instruction stringlengths 12 30k |
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For a given radius, theta, and position, can you calculate a Bezier curve? |
Max, besides the existence of a lot of literature about this subject I've written two very basic/introductive articles about it. Perhaps the better is the shorter (and newer) one, I deal with the iteration of $ f(x) = 1/(1+x) $ The idea stems originally from a consideration of continued fractions in a general form $ x;... |
I am looking for a correct proof of this statement: If $G$ is a group such that $G/Z(G)$ is cyclic, then $G$ is commutative.
Proof: $G/Z(G)$ is isomorphic to $\operatorname{Inn}(G)$ and is cyclic, and then for every $a$ and $b$ in $G$ the inner isomorphisms $\gamma_a$ and $\gamma_b$ satisfy $\gamma_a \gamma_b = \gam... |
I have a radius, `R`, for an aircraft traveling at velocity, `V`. If we start at point, `(X,Y)`, what is the position of the point at time, `t`.
For example:
The aircraft is at point `(0,0)` and traveling at `250` knots and initiates a turn with a bank angle, phi, of `5` degrees. Assume that the aircraft can inst... |
I have a radius, `R`, for an aircraft traveling at velocity, `V`. If we start at point, `(X,Y)`, what is the position of the point at time, `t`.
For example:
The aircraft is at point `(0,0)` and traveling at `250` knots and initiates a turn with a bank angle, phi, of `5` degrees. Assume that the aircraft can inst... |
Let $F_n$ be an $n$-generator free group with a free basis $x_1,\ldots,x_n.$ Is it true that the stabilizer of $x_1$ in $\mathrm{Aut}(F_n)$ is generated by all left and right Nielsen moves $\lambda_{ij}$ and $\rho_{ij}$ such that $i \ne 1?$
Let $i \ne j$ and $1 \le i,j \le n.$ The left Nielsen move $\lambda_{ij}$ ta... |
Problem: The series $\sum_{n=1}^{\infty} a_n$ diverge and positive. What can be said on the series $\sum_{n=1}^{\infty} \frac{a_n}{1+n^{2}a_n}$ ?
Approach:
1. First, I tried separating it into two cases: $a_n \to \infty$, $a_n \to A$ (where $A$ is some const value).
2. Tried the ratio test and got nowhere wit... |
Max, besides the existence of a lot of literature about this subject I've written two very basic/introductive articles about it. Perhaps the better is the shorter (and newer) one, I deal with the iteration of $ f(x) = 1/(1+x) $ The idea stems originally from a consideration of continued fractions in a general form $ x;... |
A [regular sequence][1] is an $n$-fold collection $\{r_1, \cdots, r_n\} \subset R$ of elements of a ring $R$ such that for any $2 \leq i \leq n$, $r_i$ is not a zero divisor of the quotient ring
$$ \frac R {\langle r_1, r_2, \cdots, r_{i-1} \rangle}.$$
> Does the order of the $r_i$'s matter? That is, is any permuta... |
Is a regular sequence ordered? |
Consider a particle starting at the the origin and moving along the positive real line at a constant speed of 1. Suppose there is a counter which clicks at random time intervals following the [exponential distribution](http://en.wikipedia.org/wiki/Exponential_distribution) with parameter $\lambda$ and whenever the cou... |
Currently taking a logic class and trying to understand this.
You have two set A and B.
Both sets are empty sets.
Is set A a subset of the complement of set B?
Assume the context is the universal set. |
Is there and empty set in the complement of an empty set? |
I am in the process of proving that if a space curve (in R^3) has infinite length and the curvature tends towards 0 as the natural parameter s tends to infinity, the curve must be unbounded - i.e. not contained in any sphere of finite radius. This seems correct intuitively, but I have no guarantee it is correct, unless... |
What you are asking for is a special case of the joint distribution of the maximum and sum of a sequence of random variables. The fact that the variables are stable simplifies this and allows us to compute the joint distribution in the limit as n goes to infinity. This limit can be described in terms of the joint distr... |
{An} is a sequence of events which may NOT be independent. Also A is any event such that P(A) greater than zero. Again the question says nothing about whether {An} is an independent sequence and also it says nothing about whether A or{An} are independent (we assume no independence).
we are asked to prove P[An i.o.]... |
I am an ESL learner and most of the time I am able to understand English but In my courses, we don't study mathematics, and I think that it is a mistake or wrong.
1-What are really the best sources (book or books) to study and review completely high school mathematics?
-titles
-authors
-edition
-publishers
2-... |
{An} is a sequence of events which may NOT be independent. Also A is any event such that P(A) greater than zero. Again the question says nothing about whether {An} is an independent sequence and also it says nothing about whether A or{An} are independent (we assume no independence).
we are asked to prove P[An i.o.]... |
Step 1: Show that any homomorphism from a cyclic group is determined by its image on a generator of the group.
Step 2: Show that the image of the homomorphism is generated by the image of that generator.
Step 3: What can you say about the kernel of such a homomorphism? What powers of the generator does it send to... |
Several concrete examples give rise to the following convolution identity:
$$
(f_2 * g)(t) = at(f * g)(t),
$$
where $f_2 (x) = xf(x)$ and $a \in (0,1)$ (constant).
(This probably holds also with respect to discrete convolution.)
Is this identity known (assuming it's true)?
If not, can you prove it?
Can you... |
Is this convolution identity known? |
I greatly enjoyed the Lights Out game described [here][1] (I am sorry I had to link to an older page because some wikidiot keeps deleting most of the page).
Its mathematical analysis is [here][2] (it's just linear algebra)
Now I just discovered a hexagonal version:
>http://cwynn.com/games/lightsoutmobile.htm ... |
Can someone please guide me to a way by which I can solve the following problem.
There is a dice and 2 players. Rolling stops as soon as some exceeds 100(not including 100 itself). Hence you have the following choices: 101, 102, 103, 104, 105, 106.
Which should I choose given first choice.
I'm thinking Markov chai... |
Several concrete examples give rise to the following convolution identity:
$$
(f_2 * g)(t) = at(f * g)(t),
$$
where $f_2 (x) = xf(x)$ and $a \in (0,1)$ (constant).
(This probably holds also with respect to discrete convolution.)
Is this identity known (assuming it's true)?
If not, can you prove it?
Can you... |
If $(x_n)$ and $(y_n)$ are sequences such that $(y_n)$ is bounded and $lim\frac{(x_n)}{(y_n)} = 0$, prove that $lim(x_n) = 0$
So since $(y_n)$ is bounded, there exists a real number $M$ such that for all n natural numbers, $(y_n) < |M|$
So by the limit definition :
$lim|\frac{(x_n)}{(y_n)}-0| = lim\frac{(x... |
If $(x_n)$ and $(y_n)$ are positive real sequences such that $(y_n)$ is bounded and $lim\frac{(x_n)}{(y_n)} = 0$, prove that $lim(x_n) = 0$
So since $(y_n)$ is bounded, there exists a real number $M$ such that for all n natural numbers, $(y_n) < |M|$
So by the limit definition :
$lim|\frac{(x_n)}{(y_n)}-0|... |
Several concrete examples give rise to the following convolution identity:
$$
(f_2 * g)(t) = at(f * g)(t),
$$
where $f_2 (x) = xf(x)$ and $a \in (0,1)$ (constant).
(This probably holds also with respect to discrete convolution.)
Is this identity known (assuming it's true)?
If not, can you prove it?
Can you... |
Let $H=\left\{ z\in\mathbb{C}\mid\Im\left(z\right)>0\right\}$ be the upper-half Poincare plane. Let $GL\left(2,\mathbb{R}\right)$ be the general linear group, $Z\left(GL\left(2,\mathbb{R}\right)\right)$ be the center of the general linear group and $O\left(2,\mathbb{R}\right)$ be the orthogonal subgroup of $GL\left(2,... |
Can someone please guide me to a way by which I can solve the following problem.
There is a die and 2 players. Rolling stops as soon as some exceeds 100(not including 100 itself). Hence you have the following choices: 101, 102, 103, 104, 105, 106.
Which should I choose given first choice.
I'm thinking Markov chain... |
I would approach this by contradiction:
If the sequence does not converge to 0, then there is a subsequence that is bounded away from 0. So we may fix $a>0$ such that $|x_n|>a$ for infinitely many values of $n$.
Since the sequence of $y_n$ is bounded, say by $M$, we have $|y_n|<M$ for all $n$, so if $n$ is one ... |
How is GL(1,C) related to GL(2,R)? |
Let $H=\left\{ z\in\mathbb{C}\mid\Im\left(z\right)>0\right\}$ be the upper-half Poincare plane. Let $GL\left(2,\mathbb{R}\right)$ be the general linear group, $Z\left(GL\left(2,\mathbb{R}\right)\right)$ be the center of the general linear group and $O\left(2,\mathbb{R}\right)$ be the orthogonal subgroup of $GL\left(2,... |
What does $H=GL\left(2,\mathbb{R}\right)/\left(Z\left(GL\left(2,\mathbb{R}\right)\right)\cdot O\left(2,\mathbb{R}\right)\right)$ mean? |
What does $H=GL(2,\mathbb{R})/(Z(GL(2,\mathbb{R}))\cdot O(2,\mathbb{R}))$ mean? |
I started reading about uniform spaces in Bourbaki, and the closure operation has been defined as such:
$$\bar{A}=\bigcap_{V\in\mathcal{U}}V(A)$$
Where $\mathcal{U}$ is the uniform structure of the space, and $V(A)$ is the set of all left-relatives of $A$. One thing I don't understand is why $\bar{A}=\bar{\bar{A}... |
If $(x_n)$ and $(y_n)$ are positive real sequences such that $(y_n)$ is bounded and $\lim\frac{(x_n)}{(y_n)} = 0$, prove that $\lim(x_n) = 0$
So since $(y_n)$ is bounded, there exists a real number $M$ such that for all n natural numbers, $(y_n) < |M|$
So by the limit definition :
$\lim|\frac{(x_n)}{(y_n)}... |
Can someone please share references (websites or books) where I can find problems related with Fubini's theorem and applications of Radon-Nikodym theorem? I have googled yes and don't find many problems. What are the "typical" problems (if there are any) related with these topics? [Yes, exam is coming soon so I don't... |
I heard that using some relatively basic differential geometry, you can show that the only spheres which are Lie groups are $S^0$, $S^1$, and $S^3$. My friend who told me this thought that it involved de Rham cohomology, but I don't really know anything about the cohomology of Lie groups so this doesn't help me much. ... |
Is there an easy way to show which spheres can be Lie groups? |
I heard that using some relatively basic differential geometry, you can show that the only spheres which are Lie groups are $S^0$, $S^1$, and $S^3$. My friend who told me this thought that it involved de Rham cohomology, but I don't really know anything about the cohomology of Lie groups so this doesn't help me much. ... |
A convolution identity (revision of the question "Is this convolution identity known?") |
On a 2D plane, how to construct a graph with 7 vertexes and 21 edges? I tried various combination but couldn't seem to draw that kind of contrived graph on a paper.
But my understanding is that [it is possible][1]. So anyone can help me with it?
[1]: http://mathoverflow.net/questions/30759/in-a-graph-is-it-al... |
I have a network, with $V$ vertex, $E$ edges and $C$ cycles, which has "current" flowing in and out from the network, as shown below:
<img src="https://docs.google.com/drawings/pub?id=1Micr1JiaNxGw3FTF-vJOR3h94IjNPjq1mQqEBS7QkGY&w=960&h=720">
The equations governs the current flow in the network is the so... |
How to Set the Redundant Current in a Network? |
I have a network, with $V$ vertex, $E$ edges and $C$ cycles, which has "current" flowing in and out from the network, as shown below:
<img src="https://docs.google.com/drawings/pub?id=1Micr1JiaNxGw3FTF-vJOR3h94IjNPjq1mQqEBS7QkGY&w=960&h=720">
Each edge has a current value associated to it.
The equation... |
Is Knopp's _Theory and Application of Infinite Series_ out of date? It's looks terrific to me, but the Dover edition I bought new maybe a year ago: http://preview.tinyurl.com/2eprqps seems to be the same as an edition published in 1951 and may go back as far as 1921. 60 or 90 years is a lot of math years. How about it?... |
Is Knopp's "Theory and Application of Infinite Series" out of date? |
I have a network, with $V$ vertex, $E$ edges and $C$ cycles, which has "current" flowing in and out from the network, as shown below:
<img src="https://docs.google.com/drawings/pub?id=1Micr1JiaNxGw3FTF-vJOR3h94IjNPjq1mQqEBS7QkGY&w=960&h=720">
Each edge has a current value associated to it.
The equation... |
I have a network, with $V$ vertex, $E$ edges and $C$ cycles, which has "current" flowing in and out from the network, as shown below:
<img src="https://docs.google.com/drawings/pub?id=1Micr1JiaNxGw3FTF-vJOR3h94IjNPjq1mQqEBS7QkGY&w=960&h=720">
Each edge has a current value associated to it.
The equation... |
I've never actually done a delta-epsilon proof, so I thought I'd try my hand at one. I decided to try it out for $f(x)=1/x$. If I understand correctly from the wikipedia article, I want to show for any $\epsilon>0$, there exists a $\delta>0$ such that if $|x-c|<\delta$, then $|f(x)-f(c)|<\epsilon$.
Anyway, I noticed... |
Is $f(x)=1/x$ continuous on $(0,\infty)$? |
Let $Z$ be a random variable with finite mean and consider $$ F(x) = E[\max(x+Z,0)]$$ How to prove that $F(x)$ is a continuous function of $x$?
This is not homework. A textbook I am reading states that $F(x)$ is continuous without providing any explanation, and I can't seem to fill the gap. |
This is a slightly different question than the one asked [here][1].
I understand that for non-planar graph there are edges in the graph which has more than 2 cycles sharing on it. But for [planar graph][2], can we prove that *it is always possible to find a complete set of cycle basis that each edge is shared by at... |
For a Planar Graph, is it always possible to construct a set of cycle basis, with each and every edge Is shared by at most 2 cycle bases? |
Let $Z$ be a random variable with finite mean and consider the function $F: R \rightarrow R$ defined by $$ F(x) = E[\max(x+Z,0)]$$ How to prove that $F(x)$ is a continuous function of $x$?
This is not homework. A textbook I am reading states that $F(x)$ is continuous without providing any explanation, and I can't se... |
If $\lvert x-y\rvert\leq\delta$, then $\lvert (x+Z)-(y+Z)\rvert\leq\delta$, and also $$\lvert \mathrm{max}(x+Z,0)-\mathrm{max}(y+Z,0)\rvert\leq\delta,$$
right? (Notice that $\mathrm{max}(\bullet+Z,0)$ is 1-Lipschitz.)
Hence we also have $\lvert F(x)-F(y)\rvert\leq\delta$.
Remark: In order for $\mathrm{max}(x+Z,0... |
I have a network, with $V$ vertex, $E$ edges and $C$ cycles, which has "current" flowing in and out from the network, as shown below:
<img src="https://docs.google.com/drawings/pub?id=1Micr1JiaNxGw3FTF-vJOR3h94IjNPjq1mQqEBS7QkGY&w=960&h=720">
Each edge has a current value associated to it.
The equation... |
If $\lvert x-y\rvert\leq\delta$, then $\lvert (x+Z)-(y+Z)\rvert\leq\delta$, and also $$\lvert \mathrm{max}(x+Z,0)-\mathrm{max}(y+Z,0)\rvert\leq\delta,$$
right? (Notice that $\mathrm{max}(\bullet+Z,0)$ is 1-Lipschitz.)
Hence we also have $\lvert F(x)-F(y)\rvert\leq\delta$.
<s>Remark: In order for $\mathrm{max}(x+... |
This is of course true, with the basis cycles being boundaries of the bounded faces (after killing off all pairs {edge e, reverse of edge e}).
Are you looking for a rigorous proof? It should follow from the argument establishing the fact that there are such things as faces etc (see for example chapter 4 [here][1]).... |
Can this equation be simplified?
$ \sum_{i=1}^n n!/(n-i)!(i)! $
|
Can this be simplified? |
Can this equation be simplified?
$ \sum_{i=1}^n (k^i)/i! $
|
I have a radius, `R`, for an aircraft traveling at velocity, `V`. If we start at point, `(X,Y)`, what is the position of the point at time, `t`.
For example:
The aircraft is at point `(0,0)` and traveling at `250` knots and initiates a turn with a bank angle, phi, of `5` degrees. Assume that the aircraft can inst... |
This is just a completely random question of the top of my head.
Let $\Omega$ be the set of bijections $\phi:\mathbb{N} \cup \{0\} \rightarrow \mathbb{Q}$.
We can imagine any $\phi \in \Omega$ to be equivalent to a wacky "staircase", where for $x \in \mathbb{R}^+$ we have $x \mapsto \phi(\lfloor x \rfloor)$.
I... |
Which bijection from the natural numbers to the rationals has the smallest steps? |
Many introductory books on vector spaces mention that the scalars need not be reals, and might even have sections discussing complex vector spaces or vector spaces over the integers mod 2. I have never seen any such book mention that all of the theory goes through as well if one restricts the scalars to be just ration... |
An abelian group is torsionfree and divisible iff it has the (unique) structure of a vector space over $\mathbb{Q}$.
Using this observation the classification of vector spaces over a field in terms of their dimension gives rise to a classification of torsionfree divisible abelian groups. In particular, since if $... |
An abelian group is torsionfree and divisible iff it has the (unique) structure of a vector space over $\mathbb{Q}$.
Using this observation the classification of vector spaces over a field in terms of their dimension gives rise to a classification of torsionfree divisible abelian groups. In particular, since if $... |
Can this expression be simplified?
$ \sum_{i=0}^n (k^i)/i! $
EDIT:
This expression I have got as number of possible ways to select n objects or less from k different infinite objects...
I believe it must be equal number of possible ways to select n objects from k+1 different infinite objects, the +1 is a dum... |
I've run across the recurrence $a_{n+1} = (a_n)^2 + 1$ in the past. Unfortunately, the referrence escapes me. However, my impression was that recurrences involving the product of previous terms (such as $a_{n+1} = (a_n)(a_{n-1})$) are difficult to solve. I'm wondering what is known for this very general problem.
... |
What's known about recurrences involving $(a_n)^2$? |
A general problem for the Calculus of Variations asks us to minimize the value of a functional $A[f]$, where $f$ is usually a differentiable function defined on $\mathbb{R}^n$.
What if, however, the domain of $A$ is not actually all differentiable functions. Suppose there is a constraint equation on $f$, such as (fo... |
There are presumably three ways of characterizing the abstract Euclidean space $E^n$ that are quite different in spirit:
1. axiomatically (with axioms concerning dimension)
2. by the abstract Euclidean group $E(n)$ (as its symmetry group, determining $E^n$ uniquely)
3. by presupposing a metric and requiring tha... |
In a planar graph $G$, one can easily find all the cycle basis by first finding the minimum spanning tree, and then use the remaining edge to complete cycles. Given Vertex $V$, edge $E$, there are $C=E-V+1$ number of cycles, and there are $C$ number of edges that are inside the graph, but not inside the spanning tree.
... |
(Pun definitely intended.)
Dear MSE-Community,
If I were to choose one artist that has made interesting works of art not only because of their beauty but also because of their connections to Mathematics, I would choose Escher. His works have always intrigued me. Some of his paintings are mind-boggling to look at ... |
Can we put the mathematics of paradoxes in visual art into perspective? |
How to Set the Redundant Current Value in a Network? |
There are presumably three ways of characterizing the abstract Euclidean space $E^n$ that are quite different in spirit:
1. axiomatically (with axioms concerning dimension)
2. by the abstract Euclidean group $E(n)$ (as its symmetry group, determining $E^n$ uniquely)
3. by presupposing a metric and requiring *th... |
What are the weights you were planning to use to decide what tree is minimal?
In fact you can certainly take any spanning tree $T$. The point is that any edge in the complement of $T$ can be completed to a cycle using edges in $T$ in a unique way, and the resulting cycles form a basis of the cycle space. This is you... |
There are presumably three ways of characterizing the abstract Euclidean space $E^n$ that are quite different in spirit:
1. axiomatically (with axioms concerning dimension)
2. by the abstract Euclidean group $E(n)$ (as its symmetry group, determining $E^n$ uniquely)
3. by presupposing a metric and requiring *th... |
I have a network, with $V$ vertex, $E$ edges and $C$ cycles, which has "current" flowing in and out from the network, as shown below:
<img src="https://docs.google.com/drawings/pub?id=1Micr1JiaNxGw3FTF-vJOR3h94IjNPjq1mQqEBS7QkGY&w=960&h=720">
Each edge has a current value associated to it.
The equation... |
(Pun definitely intended.)
Dear MSE-Community,
If I were to choose one artist that has made interesting works of art not only because of their beauty but also because of their connections to Mathematics, I would choose Escher. His works have always intrigued me. Some of his paintings are mind-boggling when looked... |
In a planar graph $G$, one can easily find all the cycle basis by first finding the spanning tree ( any spanning tree), and then use the remaining edge to complete cycles. Given Vertex $V$, edge $E$, there are $C=E-V+1$ number of cycles, and there are $C$ number of edges that are inside the graph, but not inside the sp... |
(Pun definitely intended.)
Dear MSE-Community,
If I were to choose one artist that has made interesting works of art not only because of their beauty but also because of their connections to Mathematics, I would choose Escher. His works have always intrigued me. Some of his paintings are mind-boggling when looked... |
Question 1: this is <a href="http://en.wikipedia.org/wiki/Projective_geometry">projective geometry</a>. In this context the horizon line is known as the line at infinity, and the drawings illustrate that "parallel lines" in the projective plane meet at the line at infinity while non-parallel lines meet at a finite poi... |
I have a network, with $V$ vertex, $E$ edges and $C$ cycles, which has "current" flowing in and out from the network, as shown below:
<img src="https://docs.google.com/drawings/pub?id=1Micr1JiaNxGw3FTF-vJOR3h94IjNPjq1mQqEBS7QkGY&w=960&h=720">
Each edge has a current value associated to it.
The equation... |
There have been some confusing comments regarding dependence on $x$ or on $c$, so let me try to put it all together.
You are correct that $\delta$ should not depend on $x$. However, when one is proving that $f(x)$ is continuous *at $c$*, then $\delta$ is allowed to depend on both $\epsilon$ *and* $c$.
Remember t... |
There are presumably three ways of characterizing the abstract Euclidean space $E^n$ that are quite different in spirit:
1. axiomatically (with axioms concerning dimension)
2. by the abstract Euclidean group $E(n)$ (as its symmetry group, determining $E^n$ uniquely)
3. by presupposing a metric and requiring *th... |
Many introductory books on vector spaces mention that the scalars need not be reals, and might even have sections discussing complex vector spaces or vector spaces over the integers mod 2. I have never seen any such book mention that all of the theory goes through as well if one restricts the scalars to be just ration... |
A general problem for the Calculus of Variations asks us to minimize the value of a functional $A[f]$, where $f$ is usually a differentiable function defined on $\mathbb{R}^n$.
What if, however, the domain of $A$ is not actually all differentiable functions. Suppose there is a constraint equation on $f$, such as (fo... |
There have been some confusing comments regarding dependence on $x$ or on $c$, so let me try to put it all together.
You are correct that $\delta$ should not depend on $x$. However, when one is proving that $f(x)$ is continuous *at $c$*, then $\delta$ is allowed to depend on both $\epsilon$ *and* $c$.
Remember t... |
I have the following constraint functions:
$$g_{i_{min}} \leq y_{i+1}-y_{i} \leq g_{i_{max}}$$
$$y_{i_{max}}-y_{i} \geq h_{i}$$
$$v_{i_{min}} \leq \Biggl[\frac{(y_{i+1}-y_{i})^{3} (h_{i}+a)^{4}}{(h_{i}+b)^{4}}\Biggr] \leq v_{i_{max}}$$
$$Q_{i_{min}} \leq \Biggl[\frac{(y_{i+1}-y_{i})^{3} (h_{i}+a)^{10}}{(h_{... |
There have been some confusing comments regarding dependence on $x$ or on $c$, so let me try to put it all together.
You are correct that $\delta$ should not depend on $x$. However, when one is proving that $f(x)$ is continuous *at $c$*, then $\delta$ is allowed to depend on both $\epsilon$ *and* $c$.
Remember t... |
For a constant, `N`, what value of `x` will maximize the cosine (or any trig) function?
\begin{equation}
1 = \cos{(Nx)}
\end{equation}
I am looking for the exact form, not the approximation because,
\begin{equation}
\frac{\arccos{(1)}}{N} = x = 0
\end{equation}
For example, [WolframAlpha.com][1] states th... |
Can this expression be simplified?
$ \sum_{i=0}^n (k^i)/i! $
EDIT:
This expression I have got as the number of possible ways to select n objects or less from k different infinite objects (you can select as many as you can from any type)...
I believe it must be equal to the number of possible ways to select n... |
I rather enjoyed Professor Stewart's book [1]. Take a look at it; I hope you enjoy it.
I have blogged about a selection from his book, you can view it at [2]. This is just one of the many different mathematical concepts covered in the book. It is more of a fun book than lots of theory. It will get you to think.
[... |
This is a follow-up to the question
[on field reductions][1].
(EDIT: Originally this question used the notation $\{\mathbb{R}(\setminus a)\}$, but now uses $\mathbb{R}(\setminus a)$ instead.)
There isn't a unique largest subfield of $\mathbb{R}$ not containing $a$,
so let $\mathbb{R}(\setminus a)$ denote the ... |
[How to solve it](http://books.google.com/books?id=Oac_PgAACAAJ&dq=how+to+solve+it&hl=en&ei=0S71TPb-BIX7lweo3viUBg&sa=X&oi=book_result&ct=result&resnum=2&ved=0CD4Q6AEwAQ) by G. Polya.
[The Value of Science](http://books.google.com/books?id=NNQEAAAAYAAJ&printsec=frontcover&dq=value+of+science&hl=en&ei=LS_1TO6XG8SAlAf... |
[How to solve it](http://books.google.com/books?id=Oac_PgAACAAJ&dq=how+to+solve+it&hl=en&ei=0S71TPb-BIX7lweo3viUBg&sa=X&oi=book_result&ct=result&resnum=2&ved=0CD4Q6AEwAQ) by G. Polya.
[The Value of Science](http://books.google.com/books?id=NNQEAAAAYAAJ&printsec=frontcover&dq=value+of+science&hl=en&ei=LS_1TO6XG8SAlAf... |
I've been wondering about the following question:
Suppose that $P$ is a polynomial of degree $n$ with complex coefficients. Assume that $a_0, a_1, \dots, a_n \in \mathbb{C}$ are distinct. Are the polynomials $$P(x + a_0), P(x + a_1), \dots, P(x + a_n)$$ linearly independent? |
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