instruction stringlengths 12 30k |
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**If $f(x)$ is a function satisfying $ \displaystyle f(x+y) = f(x) \cdot f(y) \text { for all } x,y \in \mathbb{N} \text{ such that } f(1) = 3 \text { and } $ $ \sum_{x=1}^{n} f(x) = 120 $, then find the value of $n$.**
How to approach this one?
|
Let $(X,M,\mu)$ be a measure space. Why are the following statements equivalent?
i) There exists a sequence of pairwise disjoint measurable sets $\{A_{n}\}$ in $M$ , each of finite measure, such that $\mu(A) = \sum_{n=1}^{\infty} \mu(A \cap A_{n})$ for every measurable set $A \in M$.
ii) $\mu$ is a countable su... |
(ii) $\implies$ (i)
Suppose $\mu = \sum_{n=1}^\infty \mu_n$, where $\mu_i \perp \mu_j$ for all $i \neq j$. By definition of "mutually singular," there exist pairwise disjoint $A_n \in M$ such that $\mu_n(A) = \mu_n(A \cap A_n)$ for all $A \in M$. Therefore, $\mu(A_k) = \sum_{n=1}^\infty \mu_n(A_k) = \mu_k(A_k) < \in... |
From Statistical Inference Second Edition (George Casella, Roger L. Berger)
"My telephone rings 12 times each week, the calls being randomly distributed among the 7 days. What is the probability that I get a least one call each day?"
The answer is .2285, but I don't know how they got it. My reasoning was as follo... |
(ii) $\implies$ (i)
Suppose $\mu = \sum_{n=1}^\infty \mu_n$, where $\mu_i \perp \mu_j$ for all $i \neq j$. By definition of "mutually singular," there exist pairwise disjoint $A_n \in M$ such that $\mu_n(A) = \mu_n(A \cap A_n)$ for all $A \in M$. Therefore, $\mu(A_k) = \sum_{n=1}^\infty \mu_n(A_k) = \mu_k(A_k) < \in... |
From Statistical Inference Second Edition (George Casella, Roger L. Berger)
"My telephone rings 12 times each week, the calls being randomly distributed among the 7 days. What is the probability that I get a least one call each day?"
The answer is .2285, but I don't know how they got it. My reasoning was as follo... |
(ii) $\implies$ (i)
Suppose $\mu = \sum_{n=1}^\infty \mu_n$, where $\mu_i \perp \mu_j$ for all $i \neq j$. By definition of "mutually singular," there exist pairwise disjoint $A_n \in M$ such that $\mu_n(A) = \mu_n(A \cap A_n)$ for all $A \in M$. Therefore, $\mu(A_k) = \sum_{n=1}^\infty \mu_n(A_k) = \mu_k(A_k) < \in... |
Find the complete integral of partial differential equation
`z^2 = pqxy` ?
I have solved this equation till auxiliary equation:
dp/(-pqy+2pz) = dq/(-pqx+2qz) = dz/(2pqxy) = dx/(qxy) = dy/(pxy)
But I have unable to find value of p and q. |
Let $\mathbf G$ be a given $m \times n$ matrix, $\mathbf y$ a given $m \times 1$ column vector and $\mathbf x$ an unknown $n \times 1$ column vector such that $\mathbf x \ge 0$. How do you find $\displaystyle \min_{\mathbf x} (\mathbf y - \mathbf G\mathbf x)^T(\mathbf y - \mathbf G\mathbf x)$? |
What does D = max[0; M(x)] mean?
M(x) is a function. |
What does max[] mean? |
Can somebody prove that when we add 2 bit integers and if there is an overflow then the result always will be lesser than the 2 operands used?? |
We know that the set of continuous functions and the set of smooth functions are dense in $L^1$ and $L^2$. Is the set of $C^k$ functions dense in $L^1$ or $L^2$ for any $k\in\mathbb{N}$.
**EDIT**
Is the set $C^k-C^\infty$ dense in $L^2$ or $L^1$. How to prove ? |
I recently made a model to solve a conflict problem in sports.
The model is based on Bayesian Games: a complete definition can be found [here][1].
The thing is that I am now looking to compute the equilibria of the system, but I don't know any algorithm to do so.
Do you know any?
Which language do you think... |
We know that the set of continuous functions and the set of smooth functions in $L^1$ or $L^2$ are dense in $L^1$ and $L^2$. Is the set of $C^k$ functions dense in $L^1$ or $L^2$ for any $k\in\mathbb{N}$.
**EDIT**
Is the set $C^k-C^\infty$ in $L^1$ or $L^2$ dense in $L^1$ or $L^2$. How to prove ? |
Let $G$ and $H$ be two possibly directed, non necessarily simple, vertex-labelled graphs with respective adjacency matrices $A_G$ and $A_H$. What is the name of the graph $M$ with adjacency matrix $A_M=A_HA_G$? |
I'm having trouble understanding this proof:
> **Lemma:** Let $\lambda$ be the Lebesgue measure restricted to the Borel sets of $[0,1]$. Then there exists a decreasing sequence $(E_n)_{n\ge0}$ of subsets of $[0,1]$ such that each of them is $\lambda$-thick and $\bigcap E_n = \emptyset$.
> **Proof:** Let $E_n = [0... |
We know that the set of continuous functions and the set of smooth functions in $\mathcal{L}^1$ or $\mathcal{L}^2$ are dense in $\mathcal{L}^1$ and $\mathcal{L}^2$. Is the set of $\mathcal{C}^k$ functions dense in $\mathcal{L}^1$ or $\mathcal{L}^2$ for any $k\in\mathbb{N}$?
**EDIT**
Is the set $\mathcal{C}^k\... |
Are the $\mathcal{C}^k$ functions dense in either $\mathcal{L}^2$ or $\mathcal{L}^1$ ? |
For any positive integer $N$, the binomial$(N!,p)$ distribution has the following property: for any $1 \leq n \leq N$,
there exist i.i.d. random variables $X_1,\ldots,X_n$ such that $X_1 + \cdots + X_n \sim {\rm binomial}(N!,p)$ (specifically, we take $X_1,\ldots,X_n$ to be i.i.d. binomial$(N!/n,p)$ rv's). It may be i... |
I'm having trouble understanding this proof:
> **Lemma:** Let $\lambda$ be the Lebesgue measure restricted to the Borel sets of $[0,1]$. Then there exists a decreasing sequence $(E_n)_{n\ge0}$ of subsets of $[0,1]$ such that each of them is $\lambda$-thick and $\bigcap E_n = \emptyset$.
> **Proof:** Let $E_n = [0... |
Find the complete integral of partial differential equation
`z^2 &= pqxy` ?
I have solved this equation till auxiliary equation:
$dp/(-pqy+2pz) = dq/(-pqx+2qz) = dz/(2pqxy) = dx/(qxy) = dy/(pxy)$
But I have unable to find value of p and q.
**EDIT:**
p = ∂z/∂x
q = ∂z/∂y
r = ∂²z/∂... |
Find the complete integral of partial differential equation
`z^2 = pqxy` ?
I have solved this equation till auxiliary equation:
dp/(-pqy+2pz) = dq/(-pqx+2qz) = dz/(2pqxy) = dx/(qxy) = dy/(pxy)
But I have unable to find value of p and q.
**EDIT:**
p = ∂z/∂x
q = ∂z/∂y
r = ∂²z/∂x² ... |
We know that $$e^{i \pi} = -1 .$$ We can transform all of the components of this identity into (generalized) continued fractions. When we start of with $\pi$, we see that $$ \Big(3+ \mathbf{K}_{i=1}^{\infty} \frac{(\sqrt{a_{i-1}}+2)^2}{6}\Big)^{e \cdot i} = -1 $$ When we take the $\frac{1}{i}$'th power, we see that
... |
What can Euler's identity learn us about (generalised) continued fractions? |
We know that $$e^{i \pi} = -1 .$$ We can transform all of the components of this identity into (generalized) continued fractions. When we start of with $\pi$, we see that $$ \Big(3+ \mathbf{K}_{i=1}^{\infty} \frac{(\sqrt{a_{i-1}}+2)^2}{6}\Big)^{e \cdot i} = -1 $$ When we take the $\frac{1}{i}$'th power, we see that
... |
What can Euler's identity teach us about (generalised) continued fractions? |
I'm having trouble understanding this proof:
> **Lemma:** Let $\lambda$ be the Lebesgue measure restricted to the Borel sets of $[0,1]$. Then there exists a decreasing sequence $(E_n)_{n\ge0}$ of subsets of $[0,1]$ such that each of them is $\lambda$-thick and $\bigcap E_n = \emptyset$.
> **Proof:** Let $E_n = [0... |
I'm currently reading Artificial Intelligence, by Russel & Norvig. They state that:
A) A sentence is valid if it is true in all models
B) The deduction theorem: "For any sentences $\alpha$ and $\beta$, $\alpha \models \beta$ if and only if ($\alpha \implies \beta$) is valid.
My mental blockage consists of the ... |
It is unclear to me on what grounds you claim that $g(x)b=f(x)$ will have more than one solution for $x$. Even assuming that $K$ were algebraically closed, how do you know that you don't have one solution which is a multiple solution? For a very simple example, consider the case of $K=\mathbf{F}_2$, the field with two ... |
Let $G$ and $H$ be two possibly directed, non necessarily simple, vertex-labelled graphs with respective adjacency matrices $A_G$ and $A_H$.
1) What is the name of the graph $M$ with adjacency matrix $A_M=A_HA_G$?
2) In the unlikely event that I am the first to think about using that operation, which symbols shou... |
As we saw [here][1], the minimum of two variables can be written using elementary functions and the absolute value function.
$min(a,b)=\frac{a+b}{2} - \frac{|a-b|}{2}$
There's even a nice intuitive explanation to go along with this: If we go to the point half way between two numbers, then going down by half their... |
Nice expression for minimum of three variables? |
As we saw [here][1], the minimum of two variables can be written using elementary functions and the absolute value function.
$\min(a,b)=\frac{a+b}{2} - \frac{|a-b|}{2}$
There's even a nice intuitive explanation to go along with this: If we go to the point half way between two numbers, then going down by half thei... |
As we saw [here][1], the minimum of two quantities can be written using elementary functions and the absolute value function.
$\min(a,b)=\frac{a+b}{2} - \frac{|a-b|}{2}$
There's even a nice intuitive explanation to go along with this: If we go to the point half way between two numbers, then going down by half the... |
Let $G$ and $H$ be two possibly directed, non necessarily simple, vertex-labelled graphs with respective adjacency matrices $A_G$ and $A_H$ and $V(G)=V(H)$.
1) What is the name of the graph $M$ with adjacency matrix $A_M=A_HA_G$?
2) In the unlikely event that I am the first to think about using that operation, wh... |
We know that $$e^{i \pi} = -1 .$$ We can transform all of the components of this identity into (generalized) continued fractions. When we start of with $\pi$, we see that $$ \Big(3+ \mathbf{K}_{i=1}^{\infty} \frac{(\sqrt{a_{i-1}}+2)^2}{6}\Big)^{e \cdot i} = -1 $$ When we take the $\frac{1}{i}$'th power, we see that
... |
I am stuck in trying to solve the following.
Given two points $(x_{1}; y_{1})$ and $(x_{2}; y_{2})$, how can I determine the parameters $a$ and $b$ in the equation:
$y=\frac{e^{a+bx} - e^{a}}{1+e^{a+bx}}$
In other words, I have two unknowns and two set of points, so there's for sure a solution, but how c... |
As described in Brualdi and Cvetkovic's <a href="http://books.google.com/books?id=pwx6t8QfZU8C&printsec=frontcover&dq=combinatorial+approach+to+matrix+theory">Combinatorial Approach to Matrix Theory</a>, rather than associating a directed graph in the usual way to a matrix, it's more natural to associate the **König d... |
I hope you can help me with this excersise:
Lets have finite sequence of characters 0 and 1. We are doing following algorithm:
<br>If the sequence starts with 0, we add 00 to the end.
<br>If the sequence starts with 1, we add 1101 to the end.
<br>Then we remove first three characters.
<b>Prove, that if the seq... |
Cycling sequence of 0 and 1 - is cycle length even? |
I'm interested in methods of computing the homology groups and Euler characteristics of the classical Lie groups ($GL(n,\mathbb{R}), SL(n,\mathbb{R})$, etc.). (But I'd be interested in techniques which are more generally applicable to arbitrary Lie groups as well.)
Here's the current state of my knowledge on this is... |
I am stuck in trying to solve the following:
Given two points $(x_{1}; y_{1})$ and $(x_{2}; y_{2})$, to determine the parameters $a$ and $b$ in the equation:
$$y=\frac{e^{a+bx} - e^{a}}{1+e^{a+bx}}.$$
In other words, I have two unknowns and two set of points, so there's for sure a solution, but how can the fol... |
I'm doing a raytracing exercise. I have a vector representing the normal of a surface at an intersection point, and a vector of the ray to the surface. How can I determine what the reflection will be?
In the below image, I have `d` and `n`. How can I get `r`?
![Vector d is the ray; n is the normal; t is the refra... |
How to get a reflection vector? |
Maybe it would be helpful... from position that $M$ is a hypergraph, value of matrix element $A_M(i,j)$ seems to be equal to the number of edges from vertex i to vertex j with length 2 (minus edges from j to i). But I'm not sure, if $A_M$ is adjacency matrix for hypergraph $M$... Especially after googling ["hypergraph ... |
Let $A$ and $B$ be two symmetric matrices of the same size, and let $\alpha_k$ and $\beta_k$ be the $k$th largest eigenvalues of $A$ and $B$ respectively. What can we say about $|\beta_k - \alpha_k|$ in terms of $||B - A||_2$? |
Since I started university, almost every exam I've had in mathematics have ended up as failed mostly due to my lack of using proper "syntax" on paper, written by hand.
Where can I learn the standard way of writing mathematics, beginning with the basics? Does this make sense? Is there a standard "uniform" syntax, or... |
Where to learn the syntax of mathematics? |
Since I started university, almost every exam I've had in mathematics have ended up as failed mostly due to my lack of using proper "syntax" on paper, written by hand.
Where can I learn the standard way of writing mathematics, beginning with the basics? Does this make sense? Is there a standard "uniform" syntax, or... |
Let $A$ and $B$ be two symmetric matrices of the same size, and let $\alpha_k$ and $\beta_k$ be the $k$th largest eigenvalues of $A$ and $B$ respectively. What can we say about $|\beta_k - \alpha_k|$ in terms of $||B - A||_2$? To be clearer, I want the tightest possible upper bound on $|\beta_k - \alpha_k|$ using onl... |
How would I create a rotation matrix that rotates X by a, Y by b, and Z by c?
I need to formulas, unless you're using the ardor3d api's functions/methods.
Matrix is set up like this
xx, xy, xz,
yx, yy, yz,
zx, zy, zz
A Quaternion is fine too.
|
How would I create a rotation matrix that rotates X by a, Y by b, and Z by c? |
I recently learned about finding density functions of functions of random variables using the cumulative distribution function. For example, computing the density function for the difference of two uniform random variables on [0,1] (as in Grinstead & Snell's free online book). I gave myself the following problem.
... |
If the equation is $z^2=xy\frac{\partial z}{\partial x}\frac{\partial z}{\partial y}$, I would be tempted to see the symmetry in $x$ and $y$ and try solutions of the form $z=(xy)^n$. What happens then? |
Since I started university, almost every exam I've had in mathematics have ended up as failed mostly due to my lack of using proper "syntax" on paper, written by hand.
Where can I learn the standard way of writing mathematics, beginning with the basics? Does this make sense? Is there a standard "uniform" syntax, or... |
Since I started university, almost every exam I've had in mathematics have ended up as failed mostly due to my lack of using proper "syntax" on paper, written by hand.
Where can I learn the standard way of writing mathematics, beginning with the basics? Does this make sense? Is there a standard "uniform" syntax, or... |
I'm trying to create a program, that will decompose a matrix using the Cholesky decomposition.
The decomposition itself isn't a difficult algorithm, but a matrix, to be eligible for Cholesky decomposition, must be square and positive-definite. Checking whether a matrix is square is easy, but the positive part proves... |
I am stuck in trying to solve the following:
Given two points $(x_{1}; y_{1})$ and $(x_{2}; y_{2})$, to determine the parameters $a$ and $b$ in the equation:
$$y=\frac{e^{a+bx} - e^{a}}{1+e^{a+bx}}.$$
In other words, I have two unknowns and two set of points, so there's for sure a solution, but how can the fol... |
I'm trying to create a program, that will decompose a matrix using the Cholesky decomposition.
The decomposition itself isn't a difficult algorithm, but a matrix, to be eligible for Cholesky decomposition, must be symmetric and positive-definite. Checking whether a matrix is symmetric is easy, but the positive part ... |
In logic theory, if we have a finite domain A of k elements, then we can construct only finitely many structures (finite structures) each of which has A as domain. I think the number of structures will be 2^k. How to prove it - if my claim is right? |
I recently learned about finding density functions of functions of random variables using the cumulative distribution function. For example, computing the density function for the difference of two uniform random variables on [0,1] (as in Grinstead & Snell's free online book). I gave myself the following problem.
... |
I have some kind of "homework" question. I have the following theorem:
**Theorem** (Transfinite Recursion over the class of of ordinals $\mathbf{ON}$: Let $\mathbf{V}$ be the class of all sets. If $\mathbf{F} : \mathbf{V} \to \mathbf{V}$ then there exists a unique $\mathbf{G} : \mathbf{ON} \to \mathbf{V}$ such that ... |
I'm trying to create a program, that will decompose a matrix using the Cholesky decomposition.
The decomposition itself isn't a difficult algorithm, but a matrix, to be eligible for Cholesky decomposition, must be symmetric and positive-definite. Checking whether a matrix is symmetric is easy, but the positive part ... |
How would I go about it?
|
figureing out the x,y, and z rotation of a right triangle? |
I have some kind of "homework" question. I have the following theorem:
**Theorem** (Transfinite Recursion over the class of of ordinals $\mathbf{ON}$: Let $\mathbf{V}$ be the class of all sets. If $\mathbf{F} : \mathbf{V} \to \mathbf{V}$ then there exists a unique $\mathbf{G} : \mathbf{ON} \to \mathbf{V}$ such that ... |
If you can use the transfinite recursion theorem you stated, why not simply apply it to $F$, and then use an isomorphism between $X$ and some ordinal to transfer the function $G$ from $ON$ to $X$? It seems like you are trying to reprove the transfinite recursion theorem rather than just applying it. |
I recently learned about finding density functions of functions of random variables using the cumulative distribution function. For example, computing the density function for the difference of two uniform random variables on [0,1] (as in Grinstead & Snell's free online book). I gave myself the following problem.
... |
So you have the co-ordinates $P_i = (x_i, y_i, z_i), \ \ i=1,2,3$ of the vertices and you want to find the angles?
Let the angle at $P_1$ be $\theta_1$.
If
$\displaystyle \vec{a} = (x_3 -x_1, y_3 - y_1, z_3 - z_1)$ <br/>
$\displaystyle \vec{b} = (x_2 - x_1, y_2 - y_1, z_2 -z_1)$
The we have that
$\dis... |
I have a function $S(x)$ which is bounded above and below as follows:
$f(x) + C_1 + \mathcal{O}(g(x)) < S(x) < f(x) + C_2 + \mathcal{O}(g(x))$ as $x \rightarrow \infty$
Can I conclude that $$S(x) = f(x) + C + \mathcal{O}(g(x))$$ as $x \rightarrow \infty$? Can anyone give a short proof for this fact? |
I don't think so because $g(x)$ might be too small. Take $f(x)=0, C_1=-1.5, C_2=1.5, g(x)=\exp(-x)$. Then doesn't $S(x)=\sin(x)$ violate it? |
So you have the co-ordinates $P_i = (x_i, y_i, z_i), \ \ i=1,2,3$ of the vertices and you want to find the angles of the triangle formed by $P_1, P_2, P_3$?
Let the angle at $P_1$ be $\theta_1$.
If
$\displaystyle \vec{a} = (x_3 -x_1, y_3 - y_1, z_3 - z_1)$ <br/>
$\displaystyle \vec{b} = (x_2 - x_1, y_2 - y_1... |
figuring out the x,y, and z rotation of a right triangle? |
I have a function $S(x)$ which is bounded above and below as follows:
$f(x) + C_1 + \mathcal{O}(g(x)) < S(x) < f(x) + C_2 + \mathcal{O}(g(x))$ as $x \rightarrow \infty$
Can I conclude that $$S(x) = f(x) + C + \mathcal{O}(g(x))$$ as $x \rightarrow \infty$? Can anyone give a short proof for this fact?
**EDIT:**
... |
I have a function $S(x)$ which is bounded above and below as follows:
$f(x) + C_1 + \mathcal{O}(g(x)) < S(x) < f(x) + C_2 + \mathcal{O}(g(x))$ as $x \rightarrow \infty$
Can I conclude that $$S(x) = f(x) + C + \mathcal{O}(g(x))$$ as $x \rightarrow \infty$? Can anyone give a short proof for this fact?
**EDIT:**
... |
1) Why does a small number of states suffice?
Regardless of whether the constant is 5 or 500, its still very surprising. Thankfully, it's fairly to prove if you allow the counter to be `{1...8}` instead of `{1...5}`. [This proof is by Ben-Or and Cleve.]. Start by representing the computation as a circuit, and ignore t... |
If I have the Keplerian elements for two orbits, how do I compute the shortest distance between these two orbits in 3D space? Is there any analytical expression to compute that? |
Analytical Expression to find the Shortest Distance between Two Ellipses? |
It all about maths I dont understand, how can I solve this exercise, someone say that is a 2 dimensional problem, but I can not figure out for myself, I already understand what to do, but I dont know why is the formula, can someone explain this exercise for a dummie? http://www.codeforces.com/contest/1/problem/A
why... |
math in this problem I dont understand where to get the formula? |
It all about maths I dont understand, how can I solve this exercise, someone say that is a 2 dimensional problem, but I can not figure out for myself, I already understand what to do, but I dont know why is the formula, can someone explain this exercise for a dummie? http://www.codeforces.com/contest/1/problem/A
**w... |
In logic theory, if we have a finite domain A of k elements, then we can construct only finitely many structures (finite structures) each of which has A as domain. I think the number of structures will be $2^{k}$. How to prove it - if my claim is right? |
It all about maths I don't understand, how can I solve this exercise, someone say that is a 2 dimensional problem, but I can not figure out for myself, I already understand what to do, but I don't know why is the formula, can someone explain this exercise for a dummy? http://www.codeforces.com/contest/1/problem/A
**... |
So break it into one dimensional problems. How many flagstones of length a does it take to cover a path of m meters? If you can trust a and m to be integers, m/a is correct if a divides m. Otherwise you need floor (m/a) +1. One way to combine these is floor((m+a-1)/a). Try it with some small numbers to see how it ... |
Since you wanted a diiferent proof approach, you can try using [Abel's Identity][1], which has turned out to be quite useful in analytic number theory.
For instance see this: [An estimate for sum of reciprocals for square roots of prime numbers][2].
To apply this to your question:
Since we know that
$\dis... |
Since you wanted a diiferent proof approach, you can try using [Abel's Identity][1], which has turned out to be quite useful in analytic number theory.
For instance see this: [An estimate for sum of reciprocals of square roots of prime numbers][2].
To apply this to your question:
Since we know that
$\disp... |
I'm given a number (n), an interval (i), and the number of steps (s). I want to distribute these numbers over that criteria as best as possible. The upper bounds would be n and lower bounds is 0.75 * n. So, if n=100, i=5, and s=6, you'd have [100, 95, 90, 85, 80, 75]. This one is easy because 100*.75 is <= s*i. Ho... |
How do I distribute these numbers? |
I'm given a number (n), an interval (i), and the number of steps (s). I want to distribute these numbers over that criteria as best as possible. The upper bounds would be n (which is critical) and lower bounds is greater than or equal to 0.75 * n (but divisible by the interval). So, if n=100, i=5, and s=6, you'd hav... |
I interpret the question as "how do I know how to attack a Diophantine problem". The question is in fact not easy and it is this skill that algebraic number theorists hone. Moreover, the methods that you listed are where the story was about 300 years ago. Today, there are more sophisticated and more general techniques ... |
Okay, so almost 3 months into my abstract algebra, we just started rings. I have a few questions.
A "trivial ring" is a ring with only one element. So $R={0}$ is a trivial ring. Understandable.
Then a definition states: Let $R$ be a ring. If there is an element $x \in R$ s.t. for all $a \in R$ you have $a * x = ... |
I am too untrusted to add a comment so i'm spamming this as an answer.
J.M: I think you are considering the converse of the question. If you have two very different-looking matrices it is possible that they have the same spectrum. But if you have two similar-looking matrices, then is it possible that their spectra... |
This might be an easy question, but I haven't been able to up come up with a solution.
The image of the map $$f : \mathbb{R} \to \mathbb{R}^2, a \mapsto (\frac{2a}{a^2+1}, \frac{a^2-1}{a^2+1})$$
is the unit circle take away the north pole. $f$ extends to a function $$g: \mathbb{C} \backslash \{i, -i \} \to \mathb... |
It is a local constant sheaf, but why we say the local system on topological space $X$ is one-to-one correspondes to $\tilde(X)\times_G V$, where $G$is fundamental group of $X$ and $\tilde(X)$ is the universal space of $X$, and $V$ is a $G$-module, does even #\tilde(X)\times_G V$ a local system(local constant sheaf)? |
It is a local constant sheaf, but why the local system on topological space $X$ is one-to-one correspondes to $\tilde(X)\times_G V$, where $G$is fundamental group of $X$ and $\tilde(X)$ is the universal space of $X$, and $V$ is a $G$-module, I am wondering why $\tilde(X)\times_G V$ is a local system(local constant she... |
What theorem of Serre's is being referenced? |
Let $R$ be a non-zero commutative ring. Prove that the ideal $(x)$ of $R[x]$ is prime if and only if $R$ is an integral domain.
I'm working on the left-to-right direction right now. I know that $R[x]/(x)$ is an integral domain since $(x)$ is prime. So I want to fix $r\in R$ and use the evaluation map $e_r:R[x]\to R$... |
I believe theorem 1.2 [here][1]
[1]: http://books.google.com/books?id=ZhzXJHUgcRUC&pg=PA28&lpg=PA28&dq=serre's+theorem+local&source=bl&ots=aVRphSsxvf&sig=ko1Ej39JEaVCdRAdMBFHI8WWbxc&hl=en&ei=mMH9TP-xLcOclgeR3pnICA&sa=X&oi=book_result&ct=result&resnum=4&ved=0CDQQ6AEwAw#v=onepage&q=serre's%20theorem%20local&f=fals... |
It's worth emphasizing that, in addition to unique factorization, there are also somewhat deeper structural reasons underlying the convention that $\:1\:$ is neither prime nor a prime ideal, but $\:0\:$ is. Below I discuss the motivations for these differing conventions.
One important motivation for including the ze... |
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