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Let $$D_N(x)=\frac{\sin [(N+(1/2))t]}{\sin (t/2)}$$ be the Dirichlet kernel. Let $x(N)$ be the number in $0<x<\pi/N$ such that $D_N(x)=1$. Is $$\left|\int_{x(N)}^{\pi/N} D_N(t)\mathrm dt \right|=O\left(\frac1{N}\right)$$ true?
This question arises from my attempt to give a rigorous proof of Gibbs phenomenon. In fact, ... |
It's a well known result and a proof can be found in many books, e.g, [here][1]. It's lemma 7.56.
[1]: http://books.google.com.br/books?id=lYrsiaHSHKcC&lpg=PP1&hl=pt-br&pg=PA198#v=onepage&q&f=false |
When I read some issues here I see from time to time incorrect references to the field *special functions*, it might e.g. be a discussion around Dirac's $\delta$-function which is tagged (special-functions) or a discussion around some function reminding of the Weierstrass no where differentiable continuous function. Th... |
What is a special function? |
[Non Linear Dynamics and Chaos](http://books.google.com/books?id=FIYHiBLWCJMC&printsec=frontcover&dq=nonlinear+dynamics&hl=en&ei=2kX7TKeKFISClAeVs5GlDA&sa=X&oi=book_result&ct=result&resnum=1&sqi=2&ved=0CCoQ6AEwAA#v=onepage&q&f=false) by Strogatz, page 322. I read this a long time ago, and the whole book is worth readi... |
I have a consequence of points:
(x1, y1), (x2, y2), (x3, y3)....
I would like to know is it exponential consequence and how to get expression of it? |
I have a sequence of points:
(x1, y1), (x2, y2), (x3, y3)....
I would like to know is it exponential sequence and how to get expression of it? |
I have a sequence of points:
(x1, y1), (x2, y2), (x3, y3)....
I would like to know is it exponential sequence and how to get expression of it?
![alt text][1]
[1]: https://i.stack.imgur.com/0PHR0.jpg |
Significance of $\displaystyle\sqrt[n]{a^n} $?! |
I have two sets $M$ and $H$. $M$ is an arbitrary string of length $k$ and $H$ is an string of length $p$. Both are constructed from a charset of length $r$. And $p<k$.
Hash function $f(m)=h$.
I understand that $r^p$ hashes are possible and that $r^k-r^p$ collisions occur for the complete set $M$.
But how can ... |
There's the obvious approach of proceeding based on taking the logarithm of both sides of your regression formula:
$$\ln\;y=\ln\left(ax^b\right)$$
$$\ln\;y=\ln\;a+b\ln\;x$$
which can be seen to be an expression of the form
$$v=k+bu$$
where $v=\ln\;y$, $u=\ln\;x$, and $k=\ln\;a$.
Now a linear regressio... |
***If the $5$-th term of the G.P is its middle term and its value is 2,then find the product of all terms of this GP?***
**My approach:**
Since $a_5$ is the middle term, so the GP contains 9 terms.
$$P_9 = a\cdot ar\cdot ar^2 \cdot ar^3 \cdot 2\cdot ar^5 \cdots ar^8$$
which leaves us $P_9 = 2\cdot (ar^4)^8$... |
The fourth term is $2/r$ and the sixth term is $2r$. What happens when you multiply these? What about the third and seventh terms? |
I have two sets $M$ and $H$. $M$ is an arbitrary string of length $k$ and $H$ is an string of length $p$. Both are constructed from a charset of length $r$. And $p<k$.
Hash function $f(m)=h$.
I understand that $r^p$ hashes are possible and that $r^k-r^p$ collisions occur for the complete set $M$.
But how can ... |
Let $G$ is given $m$ x $n$ matrix, $y$ is given $n$ x $1$ column vector and $x$ is unknown $m$ x $1$ column vector such that $x \ge 0$. How to find $\min_x (y - Gx)^T(y - Gx)$? |
Let $p(n)$ be a polynomial of degree $a$. Start of with plunging in arguments from zero and go up one integer at the time. Go on until you have come at an integer argument $n$ of which $p(n)$'s value is not prime and count the number of distinct primes your polynomial has generated.
>Question: what is the maximum n... |
What is the maximum number of primes generated consecutively generated by a polynomial of degree $a$ ? |
Is there a way to verify whether a non-empty intersection exists between two simple polygons (not necessarily convex) using the Chazelle's simplicity test ? |
Louis de Branges had a paper on his [homepage] [1] claiming a solution for the [Invariant Subspace Problem] [2].
But I don't see that paper anymore, though he still has a "proof" of Riemann Hypothesis on his website.
Does anyone know whether he actually admitted his proof of the Invariant Subspace Problem was w... |
Let $p(n)$ be a polynomial of degree $a$. Start of with plunging in arguments from zero and go up one integer at the time. Go on until you have come at an integer argument $n$ of which $p(n)$'s value is not prime and count the number of distinct primes your polynomial has generated.
>Question: what is the maximum n... |
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 have heard that given two sheaves $A$ and $B$ on a variety, one can identify elements of $Ext^d(A,B)$ with complexes of sheaves
$$0\to B \to C_1 \to \cdots \to C_d \to A \to 0.$$
My questions are,
> How do I see that this is true?
and
> If I have obtained an element of $Ext^n$ by some other method, can ... |
On $\mathbb{R}^n$ and $p\ge 1$ the $p$-norm is defined as $$\|x\|_p=\left ( \sum _{j=1} ^n |x_j| ^p \right ) ^{1/p}$$
and there is the $\infty$-norm which is $\|x\|_\infty=\max _j |x_j|$. It's called the $\infty$ norm because it is the limit of $\|\cdot\|_p$ for $p\to \infty$.
Now we can use the definition above fo... |
What is the 0-norm? |
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... |
OK, I think I've got one for you, but I have to admit it's a bit of a cheat. If you replace independence in your requirements by [free independence](http://en.wikipedia.org/wiki/Free_probability) then there exists a continuous bounded and infinitely divisible distribution, namely the [Wigner semi-circle distribution](h... |
Hints: Injectivity: Suppose $f(x)=f(y)$, what can you say about $f(f(x))$ and $f(f(y))$.
Surjectivity: Given $a\in A$, there exists $b\in A$ such that $f(b)=a$ since, $f(f(a))=id(a)=a$ so b=? |
If A is a subset of R with Lebesgue measure > 0 then are there a,b such that the measure of $[a,b]\cap A$ is b-a? |
I have a school assigment to do, but I have no idea, where to start. I hope you can help..here it is:
We have a finite sequence A = (a1, a2,..., an), length of A = n, elements of A are natural numbers.
Let S(i,j) be partial sum of this sequence from element[i] to element[j], where 1<=i<=j<=n. <br>
Formaly writ... |
**Problem:**
**If the sum of p terms of an AP is q and the sum of q terms is P,then find the sum of p+q terms.**
For the problem we can write (considering $a$ is the first term and $d$ is the common difference):
$$\frac{p}{2}\cdot \biggl[2a + (p-1)d \biggr] = q \qquad \cdots (1)$$
$$\frac{q}{2}\cdot \biggl[... |
**Problem:**
**If the sum of the first $p$ terms of an arithmetic progression is $q$ and the sum of the first $q$ terms is $p$, then find the sum of $p+q$ terms.**
For the problem we can write (considering $a$ is the first term and $d$ is the common difference):
$$\frac{p}{2}\cdot \biggl[2a + (p-1)d \biggr] = ... |
**Problem:**
**If $a_1,a_2,a_3 \cdots a_n$ are in HP then find the value of $ a_1 \cdot a_2 + a_2 \cdot a_3 + a_3 \cdot a_4 + \cdots + a_{n-1} \cdot a_n$ **
----------
My initial approach,using the property of HP that $ \frac{1}{a_1}, \frac{1}{a_2}, \frac{1}{a_3},\cdots ,\frac{1}{a_n}$ is in AP, I am ge... |
I have a school assignment to do, but I have no idea, where to start. I hope you can help. Here it is:
We have a finite sequence $A = (a_1, a_2,\ldots, a_n)$, length of $A$ is $n$, elements of $A$ are natural numbers.
Let $S(i,j)$ be partial sum of this sequence from $a_i$ to $a_j$, where $1\leq i\leq j\leq n$.... |
**Problem:**
If $a_1,a_2,a_3 \cdots a_n$ are in HP then find the value of $ a_1 \cdot a_2 + a_2 \cdot a_3 + a_3 \cdot a_4 + \cdots + a_{n-1} \cdot a_n$
----------
My initial approach,using the property of HP that $ \frac{1}{a_1}, \frac{1}{a_2}, \frac{1}{a_3},\cdots ,\frac{1}{a_n}$ is in AP, I am getting... |
Your answers to (b) and (c) are correct.
For (a), you need to prove that if a=b mod 16, i.e. $a-b=16n $ for some integer $n$, then $3a=3b$ mod 24, i.e. $3a-3b=24m$ for some integer $m$. Can you do that? |
How do you show that the convex hull of a given set of points S, always has the minimum perimeter ? By perimeter i mean the length of the boundary of the hull |
The rough question is:
> **How do the finite subgroups of SL(2,R/I) vary as I varies, where R is a nice local ring?**
Two cases seem basic and approachable but already very interesting:
* R the 2-adic integers, and R/I = **Z**/2<sup>n</sup>**Z** with n varying, and
* R the power series over the field with 2 ... |
how can i extract rotation, scale values from 2d transformation matrix? i mean a have a 2d transformation
matrix = [1, 0, 0, 1, 0, 0]
matrix.rotate(45 / 180 * PI)
matrix.scale(3, 4)
matrix.translate(50, 100)
matrix.rotate(30 / 180 * PI)
matrix.scale(-2, 4)
now my matrix have v... |
The rough question is:
> **How do the finite subgroups of SL(2,R/I) vary as I varies, where R is a nice local ring?**
SL(2,A) = { [a,b;c,d] : ad-bc=1, a,b,c,d in A } is the group of a 2×2 matrices with entries from A and determinant 1.
Two cases seem basic and approachable but already very interesting:
... |
Question: Calculate the rank of the following matrices:
$A = \left( \begin{array}{cc} 1 & n \\ n & 1 \end{array} \right), n \in \mathbb{Z}$ \ \ and \ \ $B = \left( \begin{array}{ccc} 1 & x & x^{2} \\ 1 & y & y^{2} \\ 1 & z & z^{2} \end{array} \right), x,y,z \in \mathbb{R}$.
So the way I understand rank($A$), is t... |
how do I found the resolvent of this formula?
$(A \vee C) \wedge ( \neg A \vee B) \wedge C$
Is it as easy as take every clause with the same statements?
$(A \vee C)$ and $( \neg A \vee B)$ and put out the contradiction (in this case $A$ and $\neg A$)
Could you recommend me some source for resolution method?... |
In most undergraduate differential geometry courses -- I am thinking of do Carmo's "Differential Geometry of Curves and Surfaces" -- the topic of study is curves and surfaces in $\mathbb{R}^3$. However, the definition of "curve" and "surface" are usually presented in very different ways.
A curve is defined simply a... |
Why study "curves" instead of 1-manifolds? |
A friend and I were doing some math together when we had to integrate 2(100 + t).
I just multiplied it out and integrated 200 + 2t, which should be 200t + t^2.
He did u-substitution and got (100 + t)^2
When I take the derivative of both of ours, I seem to get the same thing. But (100 + t)^2 = 10,000 + 200t + t... |
The only problem you have is that your expression for the vector $\overrightarrow{P_1P_2}$ is slightly off. As you say, let $P_1 = (x_1,y_1)$ and $P_2 = (x_2,y_2)$ be points on the line. Then what are the coordinates of the vector $\overrightarrow{P_1P_2}$?
Once you calculate that, then notice that the coordinates o... |
If $A$ is a subset of $\mathbb{R}$ with Lebesgue measure strictly greater than $0$, does it follow then that are there $a$ and $b$ such that the measure of $[a,b]\cap A$ is $b-a$?
Thank you. |
Question: Calculate the rank of the following matrices:
$A = \left( \begin{array}{cc} 1 & n \\ n & 1 \end{array} \right), n \in \mathbb{Z}$ \ \ and \ \ $B = \left( \begin{array}{ccc} 1 & x & x^{2} \\ 1 & y & y^{2} \\ 1 & z & z^{2} \end{array} \right), x,y,z \in \mathbb{R}$.
So the way I understand rank($A$), is t... |
I read to solve Pells equation you need a fraction close to the square roots of d and to solve ax + by = 1 you have use the greatest common divisor function. To solve Fermats Last theorem you need elliptic curves.
* the irrational number
* a special function
* a strange curve
I have two questions, Where can I f... |
What are diophantine equations REALLY? |
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 ... |
Sometimes I had a pair of equations that turned out to be easy if you read it as a real and imaginary equation, I read to solve Pells equation you need a fraction close to the square roots of d and to solve ax + by = 1 you have use the greatest common divisor function. To solve Fermats Last theorem you need elliptic cu... |
Let $X = \textrm{Spec} \mathbf{C}[[x,y]]/(y^{2} - x^{3} - x^{2})$. I would like to describe X set-theoretically. My questions are: Can one explicitly say what the elements in X are? Is it possible to interpret them geometrically? And is X irreducible?
I am not really sure where to get started. Any help would be muc... |
What is Spec $\mathbf{C}[[x,y]]/(y^{2} - x^{3} - x^{2})$? |
A lot of equations can be proved impossible if the left hand side is odd and the right hand side is even, sometimes I had a pair of equations that turned out to be easy if you read it as a real and imaginary equation, I read to solve Pells equation you need a fraction close to the square roots of d and to solve ax + by... |
Question: Calculate the rank of the following matrices:
$A = \left( \begin{array}{cc} 1 & n \\ n & 1 \end{array} \right), n \in \mathbb{Z}$ and $B = \left( \begin{array}{ccc} 1 & x & x^{2} \\ 1 & y & y^{2} \\ 1 & z & z^{2} \end{array} \right)$, $x,y,z \in \mathbb{R}$.
So the way I understand rank($A$), is the nu... |
You seem to be assuming that because "$1-n^2$" doesn't look like $0$, then it cannot be zero. That is a common, but often fatal, mistake.
Remember that $n$ stands for *some* integer. Once you get to
$$A = \left(\begin{array}{cc}
1 & n\\
0 & 1-n^2
\end{array}\right),$$
you cannot just jump to saying there are tw... |
You seem to be assuming that because "$1-n^2$" doesn't look like $0$, then it cannot be zero. That is a common, but often fatal, mistake.
Remember that $n$ stands for *some* integer. Once you get to
$$A = \left(\begin{array}{cc}
1 & n\\
0 & 1-n^2
\end{array}\right),$$
you cannot just jump to saying there are tw... |
You seem to be assuming that because "$1-n^2$" doesn't look like $0$, then it cannot be zero. That is a common, but often fatal, mistake.
Remember that $n$ stands for *some* integer. Once you get to
$$A = \left(\begin{array}{cc}
1 & n\\
0 & 1-n^2
\end{array}\right),$$
you cannot just jump to saying there are tw... |
You seem to be assuming that because "$1-n^2$" doesn't look like $0$, then it cannot be zero. That is a common, but often fatal, mistake.
Remember that $n$ stands for *some* integer. Once you get to
$$A = \left(\begin{array}{cc}
1 & n\\
0 & 1-n^2
\end{array}\right),$$
you cannot just jump to saying there are tw... |
I have the following system of equations where the $m$'s are known but $a, b, c, x, y, z$ are unknown. How does one go about solving this system? All the usual linear algebra tricks I know don't apply and I don't want to do it through tedious substitutions.
\begin{aligned}
a + b + c &= m_0 \\\\
ax + by + cz &= m_2... |
Given the line $x+2=\frac{y+4}{2}=\frac{z-5}{-2}$
I want to find the closest point on this line to $(1,1,1)$
I suppose the details here don't matter but in general how is this done? We need a vector perpendicular to the line but that also reaches $(1,1,1)$. |
How to find a point on a line closest to another given point? |
A lot of equations can be proved impossible if the left hand side is odd and the right hand side is even, sometimes I had a pair of equations that turned out to be easy if you read it as a real and imaginary equation, I read to solve Pell's equation you need a fraction close to the square roots of $d$ and to solve $ax ... |
What are Diophantine equations REALLY? |
Sometimes when you want to solve an equation you can just use algebra and rearrange it then you are done. But sometimes no amount of algebra can ever prove the equation, and then you need an idea, here is what I mean by the idea:
* parity (or modular arithmetic for higher numbers)
* complex integers.
* the irratio... |
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 ... |
I was working through a textbook on topology and I came across a problem I couldn't solve.
<p>1) It is known that <b>if a space is T1, it is countably compact if and only if every <i>countable</i> open cover has a <i>finite</i> subcover.</b> (See below for the definition of countably compact, which might be differe... |
How does the set of complex numbers (resp., nonzero complex numbers) form a group under addition (resp., multiplication)? |
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 ... |
On a practice final exam (for a Computer Security course), I am given the following equation to solve, but I have no idea how to solve it:
324x mod 121 = 1
Any direction? |
I was working through a textbook on topology and I came across a problem I couldn't solve.
<p>1) It is known that <b>if a space is T1, it is countably compact if and only if every <i>countable</i> open cover has a <i>finite</i> subcover.</b> (See below for the definition of countably compact, which might be differe... |
I was working through a textbook on topology and I came across a problem I couldn't solve.
<p>1) It is known that <b>if a space is T1, it is countably compact if and only if every <i>countable</i> open cover has a <i>finite</i> subcover.</b> (See below for the definition of countably compact, which might be differe... |
How many residue classes satisfy the congruence x^3 = 3 mod 21?
I don't understand what this question is asking me to do.
Can someone simplify the question for me, thanks.
|
How many residue classes satisfy the congruence x3 = 3 mod 21? |
How many residue classes satisfy the congruence $x^3 \equiv 3 \pmod{21}$?
I don't understand what this question is asking me to do.
Can someone simplify the question for me, thanks.
|
How many residue classes satisfy the congruence $x^3 \equiv 3 \pmod{21}$? |
How many residue classes satisfy the congruence x^3 = 3 mod 21?
I don't understand what this question is asking me to do.
Can someone simplify the question for me, thanks.
|
The question is asking you to check which of the 21 residue classes of integers modulo $21$ (to wit, the class of $0$, the class of $1$, the class of $2$, etc) are solutions to $x^3\equiv 3\pmod{21}$. If nothing else occurs to you, you can certainly plug and chug and figure out which ones are solutions and which ones a... |
How many residue classes satisfy the congruence $x^3 = 3 \mod 21$?
I don't understand what this question is asking me to do.
Can someone simplify the question for me, thanks.
|
How many residue classes satisfy the congruence $x^3 = 3 \mod{21}$?
I don't understand what this question is asking me to do.
Can someone simplify the question for me, thanks.
|
I have the following system of equations where the $m$'s are known but $a, b, c, x, y, z$ are unknown. How does one go about solving this system? All the usual linear algebra tricks I know don't apply and I don't want to do it through tedious substitutions.
\begin{aligned}
a + b + c &= m_0 \\\\
ax + by + cz &= m_1... |
On a practice final exam (for a Computer Security course), I am given the following equation to solve, but I have no idea how to solve it:
$$324x \mod{121} = 1.$$
Any direction? |
How many residue classes satisfy the congruence $x^3 \equiv 3 \pmod{21}$?
I don't understand what this question is asking me to do.
Can someone simplify the question for me, thanks.
|
Given that $f:[-1,1] \to \mathbb{R}$ is a continuous function such that $ \int\limits_{-1}^{1} f(t) \ dt =1$, how do i evaluate the limit of this integral: $$\lim_{n \to \infty} \int\limits_{-1}^{1} f(t) \cos^{2}{nt} \ dt$$
What i did was to write $\cos^{2}{nt} = \frac{1+\cos{2nt}}{2}$ and subsitute it in the integr... |
Given that $f:[-1,1] \to \mathbb{R}$ is a continuous function such that $ \int_{-1}^{1} f(t) \ dt =1$, how do I evaluate the limit of this integral: $$\lim_{n \to \infty} \int_{-1}^{1} f(t) \cos^{2}{nt} \,dt$$
What I did was to write $\cos^{2}{nt} = \frac{1+\cos{2nt}}{2}$ and substitute it in the integral so that i ... |
Given that $f\colon [-1,1] \to \mathbb{R}$ is a continuous function such that $ \int_{-1}^{1} f(t) \ dt =1$, how do I evaluate the limit of this integral: $$\lim_{n \to \infty} \int_{-1}^{1} f(t) \cos^{2}{nt} \,dt$$
What I did was to write $\cos^{2}{nt} = \frac{1+\cos{2nt}}{2}$ and substitute it in the integral so t... |
This is probably one of the more embarrassing moments of my life, but I will soldier on. Having not dealt with inequalities for quite some time, I was rather befuddled by a question from a friend of mine regarding solving for two variables given a pair of inequalities. The problem goes like this:
Given `x + y < 11` ... |
This is probably one of the more embarrassing moments of my life, but I will soldier on. Having not dealt with inequalities for quite some time, I was rather befuddled by a question from a friend of mine regarding solving for two variables given a pair of inequalities. The problem goes like this:
Given $x + y \lt 11... |
Are $x$ and $y$ required to be *integers*? Because if they are not, then the answer cannot be $4$: take $x=6.5$ and $y=4$. Then $x+y = 10.5\lt 11$, but $x-y = 2.5 \lt 4$. In fact, taking $x=6+\epsilon$ and $y=5-(\epsilon+\delta)$ with $\epsilon\gt 0$ and $\delta\gt 0$ gives you $x+y = 11-\delta\lt 11$, but $x-y = 1+2\e... |
Are $x$ and $y$ required to be *integers*? Because if they are not, then the answer cannot be $4$: take $x=6.5$ and $y=4$. Then $x+y = 10.5\lt 11$, but $x-y = 2.5 \lt 4$. In fact, taking $x=6+\epsilon$ and $y=5-(\epsilon+\delta)$ with $\epsilon\gt 0$ and $\delta\gt 0$ arbitrary but positive, and this gives you $x+y = 1... |
Are $x$ and $y$ required to be *integers*? Because if they are not, then the answer cannot be $4$: take $x=6.5$ and $y=4$. Then $x+y = 10.5\lt 11$, but $x-y = 2.5 \lt 4$. In fact, taking $x=6+\epsilon$ and $y=5-(\epsilon+\delta)$ with $\epsilon\gt 0$ and $\delta\gt 0$ arbitrary but positive, and this gives you $x+y = 1... |
Are $x$ and $y$ required to be *integers*? Because if they are not, then the answer cannot be $4$: take $x=6.5$ and $y=4$. Then $x+y = 10.5\lt 11$, but $x-y = 2.5 \lt 4$. In fact, taking $x=6+\epsilon$ and $y=5-(\epsilon+\delta)$ with $\epsilon\gt 0$ and $\delta\gt 0$ arbitrary but positive, and this gives you $x+y = 1... |
Alright, so I'm still trying to make my players movements relative to the camera. I got a good answer [here][1] But he ends it with: s(Jx x̂ + Jy ŷ)
So how can I turn that equation into a x and y?
[1]: http://stackoverflow.com/questions/4321382?tab=votes#tab-top |
Splitting this into a x and y point? |
Alright, so I'm still trying to make my players movements relative to the camera. I got a good answer [here][1] But he ends it with: $s(Jx \widehat{\mathbf{x}} + Jy \widehat{\mathbf{y}}$)
So how can I turn that equation into an $x$ and $y$?
[1]: http://stackoverflow.com/questions/4321382?tab=votes#tab-top |
I would like to know why $\mathbb{C}[x,y]$ is not isomorphic to $\mathbb{C}[x] \otimes _{\mathbb{Z}} \mathbb{C}[y]$?
Thank you! 1 |
Why is $\mathbb{C}[x,y]$ not isomorphic to $\mathbb{C}[x] \otimes _{\mathbb{Z}} \mathbb{C}[y]$? |
Find the complete integral of differential equation
`z^2 = pqxy` ?
I have solved this equation till auxillary equation:
dp/(-pqy+2pz) = dq/(-pqx+2qz) = dz/(2pqxy) = dx/(qxy) = dy/(pxy)
But I have anable to find value of p and q. |
Why is $\mathbb{C}[x,y]$ not isomorphic to $\mathbb{C}[x] \otimes _{\mathbb{Z}} \mathbb{C}[y]$ as rings? |
This is probably one of the more embarrassing moments of my life, but I will soldier on. Having not dealt with inequalities for quite some time, I was rather befuddled by a question from a friend of mine regarding solving for two variables given a pair of inequalities. The problem goes like this:
Given $x + y \lt 11... |
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