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A question from Dugundji's book which I don't even understand the statement.
Describe the relative topology of $\{z: |z|=1\}$ as a subspace of ${\mathbb{R}}^{2}$.
What do they mean by "describe"? I don't understand what they are asking for. Can you please clarify? |
1/8(3m-5)+6/5(6m-2)+5...Type your answer as a fraction. Do not use decimals!
**i got 129/273 is this correct ? |
Two questions from Dugundji's book (not hw, just practice).
1) Let $Y_{1}, Y_{2}$ be subspaces of $X$ and $A \subset Y_{1} \cup Y_{2}$. Assume that $A$ is open in $Y_{1}$ and open in $Y_{2}$. Prove A is open in $Y_{1} \cup Y_{2}$.
Can you please give a hint for this one?
2) a. Let $D$ be dense in $X$. Give an ... |
First off: I barely have any set theoretic knowledge, but I read a bit about cardinal arithmetic today and the following idea came to me, and since I found it kind of funny, I wanted to know a bit more about it.
If $A$ is the set of all real positive sequences that either converge to $0$ or diverge to $\infty$, we p... |
"Homomorphism" from set of sequences to cardinals? |
Two questions from Dugundji's book (not hw, just practice).
1) Let $Y_{1}, Y_{2}$ be subspaces of $X$ and $A \subset Y_{1} \cap Y_{2}$. Assume that $A$ is open in $Y_{1}$ and open in $Y_{2}$. Prove A is open in $Y_{1} \cap Y_{2}$.
Can you please give a hint for this one?
2) a. Let $D$ be dense in $X$. Give an ... |
1/8 (3m - 5) = 6/5 (6m - 2) + 5...Type your answer as a fraction. Do not use decimals!
**i got 129/273 is this correct ? |
(1/8)(3m - 5) = (6/5)(6m - 2) + 5...Type your answer as a fraction. Do not use decimals!
**i got 129/273 is this correct ? |
Two questions from Dugundji's book (not hw, just practice).
1) Let $Y_{1}, Y_{2}$ be subspaces of $X$ and $A \subset Y_{1} \cap Y_{2}$. Assume that $A$ is open in $Y_{1}$ and open in $Y_{2}$. Prove A is open in $Y_{1} \cup Y_{2}$.
Can you please give a hint for this one?
2) a. Let $D$ be dense in $X$. Give an ... |
This [is a similar question][1] to this one, but slightly different.
The question is given two edges ($e_1$ and $e_2$, with the vertex coordinates known), how to find the angles from $e_1$ to $e_2$, with the angles measured in anti clock wise direction?
A diagram is shown below:
<img src="https://docs.google.... |
(1/8)(3m - 5) = (6/5)(6m - 2) + 5...Type your answer as a fraction. Do not use decimals!
**i got -129/273 is this correct ? |
Could you a give a geometrical interpretation of curl?
Thanks. |
Could you a give a geometrical interpretation of curl? |
I think that there are a handful of points that might need clarification here.
- There is no "set of all infinite cardinals", the family of all infinite cardinals is a proper class, and if it were a set it would imply the [Burali-Forti paradox][1]
- Secondly, consider the meaning of this equivalence relation: $a\... |
Can the number of solutions $xy(x-y-1)=n$ for $x,y,n \in Z$ be unbounded as n varies?
x,y are integral points on an Elliptic Curve and are easy to find using enumeration of divisors of n (assuming n can be factored).
If yes, will large number of solutions give moderate rank EC?
If one drops $-1$ i.e. $xy(x-y)=... |
Could you a give a intuitive interpretation of curl, geometrical interpretation, real-world example or physical interpretation would be ok.
EDIT: Consider a specific vector field
$$\mathbf{v} = -y\mathbf{\hat{x}} + x\mathbf{\hat{y}}$$
$$\nabla\times\mathbf{v} = 2\mathbf{\hat{z}}$$
Could you give a intuitive inte... |
Could you a give a intutive interpretation of curl? |
Can the number of solutions $xy(x-y-1)=n$ for $x,y,n \in Z$ be unbounded as n varies?
x,y are integral points on an Elliptic Curve and are easy to find using enumeration of divisors of n (assuming n can be factored).
If yes, will large number of solutions give moderate rank EC?
If one drops $-1$ i.e. $xy(x-y)=... |
How do I find generators of group of rotations of d-dimensional cube into itself when viewed as a subgroup of symmetric group of order $2^d$?
This is related to earlier <a href="http://math.stackexchange.com/questions/15022/how-to-get-generators-for-a-finite-point-group">question</a> but not specializing to $d=3$ |
How to get group of hypercube rotations as a subgroup of symmetric group? |
Terence Tao has set up a "polymath project" which addresses exactly your question. Here is the relevant page:
http://michaelnielsen.org/polymath1/index.php?title=Finding_primes
It contains a list of conjectures, partial results and references. To sum it all up: At the moment the answer to your question is No.
... |
How do I find generators of group of rotations of d-dimensional cube into itself when viewed as a subgroup of symmetric group of order $2^d$?
This is related to my previous <a href="http://math.stackexchange.com/questions/15022/how-to-get-generators-for-a-finite-point-group">question</a> but not specializing to $d=3... |
I have two numbers _x_ and _y_ which are $log_2$ of two other numbers _P_ and _V_. I'm trying to calculate $log_2 P-V$ without transforming _x_ and _y_ back to the linear world.
If _P = AV_, then _P-V = V(A-1)_, and $x-y = \log_2 A$. So, if I could calculate $\log_2 A-1$ I could just add it to _y_ and be done. Is... |
Given $log_2 A$, is there an easy way to calculate $log_2 A-1$? |
Im currently creating a program to plot simple graphs in 2D. In order to draw those properly i have to know if a function graph is continuous between 2 points on the graph.
Assuming the graph for f(x)=tan(x) and two points on the graph P1(1,tan(1)) and P2(2,tan(2)). Is there an algorithmic way to tell if the graph i... |
I have two numbers _x_ and _y_ which are $log_2$ of two other numbers _P_ and _V_. I'm trying to calculate $log_2 (P-V)$ without transforming _x_ and _y_ back to the linear world.
If _P = AV_, then _P-V = V(A-1)_, and $x-y = \log_2 A$. So, if I could calculate $\log_2 (A-1)$ I could just add it to _y_ and be done.... |
Given $log_2 A$, is there an easy way to calculate $log_2 (A-1)$? |
I have two numbers $x$ and $y$ which are $\log_2$ of two other numbers $P$ and $V$. I'm trying to calculate $\log_2 (P-V)$ without transforming $x$ and $y$ back to the linear world.
If $P = AV$, then $P-V = V(A-1)$, and $x-y = \log_2 A$. So, if I could calculate $\log_2 (A-1)$ I could just add it to $y$ and be don... |
Given $\log_2 A$, is there an easy way to calculate $\log_2 (A-1)$? |
I'm currently creating a program to plot simple graphs in 2D. In order to draw those properly I have to know if a function graph is continuous between 2 points on the graph.
Assuming the graph for f(x)=tan(x) and two points on the graph P1(1,tan(1)) and P2(2,tan(2)). Is there an algorithmic way to tell if the graph ... |
For example, I have set of points in 3D. Points lie on straight line. Transformed set of points lies on straight line too. How to check if transformation is affine? |
How to check if transformation is affine? |
According to the fundamental theorem of affine geometry, it suffices to check that the transformation is a bijective *collineation*:
>**Theorem.** Let $\mathbb A^{n}$ be an affine space over $\mathbb R$ with $n > 2$ and fix
$a \in A$. Let $\phi :\mathbb A^{n}\to \mathbb A^{n}$ be a bijection which takes each three co... |
According to the fundamental theorem of affine geometry, it suffices to check that the transformation is a bijective *collineation*:
>**Theorem.** Let $\mathbb A^{n}$ be an affine space over $\mathbb R$ with $n > 2$ and fix
$a \in A$. Let $\phi :\mathbb A^{n}\to \mathbb A^{n}$ be a bijection which takes each three co... |
Can the number of solutions $xy(x-y-1)=n$ for $x,y,n \in Z$ be unbounded as n varies?
x,y are integral points on an Elliptic Curve and are easy to find using enumeration of divisors of n (assuming n can be factored).
If yes, will large number of solutions give moderate rank EC?
If one drops $-1$ i.e. $xy(x-y)=... |
[Crossposted on Mathoverflow.][1]
Given a natural number $a$, are there infinitely many natural numbers <i>not</i> of the form $anm \pm m \pm n$, $n, m$ positive natural?
I give a proof that for $a=6$ the question is equivalent to the twin prime conjecture so it is known that we don't have any proof. But what abo... |
Let $A,B \text{ and } C$ are three sets then if $ A \subset B, B \subset C, C \subset A \Rightarrow B = C $
How could we prove this axiom? |
Let $A,B \text{ and } C$ are three sets then if $ A \subset B, B \subset C, C \subset A \Rightarrow B = C $
How could we prove this ? |
i have P1(x1, y1, z1) and P2(x2, y2, z2) that lies on a line, L,
and another point P0(x0, y0, z0).
i want to fine the distance between P0 and L..
some helps? |
let $X$ be some Banach space. Let $L(X)$ be the set of continuous operators on $X$ into $X$.
Let $(\tau_i)_i$ be a set of topologies on $L(X)$ s.t. $L(X)$ is topological vector space (i.e. addition and scalar multiplication continuous, not necessarily Hausdorff).
Let $\tau$ be the initial topology on $L(X)$ induc... |
Apologies if the title is unclear, I couldn't think of a good way to express what I want.
Given the usual 2D representation of the complete graph $K_n$ as a regular $n$-gon with each pair of vertices joined, let $U_n$ be the set of the disjoint regions of space delimited by the edges of the graph (so $|U_3|=1, |U_4|... |
A while back, I wanted to see if the notion of the [arithmetic-geometric mean][1] could be extended to a pair of symmetric positive definite matrices. (I considered SPD matrices only since the notion of the matrix square root is a bit intricate for other kinds of matrices.)
I expected that some complications would a... |
i have $P_1(x_1, y_1, z_1)$ and $P_2(x_2, y_2, z_2)$ that lies on a line, $L$,
and another point $P_0(x_0, y_0, z_0)$.
i want to fine the distance between $P_0$ and $L$.
some helps? |
In the definition of the complexity class NP, we allow the certificate to be polynomial in the input size. But do we ever need a certificate that is longer than the input?
In all the NP problems I could think of, we don't. The "worst" problem was factorization, where the obvious incoding is $n+O(\lg(n))$ (where n i... |
Three fellows accused of stealing CDs make the following statements:
(1) Ed: “Fred did it, and Ted is innocent.”
(2) Fred: “If Ed is guilty, then so is Ted.”
(3) Ted: “I’m innocent, but at least one of the others is guilty.”
If the innocent told the truth and the guilty lied, who is guilty?
(Remember that false ... |
What is your recommendation for an in-depth introductory combinatoric book? A book that doesn't just tell you about the multiplication principle, but rather shows the whole logic behind the questions with full proofs. The book should be for a first-year-student in college. Do you know a good book on the subject?
Tha... |
Please forgive me if this is not the right Stack Exchange (I also posted it at <a href="http://stats.stackexchange.com/questions/5698/can-the-mutual-information-of-a-cell-be-negative">Cross Validated</a> and <a href="http://cstheory.stackexchange.com/questions/3939/can-the-mutual-information-of-a-cell-be-negative">Theo... |
Can the mutual information of a "cell" be negative? |
Do we have a proof of the infiniteness? |
Three fellows accused of stealing CDs make the following statements:
(1) Ed: “Fred did it, and Ted is innocent.”
(2) Fred: “If Ed is guilty, then so is Ted.”
(3) Ted: “I’m innocent, but at least one of the others is guilty.”
If the innocent told the truth and the guilty lied, who is guilty?
(Remember that ... |
Are there any good algorithms which can be used to construct a prime greater than $n$, for arbitrary $n$?
There are some brute force approaches: for example, factoring $n!+1$. However, I'm looking for something significantly more efficient; preferably, polynomial in $n$.
Also, if anyone knows of a proof showing a... |
When the roulette has hit 5 reds why shouldn't I bet to black? |
Greetings,
I'm performing research that will help determine the size of observed space and the time elapsed since the big bang. Hopefully you can help!
I have data conforming to a piecewise linear function on which I want to perform two linear regressions. There is a point at which the slope and intercept change,... |
Why is it so unlikely to get 6 reds in a row? Well, you have to cross your fingers and say "Hope the first one's red!" And if you get lucky and that one comes up red, then you have to cross your fingers and say "Hope the second one's red!" And if you get really lucky and that one also comes up red, then you have to ... |
Why is it so unlikely to get 6 reds in a row? Well, you have to cross your fingers and say "Hope the first one's red!" And if you get lucky and that one comes up red, then you have to cross your fingers and say "Hope the second one's red!" And if you get really lucky and that one also comes up red, then you have to ... |
I'm sorry that I'm putting this in an answer rather than a comment, but I'm too new to comment on your question. My thoughts for proving this process converges (question 1.) were along the lines of:
1. $A\,\&\,B$ both SPD $\implies AB$ and $A+B$ are also SPD
2. If $A$ and $B$ are symmetric $n\times n$ matrices, $... |
I was trying to write a poker program yesterday, and trying to figure out a way to tell the computer how to detect a straight. I realized that if you add 5 consecutive numbers and divide by 5, you'll get no remainder. I tried for 6 consecutive numbers, but it didn't work. It turns out it only works for odd numbers. Rea... |
I have asked this question on [cstheory][1].
Let $\Pi$ be NP-complete problem.
Can we partition the set of instances of $\Pi$ into finite number of subsets (subproblems) each of which is polynomially solvable (and not necessarily polynomially recognizable)?
On cstheory I have obtained two answers on my questio... |
Can we partition NP-complete problem into finite number of polynomially solvable problems? |
let $X$ be some Banach space. Let $L(X)$ be the set of continuous operators on $X$ into $X$.
Let $(\tau_i)_i$ be a set of topologies on $L(X)$ s.t. $L(X)$ is topological vector space (i.e. addition and scalar multiplication continuous, not necessarily Hausdorff).
Let $\tau$ be the initial topology on $L(X)$ induc... |
I have asked this question on [cstheory][1].
Let $\Pi$ be NP-complete problem.
Can we partition the set of instances of $\Pi$ into finite number of subsets (subproblems) each of which is polynomially solvable (and not necessarily polynomially recognizable)?
On cstheory I have obtained two answers on my questio... |
Suppose we have a square real $n\times n$ matrix $X=[x_1,...,x_n]$, where $x_i$ is $i$-th column of the matrix.
Now define $X_k=[x_1,..,x_k]$, i.e. matrix $X_k$ columns are the first $k$ columns of the matrix $X$. Define $\lambda_k$ as the maximal eigenvalue of $(X_k^TX_k)^{-1}$.
Is it possible to prove that $... |
Do the maximal eigenvalues of $(X^TX)^{-1}$ matrix increase when number of columns of $X$ increases? |
Suppose you have a double sequence $a_{nm}$. What are sufficient conditions for you to be able to say that $\lim_{n\to \infty}\lim_{m\to \infty}{a_{nm}} = \lim_{m\to \infty}\lim_{n\to \infty}{a_{nm}}$? Bonus points for necessary and sufficient conditions.
For an example of a sequence where this is not the case, co... |
When can you switch the order of limits? |
Suppose you have a double sequence $\displaystyle a_{nm}$. What are sufficient conditions for you to be able to say that $\displaystyle \lim_{n\to \infty}\lim_{m\to \infty}{a_{nm}} = \lim_{m\to \infty}\lim_{n\to \infty}{a_{nm}}$? Bonus points for necessary and sufficient conditions.
For an example of a sequence wh... |
Wikipedia [says][1] the midpoint formula for numerical integration has error of order $h^3 f''(\xi)$. I am trying to replicate this result.
I'm guessing that I want to use Lagrange's formulation of the remainder for Taylor series. Let $x_0=\frac{a+b}{2}$ (i.e. the midpoint). I want to find:
$ \begin{aligned}
err... |
Wikipedia [says][1] the midpoint formula for numerical integration has error of order $h^3 f''(\xi)$. I am trying to replicate this result.
I'm guessing that I want to use Lagrange's formulation of the remainder for Taylor series. Let $x_0=\frac{a+b}{2}$ (i.e. the midpoint). I want to find:
$ \begin{aligned}
err... |
Suppose $w(z)$ is an analytic function in a domain $G$ which is symmetric with respect to the real axis. Show that $f(z) = \overline{w(\bar{z})}$ is then an analytic function of $z$ in $G$.
So this is interesting because we are looking at the conjugate of a function of a conjugate. Let $w(z) = u(x)+iv(y)$. Then $f(... |
Wikipedia [says][1] the midpoint formula for numerical integration has error of order $h^3 f''(\xi)$. I am trying to replicate this result.
I'm guessing that I want to use Lagrange's formulation of the remainder for Taylor series. Let $x_0=\frac{a+b}{2}$ (i.e. the midpoint).
The midpoint method says $\int_a^b f(... |
Suppose $w(z)$ is an analytic function in a domain $G$ which is symmetric with respect to the real axis. Show that $f(z) = \overline{w(\bar{z})}$ is then an analytic function of $z$ in $G$.
So this is interesting because we are looking at the conjugate of a function of a conjugate. Let $w(z) = u(x)+iv(y)$. Then $f(... |
Suppose you have a double sequence $\displaystyle a_{nm}$. What are sufficient conditions for you to be able to say that $\displaystyle \lim_{n\to \infty}\,\lim_{m\to \infty}{a_{nm}} = \lim_{m\to \infty}\,\lim_{n\to \infty}{a_{nm}}$? Bonus points for necessary and sufficient conditions.
For an example of a sequenc... |
Suppose $\Omega$ is a bounded domain in the plane whose boundary consist of $m+1$ disjoint analytic simple closed curves.
Let $f$ be holomorphic and nonconstant on a neighborhood of the closure of $\Omega$ such that
$$|f(z)|=1$$ for all $z$ in the boundary of $\Omega$.
If $m=0$, then the maximum principle appl... |
I read somewhere that a set is infinite if and only if it has a proper infinite subset. I also remember seeing someones name attached to this theorem on Wikipedia once, but I can't even find that now. I haven't been able to find a proof of this theorem, nor been able to generate one myself.
I can prove that if a... |
Suppose $w(z)$ is an analytic function in a domain $G$ which is symmetric with respect to the real axis. Show that $f(z) = \overline{w(\bar{z})}$ is then an analytic function of $z$ in $G$.
So this is interesting because we are looking at the conjugate of a function of a conjugate. Let $w(z) = u(x,y)+iv(x,y)$. Then... |
These are questions given for an assignment and I REALLY need to get a good grade so please could you guys check over them and give some pointers where I've gone wrong?
thanks.
ed: I hope it's ok to overwrite the original post:
$$\sum_0^\infty {\frac{(2x-8)^{n}}{4^{n}(n^{2}+2)}}$$
\begin{align}
a_n &= ... |
Doing a little reading over the break (The Probabilistic Method by Alon and Spencer); can't come up with the solution for this seemingly simple (and perhaps even a little surprising?) result:
(A-S 1.6.3) Prove that for every two independent, identically distributed real random variables X and Y,
$Pr[|X-Y| \leq 2]... |
Are there any analogues of Galois Theory for complex numbers (with non-zero imaginary part)? This is motivated by [Schwartz Principle][1]. It says that if $f$ is analytic, $f$ is defined in the upper-half disk, and $f$ extends to a continuous function on the real axis, then $f$ can be extended to an analytic function o... |
I am under the impression that the standard convention for the homology (singular) of the empty set is 0 in all non negative degrees and $\mathbb{Z}$ in degree $-1$. I have no problem with this convention, I am just curious what role it plays. Most conventions helps something sensible remain true in a particular case, ... |
This is really a continuation of BlueRaja's answer, where I'm changing the question slightly to give another illustration of the underlying point. Consider the following two procedures:
Procedure A: I flip two coins, a nickel and a penny. After the flip, I
pick one at random, and report its value to you.
Proce... |
for an university assignment i have to visualize some curves in 3d space.
till now i've implemented bezier, helix and conical spiral.
some advice about some famous curves in geometry history? |
famous curves in geometry history? |
For improper single integrals with positive integrands of the first, second or mixed type there are the comparison and the limit tests to determine their convergence or divergence. There is also the absolute convergence theorem.
For multiple integrals I know that the comparison test can be used as well.
**Qu... |
Are there convergence tests for improper multiple integrals? |
How about [Viviani's curve][1]? It is the intersection of a sphere with a cylinder. Viviani was a student (or disciple) of Gallileo.
Wikipedia has a beautiful image:
<img src="http://upload.wikimedia.org/wikipedia/commons/thumb/0/05/Viviani_curve.png/200px-Viviani_curve.png" width="300" />
[1]: http://en.wiki... |
For an university assignment I have to visualize some curves in 3 dimensional space.
Until now I've implemented bezier, helix and conical spiral.
Could you give me some advice about some famous curves in geometry history? |
For improper single integrals with positive integrands of the first, second or mixed type there are the comparison and the limit tests to determine their convergence or divergence. There is also the absolute convergence theorem.
For multiple integrals I know that the comparison test can be used as well.
**Qu... |
A while back, I wanted to see if the notion of the [arithmetic-geometric mean][1] could be extended to a pair of symmetric positive definite matrices. (I considered SPD matrices only since the notion of the matrix square root is a bit intricate for other kinds of matrices.)
I expected that some complications would a... |
Which are the convergence tests for improper multiple integrals? |
A while back, I wanted to see if the notion of the [arithmetic-geometric mean][1] could be extended to a pair of symmetric positive definite matrices. (I considered positive definite matrices only since the notion of the matrix square root is a bit intricate for other kinds of matrices.)
I expected that some complic... |
Suppose there is a birthday party to which all of the girls (and none of the boys) in a small town are invited. If you run into a mother who has dropped off a kid at this birthday party and who has two children, the chance that she has two girls is $1/3$. Why? $3/4$ of the mothers with two children will be at the birth... |
How to prove that the Sorgenfrey line is hereditarily separable? |
I'm going through Enderton's Elements of Set Theory, as I heard it is a gentle introduction to set theory. I'm a little confused on how the subset axioms are used. In the text, the axiom is given as:
> **Subset Axioms** For each formula $\underline{\hspace{1cm}}$ not containing $B$, the following is an axiom:
$$
\... |
Almost everyone is familiar with the famous Taylor Series:
$ f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x-a)^n $
which, if it converges at more than one point, will converge in some interval about $a$. Has anyone considered the "Reverse" Taylor Series:
$ g(x) = \sum_{n=0}^\infty \frac{f^{(n)}(x-a)}{n!} ... |
I have an engineering background but am interested in learning abstract harmonic analysis. From my readings so far, it's become clear to me that this will require a significant but interesting sojourn into a wide swathe of mathematics. My background in classical analysis, linear algebra, and probability is rather good ... |
Fundamental Math Theory Resources? |
It's famously unknown whether the natural log of 2 is rational or not.
How about the natural log of other numbers. Is it known/unknown whether these are rational?
Obviously ln(1) is 0, and ln(2^n) is n*ln(2) (and is thus rational iff ln(2) is rational), but how about other cases?
|
Is the natural log of n rational? |
Let $T: E \rightarrow E$ be an endomorphism of a finite-dimensional vector space, and let $S$ be a circle in the complex plane that does not intersect any eigenvalues of $T$. Now let $Q = \frac{1}{2\pi i} \int_S (z-T)^{-1} \, dz$.
Why is $Q$ a projection operator?
The motivation behind this question is that the ... |
why is the following thing a projection operator? |
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