very sorry for the noobish post, I'm studying math on my own and lot of times im very confused about little things.
Here is my question, the original problem starts out like this $\vert x -2 \vert \geq \vert 2x -3\vert $ all of I get until this step...
How does: $\vert (x-1)(3x-5)\vert \leq 0$.
Become solved a... |
Young's inequality for convolutions states that if $1 \leq p, q, r \leq \infty$. Then for all $f \in L^p(G)$ and all $g \in L^r(G)$ satisfying $\|g\|_{L^r(G)} = \|g'\|_{L^r(G)}$ where $g'(x) = g(x^{-1})$.
Now Grafakos claims we can use this to prove the following inequality due to Hardy:
$$\left ( \int_0^\infty \... |
Young's inequality for convolutions states that if $1 \leq p, q, r \leq \infty$ satisfy
$$\frac{1}{q} + 1 = \frac{1}{p} + \frac{1}{r}.$$
Then for all $f \in L^p(G)$ and all $g \in L^r(G)$ where $g$ and $g'$ have the same $L^r$-norm and $g'(x) = g(x^{-1})$ ($G$ is a topological group). We have that:
$$\|f * g\|_... |
Young's inequality for convolutions states that if $1 \leq p, q, r \leq \infty$ satisfy
$$\frac{1}{q} + 1 = \frac{1}{p} + \frac{1}{r}.$$
Then for all $f \in L^p(G)$ and all $g \in L^r(G)$ where $g$ and $g'$ have the same $L^r$-norm and $g'(x) = g(x^{-1})$ ($G$ is a topological group). We have that:
$$\|f * g\|_... |
Let $M$ be the mean matrix of a multitype branching process.
I am trying to figure out whether $A = (I-M)^{-1}$ has any signifiance.
I think $A_{ij}$ would stand for the number of times that type $j$ would be generated from type $j$, but I can't find any material on that in the web. Anyone knows anything about th... |
what is the connection between the mean matrix and properties of a branching process? |
We have two functions:
$F(x)$ and $G(x)$
Suppose the improper integral has the value 1.
$$\int_{1}^{\infty} F(x) G(x) \mathrm{d}x = 1$$
Can we find the value of the integral, where c is a constant?
$$\int_{1}^{\infty} F(x+c) G(x) \mathrm{d}x$$
I am interested in generalized answers and not specific exampl... |
Young's inequality for convolutions states that if $1 \leq p, q, r \leq \infty$ satisfy
$$\frac{1}{q} + 1 = \frac{1}{p} + \frac{1}{r}$$
for all $f \in L^p(G)$ and all $g \in L^r(G)$ where $g$ and $g'$ have the same $L^r$-norm and $g'(x) = g(x^{-1})$ ($G$ is a topological group) we have that:
$$\|f * g\|_q \leq ... |
We have two functions:
$F(x)$ and $G(x)$
Suppose the improper integral has the value 1.
$$\int_{1}^{\infty} F(x) G(x) \mathrm{d}x = 1$$
Can we find the value of the integral, where c is a constant?
$$\int_{1}^{\infty} F(x+c) G(x) \mathrm{d}x$$
I am interested in generalized answers and not specific exampl... |
very sorry for the noobish post, I'm studying math on my own and lot of times im very confused about little things.
Here is my question, the original problem starts out like this: Solve the inequality $\vert x -2 \vert \geq \vert 2x -3\vert $
Here is the solved problem from my book
1. $\vert x -2 \vert \geq... |
Do you know anything about $F$ and $G$? If not, then certainly no. Let $G(x) = 1$ if $x \leq 2$ and $0$ otherwise. Let $F(x) = 1$ if $x< 2$; we leave $F(x)$ undefined elsewhere for now. Then $\int_1^{\infty} F(x)G(x) = 1$.
Consider the function $h(c) = \int_1^\infty F(x+c)G(x)dx$ defined on $0 < c$. I claim that fo... |
We have two functions:
$F(x)$ and $G(x)$
Suppose the improper integral has the value 1.
$$\int_{1}^{\infty} F(x) G(x) \mathrm{d}x = 1$$
Can we find the value of the integral, where c is a constant?
$$\int_{1}^{\infty} F(x+c) G(x) \mathrm{d}x$$
I am interested in generalized answers and not specific exampl... |
We have two functions:
$F(x)$ and $G(x)$
Suppose the improper integral has the value 1.
$$\int_{1}^{\infty} F(x) G(x) \mathrm{d}x = 1$$
Can we find the value of the integral, where c is a constant?
$$\int_{1}^{\infty} F(x+c) G(x) \mathrm{d}x$$
I am interested in generalized answers and not specific exampl... |
Here is another: [Handbook of Writing for the Mathematical Sciences - Nicholas J. Higham](http://books.google.com/books?id=9gQd2fJA7Y4C&printsec=frontcover&dq=Nicholas+J.+Higham+-+Handbook+of+Writing+for+the+Mathematical+Sciences&hl=pt-PT&ei=IQy7TJXNJM7rsgaq-PHIDQ&sa=X&oi=book_result&ct=result&resnum=1&ved=0CCcQ6AEwAA#... |
[How to write mathematics][1] is a free essay by Paul Halmos which fulfills your first three criteria. It is short, but has several good points.
[1]: http://retro.seals.ch/digbib/view?rid=ensmat-001:1970:16::59&id=browse&id2=browse2 |
Why does $ \text{Var}(Y) = E(\text{Var}(Y|X))+ \text{Var}(E(Y|X))$? What is the intuitive explanation for this? In laymen's terms it seems to say that the variance of $Y$ equals the expected value of the conditional variance plus the variance of the conditional expectation. |
How to assess a non-natively english speaking high-schooler's mathematical ability? |
Well done! Your observations are correct and can be completed to give a proof that the sequence is convergent to $\frac{4}{5}$.
The tough part in proving your observations it to prove that the odd/even sub-sequences are monotonic and appropriately bounded (what do I mean by this?).
Here is a hint:
Show that ... |
Here is a more explicit version of my hint. I said it would be easier to show that $d_n = c_n - \frac{4}{5}$ tends to zero as $n \to \infty$. This gives $d_1 = \frac{1}{5}$ and the recurrence
$$d_{n+1} + \frac{4}{5} = \frac{4}{1 + 5(d_n + \frac{4}{5})} = \frac{4}{5} \frac{1}{1 + d_n}$$
hence
$$d_{n+1} = - \f... |
Here's one way to prove it: let $f(x) = 4/(1+5x)$. Say $|x-4/5| \le C$ for some constant $C$. Can you find $C$ and some constant $0 \le k < 1$ so that if $|x-4/5| \le C$, then $|f(x)-4/5| \le k|x-4/5|$?
If you do this, then you can iterate to get $|f^j(x)-4/5| \le k^j |x-4/5|$, for all $j$, and so if you make $j$ l... |
A rigorous proof is [here](http://en.wikipedia.org/wiki/Law_of_total_variance); it relies on the law of total expectation, which says that $E(E(X|Y))=E(X)$. The intuitive explanation of _that_ is that $E(X|Y)$ is the expected value of $X$ given a particular value of $Y$, and that $E(E(X|Y))$ is the expected value of t... |
Well done! Your observations are correct and can be completed to give a proof that the sequence is convergent to $\frac{4}{5}$.
The tough part in proving your observations it to prove that the odd/even sub-sequences are monotonic and appropriately bounded (what do I mean by this?).
Here is a hint:
Show that ... |
Well done! Your observations are correct and can be completed to give a proof that the sequence is convergent to $\frac{4}{5}$.
The tough part in proving your observations it to prove that the odd/even sub-sequences are monotonic and appropriately bounded (what do I mean by this?).
Here is a hint:
Show that ... |
Well, I'm a non-native English speaker, so I can provide some insight.
I think that one of the first problems would be names and notation in general. As far as I can see, there is no easy way to solve this unless you already know the customs of the country the student is coming from, but your education will probabl... |
Well done! Your observations are correct and can be completed to give a proof that the sequence is convergent to $\frac{4}{5}$.
The tough part in proving your observations it to prove that the odd/even sub-sequences are monotonic and appropriately bounded (what do I mean by this?).
Here is a hint:
Show that ... |
Well, I'm a non-native English speaker, so I can provide some insight.
I think that one of the first problems would be names and notation in general. As far as I can see, there is no easy way to solve this unless you already know the customs of the country the student is coming from, but your education will probabl... |
Prove: If X is NOT divisible by 3 and Y is NOT divisible by 3 Then X^2 - y^2 is Divisible by 3
I cannot find a good way to prove this.. Tried contrapositive and contradiction but cannot see the answer. Anyone have any ideas? |
Here is a more explicit version of my hint. I said it would be easier to show that $d_n = c_n - \frac{4}{5}$ tends to zero as $n \to \infty$. This gives $d_1 = \frac{1}{5}$ and the recurrence
$$d_{n+1} + \frac{4}{5} = \frac{4}{1 + 5(d_n + \frac{4}{5})} = \frac{4}{5} \frac{1}{1 + d_n}$$
hence
$$d_{n+1} = - \f... |
Here is a more explicit version of my hint. I said it would be easier to show that $d_n = c_n - \frac{4}{5}$ tends to zero as $n \to \infty$. This gives $d_1 = \frac{1}{5}$ and the recurrence
$$d_{n+1} + \frac{4}{5} = \frac{4}{1 + 5(d_n + \frac{4}{5})} = \frac{4}{5} \frac{1}{1 + d_n}$$
hence
$$d_{n+1} = - \f... |
Krull's Hauptidealsatz (principal ideal theorem) says that for a Noetherian ring $R$ and any $r\in R$ which is not a unit or zero-divisor, all primes minimal over $(r)$ are of height 1. How badly can this fail if $R$ is a non-Noetherian ring? For example, if $R$ is non-Noetherian, is it possible for there to be a minim... |
The [Wikipedia article](http://en.wikipedia.org/wiki/Long_division) on long division explains the different notations. I still use the European notation I learned in elementary school in Colombia. I had difficulty adapting to the US/UK notation when I moved to the US. However, I did enjoy seeing my classmates' puzzled ... |
My question is essentially this: Why does the principle of transfinite induction not suffice to show the axiom of choice when the sets to be chosen from are indexed by a well ordered set?
I have read that one can prove the axiom of finite choice from simple induction. You simple induct on the size of the system of s... |
I am trying to calculate the average distance between any two nodes in an $r$-dimensional mesh.
This is a $3$-dimensional mesh with $n=3$
![alt text][1]
To choose any two points in the mesh there are $\left(n^r \left(n^r - 1\right)\right)/2$ ways of doing this.
So if we choose 2 points in the mesh $p\lbrace... |
My question is essentially this: Why does the principle of transfinite induction not suffice to show the axiom of choice when the sets to be chosen from are indexed by a well ordered set?
I have read that one can prove the axiom of finite choice from simple induction. You induct on the size of the system of sets you... |
Let G be the group defined by generators a,b and relations a^4=e a^2*b^-2=e abab^-1=e. Since the quaternion group of order 8 is generated by elements a,b satisfying the previous relations, there is an epimorphism from G onto Q8. Let F be the free group on {a,b} and N the normal subgroup generated by { a^4 a^2*b^-2 a*b*... |
Let G be the group defined by generators a,b and relations $a^4=e$, $a^2*b^{-2}=e$, $abab^{-1}=e$. Since the quaternion group of order 8 is generated by elements $a,b$ satisfying the previous relations, there is an epimorphism from $G$ onto $Q_8$. Let $F$ be the free group on $\\{a,b\\}$ and $N$ the normal subgroup gen... |
To write down the group $F/N$, the best way in this case is to find a normal form. Note that the last relation $abab^{-1}=e$ means that $ba=a^{-1}b = a^3b$ (that is, the element $b^{-1}a^{-3}ba$ is in $N$); so any coset representative that contains the string $ba$ can be replaced with a coset representative in which th... |
Let G be the group defined by generators a,b and relations $a^4=e$, $a^2b^{-2}=e$, $abab^{-1}=e$. Since the quaternion group of order 8 is generated by elements $a,b$ satisfying the previous relations, there is an epimorphism from $G$ onto $Q_8$. Let $F$ be the free group on $\\{a,b\\}$ and $N$ the normal subgroup gene... |
To write down the group $F/N$, the best way in this case is to find a normal form. Note that the last relation $abab^{-1}=e$ means that $ba=a^{-1}b = a^3b$ (that is, the element $b^{-1}a^{-3}ba$ is in $N$); so any coset representative that contains the string $ba$ can be replaced with a coset representative in which th... |
What is the highest power of 2 dividing 100!
This is what I have so far:
50 multiples of 2
25 multiples of 4
12 multiples of 8
6 multiples of 16
3 multiples of 32
1 multiple of 64
giving a total of **2^97**
am I missing anything here? |
What is the highest power of 2 dividing 100!
This is what I have so far:
50 multiples of 2
25 multiples of 4
12 multiples of 8
6 multiples of 16
3 multiples of 32
1 multiple of 64
EDIT: giving a highest power of **2^97**
am I missing anything here? |
Kodaira defines a complex analytic family of compact complex manifolds as the data $(E,B,\pi)$, where $E$ and $B$ are complex manifolds, and $\pi$ is a surjective holomorphic submersion such that the preimage $\pi^{-1}(x)$ of any point $x \in B$ is a compact complex submanifold of $E$. (Here complex manifolds are requi... |
Calculate 1^30 + 2^30 + 3^30 + ... + 17^30 mod 31
Using Fermat's Theorem:
1^30 = 1 (mod 31) (since 31 is prime)
This implies the above is congruent to 17 (mod 31)
This is correct, right? |
There's a whole room devoted to physical demonstrations of mathematical concepts in the Museum of Science in Boston. I'll list the ones I can remember, and others who have been there please feel free to add to the list.
* A little train that goes around on a track that's on a mobius strip.
* A "Pachinko" type mac... |
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