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Suppose $E$ is a set of edges on graph $G$ over $n$ vertices, $\mathbf{x} \in \\{-1,1\\}^n$. The following seems to hold if G is <a href="http://mathworld.wolfram.com/EulerianGraph.html">Eulerian</a>, can anyone see a proof?
$$\sum_\mathbf{x} \prod_{ij\in E} x_i x_j=0$$ |
There's a book that could fit your actual level perfectly. The book is [Beginning Functional Analysis](http://books.google.co.cr/books?id=QALoZC64ea0C&pg=PP1&dq=beginning+functional+analysis&hl=es&ei=S-HBTLT6PIbGlQeX1ojTCg&sa=X&oi=book_result&ct=result&resnum=1&ved=0CCcQ6AEwAA#v=onepage&q&f=false) by Karen Saxe. It is ... |
i couldn't do the following question for hours
minimize $\sum_{i=1}^{n}x_{i}^{3}$
s.t. $\sum_{i=1}^{n}x_{i}=0$
and $\sum_{i=1}^{n}x_{i}^{2}=n$.
by Lagrange multiplier rule
? |
Suppose one is given an arbitrary moment generating function $M_{X}(t)$. How would you determine $P(X=k)$ from this? We know that $M_{X}(t) = E[e^{tX}]$ and $M_{X}(0) = 1$. |
first time poster.
I'm trying to determine the period of a repeating sequence, with noise.
I.e. given the sequence 4, 12, 3, 4, 12, 3, 4, 12, 3, ... the period is apparently 3.
Now, what if the sequence was more complicated. What if each value had added noise?
I was thinking of something with autocorrelation, ... |
Given a large univariate polynomial, say of degree 200 or more, is there a procedural way of finding all the complex roots? By "roots", I mean complex decimal approximations to the roots, though the multiplicity of the root is important. I have access to MAPLE and the closest function I've seen is:
with(RootFind... |
first time poster.
I'm trying to determine the length of a repeating sequence, with noise.
I.e. given the sequence 4, 12, 3, 4, 12, 3, 4, 12, 3, ... the period is apparently 3.
Now, what if the sequence was more complicated. What if each value had added noise?
I was thinking of something with autocorrelation, ... |
As an introductory book, "Topology without tears" by S. Morris. You can download PDF for free, but you might need to obtain a key to read the file from the author. (He wants to make sure it will be used for self-studying.)
Also, another great introductory book is Munkres, Topology.
On graduate level (non-introd... |
Suppose $E$ is a set of edges on graph $G$ over $n$ vertices, $\mathbf{x} \in \\{-1,1\\}^n$. The following seems to hold if G is not <a href="http://mathworld.wolfram.com/EulerianGraph.html">Eulerian</a>, can anyone see a proof?
$$\sum_\mathbf{x} \prod_{ij\in E} x_i x_j=0$$ |
Is most of the GM-AM Inequality in its codicil? |
Everyone's first starting point when dealing with the polynomial rootfinding problem should be a peer at [J.M. McNamee's excellent bibliography][1] and [book][2].
Now, it is a fact that polynomials of very high degree tend to make most polynomial rootfinders choke. Even the standard blackbox, the Jenkins-Traub algor... |
Is there an algorithm that will allow me to numerically compute the limit of a function f(x) in a principled way?
The most naive algorithm would be to continue to compute the function for larger values of x. The first problem is how to figure out the 'large' values for x to compute the function for. How do I know w... |
It's been a while since my last statistics class...
I have 404 files that went through some automated generation process. I would like to manually verify some of them to make sure that their data is indeed correct. I want to use probability to help me out so that I don't need to check every single file.
How would... |
It's not surprised you're a bit confused; understanding what's really going on with confidence intervals can be tricky.
The short version: If you don't want to check all the files you have to choose two different percentages: the confidence level (99% in your example), and how far off you're willing to be at that le... |
It's not surprising you're a bit confused; understanding what's really going on with confidence intervals can be tricky.
The short version: If you don't want to check all the files you have to choose two different percentages: the confidence level (99% in your example), and how far off you're willing to be at that l... |
It's not surprising you're a bit confused; understanding what's really going on with confidence intervals can be tricky.
The short version: If you don't want to check all the files you have to choose two different percentages: the confidence level (95% in your example), and how far off you're willing to be at that l... |
Are there known sharp upper bounds for sums of the form \sum_{p prime < x} 1/(p+1) in terms of x?
A related question: Suppose {p_i} is a set of n consecutive primes which minimize 1 - \sum_{i = 1}^n 1/(p_i+1) > 0. Are there known bounds for 1 - \sum_{i = 1}^{n} 1/(p_i + 1) from below in terms of n?
Thanks! |
Are there known sharp upper bounds for sums of the form $\sum_{p \, \text{prime} < x} \frac{1}{p+1}$ in terms of $x$?
A related question: Suppose $\{ p_i \}$ is a set of $n$ consecutive primes which minimize $1 - \sum_{i = 1}^n \frac{1}{p_i+1} > 0$ for a given $n > 0$. Are there known bounds for $1 - \sum_{i = 1}^{... |
Are there known sharp upper bounds for sums of the form $\sum_{p \, \text{prime} < x} \frac{1}{p+1}$ in terms of $x$?
A related question: Suppose $\{ p_i \}$ is a set of $n$ consecutive primes which minimizes $1 - \sum_{i = 1}^n \frac{1}{p_i+1} > 0$ for a given $n > 0$. Are there known bounds for $1 - \sum_{i = 1}^... |
Given a smooth vector field on a closed smooth manifold, does the flow of that vector field exists for all time $t \in R$? This may be a well-known and elementary conclusion, but I just wnat to confirm it to me. |
Are there known sharp upper bounds for sums of the form $\sum_{p \, \text{prime} < x} \frac{1}{p+1}$ in terms of $x$?
A related question: Suppose $(p_i)$ is a set of $n$ consecutive primes which minimizes $1 - \sum_{i = 1}^n \frac{1}{p_i+1} > 0$ for a given $n > 0$. Are there known bounds for $1 - \sum_{i = 1}^{n} ... |
Are there known sharp upper bounds for sums of the form $\sum_{p \, \text{prime} < x} \frac{1}{p+1}$ in terms of $x$?
A related question: Suppose $(p_i)$ is a set of $n$ consecutive primes which minimizes $1 - \sum_{i = 1}^n \frac{1}{p_i+1} > 0$ for a given $n > 1$. Are there known bounds for $1 - \sum_{i = 1}^{n} ... |
Let's forget about confidence intervals for a while and try to think of your sample size problem from first principles. This might also answer @Mike's point about finite samples.
Suppose among your $N$ files $M$ are good and $N-M$ are bad. If you pick a random sample of $n$ files the probability that all of them are... |
Let's forget about confidence intervals for a while and try to think of your sample size problem from first principles. This might also answer @Mike's point about finite samples.
Suppose among your $N$ files $M$ are good and $N-M$ are bad. If you pick a random sample of $n$ files the probability that all of them are... |
Are there known sharp upper bounds for sums of the form $\sum_{p \mid k} \frac{1}{p+1}$ for $k > 1$ subject to the constraint $\sum_{p \mid k} \frac{1}{p+1} < 1$?
A related question: Suppose $(p_i)$ is a set of $n$ consecutive primes which minimizes $1 - \sum_{i = 1}^n \frac{1}{p_i+1} > 0$ for a given $n > 1$. Are t... |
Let's forget about confidence intervals for a while and try to think of your sample size problem from first principles. This might also answer @Mike's point about finite samples.
Suppose among your $N$ files $M$ are good and $N-M$ are bad. If you pick a random sample of $n$ files the probability that all of them are... |
Let's forget about confidence intervals for a while and try to think of your sample size problem from first principles. This might also answer @Mike's point about finite samples.
Suppose among your $N$ files $M$ are good and $N-M$ are bad. If you pick a random sample of $n$ files the probability that all of them are... |
Let's forget about confidence intervals for a while and try to think of your sample size problem from first principles. This might also answer @Mike's point about finite samples.
Suppose among your $N$ files $M$ are good and $N-M$ are bad. If you pick a random sample of $n$ files the probability that all of them are... |
Let's forget about confidence intervals for a while and try to think of your sample size problem from first principles. This might also answer @Mike's point about finite samples.
Suppose among your $N$ files $M$ are good and $N-M$ are bad. If you pick a random sample of $n$ files the probability that all of them are... |
Not to scare you, but list of requirements for a first course in functional analysis is rather long:
- Basic theorems of metric spaces including, but not limited to:
- The Baire category theorem
- $\ell^p$ is complete
- [Arzelà-Ascoli](http://en.wikipedia.org/wiki/Arzel%C3%A0%E2%80%93Ascoli_theorem) (how e... |
I think one of the biggest problems is approximating multiple roots. The approach described in
*L.Brugnano, D.Trigiante. "Polynomial Roots: the Ultimate Answer?", Linear Algebra and its Applications 225 (1995) 207-219*
relies on the approximation of eigenvalues of a tridiagonal matrix, obtained via the applicatio... |
Look for a homeomorphism $f: \mathbb{R} \to (0,\infty)$ (you should know a good one). Then take the usual metric on $\mathbb{R}$ and use it to define a metric $d$ on $(0,\infty)$ by $d(x,y) = |f(x)-f(y)|$ (*). Now what can you say about the relationship between $d$ and the usual metric on $(0,\infty)$?
Now think abo... |
Look for a homeomorphism $f: \mathbb{R} \to (0,\infty)$ (you should know a good one). Then take the usual metric on $\mathbb{R}$ and use it to define a metric $d$ on $(0,\infty)$ by $d(x,y) = |f(x)-f(y)|$ (*). Now what can you say about the relationship between $d$ and the usual metric on $(0,\infty)$?
Which propert... |
Look for a homeomorphism $f: \mathbb{R} \to (0,\infty)$ (you should know a good one). Then take the usual metric on $\mathbb{R}$ and use it to define a metric $d$ on $(0,\infty)$ by $d(x,y) = |f(x)-f(y)|$ (*). Now what can you say about the relationship between $d$ and the usual metric on $(0,\infty)$?
EDIT: Made th... |
Look for a homeomorphism $f: \mathbb{R} \to \mathbb{R}^+$ (you should know a good one). Then take the usual metric on $\mathbb{R}$ and use it to define a metric $d$ on $\mathbb{R}^+$ by $d(x,y) = |f(x)-f(y)|$ (*). Now what can you say about the relationship between $d$ and the usual metric on $\mathbb{R}^+$?
EDIT: M... |
A recent [Missouri State problem][1] stated that it is easy to decompose the plane into half-open intervals and asked us to do so with intervals pointing in every direction. That got me trying to decompose the plane into closed or open intervals. The best I could do was to make a square with two sides missing (which ... |
How can I get this tricky sum? |
Sorry to give yet another answer that does not address the issue of fractional $n$ [it seems that fractional derivatives are not such a familiar topic to many research mathematicians; certainly they're not to me], but:
There is a little issue here which has not been addressed. By the context of the OP's question, I... |
I remember hearing somewhere that if $p^n$ is the largest power of a prime $p$ dividing the class number $h_K$ of a number field $K$, then there is unique factorization of the $p^n$-th powers of ideals in $\mathcal{O}_K$ (presumably, into $p^n$-th powers of primes). I can see that there can't be any $p$-torsion for a $... |
Unique factorization of powers of ideals? |
In general, how would one find the distribution of $f(X)$ where $X$ is a random variable? Or consider the inverse problem of finding the distribution of $X$ given the distribution of $f(X)$. For example, what is the distribution of $\max(X_1, X_2, X_3)$ if $X_1, X_2$ and $X_3$ have the same distribution? Likewise, if o... |
If $(X,d)$ is a connected, complete metric space and $G$ is a nonempty proper open subset of $X$, then $G$ is not complete with the restricted metric (because it is not closed), but $G$ is homeomorphic to a complete metric space, because the map $x\mapsto(x,1/d(x,X\setminus G))$ embeds $G$ onto a closed subspace of $X\... |
I remember hearing somewhere that if $p^n$ is the largest power of a prime $p$ dividing the class number $h_K$ of a number field $K$, then there is unique factorization of the $p^n$-th powers of elements in $\mathcal{O}_K$ into $p^n$-th powers of irreducibles. I can see that there can't be any $p$-torsion for a $p^n$-t... |
Recovering unique factorization of elements by only looking at $p^n$-th powers? |
How to prove the following ? |
I am trying to self-read through Ian Stewart's Galois theory and am stuck at this paragraph. I quote - "Lagrange observed that all methods for solving polynomial equations by radicals involve constructing rational functions of the roots that take a small number of values when the roots $\alpha_j$ are permuted. Prominen... |
I am trying out some problems based on this [formula][1],but am facing some issues in determing whether or not consider ordering as important.
For Example:
**In how many ways 15 different books can be divided five heaps of equal number of books ?**
So here according to the answer the ordering of groups is not ... |
This thing came up in a combinatorics course I am taking.
Choose a fixed set of primes $p_1,p_2,\dots,p_k$ and let $A_n$ be number of integers in $\\{1,2,\dots,n\\}$ which are not divisible by any $p_k$. $A_n$ is given by $ n - \sum_{1\leq i_1 \leq k} \lfloor \frac{n}{p_{i_1}} \rfloor + \sum_{1\leq i_1 < l_2 < k} \l... |
I am trying to solve some problems based on this [formula][1],but am facing some issues in determine whether or not consider ordering as important.
For Example:
**In how many ways 15 different books can be divided five heaps of equal number of books ?**
So here according to the answer the ordering of groups is... |
There are different notions of "ordered" and "unordered" that can arise in relation to set partitions.
- An unordered set of unordered sets,
- An unordered set of ordered sets,
- An ordered set of unordered sets,
- An ordered set of ordered sets,
all of which are counted differently.
> In how many way... |
I want to graph an equilateral triangle. It would be ideal if I had a set of three points: $(x_1, y_1), (x_2, y_2), (x_3, y_3)$ with $x_i, y_i \in \mathbb{Z}$ as the vertices. However, this is impossible. I am willing to "settle" for a triangle that is off by the width of a pencil lead. (That is, if we draw circles of ... |
It is a theorem that every locally free coherent sheaf on $\mathbb{P}^1$ over an algebraically closed field is isomorphic to a unique sum of sheaves $\mathcal{O}(n)$ for various integers $n$. In particular, the K-ring of locally free coherent sheaves (or all coherent sheaves, $\mathbb{P}^1$ being nonsingular) is isomor... |
Does the term Row-Complete have any synonyms? |
I use stories like these to develop intuition... or perhaps to destroy it. I have my own answers in mind, but I want to see if I have made any mistakes...
You are standing at the origin of an "infinite forest" holding an "infinite bb-gun." The "trees" in this forest are at the lattice points all around you. (The lat... |
This thing came up in a combinatorics course I am taking.
Choose a fixed set of primes $p_1,p_2,\dots,p_k$ and let $A_n$ be number of integers in $\\{1,2,\dots,n\\}$ which are not divisible by any $p_k$. $A_n$ is given by $ n - \sum_{1\leq i_1 \leq k} \lfloor \frac{n}{p_{i_1}} \rfloor + \sum_{1\leq i_1 < i_2 < k} \l... |
I use stories like these to develop intuition... or perhaps to destroy it. I have my own answers in mind, but I want to see if I have made any mistakes...
You are standing at the origin of an "infinite forest" holding an "infinite bb-gun." The "trees" in this forest are at the lattice points all around you. (The lat... |
During my study of plane algebraic curves, I got curious if there is a nontrivial example of a plane algebraic curve that has a [node][1], a [cusp][2] (for my purposes I do not care which of the two kinds of cusps would the example exhibit), a [tacnode][3], and an [isolated point][4]. By "nontrivial" I mean a curve tha... |
Or has it remained a terminal node at the frontier of mathematics?
|
Has anything further been done with Morley’s Miracle? |
B. in the annulus from $r$ to $r+dr$ there are $2*\pi *r*dr$ trees. If we move the diameter from the bullet to the trees, they subtend an angle $\delta*2*\pi*r*dr/r$. This reaches $2\pi$ at $r=1/\delta$, so that is about where we would expect to hit a tree. As the integral diverges, we are guaranteed to hit a tree.
... |
You have 3 bins. You can toss balls one at a time to the bins until any one of them has 2 balls in it, and then you stop.
The tosses are independent, and each bin is equally likely to be hit. It's impossible to not make it in a bin on any given toss.
What is the expected number of tosses so that 1 bin has 2 bal... |
Hint: So what is the chance that you get success in two throws? Call it x. In exactly three throws? Call it y. And that tells you the chance that you will need four is 1-x-y. Then what is the expected number of throws? |
I remember my professor in college challenging me with this question, which I failed to answer satisfactorily: I know there exists a bijection between the rational numbers and the natural numbers, but can anyone produce an explicit formula for such a bijection? |
Produce an explicit bijection between rationals and naturals? |
The following comes from questions comes from a recent combinatorics paper I attended :
**1.27 people are to travel by a bus which can carry 12 inside and 15 outside. In how many ways can the party be distributed between inside and outside if 5 people refuse to go outside and 6 will not go inside?**
The solution ... |
Does anyone know if there's a client or web app that generates Penrose patterns which can then be converted to a tileable rectangular background image for web site?
I found this http://stephencollins.net/penrose but I can't seem to get a rectangular shape out of it for web use. Thanks. |
I am just beginning to read about the use of "Concave Programming" methods and use of the Karush-Kuhn-Tucker conditions to identify the maximum value of a non-linear objective function subject to inequality constraints.
The examples I have seen in the text I have at hand, all involve only linear constraints. Is this... |
Let $X$ be a Poisson random variable with mean $\lambda$. How do I show that $P[X \geq \lambda] = 1/2$? Also, I was wondering what distributions have this property that the density is concentrated equally around the mean. I see that it's true for Gaussian and Uniform distributions. |
If $S$ is a surface which is the complement of finite many points in a compact surface, and the metric in $S$ is complete, then is Gauss-Bonnet theorem still valid for $S$? |
As promised in my response to Byron, here is my very partial progress to
date on this problem. Though long-winded, it doesn't amount to much.
I'm still looking for an answer to the general case, or to other special
cases ($n=2$? $c=2$?) or even just better estimates. I'm also
interested to know about any applica... |
As promised in my response to Byron, here is my very partial progress to
date on this problem. Though long-winded, it doesn't amount to much.
I'm still looking for an answer to the general case, or to other special
cases (like $n=2$ or $c=2$) or even just better estimates. I'm also
interested to know about any ap... |
I've started a little reading on quadratic reciprocity, and a reason for this has eluded me. Here's a little of what I came up with so far. I decided I want to show that for all primes $p$, if $p|x^2-2$, then $p$ does not divide $2y^3+3$. Then, by way of contradiction, if $(x^2-2)/(2y^3+3)$ is an integer, then any $p$ ... |
Why is $(x^2-2)/(2y^2+3)$ never an integer for any integers $x$ and $y$? |
Assume you want to prove an identity such as
$$\sum_{k=m+1}^{n}A(k,m)-B(k,m)=S(m)+T(n,m)\qquad\text{for } n,m\in
\mathbb{Z},n,m\geq 0.$$
Question: does application of two inductive arguments, one on $m$ and the
other on $n$, guarantee the validity of such a proof? |
I've started a little reading on quadratic reciprocity, and a reason for this has eluded me. Here's a little of what I came up with so far. I decided I want to show that for all primes $p$, if $p|x^2-2$, then $p$ does not divide $2y^3+3$. Then, by way of contradiction, if $(x^2-2)/(2y^3+3)$ is an integer, then any $p$ ... |
I've started a little reading on quadratic reciprocity, and a reason for this has eluded me. Here's a little of what I came up with so far. I decided I want to show that for all primes $p$, if $p|x^2-2$, then $p$ does not divide $2y^2+3$. Then, by way of contradiction, if $(x^2-2)/(2y^2+3)$ is an integer, then any $p$ ... |
Assume you want to prove an identity such as
$$\sum_{k=m+1}^{n}A(k,m)-B(k,m)=S(m)+T(n,m)\qquad\text{for } n,m\in
\mathbb{Z},n,m\geq 0.$$
*Added*: I applied mathematical induction on $m,n$ to prove it. I am unsure because up to now I have seen it applied to properties depending on a single variable only.
Que... |
Do somebody knows anything about the Dirca's identity?
\begin{equation} \label{Dirac}
\frac{\partial^2}{\partial x_{\mu}\partial x^{\mu}} \delta(xb_{\mu}xb^{\mu}) =
-4\pi \delta(xb_0)\delta(xb_1)\delta(xb_2)\delta(xb_3)
\end{equation}
here
- $xb$, is the 4-vector $x-b$ in Minkowsky spacetime
- $\delta... |
We know that the Mandelbrot set is derived from the iterations of z^2 + c.
Do anyone know something about magnet Mandelbrot? I found it in the software UltraFractal, and it is much more beautiful than the original Mandelbrot, in my opinion.
Do you know anything about it? What is it iterating?
EDIT: Here's a pi... |
How about $f(x) =\frac{1}{\sqrt x}$ on $(0,1)$? To have it on $\mathbb{R}$ you could just multiply $f$ with the indicator on $(0,1)$.
This function is clearly in $L^1$ and not in $L^2$ since $f^2(x) = \frac{1}{x}$. |
How many solutions does the following equation have?
$$k+l+m+n=30,$$
where $k,l,m,n\in\mathbb{Z},0\leq k,l,m,n\leq 10.$ |
If $S$ is a surface which is the complement of finitely many points in a compact surface, and the metric in $S$ is complete, then is Gauss-Bonnet theorem still valid for $S$? |
Consider commutative semigroup S and its Grothendieck completion group
G(S).Suppose I insist on defining G(S) as free abelian group on basis $[a]$ (with $a\in S$) divided out by the relations $[a+b]-[a]-[b]$. How do I show *with that definition* that if the images of $a,b\in S$ become equal in G(S), then necessaril... |
How to show directly that two elements become equal in Grothendieck group? |
I didn't get this method.
3a^2-9a+6 = 3(a^2-3a+2) = 3(a^2-1a*3+3^2-3^2+2) = 3[(a-3)^2-7] ,then? |
Is there a specific way to attack congruences with two variables? The reason I ask is because up until now, I had never encountered such a problem. While browsing tonight, I came across this challenge problem in one of my books:
Suppose that $(ab,p)=1$ and that $p>2$. Show that the number of solutions $(x,y)$ of the... |
please lisen this vedio lecture
http://www.youtube.com/watch?v=3lS5ZMsfUdQ
You will understand the geometry behind the convergence of fourier series |
How many solutions does the following equation have?
$$k+l+m+n=30,$$
where $k,l,m,n\in\mathbb{Z},0\leq k,l,m,n\leq 10.$
Edit: this is an exercise in the *inclusion-exclusion principle*.
|
Given the coordinates of two rectangles on a coordinate plane, what would be the easiest way to find the coordinates of the intersecting rectangle of the two?
I am trying to do this programatically. |
I didn't get this method.
$3a^2-9a+6 = 3(a^2-3a+2) = 3(a^2-1a*3+3^2-3^2+2)=$ $= 3[(a-3)^2-7]$ ,then? |
Are there known sharp upper bounds for sums of the form $\sum_{p \mid k} \frac{1}{p+1}$ for $k > 1$ subject to the constraint $\sum_{p \mid k} \frac{1}{p+1} < 1$? (The factor of +1 in the denominator and the constraint of the sum of inverse "shifted primes" is bounded by 1 are both necessary.)
A related question: Su... |
Are there known sharp upper bounds for sums of the form $\sum_{p \mid k} \frac{1}{p+1}$ for $k > 1$ subject to the constraint $\sum_{p \mid k} \frac{1}{p+1} < 1$? (The factor of +1 in the denominator and the constraint of the sum of inverse "shifted prime divisors" is bounded by 1 are both necessary.)
A related ques... |
**Question 1:** The volume of a parallelepiped in $\mathbb{R}^n$ with n sides given by the vectors $(x_{1_1}, x_{1_2} ... x_{1_n}), (x_{2_1}, x_{2_2} ... x_{2_n}) ... (x_{n_1}, x_{n_2} ... x_{n_n})$ and diagonal $(x_{1_1} + x_{2_1}+ \cdots + x_{n_1}, x_{1_2} + x_{2_2} + \cdots + x_{n_2}, ..., x_{1_n} + x_{2_n} + \cdots... |
Nice geometric parallelepiped proof? |
This thing came up in a combinatorics course I am taking.
Choose a fixed set of primes $p_1,p_2,\dots,p_k$ and let $A_n$ be number of integers in $\\{1,2,\dots,n\\}$ which are not divisible by any $p_k$. $A_n$ is given by $ n - \sum_{1\leq i_1 \leq k} \lfloor \frac{n}{p_{i_1}} \rfloor + \sum_{1\leq i_1 < i_2 < k} \l... |
**Question 1:** The volume of a parallelepiped in $\mathbb{R}^n$ with n sides given by the vectors $(x_{1_1}, x_{1_2} ... x_{1_n}), (x_{2_1}, x_{2_2} ... x_{2_n}) ... (x_{n_1}, x_{n_2} ... x_{n_n})$ and diagonal $(x_{1_1} + x_{2_1}+ \cdots + x_{n_1}, x_{1_2} + x_{2_2} + \cdots + x_{n_2}, ..., x_{1_n} + x_{2_n} + \cdots... |
The Rochester Institute of Technology offers a MS in applied statistics that is completely online (I think, I'm not enrolled yet.)
http://www.rit.edu/emcs/ptgrad/_program_detail.php?id=441
It's pretty expensive, since it combines the increased out-of-state tuition with the extra online cost. However an online deg... |
I'm trying to work out a basic example where formal smoothness should fail.
I'm considering $\mathbb{R} \to \mathbb{R}[x,y]/(x^2-y^2)$.
The idea is that not every $\mathbb{R}$-homomorphism $\mathbb{R}[x,y]/(x^2-y^2) \to \mathbb{R}$ should lift to a homomorphism $\mathbb{R}[x,y]/(x^2-y^2) \to \mathbb{R}[\varepsilo... |
Have I got the right definition of formal smoothness? |
I would like to find the apothem's regular pentagon. It follows from
$$\cos \dfrac{2\pi }{5}=\dfrac{-1+\sqrt{5}}{4}.$$
But how can this be proved (geometrically or trigonometrically)? |
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