instruction stringlengths 12 30k |
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For a 2D Gaussian distribution with
$$ \mu = \begin{pmatrix} \mu_x \\ \mu_y \end{pmatrix}, \quad \Sigma = \begin{pmatrix} \sigma_x^2 & \rho \sigma_x \sigma_y \\ \rho \sigma_x \sigma_y & \sigma_y^2 \end{pmatrix},
$$
its probability density function is
$$
f(x,y) = \frac{1}{2 \pi \sigma_x \sigma_y \sqrt{1-\rho^... |
Let $\hat{\mathbb{C}} = \mathbb{C} \cup \{ \infty \}$ denote the [Riemann Sphere](http://en.wikipedia.org/wiki/Riemann_sphere). While reading the wikipedia article about it I found a passage that said that [every rational function on the complex plane can be extended to a continuous function on the Riemann Sphere](http... |
How to find center of ellipse from two points(these are just points on the ellipse, not related to foci), and two radiuses(rx and ry, from standart definition of the ellipse x^2/rx^2 + y^2/ry^2=1) of that ellipse? (there will be two centers, I assume) |
I was wondering if f(x)=O(x^(c+a)) for all a>0 then is it necessarily true that f(x)=O((x^c)logx)?
I suspect it's not true but want to know why. (I know the converse is true.)
Any help is much appreciated,
Thank you. |
If f is big O of x^c+a for all a>0 then is f big O of (x^c)logx? |
I have two questions and I hope it is alright to ask them both at the same time. I'm currently trying to solve an online-exercise about differential calculus and the Taylor series and I'm having some trouble there.
> Let $f: (a,b) \rightarrow \mathbb{R}$ be differentiable at the point $x_0$. Which of the following s... |
What is your recommended symbolic computation program/software for free or commercial respectively?
What are its strength and weakness?
Thanks! |
What is your recommended symbolic computation program/software for free and commercial respectively?
What are its strength and weakness? For example, efficiency, comprehensiveness, etc
Thanks! |
John Von Neumann once said to Felix Smith, "Young man, in mathematics you don't understand things. You just get used to them." This was a response to Smith's fear about the method of characteristics.
Did he mean that with experience and practice, one obtains understanding? |
What are some interpretations of Von Neumann's quote? |
Let $X$, $Y$ be pointed CW complexes, $Y$ connected and $f:X\to Y$ a mapping.
Does the assertion '$\Sigma f:\Sigma X\to\Sigma Y$ is a homotopy equivalence' imply that $f$ is a homotopy equivalence? '$\Sigma$' is the reduced suspension.
If not, is it true with some additional hypotheses on $Y$? |
Is a map a homotopy equivalence if its suspension is so? |
Yes there are linear algebra and graph theoretic results for this game, for any arrangement of switches and starting from all off position.
Notice that you can construct a simple graph out of the arrangement of lights, by taking the lights to be vertices, and making them adjacent in the graph if they are neighbours.... |
Let's say you have a single-lane road (single in this case meaning single lane in each direction, or what you could also call two-lane). Let's say you have a random number of vehicles on a given 10-mile stretch of road. The speed limit is 45 mph, but some people go faster and some go slower.
Is there any math that d... |
In Calculus and Analysis, there is no good answer as to what $0^0$ "should" be, so it is usually left undefined.
Basically, if you consider $x^y$ as a function of two variables, then there is no limit as $(x,y)\to(0,0)$ (with $x\geq 0$): if you approach along the line $y=0$, then you get $\lim\limits_{x\to 0^+} x^0 ... |
Let $\ell^n_p$ denote $\mathbb{R}^n$ with the $p$-norm; assume $n>1$. Now it's well-known that for $p\ne q$, $\ell^n_p$ and $\ell^n_q$ are never isometric unless $n=2$ and $\{p,q\}=\{1,\infty\}$, but I realized recently I didn't actually know how to prove this.
I can prove most of it - the only case I still need to... |
(i) You are sort of on the right track, but you are trying to (1) prove too much; and (2) drawing the wrong conclusion.
If you have a bunch of linearly independent vectors, then you can *always* find (at least one) linear transformation that maps them to whatever you want. This because we can always do it with a bas... |
In Calculus and Analysis, there is no good answer as to what $0^0$ "should" be, so it is usually left undefined.
Basically, if you consider $x^y$ as a function of two variables, then there is no limit as $(x,y)\to(0,0)$ (with $x\geq 0$): if you approach along the line $y=0$, then you get $\lim\limits_{x\to 0^+} x^0 ... |
Is there an easy way to see that the $n$-dimensional Fourier transform of $1/|x|^a$ is equal to $1/|x|^{n-a}$ (up to multiplicative constant) where $x$ is an $n$ dimensional vector (assuming that $ 0 < a < n $) ?
|
Let $X$, $Y$ be pointed CW complexes, $Y$ connected and $f:X\to Y$ a mapping.
Does the assertion '$\Sigma f:\Sigma X\to\Sigma Y$ is a homotopy equivalence' imply that $f$ is a homotopy equivalence? '$\Sigma$' is the reduced suspension.
If not, is it true with some additional hypotheses on $Y$?
*Addition:*
Doe... |
In Calculus and Analysis, there is no good answer as to what $0^0$ "should" be, so it is usually left undefined.
Basically, if you consider $x^y$ as a function of two variables, then there is no limit as $(x,y)\to(0,0)$ (with $x\geq 0$): if you approach along the line $y=0$, then you get $\lim\limits_{x\to 0^+} x^0 ... |
I'm reading Serre's "Lie algebra and Lie groups" now, and I found a description of Lie algebra of an algebraic matrix group, given for a free algebra (free as a module) $k'$ with a basis $\lbrace 1, \varepsilon\rbrace$, $\varepsilon^2 = 0$ as all $X \in M(n,k)$ such that $1 + \varepsilon X \in G(k')$, where $G(k')$ is ... |
I'm reading Serre's "Lie algebra and Lie groups" now, and I found a description of Lie algebra of an algebraic matrix group, given for a free algebra (free as a module) $k'$ with a basis $\lbrace 1, \varepsilon\rbrace$, $\varepsilon^2 = 0$ as all $X \in M(n,k)$ such that $1 + \varepsilon X \in G(k')$, where $G(k')$ is ... |
$\lim\limits_{x\rightarrow 0 } \frac{x}{ln x}$.
This was wrong, I got a big red wrong! Why doesn't L'Hopital work on this one?
The problem is that ln is not defined for 0. It needs to be rewritten?
(Thanks to everyone helping me out with my homework, due anxiety I'm not able attend the class workshops.)
Edi... |
How can we prove the following trigonometric identity?
$$\displaystyle \tan(3\pi/11) + 4\sin(2\pi/11) =\sqrt{11}$$ |
Let $\ell^n_p$ denote $\mathbb{R}^n$ with the $p$-norm; assume $n>1$. Now it's well-known that for $p\ne q$, $\ell^n_p$ and $\ell^n_q$ are never isometric unless $n=2$ and $\{p,q\}=\{1,\infty\}$, but I realized recently I didn't actually know how to prove this.
I can prove most of it - the only case I still need to... |
The integral closure of a complete Noetherian local domain is also complete, local and Noetherian. So the integral closure of $R/p$ is a complete regular local ring containing a field (normal rings in dimension $1$ are regular by Serre criterion), and you win!
EDIT: add more details, by Tymothy's comments: So let $... |
How to find center of ellipse from two points(these are just points on the ellipse, not related to foci), and two radiuses (rx and ry, from standard definition of the ellipse x^2/rx^2 + y^2/ry^2=1) of that ellipse? (there will be two centers, I assume) |
Gaussian integers are the set:
$$\mathbb{Z}[\imath] =\{a+b\imath : a,b \in\mathbb{Z} \}$$
With norm:
$$\mathrm{N}(a+b\imath)=a^{2}+b^{2}$$
It satisfies $\mathrm{N}(\alpha\cdot \beta )=\mathrm{N}(\alpha)\cdot\mathrm{N}( \beta )$
The units of $\mathbb{Z}[\imath]$ are precisely: $1,-1,\imath,-\imath$
Res... |
Gaussian integers are the set:
$$\mathbb{Z}[\imath] =\{a+b\imath : a,b \in\mathbb{Z} \}$$
With norm:
$$\mathrm{N}(a+b\imath)=a^{2}+b^{2}$$
It satisfies $\mathrm{N}(\alpha\cdot \beta )=\mathrm{N}(\alpha)\cdot\mathrm{N}( \beta )$
The units of $\mathbb{Z}[\imath]$ are precisely: $1,-1,\imath,-\imath$
Res... |
Background: Let $k$ be an even integer. The Eisenstein series are defined by $E_{k} = 1 - \frac{2k}{B_{k}}\sum_{n=1}^{\infty} \sigma_{k-1}(n)q^{n}$ where $\sigma_{k-1}(n)= \sum_{d \mid n, d \geq 1} d^{k-1}$ and $B_{k}$ is the $k$-th Bernoulli number. If $k > 2$ then $E_{k}$ is a modular form of weight $k$. In the case... |
why does a certain formula hold in Lang's book on modular forms? |
I'm working through a little proof of this statement, and here is an excerpt of the problem I'm following:
> Suppose $p \equiv 1\pmod{4}$. Show that if $x^2-py^2=1$ the $x$ is odd and $y$ is even. Suppose that ${x_0}^2-p{y_0}^2=1$ with $y_0>0$, $y_0$ minimal. Show that $gcd(x_0+1,x_0-1)=2$. Deduce that one of two ca... |
Motivation
===
Suppose that $u \in \mathbb{R}^d$ is a unit-norm vector, $\|u\| = 1$, $a, b, c$ are some positive constants and $\xi \in [0,1]$ is another constant (usually chosen close to 1). I am interested in solving the following problem
$$
\sup_{v \in \mathbb{R}^d}\ (\xi + (1-\xi)\|v\|_1^2)\left(\sqrt{
\f... |
How to compute the change in the angle between two unit norm vectors as the $\ell_1$ norm of one vector changes? |
I'm working through a little proof of this statement, and here is an excerpt of the problem I'm following:
> Suppose $p \equiv 1\pmod{4}$. Show that if $x^2-py^2=1$ the $x$ is odd and $y$ is even. Suppose that ${x_0}^2-p{y_0}^2=1$ with $y_0>0$, $y_0$ minimal. Show that $gcd(x_0+1,x_0-1)=2$. Deduce that one of two ca... |
Gaussian integers are the set:
$$\mathbb{Z}[\imath] =\{a+b\imath : a,b \in\mathbb{Z} \}$$
With norm:
$$\mathrm{N}(a+b\imath)=a^{2}+b^{2}$$
It satisfies $\mathrm{N}(\alpha\cdot \beta )=\mathrm{N}(\alpha)\cdot\mathrm{N}( \beta )$
The units of $\mathbb{Z}[\imath]$ are precisely: $1,-1,\imath,-\imath$
Res... |
If someone could give answers and explain, it would be greatly appreciated. Help required studying for a final.
One hundred tickets, numbered 1,2,3,…,100, are sold to 100 different people for a drawing.
Four different prizes are awarded, including a grand prize (a trip to Tahiti).
A) How many ways are there to ... |
How do I do Combinatorics / Counting? |
The German mathematician [Martin Eichler][1] once stated that there were five fundamental operations of mathematics: addition, subtraction, multiplication, division, and modular forms. This was also mentioned in Simon Singh's book on Fermat's last theorem.
I don't know much about modular forms beyond the [Wikipedia ... |
The basic rules of counting are:
1. *Sum rule:* If one event can occur in $n$ different ways, and another, independent, event can occur in $m$ different ways, then the number of different ways in which either one or the other event can occur is $n+m$.
2. *Product rule:* If one even can occur in $n$ differen... |
Try to solve the questions one award at a time, for each award asking yourself how many people could receive this award? Note that this approach may lead to branching, question C for instance you would get 100*99*98 for the first 3 awards, but the number of persons who may receive the final award is either 1 or 97 depe... |
Gaussian integers are the set:
$$\mathbb{Z}[\imath] =\{a+b\imath : a,b \in\mathbb{Z} \}$$
With norm:
$$\mathrm{N}(a+b\imath)=a^{2}+b^{2}.$$
It satisfies $\mathrm{N}(\alpha\cdot \beta )=\mathrm{N}(\alpha)\cdot\mathrm{N}( \beta )$
The units of $\mathbb{Z}[\imath]$ are precisely: $1,-1,\imath,-\imath$
Result: ... |
why does a certain formula in Lang's book on modular forms hold? |
There is a nice proof using <a href="http://en.wikipedia.org/wiki/Minkowskis_theorem">Minkowski's theorem</a> which also proves the four-square theorem. It is Proof #2 <a href="http://www.artofproblemsolving.com/Forum/blog.php?u=14052&b=10589&">here</a>. |
The basic rules of counting are:
1. *Sum rule:* If one event can occur in $n$ different ways, and another, independent, event can occur in $m$ different ways, then the number of different ways in which either one or the other event can occur is $n+m$.
2. *Product rule:* If one event can occur in $n$ differe... |
Background: Let $k$ be an even integer. The Eisenstein series are defined by $$E_{k} = 1 - \frac{2k}{B_{k}}\sum_{n=1}^{\infty} \sigma_{k-1}(n)q^{n}$$ where
$$\sigma_{k-1}(n)= \sum\limits_{d \mid n,\;d \geq 1} d^{k-1}$$ and $B_{k}$ is the $k$-th Bernoulli number. If $k \gt 2$ then $E_{k}$ is a modular form of weight ... |
Suppose you are given a question that goes like this:
> Consider three planets that revolve
> around a star on separate circular
> orbits sharing the same orbital plane.
> The first planet, A, takes 385 days to
> complete its orbit. The second planet,
> B, takes 814 days, while the third
> planet, C, takes 167... |
Is there official name and proof for this formula/theorem (for now I'm calling it Orbital Collinearity Theorem)? |
Your question is an interesting one, but I think the reason you haven't received an answer to it yet is that it's ill-constrained. That's not a criticism but a technical term. Basically it means that the problem isn't well-defined enough for there to be a clear, distinct answer. (This is what the second part of Rahu... |
The question :
Prove that :$$ \text{ if } y = 2x^2 - 1,\text{ then } \biggl[ \frac{1}{y} + \frac{1}{3y^3} + \frac{1}{5y^5}+ \cdots \biggr]$$ is equal to $$\frac{1}{2} \biggl[ \frac{1}{x^2} + \frac{1}{2x^4} + \frac{1}{3x^6} + \cdots \biggr]$$
Here I have modified the question, in the actual question (from my paper)... |
Any tricky method to solve this one ? |
This question might turn out to be really trivial.
$f$ is a one-to-one function from the interval $[0,1]$ to $\mathbb{R}$. Is it necessary that $\exists q \in \mathbb{Q}$ such that $f(x) = q$ for some $x \in [0,1]$ i.e. is it necessary that the image of $f$ contains a rational number?
I came across this question ... |
One-to-one function from the interval $[0,1]$ to $\mathbb{R}$? |
I'm interested because I want to show that $x^2-34y^2\equiv -1\pmod{m}$ for all integers $m$. I started by using the following reasoning:
If $3\nmid m$, then $gcd(m,3)=1$. Then there exists a multiplicative inverse $\bar{3}$ modulo $m$. I note that $5^2-34=-(3^2)$, and thus $\bar{3}^2(5^2-34)\equiv (\bar{3}\cdot 5)^... |
How does the Chinese Remainder Theorem behave with polynomials with multiple variables? |
The basic rules of counting are:
1. *Sum rule:* If one event can occur in $n$ different ways, and another, independent, event can occur in $m$ different ways, then the number of different ways in which either one or the other event can occur is $n+m$.
2. *Product rule:* If one event can occur in $n$ differe... |
I'm interested because I want to show that $x^2-34y^2\equiv -1\pmod{m}$ has solutions for all integers $m$. I started by using the following reasoning:
If $3\nmid m$, then $gcd(m,3)=1$. Then there exists a multiplicative inverse $\bar{3}$ modulo $m$. I note that $5^2-34=-(3^2)$, and thus $\bar{3}^2(5^2-34)\equiv (\b... |
The basic rules of counting are:
1. *Sum rule:* If one event can occur in $n$ different ways, and another, independent, event can occur in $m$ different ways, then the number of different ways in which either one or the other event can occur is $n+m$.
2. *Product rule:* If one event can occur in $n$ differe... |
One-to-one function from the interval $[0,1]$ to $\mathbb{R}\setminus\mathbb{Q}$? |
This question might turn out to be really trivial.
$f$ is a one-to-one function from the interval $[0,1]$ to $\mathbb{R}$. Is it necessary that $\exists q \in \mathbb{Q}$ such that $f(x) = q$ for some $x \in [0,1]$ i.e. is it necessary that the image of $f$ contains a rational number?
I came across this question ... |
I've got a multiplicative-with-noise model $F(x,y)=S(x)*R(y)*D(x,y)+N$, where $S(x)$ and $R(y)$ are unknown functions, $D(x,y)$ is a distance function, that is, a function that depends only on $|x-y|$ and decreases quickly when distance increases. $N$ is an uncorrelated "small random noise" function. All of the functi... |
How to decompose a result of a multiplication? |
Is there a proof in the spirit of Euclid to prove Dirichlet's theorem on primes in arithmetic progression? (By the spirit of Euclid, I mean assuming finite number of primes we try to construct another number which has a prime factor which falls in the same equivalence class as the other primes but the number is not div... |
Suppose that a in the plane a given conic curve is compelled to pass through two fixed points of that plane.
What are the curves covered by a fixed point of the conic, its center (for an ellipse), its focus, etc ?
(I apologize for the bad english ...) |
Suppose $T_1, T_2 : l^2(Z_N)-> l^2(Z_N)$ are translation invariant linear transformation. How to prove that the composition $T_2 o T_1$ is translation invariant?
Recall the translation operator $R_k$ defined by
$(R_k z)(n)=z(n-k) $
Also recall definition:
Let $T : l^2(Z_N)-> l^2(Z_N)$ be a linear transformation. $... |
How to prove that the composition $T_2 o T_1$ is translation invariant? |
I ask this in a partly recreational, and partly research-related spirit, and I realize my problem might be ill-posed, so any suggestions for clarification might go a long way.
Succinctly, my problem can be stated as
> Find the period of a "black box"
> pseudorandom number generator
but I suppose the following... |
How to prove that convolution is associative? |
I ask this in a partly recreational, and partly research-related spirit, and I realize my problem might be ill-posed, so any suggestions for clarification might go a long way.
Succinctly, my problem can be stated as
> Find the period of a "black box"
> pseudorandom number generator
but I suppose the following... |
Suppose $b_1, b_2 \in l^2(Z_N)$. How to prove that the composition $T_{b_2} o T_{b_1} $ of the convolution operators $T_{b_2}$ and $T_{b_1}$ is the convolution operator $T_b$ with $b=b_2 * b_1$? Hint use previous question where I asked: How to prove that convolution is associative? and also
$b_2 * b_1(m) = \sum_{n=0}... |
How to prove that the composition $T_{b_2} o T_{b_1} $ of the convolution operators $T_{b_2}$ and $T_{b_1}$ is the convolution operator $T_b$ ? |
how to find bounds of the series? |
Given two vectors $u, v \in \mathbb{R}^d$ represented the spherical coordinates is there a simple formula to compute the angle between the two vectors? Without loss of generality, we can assume that the vectors $u$ and $v$ have unit norm.
One way to find the angle is to represent the two vectors in the Euclidean coo... |
How to compute the angle between two vectors expressed in the spherical coordinates? |
Let's say you have a single-lane road (single in this case meaning single lane in each direction, or what you could also call two-lane). Let's say you have a random number of vehicles on a given 10-mile stretch of road. The speed limit is 45 mph, but some people go faster and some go slower.
Is there any math that d... |
IS the values generated by non-linear equations truly random ? |
Program for working with DFA/NFA/PDA ? |
Are the values generated by non-linear equations truly random ? |
Suppose $b_1, b_2 \in l^2(Z_N)$. How to prove that the composition $T_{b_2} \circ T_{b_1} $ of the convolution operators $T_{b_2}$ and $T_{b_1}$ is the convolution operator $T_b$ with $b=b_2 \cdot b_1$? Hint use previous question where I asked: How to prove that convolution is associative? and also
$b_2 \cdot b_1(m) ... |
How to prove that the composition $T_{b_2} \circ T_{b_1} $ of the convolution operators $T_{b_2}$ and $T_{b_1}$ is the convolution operator $T_b$ ? |
Suppose $T_1, T_2 : \ell^2(Z_N)-> \ell^2(Z_N)$ are translation invariant linear transformation. How to prove that the composition $T_2 \circ T_1$ is translation invariant?
Recall the translation operator $R_k$ defined by
$(R_k z)(n)=z(n-k) $
Also recall definition:
Let $T : \ell^2(Z_N)-> \ell^2(Z_N)$ be a linear t... |
How to prove that the composition $T_2 \circ T_1$ is translation invariant? |
how to find bounds of the series $$\sum_{i=1}^n \frac1{4i-1}$$ ? |
It is known that for any locally compact Hausdorff space X, we can define a Hausdorff one-point compactification containing X.
In the case of the (differentiable) manifold $\mathbb R^n$ this one-point compactification turns out to be (homeomorphic to) $\mathbb S^n$, which is again a (differentiable) manifold.
This ... |
I've come across a problem with the size of coefficients of a series expansion. Here's an example of an expression that I expand into a power series:
$\displaystyle\frac{1-e^{i3n\cdot t}}{1-e^{i\cdot n\cdot t}}$ where $n \in \mathbb{N}$
The problem is that I'd like to work with small coefficient values, but the ... |
Do the Mobius function, totient function, sum of divisors and number of divisors uniquely specify a number? |
I've come across a problem with the size of coefficients of a series expansion. Here's an example of an expression that I expand into a power series:
$\displaystyle\frac{1-e^{i3n\cdot t}}{1-e^{i\cdot n\cdot t}}$ where $n \in \mathbb{N}$ and $t \in \mathbb{R}$
The problem is that I'd like to work with small coeff... |
I'm searching for a Statistical method to determine if a player is cheating in an online game.
Given a number of positive points and a number of negative points per player (score) Given n players (n <= 64). Each player can cheat independant of each other. I used stddev, which works great on a single player but fai... |
I'm searching for a statistical method to determine if a player is cheating in an online game.
Given a number of positive points and a number of negative points per player (score) and given $n$ players ($n <= 64$). Each player can cheat independent of each other. I used standard deviation, which works great on a s... |
I would be extremely surprised if there existed a translation or a LaTeX version. I have a searchable pdf of it though. Annoying Megaupload link: http://www.megaupload.com/?d=F6GBYUBU |
Let $s = (\frac{1}{\sqrt{d}}, \ldots, \frac{1}{\sqrt{d}})$ and $u \in \mathbb{R}^d$ be two distinct unit norm vectors in the first orthant. Consider moving along the great circle defined by $s$ and $u$ (in the direction from $s$ to $u$) until the first intersection of the great circle and one of the following hyperplan... |
How to find the intersection between the great circle and a hypeplane? |
I just want to make sure that I understand these two objects correctly.
Let $R$ be a ring. A subset $R'\subseteq R$ is a subring provided $R'$ is an additive subgroup that is closed with respect to multiplication.
A subset $I\subseteq R$ is an ideal provided $I$ is an additive subgroup such that $rI\subseteq I$ ... |
Suppose you have two options for making a payment:
(A) Pay $90 \%$ of the purchase price two months after the date of the sale.
(B) Deduct $X \%$ off the purchase price and pay cash on the date of sale.
Determine $X$ so that a customer is indifferent between the two options when valuing them using an effecti... |
How to find the intersection between the great circle and a hyperplane? |
How to prove that a power series in more than one variable is invertible?
Is this still true if the variables are noncommutative?
|
I'm hoping that this isn't such a basic question that it gets completely laughed off the site, but why would I want to multiply two polynomials together?
I flipped through some algebra books and have googled around a bit, and whenever they introduce polynomial multiplication they just say 'Suppose you have two poly... |
Why would I want to multiply two polynomials? |
What is the order of the set of distinct (up to similarity) nxn matrices over $\mathbb{R}$ with determinant equal to some non-zero scalar... say 6? (eg. countable, uncountable etc.)
The set of matrices over $\mathbb{R}$ is uncountable. So is the order of the set consisting of classes of matrices with the same determ... |
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