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Hint: What does the hypothesis of [Green's theorem][1] say about the requirements on functions $P$ and $Q$?
[1]: http://en.wikipedia.org/wiki/Green's_theorem |
The problem is "Calculating $\oint_{L} \frac{xdy - ydx}{x^2 + y^2}$, where L is a simple smooth and closed curve that does not pass throught the orignal point".
Here is my solution:
Let $$P(x,y) = \frac{-y}{x^2 + y^2}, Q(x,y) = \frac{x}{x^2 + y^2}$$
Get $$\frac{\partial{P}}{\partial{y}} = \frac{y^2 - x^2}{(x^2 +... |
The problem is "Calculating $\oint_{L} \frac{xdy - ydx}{x^2 + y^2}$, where L is a smooth, simple closed, and postive oriented curve that does not pass throught the orignal point".
Here is my solution:
Let $$P(x,y) = \frac{-y}{x^2 + y^2}, Q(x,y) = \frac{x}{x^2 + y^2}$$
Get $$\frac{\partial{P}}{\partial{y}} = \fra... |
The problem is "Calculating $\oint_{L} \frac{xdy - ydx}{x^2 + y^2}$, where L is a smooth, simple closed, and postive oriented curve that does not pass through the orgin".
Here is my solution:
Let $$P(x,y) = \frac{-y}{x^2 + y^2}, Q(x,y) = \frac{x}{x^2 + y^2}$$
Get $$\frac{\partial{P}}{\partial{y}} = \frac{y^2 - x... |
The problem is "Calculating $\oint_{L} \frac{xdy - ydx}{x^2 + y^2}$, where L is a smooth, simple closed, and postively oriented curve that does not pass through the orgin".
Here is my solution:
Let $$P(x,y) = \frac{-y}{x^2 + y^2}, Q(x,y) = \frac{x}{x^2 + y^2}$$
Get $$\frac{\partial{P}}{\partial{y}} = \frac{y^2 -... |
In [this][1] problem, I know that the hypothsis of Green theorem must ensure that the simple closed curve is smooth, but what is smooth? Could you give a definition and an intuitive explaination?
[1]: http://math.stackexchange.com/questions/12022/calculating-oint-l-fracxdy-ydxx2-y2 |
What does smooth curve mean? |
I think I got it. It seems to be problem 10360, originally published in 1994 (volume 101, issue 1, page 76), proposed by Richard Stanley. I'm looking at <a href="http://biere.louisiana.edu:2095/stable/2589004?&Search=yes&term=tree&term=lattice&list=hide&searchUri=%2Faction%2FdoBasicSearch%3Ffilter%3Djid%253A10.2307%252... |
[The orginal problem][1] is "Calculating $\oint_{L} \frac{xdy - ydx}{x^2 + y^2}$, where L is a smooth, simple closed, and postively oriented curve that does not pass through the orgin".
But what if I modify the hypothesis and allow non-simple closed curve? I mean is there something like green formula that allows us ... |
I think I got it. It seems to be problem 10360, originally published in 1994 (volume 101, issue 1, page 76), proposed by Richard Stanley. I'm looking at <a href="http://www.jstor.org/pss/2589004">the JSTOR page of the solution</a> (by <a href="http://math.stackexchange.com/users/226/robin-chapman">Robin Chapman</a>), w... |
Was trying to calculate $$\int_{0}^{\infty}e^{-x}\ln(x) dx=-\gamma$$ and I found this question:
I want to analyze $$\int\frac{e^{-x}}{x}dx$$
With $u=\displaystyle\frac{1}{x} \Rightarrow du = \displaystyle\frac{-1}{x^{2}} dx $, and $dv=e^{-x} \Rightarrow v=-e^{-x}$
Then
$$\int\frac{e^{-x}}{x}dx = \display... |
These calculations are correct ? About $\int\frac{e^{-x}}{x}dx$ |
Let $L(n)=\lfloor(1+\sqrt{5})^n\rfloor$. What kind of a linear recurrence is satisfied by $L(n)$? I have no idea how to go about this, because of the presence of the greatest integer function.
Please feel free to retag it as I kept getting an error on every tag I thought was appropriate. |
[The orginal problem][1] is "Calculating $\oint_{L} \frac{xdy - ydx}{x^2 + y^2}$, where L is a smooth, simple closed, and postively oriented curve that does not pass through the orgin".
But what if I modify the hypothesis and allow non-simple closed curve? I mean is there something like green formula that allows us ... |
How to prove the following trigonometric identities ?
1. If $\displaystyle \tan \alpha \cdot \beta = 1 \text{ then } \alpha + \beta = \frac{\pi}{2}$
2. $\displaystyle\sec\theta + \tan \theta = \frac{1}{ \sec\theta + \tan \theta}, \theta \neq (2n+1)\frac{\pi}{2}, n \in \mathbb{Z} $ |
How to prove the following trigonometric identities ?
1. If $\displaystyle \tan (\alpha) \cdot \tan(\beta) = 1 \text{ then } \alpha + \beta = \frac{\pi}{2}$
I tried to prove it by using the the formula $\tan(\alpha + \beta)$ but isn't it valid only when $\alpha + \beta \neq \frac{\pi}{2}$ ?
2. $\displaystyle... |
How to prove the following trigonometric identities ?
1. If $\displaystyle \tan (\alpha) \cdot \tan(\beta) = 1 \text{ then } \alpha + \beta = \frac{\pi}{2}$
I tried to prove it by using the the formula $\tan(\alpha + \beta)$ but isn't it valid only when $\alpha + \beta \neq \frac{\pi}{2}$ ?
2. $\displaystyle... |
How to prove the following trigonometric identities ?
1. If $\displaystyle \tan (\alpha) \cdot \tan(\beta) = 1 \text{ then } \alpha + \beta = \frac{\pi}{2}$
I tried to prove it by using the the formula for $\tan(\alpha + \beta)$ but ain't it valid only when $\alpha + \beta \neq \frac{\pi}{2}$ ?
2. $\displays... |
How to prove the following trigonometric identities ?
1. If $\displaystyle \tan (\alpha) \cdot \tan(\beta) = 1 \text{ then } \alpha + \beta = \frac{\pi}{2}$
I tried to prove it by using the the formula for $\tan(\alpha + \beta)$ but ain't it valid only when $\alpha + \beta \neq \frac{\pi}{2}$ ?
2. $\displays... |
Given the following series: $\sum_{n=1}^{\infty} (\sqrt[3]{n+1} - \sqrt[3]{n-1})^{\alpha }$
where $\alpha \in \mathbb{R}$. Does the series converge or diverge?
Attempts to solve the problem:
1) $\lim_{n\to\infty} (\sqrt[3]{n+1} - \sqrt[3]{n-1})^{\alpha } = 0$ - not helpful.
2) Used the formula $a^{3} - b^{3}... |
Using Lagrange's remainder, I have to prove that:
$\log(1+x) = \sum\limits_{n=1}^\infty (-1)^{n+1} \cdot \frac{x^n}{n}, \; \forall |x| < 1$
I am not quite sure how to do this. I started with the Taylor series for $x_0 = 0$:
$f(x_0) = \sum\limits_{n=0}^\infty \frac{f^{(n)}(x_0)}{n!} \cdot x^n + r_n$, where $r_n... |
How to use the Lagrange's remainder to prove that log(1+x) = sum(...)? |
Do you have any reason to suspect $L(n)$ should satisfy a linear recurrence? Here's a way you can prove that it doesn't satisfy a linear recurrence of depth 2 (which can be generalised to any depth).
*Step 1*: Compute $L(n)$ for small $n$: 3, 10, 33, 109, 354, 1148, 3716, ...
*Step 2*: Assume $b L(n-2)+a L(n-1)... |
Let $p,q$ belong to $\mathbb{N}$ and are relatively prime to each other. If $\alpha,\beta$ belong to $\mathbb{N}$, are also relatively prime to each other,then are $(p\beta+q\alpha)$ and $q\beta$ always relatively prime ? |
Let $p,q$ belong to $\mathbb{N}$ and are relatively prime to each other. If $\alpha,\beta$ belong to $\mathbb{N}$, are also relatively prime to each other,then are $(p\beta+q\alpha)$ and $q\beta$ always relatively prime ?
**EDIT**
there is a condition that p is not equal to α and q not equal to β . |
With help from Maple, I got
$$\left(\frac{ax+by+cz}{x-y}\right)^2+\left(\frac{ay+bz+cx}{y-z}\right)^2+\left(\frac{az+bx+cy}{z-x}\right)^2-(c-a)^2-(c-b)^2$$
equal to
$$\frac{(c(x^3+y^3+z^3)+(a-c)(x^2y+y^2z+z^2x)+(b-c)(x^2z+y^2x+z^2y)-3(a+b-c)xyz)^2}{(x-y)^2(y-z)^2(x-z)^2}$$ which of course is $\ge 0$.
But with n... |
How to prove this inequality without use of computers? |
Let $p,q$ belong to $\mathbb{N}$ and are relatively prime to each other. If $\alpha,\beta$ belong to $\mathbb{N}$, are also relatively prime to each other,then are $(p\beta+q\alpha)$ and $q\beta$ always relatively prime ?
|
Recently,i consider a problem:Let E/F is a Galois extension of number field,denote the idele group of E(resp F) by I_E(resp I_F).there is a homomorphism induecd by norm map N_E/F:
N: I_E/E* ---> I_F/F*
If there is a finite abel extension K/E,from Global Class Field Theory,the... |
Recently, I consider a problem: Let $E/F$ is a Galois extension of number field, denote the idele group of $E$ (resp $F$) by $I_E$ (resp $I_F$). There is a homomorphism induced by norm map $N_{E/F}$:
$$N: I_E/E^{\ast} \to I_F/F^{\ast}$$
If there is a finite abel extension $K/E$ ,from Global Class Field Theory,the... |
How to prove the following trigonometric identities ?
1) If $\displaystyle \tan (\alpha) \cdot \tan(\beta) = 1 \text{ then } \alpha + \beta = \frac{\pi}{2}$
I tried to prove it by using the the formula for $\tan(\alpha + \beta)$ but ain't it valid only when $\alpha + \beta \neq \frac{\pi}{2}$ ?
2) $\displaysty... |
Was trying to calculate $$\int_{0}^{\infty}e^{-x}\ln(x) dx=-\gamma$$ and I found this question:
I want to analyze $$\int\frac{e^{-x}}{x}dx$$
With $u=\displaystyle\frac{1}{x} \Rightarrow du = \displaystyle\frac{-1}{x^{2}} dx $, and $dv=e^{-x} \Rightarrow v=-e^{-x}$
Then
$$\int\frac{e^{-x}}{x}dx = \display... |
How can we prove that the product of n consecutive integers is divisible by n factorial? |
I'll do one better: I'll write you one (mostly because I actually have already written one). Recall that a first course in analysis typically asserts that there exists a field with an order and the least upper bound property, and you then develop the theory of sequences and series from there. Thus our goal should be to... |
How can we prove, without using the properties of binomial coefficients, the product of n consecutive integers is divisible by n factorial?
|
###Update###
This *should* be the final edit. Lots of typos have been corrected and the presentation in the section on existence has been greatly improved. It should be very close to a form that can be used as a basis for a project for students.
##Introduction##
I'll do one better: I'll write you one (mostly becau... |
###Update###
This *should* be the final edit. Lots of typos have been corrected and the presentation in the section on existence has been greatly improved. It should be very close to a form that can be used as a basis for a project for students.
##Introduction##
I'll do one better: I'll write you one (mostly becau... |
Was trying to calculate $$\int_{0}^{\infty}e^{-x}\ln x dx=-\gamma$$ and I found this question:
I want to analyze $$\int\frac{e^{-x}}{x}dx$$
With $u=\displaystyle\frac{1}{x} \Rightarrow du = \displaystyle\frac{-1}{x^{2}} dx $, and $dv=e^{-x} \Rightarrow v=-e^{-x}$
Then
$$\int\frac{e^{-x}}{x}dx = \displays... |
For example, if $f(x) = \sin x$ and $x$ is uniformly distributed on $[0, \pi]$, how is the equation found that satisfies the probability distribution function of $f(x)$? I imagine the distribution function will be greater when the derivative of $f(x)$ is closer to zero, but this is just a guess.
I apologize if this ... |
Given a function $f(x)$ where $x$ is uniformly distributed between $a$ and $b$, how do I find the probability density function of $f$? |
Let $(X,d)$ be a metric space and $S \subset X$. Show that $d_S(x):=\text{inf}\{d(x,s): s \in S\}=0 \Leftrightarrow x \in \overline S .$
Notes: $\overline S$ is the closure of S. Maybe you can use that a closed set is also closed for sequences in the set? I think the difficult part is when S is open, otherwise its t... |
If $x \in \overline{S}$, then there is a sequence $(s_n)$ in $S$ converging to $x$ (e.g. take $s_n$ to be some element in $S \cap B(x;\tfrac{1}{n})$). Then $d_S(x) \leq \inf\lbrace d(x,s_n) \;\vert\; n \in \mathbb{N}\rbrace = 0$ (why?).
For the other directtion, we prove the contrapositive: If $x \notin \overline{S}... |
In my website, users can either "like" or "dislike" a posted comment. I want to put a link to sort comments by liking such that the most liked ones becomes on top.
Of course I cannot just sort by the number of likes only. I have to subtract the dislikes. But what if the difference (likes - dislikes) are the same. e... |
If there is a field F that is a field reduction of the real numbers, that is F(a)=R for some a, let's also denote this F=R(\a), then given x in R is there a general method to determine whether x is in F or x is in R\F ? |
I guess your question is: Why is $E(X)=\int_0^\infty 1-F(x) dx$?
The answer is as follows: By definition $$E(X)=\int_0^\infty tf(t) dt.$$
Now \begin{eqnarray*} & & \int_0^\infty tf(t) dt\\
&=&\int_0^\infty f(t)\int_0^t 1 dx dt\\
&=&\int_0^\infty\int_0^t f(t) dx dt\\
&=&\int_0^\infty\int_x^\infty f(t) dt dx \\
&... |
The empty set is a member of $P({a,b}) x P({p,q})$. True or false?
My first instinct was false, since the empty set is a member of each power set individually, but when multiplied together, you get {0,0}, which I'm not sure represents the empty set. But my counter argument is that the empty set is a member of the po... |
This problem may be amenable to the "Purkiss Principle". I'll leave it as an exercise for you to determine if it can be applied here. But even if not you should read the following beautiful article on it by Wm. Waterhouse [Do Symmetric Problems Have Symmetric Solutions?][1] I recall thinking that this was one of the mo... |
The empty set is a member of $P({a,b}) \times P({p,q})$. True or false?
My first instinct was false, since the empty set is a member of each power set individually, but when multiplied together, you get {0,0}, which I'm not sure represents the empty set. But my counter argument is that the empty set is a member of t... |
well i was wondering if, there s a way in which one could compare these two theories on the ground that which one will provide more accurate answers to a given problem.
Fuzzy logic always focuses on analysing between 0 and 1 and logic only dolls down to 0 and 1 , so does that imply that fuzzy logic provides us with m... |
Imagine a two-dimensional grid consisting of 20 points along the x-axis and
10 points along the y-axis. Suppose the origin (0,0) is in the bottom-left corner and the
point (20,10) is the top-right corner. A path on the grid consists of a series of moves in
which each move is either one unit to the right or one unit ... |
You have 30 moves to make, 10 of which must be up and 20 of which must be right. How many ways to pick which are the 10? |
Original post on Mathoverflow [here](http://mathoverflow.net/questions/47487/probability-of-the-maximum-levy-stable-random-variable-in-a-list-being-greater).
Given a list of identical and independently distributed [Levy Stable](http://en.wikipedia.org/wiki/Stable_distribution) random variables, $(X_0, X_1, \dots, X_... |
Probability of the maximum (Levy Stable) random variable in a list being greater than the sum of the rest? |
How can we prove that the product of n consecutive integers is divisible by n factorial?
From OP's comment to this question:
> I wish to obtain a proof which does
> not use the properties of binomial
> coefficients. |
Suppose a graph with 12 vertices is colored with exactly 5 colors. By the pigeonhole principle, each color appears on at least two vertices. True or false?
The correct answer is false, but I assumed it to be true. Why is this so? |
The Pigeonhole Principle implies that there is *at least one* color which appears on at least $2$ vertices, not that *each* color appears. Is it possible to color $12$ vertices with five colors in such a way that one of the colors is used only on a single vertex? |
### Update 1 ###
This *should* be the final edit. Lots of typos have been corrected and the presentation in the section on existence has been greatly improved. It should be very close to a form that can be used as a basis for a project for students.
### Update 2 ###
Even more typos corrected. Added example of non-... |
For the fifth function in the link you <a href="http://mathworld.wolfram.com/HeartCurve.html">mentioned</a> (which I thought it was the most similar to a heart):
PolarPlot[(Sin[t]Sqrt[Abs[Cos[t]]])/(Sin[t]+7/5)-2Sin[t]+2, {t, 0, 10}]
Similarly, <a href="http://www.wolframalpha.com/input/?i=PolarPlot+(Sin[t]+S... |
My TI-83 calculator doesnt allow me to do this, but using Windows calculator, I can compute the factorial of say 5.8. What does this mean and how does it work? |
This is probably simple, but I'm solving a practice problem:
$\lim_{n \to \infty}\frac{1}{n}\left( \cos{\frac{\pi}{n}} + \cos{\frac{2\pi}{n}} + \ldots \cos{\frac{n\pi}{n}} \right)$
I recognize this as the Riemann sum from 0 to $\pi$ on $\cos{x}$, i.e. I think its the integral
$\int_0^\pi{ \cos{x}dx }$
which... |
How can we prove that the product of n consecutive integers is divisible by n factorial?
>Note: In this [subsequent question][1] and the comments here the OP has clarified that he seeks a proof that "does not user the properties of binomial coefficients". Please post answers in the newer thread so that this incorrec... |
I accidentally typed `4 ** 5` into Google and received `1,024` instead of `20`. But `4^5` is also `1024`...
Does the double multiplication sign also mean the power sign? Why two symbols? |
What is the double multiplication sign for? |
![alt text][1]
1. The answer for this turns out to be only irreflexive. However, how is this not transitive? The definition I have for transitive states "whenever there is a
path from x to y then there must be a direct arrow from x to y". So for the above graph, if there exists a path from one point to another, t... |
> However, how is this not transitive?
There is an edge from Z to A and one from A to Z, but no edge from Z to itself or from A to itself.
> Does this mean that if a relation is transitive, then it is also irreflexive?
No, it means that if a transitive relation is irreflexive, it is also asymmetric and a stric... |
> However, how is this not transitive?
There is an edge from Z to A and one from A to Z, but no edge from Z to itself or from A to itself.
> Does this mean that if a relation is transitive, then it is also irreflexive?
No, it means that if a transitive relation is irreflexive, it is also asymmetric and a stric... |
How can we prove that the product of n consecutive integers is divisible by n factorial?
From OP's comment to this question:
> I wish to obtain a proof which does
> not use the properties of binomial
> coefficients. |
There's one basilar math thing that keeps bugging me: the fact that a really simple 2D geometrical figure (like a circle) might not be a function.
I know what the definition of a function is. A circle is not a function (of one variable) because it would associate two values of the codomain to a single value of the d... |
This is probably simple, but I'm solving a practice problem:
$\lim_{n \to \infty}\frac{1}{n}\left( \cos{\frac{\pi}{n}} + \cos{\frac{2\pi}{n}} + \ldots +\cos{\frac{n\pi}{n}} \right)$
I recognize this as the Riemann sum from 0 to $\pi$ on $\cos{x}$, i.e. I think its the integral
$\int_0^\pi{ \cos{x}dx }$
whic... |
Question: Let $V$ be a $K$-Vectorspace and $f: V \rightarrow V$ be linear. It holds that $f \circ f = f$. Show that $V = \mbox{ker}(f) \oplus \mbox{im}(f)$.
My attempt: So i guess that the $\oplus$ denotes a direct sum which means i have to show that (i) $V = \mbox{ker}(f) + \mbox{im}(f)$ and (ii) $\mbox{ker}(f) \ca... |
Let's see (ii) first: (Yes, your use of the implication symbol $\Rightarrow$ is appropriate.) You want to use that $f\circ f=f$, which you do not mention explicitly. The point is: In (ii) you need to show that the $v$ you are considering is $0$. You have: $f\circ f=f$, so $v=f(u)=(f\circ f)(u)$ etc, as you did. This gi... |
How can we prove that the product of n consecutive integers is divisible by n factorial?
>Note: In this [subsequent question][1] and the comments here the OP has clarified that he seeks a proof that "does not user the properties of binomial coefficients". Please post answers in the newer thread so that this incorrec... |
[Lennart Carleson][1] proved Luzin's conjecture that [the Fourier series of each $f\in L^2(0,2\pi)$ converges almost everywhere][2]. Also, [Richard Hunt][3] extended the result to [$L^p$ ($p>1$)][4].
Some time ago I tried to read Carleson's paper, but I would say it is fairly hard to assimilate.
1. Is there an e... |
Given the series: $\sum_{n=0}^{\infty}(-1)^{n}(\sqrt[n]{n} - 1)^{n}$. Does the series converge?
Attempt to solution (might be incorrect):
$(\sqrt[n]{n} - 1)^{n}> (1+\frac{1}{n})^{n}$
$(1+\frac{1}{n})^{n} \to e \Rightarrow (\sqrt[n]{n} - 1)^{n}$ lower-bounded by $e$. Based on Leibniz Criterion the sequence $\{A... |
Is $\lim_{n \to \infty}\frac{1}{n}\left( \cos{\frac{\pi}{n}} + \cos{\frac{2\pi}{n}} + \ldots + \cos{\frac{n\pi}{n}} \right)$ a Riemann sum? |
**Question 1**
Lets consider a 3x3 matrix, A.
The inverse of this matrix can be found by finding the adjoint of the matrix and dividing it by its determinant. Can someone please explain why this process work and how it can be visualized?
I know that the determinant can be thought of as the volume spanned ... |
The answers given are pretty good from a musical, mathematical, and socialogical / historical reason. But they miss the fundamental reason why there are 12 notes in a western scale (or 5 notes in an eastern pentatonic, etc.), and why it's those particular 12 notes (or 5).
Qiaochu almost nailed it by pointing out th... |
State in words the negation of the following sentence: For every martian M, if M is green, then M is tall and ticklish.
I got the right answer to this, give or take a few words, but this is a question of form more than anything. After converting this statement to symbols and negating everything, I come up with: $\e... |
State in words the negation of the following sentence: For every martian M, if M is green, then M is tall and ticklish.
I got the right answer to this, give or take a few words, but this is a question of form more than anything. After converting this statement to symbols and negating everything, I come up with: $\e... |
Is $\lim\limits_{n \to \infty}\frac{1}{n}\left( \cos{\frac{\pi}{n}} + \cos{\frac{2\pi}{n}} + \ldots + \cos{\frac{n\pi}{n}} \right)$ a Riemann sum? |
Consider a non-empty set A containing n objects. How many relations on A are both symmetric and reflexive?
The answer to this is $2^p$ where $p=$ $n \choose 2$. However, I dont understand why this is so. Can anyone explain this? |
To be reflexive, it must include all pairs $(a,a)$ with $a\in A$. To be symmetric, whenever it includes a pair $(a,b)$, it must include the pair $(b,a)$. So it amounts to choosing which $2$-element subsets from $A$ will correspond to associated pairs. If you pick a subset $\{a,b\}$ with two elements, it corresponds to... |
How can we prove that the product of n consecutive integers is divisible by n factorial?
>Note: In this [subsequent question][1] and the comments here the OP has clarified that he seeks a proof that "does not use the properties of binomial coefficients". Please post answers in said newer thread so that this incorrec... |
[Freyd-Mitchell's embedding theorem][1] states that: if A is a small abelian category, then there exists a ring R and a full, faithful and exact functor F: A → R-Mod.
This is quite the theorem and has several useful applications (it allows one to do diagram chasing in abstract abelian categories, etc.)
I have bee... |
If any edge from a given spanning tree T0 is contained by a minimum spanning tree T*, does this imply that T0 is also a minimum spanning tree ?
Right now, I'm trying to draw on paper some graphs to prove that it doesn't. Please correct me if it does, or help me find an example if it doesn't.
Thanks in advance.
... |
If any edge from a given spanning tree T0 is contained in some minimum spanning tree T*, does this imply that T0 is also a minimum spanning tree ?
Right now, I'm trying to draw on paper some graphs to prove that it doesn't. Please correct me if it does, or help me find an example if it doesn't.
Thanks in advance.... |
### Update 1 ###
This *should* be the final edit. Lots of typos have been corrected and the presentation in the section on existence has been greatly improved. It should be very close to a form that can be used as a basis for a project for students.
### Update 2 ###
Even more typos corrected. Added example of non-... |
How do I calculate the instantaneous angular speed of a spindle given a spooled radius and required feed speed? |
"When you draw a circle in a plane of radius 1 you can perfectly surround it with 6 other circles of the same radius."
BUT when you draw a circle in a plane of radius 1 and try to perfectly surround the central circle with 7 or 8 circles you have to chance the radius of the surround circles.
how can i find the r... |
The set of rationals Q has the same cardinality as
the set of integers Z. True or false?
This was a question on an old exam for our class. The correct answer is true. However, I did some additional reading and came across Cantor's transfinite numbers. In the book I'm reading, it says that "there are more real numbe... |
The set of rationals $\mathbb{Q}$ has the same cardinality as
the set of integers $\mathbb{Z}$. True or false?
This was a question on an old exam for our class. The correct answer is true. However, I did some additional reading and came across Cantor's transfinite numbers. In the book I'm reading, it says that "the... |
Here is a very prosaic answer. If you go into a grocery store with an order, you expect the cash register to read 0 before any items are rung up. This is because $0 + x = x$ for any number $x$. It is the fact that zero is the neutral element of addition that makes it a valid start for the register; starting there wil... |
"When you draw a circle in a plane of radius 1 you can perfectly surround it with 6 other circles of the same radius."
BUT when you draw a circle in a plane of radius 1 and try to perfectly surround the central circle with 7 circles you have to change the radius of the surround circles.
how can i find the radius... |
Does there exist a nonzero linear transformation $T$ such that ${\alpha}^{T} T \alpha = 0$ for all $\alpha \in \mathbb{R^n}$? |
Does there exist a nonzero linear transformation $T$ such that ${\alpha}^{T} T \alpha = 0$ for all $\alpha \in \mathbb{R}^n$? |
If there is a field $F$ that is a field reduction of the real numbers, that is $F(a)=\mathbb{R}$ for some $a$, let's also denote this $F=\mathbb{R}(\setminus a)$, then given $x \in \mathbb{R}$ is there a general method to determine whether $x$ is in $F$ or $x$ is in $\mathbb{R}\setminus F$ ? |
I'm trying to understand a proof in "Combinatorial Group Theory" by Lyndon & Schupp, page 26, Lemma 4.7.
Here's my problem:
$F$ is a free group with $X$ as basis. $x \in X$. $\alpha \in IA(F)$ - meaning it's an automorphism of $F$ which induces the identity automorphism on $F/[F,F]$. $k \in F_n$ - a term in the d... |
I'm trying to understand a proof in "Combinatorial Group Theory" by Lyndon & Schupp, page 26, Lemma 4.7.
Here's my problem:
$F$ is a free group with $X$ as basis. $x \in X$. $\alpha \in IA(F)$ - meaning it's an automorphism of $F$ which induces the identity automorphism on $F/[F,F]$. $k \in F_n$ - a term in the d... |
How does one diagonalize a large sparse symmetric matrix to get the eigenvalues and the eigenvectors?
The problem is the matrix could be very large (though it is sparse), at most 2500*2500. Is there a good algorithm to do that and most importantly, I can implement it into my own code. Thanks a lot! |
A function $f(x)$ is defined $\forall x \in \mathbb{Q}$ and $ x \in (0,1)$ as $f(x) = q $ where $x = (p/q)$ in simplest form. Find $\sup_{\tau} \quad \min_{a} \quad f(a) + f(\tau-a)$ where $ a \in \mathbb{Q}$ and $ a \in (0,1)$ and $ \tau \in (\alpha,\beta)$ where $ 0< \alpha < \beta <2 $ and $ \alpha,\beta \in \m... |
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