Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Find an orthonormal basis for a given basis and give its orthonormal projection The given basis of the vector space $V$ is $B=\{(1,1,1,1)^T,(1,0,1,0)^T,(2,1,1,2)^T\}$. Find the orthnormal basis $W=\{w_1,w_2,w_3\}$. I apply Gramm-Schmidt as follows:
Let $w_1=v_1$. Then $w_2=(1,0,1,0)^T - \frac{(1,0,1,0)^T\cdot (1,1,1,1)... | Close. You neglected to normalize the result of each iteration of the Gram-Schmidt process. It happens that two of the vectors that it generated were unit vectors, anyway, but you usually won’t be that lucky. Start with $w_1=v_1/\|v_1\|$ and normalize the output at each stage. You won’t have to divide by $w_i\cdot w_i$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2838511",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Basic math subjects book recommendations I've been wondering about what are the best books to review some of the elementary math subjects. My choices are listed below:
Real Analysis: Introduction to Real Analysis, R.G. Bartle
Multivariate Calculus: Vector Calculus, J.E. Marsden
Linear Algebra: Linear Algebra and Its Ap... | Introduction to Real Analysis by Robert Bartle
Multivariate Calculus by James Hurley
Differential Equations by Shepley Ross
Complex Fundamentals of Complex Analysis for Mathematics, Science And Engineering (2nd Edition)
by E. B. Saff (Author), A. D. Snider (Author)
Introduction to Linear Algebra (Gilbert Strang)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2838667",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Sampling with and without replacement; Overcounting I am really having a lot of trouble with counting questions in my intro Probability course. An issue I am having now:
• How many ways are there to split a dozen people into 3 teams, where one team has 2 people, and the other two teams have 5 people each?
For this ques... | For the second problem, it may be easier to find the probability that there is no conflict in the student's schedule, and then subtract from 1.
For the first class there is no possible conflict, so the probability of no conflict is 1. For the second class, there is one bad day, so the probability of no conflict is $9/... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2838805",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Markov kernel intution The usual definition of a Markov kernel (as for example the Wikipedia definition of a Markov kernel) introduces it as a map from the product space of a set (equipped with a sigma algebra) and another sigma algebra to the closed real unit interval. The common way this concept is thaught is by desc... | You should think about a Markov kernel as a non-deterministic generalized map from one space to another, that instead of assigning any element on the first space with an element on the second, it assigns any element in the first space with a probability measure on the second space, so it actually spreads each element o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2838963",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Connection between ideals in $R$ and those in $R[x]$
Let $R$ be a commutative ring, $I\subset R$ an ideal. Then $I[x]$ (all
polynomials with coefficients in $I$) is an ideal of $R[x]$.
Prove or disprove:
*
*$I$ maximal $\implies I[x]$ maximal;
*$I$ prime $\implies I[x]$ prime.
Here is a counterexampl... | We want to show that $I[x]$ is prime. For this, you want to show that $R[x]/I[x]$ is an integral domain. But $R[x]/I[x]$ is isomorphic to $(R/I)[x]$ which is an integral domain since $R/I$ is an integral domain. So you are done.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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What could be the minimum and maximum value of angle in a triangle? Sum of angles of a triangle is 180 degress. So while studying trigonometric ratios , I got surprised at cos0 and cos 180 values. Although cosine is ratio of adjecent and hypotenuse ,in case of cos0 with value 1 or cos180 with value 0, i doubt that how... | In a sense, there are two different notions of trigonometric functions — although they do agree with each other on their common domain, so to speak.
One concept is that of trigonometric functions of an acute angle in a right triangle. This definition ONLY makes sense for angles $0^{\circ}<\theta<90^{\circ}$, or $0<\the... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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show that the sequence {$b_n$}, where $b_n = ( 1 + \frac {x}{n})^{l+n}$ for $n \in \mathbb{N}$, is strictly decreasing. suppose that $x > 0$, $l \in \mathbb{N}$ and $l > x.$ show that the sequence {$b_n$}, where $b_n = ( 1 + \frac {x}{n})^{l+n}$ for $n \in \mathbb{N}$, is strictly decreasing.
My attempts : i ... | $l>x\,$ , $\,n>0\,$ , $\,\displaystyle (1+\frac{x}{n})^n<e^x\,$ :
$\displaystyle \frac{d}{dn}\ln b_n = \frac{1}{n}\left(\ln \left((1+\frac{x}{n})^n\right)-\frac{x(l+n)}{n+x}\right) <
\frac{x}{n}\left(1-\frac{n+l}{n+x}\right)<0$
$ \ln b_n $ is strictly decreasing therefore $\, b_n>0\,$ is strictly decreasing too:
$\d... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2839402",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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What conditions would guarantee $\Psi: \mathbb{R}^n \rightarrow \mathbb{R}^2$ to be non-zero in a small neighbourhood? Suppose I have $\Psi: \mathbb{R}^n \rightarrow \mathbb{R}^2$ where
$\Psi(\mathbf{0}) = \mathbf{0}$. I would like to show that there exists a small open set $U$ around $\mathbf{0}$ such that it is non-... | If $n\leq2$ and ${\rm rank}\bigl(d\Psi({\bf 0})\bigr)=n$ then $\Psi$ is injective in a neighborhood of ${\bf 0}\in{\mathbb R}^n$, by the inverse function theorem.
If $n\geq3$ and ${\rm rank}\bigl(d\Psi({\bf 0})\bigr)=2$ then $\Psi^{-1}({\bf 0})$ is an $(n-2)$-dimensional submanifold of ${\mathbb R}^n$, by the implicit ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2839521",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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A closed-form expression for $\int_0^\infty \frac{\ln (1+x^\alpha) \ln (1+x^{-\beta})}{x} \, \mathrm{d} x$ I have been trying to evaluate the following family of integrals:
$$ f:(0,\infty)^2 \rightarrow \mathbb{R} \, , \, f(\alpha,\beta) = \int \limits_0^\infty \frac{\ln (1+x^\alpha) \ln (1+x^{-\beta})}{x} \, \mathrm{... | Only a comment. We have
$$ \int_{0}^{\infty} \frac{\log(1+\alpha x)\log(1+\beta/x)}{x} \, dx = 2\operatorname{Li}_3(\alpha\beta) - \operatorname{Li}_2(\alpha\beta)\log(\alpha\beta) $$
which is valid initially for $\alpha, \beta > 0$ and extends to a larger domain by the principle of analytic continuation. Then for inte... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 5,
"answer_id": 1
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Inverse Laplace transform of a product using convolution I want to calculate $\mathcal{L}^{-1}\left\{\frac{1}{s^2(s^2+a^2)}\right\}$ using the convolution theorem $\mathcal{L}\{f*g\}=\mathcal{L}\{f\}\cdot\mathcal {L}\{g\}$. I have already calculated it using partial fraction decomposition which yielded $\frac{t}{a^2} -... | Note that we have
$$f(t)=\mathscr{L}^{-1}\left\{\frac1{s^2}\right\}=tu(t)$$
and
$$g(t)=\mathscr{L}^{-1}\left\{\frac1{s^2+a^2}\right\}=\frac{\sin(|a|t)}{|a|}u(t)$$
Then, application of the convolution theorem yields
$$\begin{align}
\mathscr{L}^{-1}\left\{\frac{1}{s^2(s^2+a^2)} \right\}&=(f*g)(t)\\\\
&=\int_{-\infty}^\i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2839734",
"timestamp": "2023-03-29T00:00:00",
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Having trouble simplyfing radicals of this sort I'm studying radicals and rational exponents. I'm having lots of hardships with problems of this sort: prove $$\sqrt{43+24\sqrt{3}}=4+3\sqrt{3}$$ I keep going around and around experimenting with factoring. I can't seem to be able to prove this one in particular. Am I mis... | HINT: Square both sides
$$ (\sqrt{43 + 24 \sqrt{3}})^2 = 16 + 9(3) + 24 \sqrt{3},$$
$$ 43 + 24 \sqrt{3} = 43 + 24\sqrt{3}.$$
NOTE $(a+b)^2 = a^2 + 2ab + b^2$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2840001",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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How do I find the last two digits $2012^{2013}$? How do I find the last two digits 20122013
My teacher said this was simple arithmetics(I still don't see how this is simple).
I thought of using Congruence equation as 2012 is congruent to 2 mod 10...but i can't get 22013
Can anyone help please?
| We work modulo $100$, so we work modulo $2^2=4$ and $5^2=25$. The given number is of course zero modulo four, so we only need it modulo $25$. The Euler indicator of $25$ is $\frac 45\cdot 25=20$, so $12^{20}=1$ modulo $25$. So
$$
2012^{2013}=12^{2013}=12^{20\cdot 100+13}=(12^{20})^{100}\cdot 12^{13}
=12^{13}=22
$$
mod... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2840093",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Factoring a convergent infinite product of polynomials. Suppose that $f(z)=\displaystyle\prod_{k=1}^\infty p_k(z)$ is a convergent product of polynomials $p_k$ such that $p_k(0)=1$. I want to know if I can "factor" $f(z)$ in the following way: if we list the roots of all the $p_k$ as $r_1, r_2, r_3, \ldots$, must the p... | $\prod_{n=1}^\infty (1-\frac{z^2}{n^2})$ converges locally uniformly on $\mathbb{C}$. We can let $p_1 = \prod_{n=1}^{N_1} (1-\frac{z^2}{n^2}), p_2 = \prod_{n=N_1+1}^{N_2} (1-\frac{z^2}{n^2}), p_3 = \prod_{n=N_2+1}^{N_3} (1-\frac{z^2}{n^2})$, etc. for positive integers $N_1 < N_2 < \dots$. We may order the roots of $p_1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2840212",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to solve $a 2^x - x = b$? I need to solve $a 2^x - x = b$ for $x$ where $a$ and $b$ are parameters. Does it have closed form solution? I need to substitute $x$ in another system of equations in Mathematica.
| $$x = -{\frac {{\rm W} \left(-\ln \left( 2 \right) a{2}^{-b}\right)}{\ln
\left( 2 \right) }}-b$$
where $\rm W$ is any branch of the Lambert W function (ProductLog in Mathematica).
But if you have Mathematica, why didn't you ask Mathematica to solve it?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2840346",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Is probability-raising closed under union? Suppose that $\Pr(X \mid A) > \Pr(X)$, and that $\Pr(X \mid B) > \Pr(X)$. Does it follow that $\Pr(X \mid A \cup B) > \Pr(X)$?
$\Pr(X \mid A \cup B) > \Pr(X)$ holds just in case
$$
[\Pr(X A) - \Pr(X) \cdot \Pr(A)] + [\Pr(XB) - \Pr(X) \cdot \Pr(B)] > \Pr(X A B) - \Pr(X) \cdot... | Here is a counterexample. Two balls are drawn, with replacement, from an urn containing
$1$ black ball and $2$ white balls. Consider the following events.
$X$: The two balls are the same color.
$A$: The first ball is white.
$B$: The second ball is white.
Then $\Pr(X)=\frac59$, $\ \Pr(X\mid A)=\Pr(X\mid B)=\frac23\gt\Pr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2840480",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Problem on the set of all orthogonal matrices If $\alpha : (-1,1) \to O(n, \mathbb{R})$ be any smooth map with $\alpha(0)=I$ then what can we say about $\alpha^{'}(0) ?$ Here $I$ is the identity matrix and $O(n,\mathbb{R})$ is the set of all $n \times n$ orthogonal matrices over $\mathbb{R}$.
*
*It is non-singular
... | You can say that $\alpha'(0)$ is ant-symmetric. You know that$$\bigl(\forall t\in(-1,1)\bigr):\alpha(t)^T.\alpha(t)=\operatorname{Id}_n.$$Therefore $\alpha'(0)^T.\alpha(0)+\alpha(0)^T.\alpha'(0)=0$. In other words, $\alpha'(0)^T+\alpha'(0)=0$, which means that $\alpha'(0)$ is anti-symmetric.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2840716",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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$\int_{0}^{2008}x|\sin\pi x| dx$ Evaluate:
$$\int_{0}^{2008}x|\sin\pi x| dx$$
That modulus sign is causing problems. How do I handle it?
I am trying integration by parts
I have even evaluated: $\int_0^1 {|\sin \pi x|}= \frac 2 \pi$. Not sure how to utilise it in the problem.
I just need help with the modulus part.
| Not sure how to bring graphs into answers, but here is a link to your function.
http://www.wolframalpha.com/input/?i=graph+%7Csin(pix)%7C
As you can see, the period is $1$, i.e. if
$f(x)=|\sin(\pi x)|$ then $f([0,1])=f([1,2])$.
So $\int_0^{2008}f(x) dx = 2008\int_0^1 f(x) dx$
${}{}{}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2840848",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 2
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Rate of weak convergence of sin(nx) Since $\sin(n\cdot)$ converges weakly to zero, we know that
$$
\lim_{n\rightarrow\infty} \int_a^b g(x)\sin(nx)\mathrm{d}x = \int_a^b g(x)\cdot 0\,\mathrm{d}x = 0
$$
holds for all $g\in L^2([a,b])$.
Is there a way to find an explicit formula for the rate of convergence in the above eq... | Consider the functions $f_k = sin(kx)$ for all integers $k$.
for every $k$:
$||f_k||_{L^\infty } = 1$
$\int_0^{2\pi} f_ksin(kx) \,dx= \pi \le C(k)$
therefore, there is no function $C(n)$ that satisfies your conditions.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2840927",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Minimization of positive quadratic function using gradient descent in at most $ n $ steps For minimization positive quadratic form $$f = \frac{1}{2}\left\langle Ax,x \right\rangle - \left\langle b,x\right\rangle \rightarrow \min_{x\in\mathbb{R}^n},$$ we use gradient descent $$x^{k+1} = x^{k} - \alpha_k \nabla f(x^k)$$... | You can assume that you have an orthonormal basis of eigenvectors so that $A$ is
diagonal, with the eigenvalues $\lambda_k$ on the diagonal. Let me use $\Lambda$
for $A$ in this basis, just for emphasis.
Note that the solution is given by $x^*= \Lambda^{-1} b$.
Then $x_{k+1} = x_k -{1 \over \lambda_{k+1}} (\Lambda x_k ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2841032",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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I need to find the real and imaginary part of this $Z_{n}=\left (\frac{ \sqrt{3} + i }{2}\right )^{n} + \left (\frac{ \sqrt{3} - i }{2}\right )^{n}$ I have a test tomorrow and i have some troubles understanding this kind of problems, would really appreciate some help with this
$$ Z_{n}=\left (\frac{ \sqrt{3} + i }{2}\r... | The above two can be written using binomial theorem
$$\sum_{k=0}^n \binom{n}{k}\left(\frac{3}{4}\right)^\frac{n-k}{2}\left(\frac{i}{2}\right)^k$$
And
$$\sum_{k=0}^n \binom{n}{k}\left(\frac{3}{4}\right)^\frac{n-k}{2}\left(\frac{-i}{2}\right)^k$$
We rewrite the second as
$$\sum_{k=0}^n \binom{n}{k}\left(\frac{3}{4}\right... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2841160",
"timestamp": "2023-03-29T00:00:00",
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About decay of Fourier coefficients of "almost" $C^2$ functions. If a function is $C^2$ outside a measure zero set in its domain then do its Fourier coeffients still decay as $O(\frac{1}{k^2})$ ? Assume that the function is continuous but not differentiable on that measure-zero set.
| No. For example, consider $f(x)=\sqrt{|x|}$ on the interval $[-\pi, \pi]$. The cosine coefficients are
$$
A_n = \frac{2}{\pi} \int_0^{\pi} \sqrt{x}\cos nx\,dx = -\sqrt{\frac{2}{\pi}}\frac{\operatorname{Si}(\sqrt{2n})}{n^{3/2}}
$$
according to Wolfram Alpha. Since the sine integral has a nonzero limit at infinity, the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2841245",
"timestamp": "2023-03-29T00:00:00",
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"question_score": "2",
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Evaluate the integral $\int \frac{\sec x}{\sqrt{3+\tan x}}dx$ Evaluate the following integral.$$\int \frac{\sec x}{\sqrt{3+\tan x}}dx$$
On putting $t=\tan x$ I am getting $$\int\frac{1}{\sqrt{(t^2+1)(t+3)}}dt$$.
How should i proceed from here.
| We are given:
$$ y = \int\frac{\sec x\,dx}{\sqrt{3+\tan x}}. $$
First, change the variable of integration from $x$ to $u=\sqrt{3+\tan x}$; we have that
\begin{align*}
du &= \frac{\sec^{2}x\,dx}{2\sqrt{3+\tan x}} \\
\therefore dy &= \frac{2\,du}{\sec x} \\
&= \frac{2\,du}{\sqrt{1+(u^{2}-3)^{2}}}.
\end{align*}
Now, if th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2841385",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
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Exponential distribution $P(Z \geq 5)$ Exponential distribution
Let $Z ∼ Exponential(4)$. Compute each of the following
(a) $P(Z \geq 5)$
$$P(Z \geq 5) = \int_{5}^{\infty} 4e^{-4x}dx$$
Let $u = -4x$, then $du = -4dx \leftrightarrow -\frac{1}{4}du = dx$
$$-\int_{-\infty}^{-20} e^{u} du = -e^{u}|_{-\infty}^{-20} = -(e^... | Just this:
$$\int_5^\infty 4 e^{-4x}\mathsf d x= \lim_{x\to\infty}(-e^{-4x})-(-e^{-4\cdot 5})$$
When you apply the substitution
$$\begin{split}\int_{-20}^{-\infty} 4e^{u}\dfrac{\mathsf d u}{-4} &= -\int_{-20}^{-\infty}e^u\mathsf d u \\ &=\int_{-\infty}^{-20}e^u\mathsf d u \\ &=e^{-20}-\lim_{u\to-\infty}e^u\end{split}$$... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Find all complex numbers satisfying $z\cdot\bar{z}=41$, for which $|z-9|+|z-9i|$ has the minimum value My first attempt was to express $z$ as $x+iy$ and minimize the expression $\sqrt{(x-9)^2+y^2}+\sqrt{x^2+(y-9)^2}$ where $x^2+y^2=41$.
That said, it seems to me that using the geometric interpretation could be easier.... | The locus of points with sum of distances $a$ from $(9,0)$ and $(0,9)$ is an ellipse. If we have $a=9\sqrt{2},$ we get a degenerate line segment between the 2 points, but as $a$ increases, the ellipse expands and then becomes tangent to the circle. Thus, you want to find the value of $a$ so that the ellipse with foci a... | {
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Prove by induction: $3^n+1$ is divisible by $2$ for $n\ge 0$ I'm going through the process of induction and when I'm attempting to prove $P_{k+1}$ I keep getting $2(3A-\frac32)$, where $A$ is an integer and $A\geq 1$, which isn't possible since $3A-\frac32$ should be an integer.
My method is to write $P_k$ as $3^k+1=2A... | For $k+1$, you already assume that $3^k + 1 = 2n$ for some integer $n$. Then, use the fact that
$$3^{k+1} + 1 = 3\cdot 3^{k} + 1$$
and substitute $3^k$ with the expression above.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 7,
"answer_id": 0
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Find symmetric matrix containing no 0's, given eigenvalues I'm preparing for a final by going through the sample exam, and have been stuck on this:
$$Produce\ symmetric\ matrix\ A ∈ R^{3×3},\ containing\ no\ zeros.\ \\
A\ has\ eigenvalues\ λ_1 = 1,\ λ_2 = 2,\ λ_3 = 3$$
I know $A = S^{-1}DS$, where A is similar to th... | You are correct in observing that "too little" information is given in the sense that there are infinitely many such matrices. But you need to produce just one. So start with the diagonal matrix $D = \operatorname{diag}(1,2,3)$ and conjugate it by a simple (but not too simple) orthogonal matrix $S$. You don't want $S$ ... | {
"language": "en",
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Order of generators in subgroup of Free Product of Cyclic Groups In this question, it is shown that a subgroup of a free product of cyclic groups is still a free product of cyclic groups. (Subgroup of free product of cyclic group is still a free product of cyclic groups?)
Suppose $G=\langle g_1,g_2,\dots\mid g_1^{k_1}=... | Without using the Kurosh subgroup theorem: In a free product, elements of finite order are conjugate into one of the factor groups. (To see this you could consider the action of $G$ on its Bass-Serre tree, where vertex stabilisers are factor groups.)
Hence, in your example if $h_i\in H$ has finite order then $h_i$ is c... | {
"language": "en",
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"question_score": "1",
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"answer_id": 1
} |
Simplify $ a = \frac{N\sum(xy)-\sum(x)\sum(y)}{N\sum(x^2)-(\sum(x))^2} $ to yield $ a = \frac{\bar{xy} - \bar{x}\bar{y}}{\bar{x^2}-\bar{x}^2}$ I'm working out some multivariable linear regression equations on paper for a class I'm taking, and I'm getting an erroneous factor of N in my solution according to the class. ... | There is no extra factor. We have $\sum\limits_{i=1}^Nx_i\sum\limits_{i=1}^Ny_i$. Now we divide it by $N$.
$\underbrace{\frac1N\sum\limits_{i=1}^Nx_i}_{=\overline x}\sum\limits_{i=1}^Ny_i$
$\overline x\sum\limits_{i=1}^Ny_i$
Dividing the term by $N$ again
$\overline x\cdot \underbrace{ \frac1N\cdot \sum\limits_{i=1}^N... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2842045",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Solution to generalized-polynomial equation? Is it possible to obtain the solution to this generalized-polynomial equation?
$$b x^a - x +c =0$$
with $-1<a<1$, $b>0$, $c>0$ and $x>0$.
| In general, only by numerical methods or series. The following series solution in powers of $b c^{a-1}$ converges if $b c^{a-1}$ is small :
$$ \eqalign{x &= c + b c^a + \frac{ca}{b} (b c^{a-1})^2 + \frac{ca(3a-1)}{2} (b c^{a-1})^3 + \ldots\cr
&= c + b c^a + c \sum_{k=2}^\infty \frac{(b c^{a-1})^k}{k!} \prod_{j=0}^{k-2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2842143",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Is there a subspace of $R^3$ of dimension $1$ that contains the vectors $v=(1,1,2)$ and $w=(1,-1,2)$?
Is there a subspace of $R^3$ of dimension $1$ that contains the vectors $v=(1,1,2)$ and $w=(1,-1,2)$?
I see that $v$ and $w$ linearly independent so I think that there isn't a subspace of dimension $1$ that contains ... | It suffices to note that since $\vec v$ and $\vec w$ are linearly independent they span, as a basis, a subspace with dimension $2$. Therefore there isn't a subspace of dimension $1$ that contains both vectors.
Indeed if such subspace would exist with basis $\{\vec u\}$ we had
*
*$\vec v=a \vec u \quad \vec w=b \vec... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2842278",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
solutions to $x^n\equiv a\pmod{p}$ I'm asked to prove that if $\gcd(n,p-1)=1$ where p is prime, then
$$x^n\equiv a \pmod{p}$$
has exactly one solution. What I've done so far is the following,
Since $\gcd(n,p-1)=1$, that means, except for $1$, there is no least residue $r$ such that $r^n\equiv 1\pmod{p}$. Moreover, it w... | We are given the fact that $gcd(n,p-1)=1$ i.e.
$$nk_1=k_2(p-1)+1 \implies n=k_2\times k_1^{-1}(p-1)+1\tag{1}$$, for some integers $k_1$ and $k_2$
Let ,we are asked to find $x^n \mod p$ i.e.
$$x^{(k_2 k_1^{-1}p-k_2 k_1^{-1}+1)}\mod p\tag{2}$$ Since $p$ is a prime,
$x^p \mod p=x$
.Put it in expression (2) to get $x \mod ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2842399",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 2
} |
Points of a dense set are not limit points I'm reading Rudin's Principles of Mathematical Analysis. Here is how the book defines the dense set:
$E$ is dense in $X$ if every point of $X$ is a limit point of E, or a point of $E$ (or both).
To fully understand the definition, here is my question:
Is there a dense set $... | Take any nonempty set $X$, and define $d(x,x)=0$, for all $x\in X$, and $d(x,y)=1$, for all $x,y\in X$ with $x\ne y$.
It's easily verified that $d$ is a metric on $X$ (it's called the discrete metric), and clearly, all points of $X$ are isolated.
Now just let $E=X$.
Then no point of $E$ is a limit point, and by def... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2842500",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
Find the last two digits of $1717^{1717}$
$1717^{1717} \mod 100$
Since $\phi(100) = 40$ , we can transform this into:
$17^{37}101^{37} \mod 100 = 17^{37} \mod 100$
How do I proceed further?
| Like What are the last two digits of $77^{17}$?,
using Carmichael Function, $\lambda(100)=20\implies**17^{**17}\equiv17^{17}\pmod{100}$
Now $17^2=290-1, 17^{17}=17(290-1)^8$
Again, $\displaystyle(290-1)^8=(1-290)^8\equiv1-\binom81290\pmod{100}\equiv1-90\cdot8\equiv-19$
Now $-19\cdot17=-323\equiv77\pmod{100}$
See also: ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2842588",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 3
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Diagonalization differential equation I have a problem in solving the diagonalization of this differential equation :
$$\frac{d}{dt}\binom{x}{f} = \left(\begin{matrix} -\frac{1}{\tau} & 1 \\ 0 & -\frac{1}{\tau_{c}}\end{matrix}\right)\binom{x}{f}$$
Can anybody help me?
| Making the change of variables
$$
p = (x,f)^{\dagger}\\
P = (X,F)^{\dagger}
$$
such that
$$
p = T\cdot P
$$
with $T$ invertible, we have
$$
\left(T\dot P\right) = A\cdot \left(T \cdot P\right)\Rightarrow \dot P = \left(T^{-1}\cdot A \cdot T\right)\cdot P
$$
now choosing
$$
T = \left(
\begin{array}{cc}
\frac{\tau \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2842699",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
integration by parts of trig function I tried to solve the integral below using integration by parts
$$\int_0^t\cos(x)\cos(t-x)dx=\frac{1}{2}(\sin(t)+t\cos(t))$$
It seemed solvable through doing integration by parts twice,
but it hasn't worked for me yet...
tcos(t) doesn't come up!
I know how it can be solved using pro... | Hint: use the formula
$$\cos(x)\cos(y)=\frac{1}{2}(\cos(x-y)+\cos(x+y))$$
for your Control:
$$1/2\,\sin \left( t \right) +1/2\,\cos \left( t \right) t$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2842801",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 0
} |
Calculus tangent line For some constant c, the line $y=4x+c$ is tangent to the graph of $f(x)=x^2+2$, what is the value of $c$?
I don’t understand how to find the value of c. Because it’s a tangent line I understand they touch at one point. Probably a dumb question, I just don’t understand.
| Say you have a tangent at point $T(a,b)$. Then $f'(a) = 4$ and $f(a)=b$ and $b=4a+c$. So we have
$\bullet \;\; 2a=4\implies a=2$
$\bullet \;\; a^2+2=b\implies b=6$
$\bullet \;\; c=-b+4a\implies c=-2$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2842897",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Find $\lim_{n \to \infty}f_n(x)$ Find $$\lim_{n \to \infty}f_n(x)$$ where $$f_n(x)=n^2x(1-x)^n$$ $0 \lt x \lt 1$
My try:
By symmetry $$\lim_{n \to \infty}f_n(x)=\lim_{n \to \infty}f_n(1-x)=\lim_{n \to \infty}n^2(1-x)x^n=(1-x)\lim_{n \to \infty}n^2 x^n$$
Now $$\lim_{n \to \infty}n^2 x^n=\lim_{n \to \infty}\frac{x^n}{\fr... | An option:
For $0< x <1$, show that
$\lim_{n \rightarrow \infty} n^2x^n =0.$
Set $x=e^{-y} , y>0$, and consider $\dfrac{n^2}{e^{ny}}$.
$e^{ny} =$
$ 1+ ny +(ny)^2/2! + (ny)^3/3! +.. \gt (ny)^3/3!$.
Hence :
$\dfrac{n^2}{e^{ny}} \lt \dfrac{(3!)n^2}{n^3y^3}= (\dfrac{3!}{y^3})(\dfrac{1}{n}).$
The limit $n \rightarrow \inf... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2843120",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 2
} |
Why is vector times vector equal to a number? It just occurred to me that we have
$$
\text{number} \cdot \text{number} = \text{number} \\
\text{matrix} \cdot \text{matrix} = \text{matrix}
$$
but
$$
\text{vector} \cdot \text{vector} = \text{number}
$$
Why is that? Why is $\text{vector} \cdot \text{vector}$ not equal to ... | Three kinds of vector products, along with what they produce:
*
*Dot product: $vector \cdot vector = scalar$
*Cross product: $vector \times vector = vector$
*Outer product: $vector \otimes vector = matrix$
So, it only produces a number (scalar) if it's a dot product.
It boils down to definitions.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2843179",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 2
} |
Probability that a random variable lies between two different random variables. Say we have three different random variables $X_{1}$, $X_{2}$, and $X_{3}$ with pdf's $f_{X_{1}}$, $f_{X_{2}}$, and $f_{X_{3}}$. Random variable $X_{2}$ is independent of random variables $X_{1}$ and $X_{3}$. But random variables $X_{1}$ an... | No! I don't know how you arrived at that equation. The correct method is this: $P\{X_1<X_2<X_3\}=E\int_{X_1}^{X_3} f_{{X_2}}(x) \, dx=\int \int\int_u^{v} f_{{X_2}}(x) \, dx \, f_{{X_1},{X_3}}(u,v) \, du \, dv$ which depends on the joint distribution of $(X_1,X_3)$. You cannot express this in terms of the marginal densi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2843296",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Solve algebraic equation with floor functions I am looking for a way to find all the solutions to this equation without guess and check:
$$\frac{-x^2+45x}{2x+1}-\left\lfloor \frac{-x^2 +45x}{2x+1} \right\rfloor + x - \lfloor x \rfloor = 0$$
I graphed it in desmos and found that the most obvious solutions are $x = -4, -... | The floor function is very discontinuous, so I wouldn't expect there to be good numerical methods for solving equations involving it. (Numerical methods assume you have an approximate answer that you can tweak to make closer to the actual answer, and you'll have trouble doing that if your function jumps around.) And I ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2843406",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
} |
Algebra with exponential functions If $f(x) = 4^x$ then show the value of $$f(x+1) - f(x)$$ in terms of $f(x)$.
I know the answer is $3f(x)$ because
$f(x+1)$ means that it is $4^x$ multiplied by 4 once more, which minus one is 3.
The question: How do I show this process algebraically? (Hints only please) I have tried ... | Hint: $$f(x+1) = 4^{x+1} = 4^x\cdot 4=...$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2843516",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Solving $y''-k^2y=0$ without substituting $e^{kx}$ As the title says I am trying to solve $y''-k^2y=0$. The method that I want to use is to assume $y'=p$ which gives us $y''=p\frac{dp}{dy}$.
Substituting above values in original equation gives me $p\frac{dp}{dy}-k^2y=0$ which further reduces to $\frac{dy}{dx}=\sqrt{k^... | $$\frac{dy}{dx}=\sqrt{k^2y^2+c}$$
with $c=k^2a$
$$\frac{dy}{dx}=|k|\sqrt{y^2+a}$$
Substitute $y=\sqrt a\sinh(t)$
$$\frac{dy}{dx}=|k|\sqrt{a\sinh^2(t)+a}=|k|\sqrt{a\cosh^2(t)}$$
$$\frac {\sqrt{a}\cosh(t)}{\cosh(t)}dt=|k|\sqrt{a}dx$$
$$\int dt=|k|\int dx$$
$$y(x)=\sqrt a \sinh(|k|x+K)$$
Using Euler's formula
$$y(x)=c_1e^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2843613",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
} |
Condition for having order 2 elements in Galois group for polynomials over $\mathbb{Q}$ Suppose we have $f \in \mathbb{Q}[X]$, with only real roots. Then the complex conjugation is not an automorphism, but is this enough to say that there exist no order two elements in $\text{Gal}(f)$?
The case I was studying is $f = x... | Here is an a;ternative example centred on algebraic extensions (as opposed to polynomials) having even ordered Galois groups.
Consider prime numbers $p$ of the form $p=4k+1$, and let $\zeta = \exp (2\pi i/p)$ a primitive $p$th root of unity.
Then the algebraic number $\alpha = \zeta + \bar \zeta$ generates a Galois ext... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2843705",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
What is the solution to this inequality: $| 2x-3| > - | x+3|?$ By using graphical method, I am getting all real numbers..
Where am I wrong in graphical method? How to solve this using calculation?
| Because absolute values are always positive we have
$$
\lvert 2x - 3 \rvert \geq 0 \geq -\lvert x + 3 \rvert
$$
for all values of $x$. Hence the only time $\lvert 2x - 3 \rvert > -\lvert x + 3 \rvert$ could potentially not hold is when both sides equal $0$. But $2x - 3 = 0$ if and only if $x = \frac{3}{2}$ and $x + 3 =... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2843802",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Example of Non-Equal Random Variables that are Identically Distributed? What is a simple example of two random variables $X$ and $Y$ that are identically distributed s.t. $X \ne Y$?
Is the only way to achieve this by changing the sample space underneath $X$ and $Y$ (i.e., $\Omega_X \ne \Omega_Y$)?
| A method to generate such examples where the underlying sample space is the same is to use transformations that leave the probability measure invariant and apply them to the random variable.
As Chris Janjigian mentions, one instance is when you have a random variable $X$ with symmetric distribution and let $Y:=-X$.
A... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2844020",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 1
} |
Is $ \lim_{x\to \infty}\frac{x^2}{x+1} $ equal to $\infty$ or to $1$? What is
$ \lim_{x\to \infty}\frac{x^2}{x+1}$?
When I look at the function's graph, it shows that it goes to $\infty$, but If I solve it by hand it shows that the $\lim \rightarrow 1$
$ \lim_{x\to \infty}\frac{x^2}{x+1} =$ $ \lim_{x\to \infty}\frac{... | Hint: Write $$x\cdot \frac{1}{1+\frac{1}{x}}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2844097",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 6,
"answer_id": 5
} |
A confused question about a partially ordered set
Let $X$ be a nonempty set and $\mathbb{F}$ be the collection of all extended nonnegative-valued functions $f \colon X \to [0, +\infty]$.
I am wondering that when this set $\mathbb{F}$ is equipped with the usual pointwise ordering $\geq$, then
is this $(\mathbb{F}, \g... | Fact:
If $(Y, \le_Y)$ is a partially ordered set, $X$ is a set and on $\mathbb{F} = \{f: X \to Y\}$ we define $$f \le_F g \iff \forall x \in X: f(x) \le_Y g(x)$$ then $(\mathbb{F}, \le_F)$ is also a partially ordered set.
The proof is just plugging in the definitions, nothing fancy.
And both $[0,+\infty)$ and $[0,+\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2844468",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Limit $\lim_{n\to\infty} n^2\left(\sqrt{1+\frac{1}{n}}+\sqrt{1-\frac{1}{n}}-2\right)$ Greetings I am trying to solve $$\lim_{n\to\infty} n^2\left(\sqrt{1+\frac{1}{n}}+\sqrt{1-\frac{1}{n}}-2\right)$$ Using binomial series is pretty easy: $$\lim_{n\to\infty}n^2\left(1+\frac{1}{2n}-\frac{1}{8n^2}+\mathcal{O}\left(\frac{1... | Making $\delta = \frac 1n$ You can arrange it as
$$
\lim_{\delta\to 0}\left(\frac{\frac{\sqrt{1+\delta}-1}{\delta}-\frac{\sqrt{1-\delta}-1}{\delta}}{\delta}\right) = \left(\frac{d^2}{dx^2}\sqrt x\right)_{x=1} = -\frac 14
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2844601",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
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Product of Uniform Distribution I know that there exists some discussions related to my question, however, I couldn't find an explanation for my question. I hope it is not a duplicate.
Let $X_n$ be sequence of i.i.d. uniform distributions on $(0,a)$, and define $Y_n = \prod^n_k X_k$.
Problem
For what values of $a$,... | Copied from the comment of Did,
If $a=e$, then $S_n = \sum_{k=1}^n \log X_k$ defines a random walk on the real line with centered integrable steps, hence $(S_n)$ is recurrent, which implies that $(Y_n)$ is almost surely unbounded. In particular, $P(Y_n \to 0) = 0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2844754",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Evaluating the integral of $\exp(-x^2) \cos(2xy)$ using power series So I am trying to compute:
$$
\int_{0}^\infty\exp(-x^2)\cos(2xy) \mathrm{d}x
$$
using the power series of $\cos$. I have done the following:
We first evaluate the integral by expanding $\cos (2xy) $ using its power series. Uniform convergence of the p... | There is a much more direct way of doing this calculation. $cos(2xy)=\frac{e^{2ixy}+e^{-2ixy}}{2}$. $\int_0^{\infty}\frac{e^{-x^2-2ixy}}{2}dx=\int_{-\infty}^0\frac{e^{-x^2+2ixy}}{2}dx$, by changing $x$ to $-x$. Therefore the final integral $=\int_{-\infty}^{\infty}\frac{e^{-x^2+2ixy}}{2}dx=\sqrt{\pi}e^{-\frac{y^2}{4}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2845006",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Morphism between thick fibers of schemes extends to a neighbourhood Let $ S $ be a locally Noetherian scheme, and $ X $, $ Y $ finite type $ S $-schemes. Let us fix $ s \in S $. Let $ \varphi : X \times _ { S } \mathcal{O}_{S,s} \to Y \times _ { S } \mathcal{O}_{S,s} $ be a morphism of $ S $-schemes. Show t... | Slightly less pedantic version of the same solution for quicker future reference.
We can assume that $ X $ and $ S $ are affine, say $ X = \text{Spec } A $ and $ S = \text{Spec } R $, since $ X $ is quasi-compact and we can shrink $ S $ to an open subset any finite number of times. Denote $ s $ by the prime $ \math... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2845143",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
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Subset of the $(x,y)$ plane satisfying $x^2 - xy + y^2 \le 0$ The problem:
Describe and Illustrate the region in the $(x,y)$ plane satisfying $x^2 - xy + y^2 \le 0$.
My thoughts:
Write the inequality as $x^2 + y^2 \le xy$.
We can associate with a point (other than the origin) in the plane the right-triangle with hypo... | $$x^2-xy+y^2=\frac{1}{2}(2x^2+2y^2-2xy)=\frac{1}{2}(x^2+y^2+(x-y)^2)\geq 0$$
with equality iff $$x=y=x-y=0$$
Your reasoning seems fine to me! You just need to be a bit careful where a triangle doesn't exist - i.e. one of the legs has length $0$.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Factor $d^k+(a-d)^k$ I was reading a number theory book and it was stated that $d^k+(a-d)^k=a[d^{k-1}-d^{k-2}(a-d)+ . . .+(a-d)^{k-1}]$ for $k$ odd. How did they arrive at this factorization? Is there an easy way to see it?
| Start by understanding how to factor
$$
x^n - y^n .
$$
Presumably you know how to do that when $n=2$. For $n=3$ you can check that
$$
x^3 - y^3 = (x-y)(x^2 + xy + y^2)
$$
Now guess for higher powers.
Then see what happens if $n$ is odd and you replace $y$ by $(-y)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2845451",
"timestamp": "2023-03-29T00:00:00",
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How To Calculate Binomial Distribution Of Really Small %? I asked this question on the Bitcoin Forum, but I think it's more appropriate for a mathematics forum.
I'm making an informative video and I need a binomial distribution calculation. I want to find out how many trials are needed to get 1%, 50% and 90% likelihood... | Using manual tinkering with R I get the following values
pbinom(0,3.365231884e48,2^(-160), FALSE) = 0.9000000000339017
pbinom(0,1.0130357393e48,2^(-160), FALSE) = 0.5000000000001161
pbinom(0,1.46885823057e46,2^(-160), FALSE) = 0.01000000000005571
As a check
pbinom(0,229,0.01, FALSE) = 0.8998941257385102
pbinom(0,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2845598",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 0
} |
Is my proof of straight line being the shortest route from one point to another correct? Every proof seems to go above my head as I'm not thorough with calculus or what is being talked about in this similar question in which the author proves it using complicated terms. As a result I tried proving it on my own.
I thin... | I think it works for polygonal paths, but this certainly does not show the result in full generality. Indeed, a different way to go, might be to define a path to be a continuous map $\lambda:[0,1] \to \mathbb R^n$.
We need a meaningful notion of "length" in this context, and if we take $d(x,y)$ to be the usual euclidi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2845679",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Discretizing a Stochastic Volatility SDE How does the discrete time stochastic volatility model arise from the continuous time one?
I have the following continuous time stochastic volatility model. $S_t$ is the price, and $v_t$ is a variance process.
$$
dS_t = \mu S_tdt + \sqrt{v_t}S_t dB_{1t} \\
dv_t = (\theta - \alp... | I guess you can discretize the raw price process too instead of the log price process. You get
$$
S_{t+1} = S_t + \mu S_t + \sqrt{v_t} S_t Z_t
$$
(where $Z_t$ is a standard normal variate), or
$$
\frac{S_{t+1}}{S_t} - 1 = \mu + \sqrt{v_t} Z_t.
$$
Got the idea from: https://arxiv.org/pdf/1707.00899.pdf
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2845825",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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Isomorphism of group algebras of the dihedral group. I'm trying to solve the following problem.
Let $m,n \in \mathbb{N}, m|n$. Prove that $\mathbb{Z[D_{n}]}/ \langle R^{m} - 1 \rangle \sim \mathbb{Z[D_{m}]}$.
I'm trying to prove it using a hands on approach exhibiting the isomorphism between both rings without much ... | Let $m | n$ be two positive integers. Consider the morphism
$$
\newcommand{\zd}[1]{\mathbb{Z}[\mathbb{D}_{#1}]}
\begin{align}
g : \zd{n}& \to\zd{m} \\
& 1 \mapsto 1 \\
& r \mapsto \rho \\
& s \mapsto \sigma
\end{align}
$$
with $r,s$ and $\rho,\sigma$ the generators of the corresponding dihedral groups. This mapping... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2845976",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Cyclic Shift of Latin Squares I'm trying to solve this following problem on Latin squares:
"Suppose that the first row of an $n \times n$ array is
\begin{align*}
x_1 \ \ x_2 \ \ x_3 \ldots x_{n-1} \ \ x_n,
\end{align*}
and suppose also that each successive row is obtained from the previous one by a cyclic shift of $r$... | All rows have distinct entries by the construction. For columns, if $\gcd(n,r)>1$ there will be a row before the bottom that is identical to the first, which is not allowed. Thus, the condition for $n$ and $r$ to generate a Latin square is $\gcd(n,r)=1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2846080",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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In $\triangle ABC$, we have $AB = 14$, $BC = 16$, and $\angle A = 60^\circ$. Find the sum of all possible values of $AC$. In $\triangle ABC$, we have $AB = 14$, $BC = 16$, and $\angle A = 60^\circ$. Find the sum of all possible values of $AC$.
When I use the Law of Cosines, I get a quadratic like expected. However, whe... | What you did wrong was forgetting that there is a negative solution!
When you use the Law of Cosines, you get
$$\begin{align}16^2 &= 14^2 + x^2 - 2(14)(x)cos(60^\circ) \\
16^2 - 14^2 &= x^2 - 28x\cdot(\frac{1}{2}) \\
16^2 - 14^2 &= x^2 - 14x \\
60 &= x^2 - 14x \\
0 &= ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2846205",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Automorphism group of tree as a topological group? I'm reading the paper (https://link.springer.com/article/10.1007%2Fs10711-006-9113-9) and in this paper, author define 'arboreal representation' as the following.
An arboreal representation of a profinite group G is a continuous homomorphism G→Aut$(T)$, where $T$ is t... | Profiniteness is a property of topological groups, not of groups! So when you say $G$ is a profinite group, that already means it has a topology. Explicitly, is profinite group is by definition a topological group that is an inverse limit (as a topological group) of a system of finite discrete groups. So, the topolo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2846332",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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About an exercise in Rudin's book In the book of Walter Rudin Real And Complex Analysis page 31, exercise number 10 said:
Suppose $\mu(X) < \infty$, $\{f_n\}$ a sequence of bounded complexes measurables functions on $X$ , and $f_n \rightarrow f $ uniformly on $X$. Prove that $$ \lim_{n \rightarrow \infty} \int_X f_n d... | Your proof assumes that $h$ is integrable. It doesn't have to be. Suppose that $X=(0,1]$ (with the Lebesgue measure) and that $f_n=\frac1{x^2}\chi_{\left(\frac1{n+1},\frac1n\right]}$. Then $(f_n)_{n\in\mathbb N}$ converges pointwise to the null function, but $h$ is not integrable, since $h(x)=\frac1{x^2}$. And we don't... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2846416",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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anti-derivative not differentiable at any point Reading about primitives and anti-derivatives, I noticed that primitive functions of non-continuous functions are not differentiable at some point, but the set of non-differentiability is often negligible.
I tried to think of a function horrible enough to get a non-diffe... | It isn't possible. Lebesgue's Differentiation Theorem states that if $f$ is integrable over $\mathbb{R}$, and we let:
$$
F(x) =
\int \limits_{(-\infty, x]} f(t) \, dt
$$
Then $F$ is almost everywhere differentiable.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2846559",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Is it possible for the sum of even and neither odd nor even function to be even or odd? Consider the functions $f(x)$, $g(x)$, $h(x)$, where $f(x)$ is neither odd nor even, $g(x)$ is even and $h(x)$ is odd. Is it possible for $f(x) + g(x)$ to be
*
*even;
*odd?
For the second case I can imagine for example $f(x)... | f(x) = f(-x)
$g(x) \ne g(-x)$ for some x
z(x) = f(x) + g(x)
then $z(-x) = f(-x) + g(-x) \ne f(x) + g(x) $ for some x
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Probability on geometry and drawing of balls .
(i) There are $4$ red and $6$ black balls. A ball is drawn at random, its colour is observed and this ball with another two balls of same colour are returned. Now, if a ball is drawn at random, what is the probability that the ball is red?
MY WORK :
If the ball drawn a... | The probability of the second ball being red is the sum of probabilities of choosing a red ball both times and choosing a black ball first then a red ball.
$\frac{4}{10}\cdot \frac{6}{12} + \frac{6}{10}\cdot \frac{4}{12} = \frac{48}{120} = \frac{2}{5}$
Your answer would be correct for the second question if you defined... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Does an alternating sequence converge or diverge or none? How come this sequence does not approach any limit?
$\{\max((-1)^n,0)\}^\infty _{n=1} : {0,1,0,1,0,1,0,1,...}$
I read that since this alternates between 0 and 1 this does not approach any limit. Hence not convergence.
Is it safe to say that it does not diverge e... | You have missed the definition of a divergent sequence.
A divergent sequence does not have to be unbounded, it simply does not have a limit.
$$ 1,0,1,0,1,0,... $$ does not converge so it is divergent.
Simply put, if a sequence is not convergent we call it divergent regardless of its other properties.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2846887",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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The Commutator Subgroup $K$ of $G$ is the "smallest" subgroup such that $G/K$ is Abelian. Let $G$ be a group. A commutator is an element of the form $aba^{^-1}b^{-1}$. The set of finite products of commutators is a normal subgroup $K$ called the commutator subgroup.
The book claims $K$ is the smallest subgroup such th... | Suppose that $G/K$ is Abelian. Note that
$$\forall a,b \in G: abK = baK \iff \forall a,b\in G:a^{-1}b^{-1}ab \in K$$ This means that $\{[a,b]: a,b \in G\} \subseteq K \subseteq G$ and hence $G'=\langle \{[a,b]: a,b \in G\} \rangle \subseteq K$.
That means that $G'$ is the smallest subgroup of $G$ such that the quotient... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Proof that it is unsolvable whether there's an infinity between countable and uncountable? I have recently watched a video by "Undefined Behavior", explaining countable and uncountable infinities, and showing why uncountable infinity is larger than countable infinity. He then stated that a question had been asked if th... | That person was talking about the continuum hyphothesis. It was proved (by Kurt Gödel and Paul Cohen) that, assuming that set theory is consistent, neither the continuum hyphothesis nor its negation can be proved from the standard set theory axioms (the Zermelo-Fraenkel axioms).
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is group cohomology killed by exponent of group? Let $G$ be a finite group, the exponent $e(G)$ is defined to be the lcm of order of elements in $G$. Let $M$ be a $G$ module, we know by restriction corestriction that $H^i(G,M)$ is annihilated by $|G|$, for positive $i$. Is there examples that $H^2(G,M)$ is not annihila... | Yep. For every finite group, there's $M$ such that $H^2(G, M) = \Bbb Z/|G|$.
For every group (not necessary finite) augmentation ideal $I$ can be covered by free module of rank equal to rank of group: suppose $G$ generated by $g_i$, then map goes like
$$\Bbb Z[G]^{\mathrm{rk} G} \to I, x_i \mapsto (g_i -1)$$
So we hav... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2847221",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Equation for generating integers for n-bit binary strings with k bits set to 1 Assume there is an $n$-bit binary string where $k$ bits of it are set to $1$. We can show that it results in $(^n_k)$ number of binary strings with k bits set to $1$.
How to define an equation to generate these $(^n_k)$ numbers (in integers... | You probably want a function $F(n,k,i)$ which gives the $i^{th}$ number in increasing order that has $n$ binary bits of which $k$ are $1$. As you say, there are $n \choose k$ of them and it will be convenient to let $i$ range from $0$ to ${n \choose k}-1$. There are ${n-1 \choose k}$ that start with a $0$ in the most... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2847310",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Dual spaces and gradients and subgradients Suppose we have some function $f:{\mathbb R}^n \rightarrow \mathbb{R}$. Its gradient is defined as the vector which gives the directional derivative via $(v,\nabla f )=D_{v}f$ for any direction $v$.
Could, or should, we think of $\nabla f$ as something belonging to the dual sp... | You are right to mention that $\nabla f$ is vector information coded in the dual.
On the level curves $f(x)=a$ for a constant value, we perform a composition $f\circ C:I\to\Bbb R^n\to\Bbb R$ with $f\circ C(t)=f(C(t))$, so you are going to get that for $x$ which are on the level curve $$f(C(t)=f(x)=a,$$ and $$\nabla f(... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Find the quotient and remainder Find the quotient and remainder when $x^6+x^3+1$ is divided by $x+1$
Let $f(x)=x^6+x^3+1$
Now $f(x)=(x+1).q(x) +R $ where r is remainder
Now putting $x=-1$ we get $R=f(-1)$
i.e $R=1-1+1=1$
Now $q(x)=(x^6+x^3)/(x+1)$
But what I want to know if there is another way to get the quotient ex... | You're on the right track:
$$
x^6+x^3=x^3(x^3+1)=x^3(x+1)(x^2-x+1)
$$
Therefore
$$
q(x)=(x^6+x^3)/(x+1)=x^3(x^2-x+1)=x^5-x^4+x^3
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2847682",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Proof Verification: Finding A Ball Strictly Contained In An Open Set Of A Metric Space Problem: Let $X$ be a metric space and let $A$ be an open set of $X$ containing a point $x \in X$. Prove that there exists an $\epsilon > 0$ such that $B_{\epsilon}(x)$ is strictly contained in $A$.
Proof Attempt:
Case 1: $\partial A... | Your proof is flawed. The part that says "the only clopen sets of a metric space are $\emptyset$ and the entire space" is true only when $X$ is connected.
Moreover, your statement works only if $|A| \geq 2$.
Here's a revised argument:
By the definition of a base for the metric space $X$, you can find an open ball $x ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Proof by Induction: If $x_1x_2\dots x_n=1$ then $x_1 + x_2 + \dots + x_n\ge n$
If $x_1,x_2,\dots,x_n$ are positive real numbers and if $x_1x_2\dots x_n=1$ then $x_1 + x_2 + \dots + x_n\ge n$
There is a step in which I am confuse. My proof is as follows (it must be proven using induction).
By induction, for $n=1$ the... | An alternative proof to AM-GM inequality is Lagrange Multiplier method.
We are minimizing $$x_1+x_2+...+x_n$$ subject to $$x_1x_2x_3...x_n=1$$
That gives us$$ <1,1,1,...,1>=\lambda<x_2x_3...x_n, x_1x_3....x_1x_2...x_{n-1}>$$
$$x_1=x_2=x_3=...=x_n=1$$
$$x_1+x_2+...+x_n=n$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to integrate $\int \frac{1}{\sqrt{1-x^2-y^2}}\,dy$? I have this integral:
$$\int \frac{1}{\sqrt{1-x^2-y^2}}\; dy$$
The way I would integrate it is:
$$\int \dfrac{1}{\sqrt{(\sqrt{1-x^2})^2-y^2}}\;dy=\sin^{-1} \dfrac{y}{\sqrt{1-x^2}}$$
$\int \dfrac{du}{\sqrt{a^2-u^2}}=\sin ^{-1} \dfrac{u}{a}, \; \text{where} \; a=\... | The two expressions are equal, as $\sinh^{-1}(z)=\frac1i\sin^{-1}(iz)$. Look here.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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(edited)Which $f$ can satisfy $f(A)=f(B) \to A=B$? As I wrote in title,
What is the necessary and sufficient relation between function $f$ and set $A,B$ which can satisfy
$f(A)=f(B) \to A = B\\ where f(A) =\{f(x)\|x\in A\}, f:X\to X$
and is it possible for above $f$ to make $f(A)$ and $f(B)$ intersect while A and B ar... | Try to prove:
$f$ is injective $ \iff$ for all subsets $A,B$ of $X$ we have that $ f(A)=f(B)$ implies $A=B$:
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2848344",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Compute $\int_{0}^{\pi/4}\ln(1-\sqrt[n]{\tan x})\frac{dx}{\cos^2(x)}$ I am trying to compute this
$$
\int_{0}^{\pi/4}\ln(1-\sqrt[n]{\tan x})\frac{\mathrm dx}{\cos^2(x)},\qquad
(n\ge1).
$$
Making a transformation of $I$ to utilise a sub of $u=1-\sqrt[n]{\tan x}$
\begin{align}
I&=\int_{0}^{\pi/4}\frac{\sec^2(x)}{n\sqrt[n... | Your integral is given by the negative of the $n$-th harmonic number:
$$ I_n \equiv n \int \limits_0^1 (1-u)^{n-1} \ln (u) \, \mathrm{d} u = - H_n = - \sum \limits_{k=1}^n \frac{1}{k} \, . $$
You can use the substitution $u = 1-t$ and then have a look at this question for a derivation. Here's an alternative route:
Us... | {
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"url": "https://math.stackexchange.com/questions/2848441",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Theory for general fractional differintegral equations? I am aware there exist ways to construct fractional calculus, fractional differential operators and integral operators, for example by using Cauchy integral theorem in complex analysis or by Fourier analysis.
But do there exist any theory for differential equation... | Diethelm, K.: The Analysis of Fractional Differential Equations. An Application-Oriented Exposition Using Differential Operators of Caputo Type. Springer, 2010:
Theory of Fractional Differential Equations
*
*Existence and Uniqueness Results for Riemann-Liouville Fractional Differential Equations
*Single-Term Caputo ... | {
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The sum of powers of $2$ that are less than or equal to $n$ is less than $2n$. I am working with an amortised analysis problem where the given solution states that
$$\sum\{2^k:0<2^k\le n\}<2n$$
I am not mathematically literate; is there a simple way to prove this or at least calculate said sum?
| If $k = \lfloor \log n \rfloor$ then you would like to prove that
$$
\sum_{i=0}^k 2^i < 2n
$$
Note that by summing the geometric series,
$$
\sum_{i=0}^k 2^i = \frac{2^{k+1}-1}{2-1} = 2\cdot 2^k - 1 = 2 \cdot 2^{\lfloor \log n \rfloor}-1 < 2n-1 < 2n.
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2848676",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Decomposition of a Matrix by Sparse Matrices Let $\mathbb{F}$ is a field. Consider an $n \times n$ matrix $\bf A$ over $\mathbb{F}$.
$\bf A$ is called sparse matrix over $\mathbb{F}$ iff the number of non-zero entities of $\bf A$ be at most $2n$.
My question:
Consider a non-zero $n \times n$ matrix $\bf M$ over $\... | Over any field, you can do the job with $2n-2$ "sparse" (according to your definition) matrices. Behind lies the decomposition LU.
Take $A_1,\cdots A_{n-1}$ lower triangular with $2$ bands and diagonal vector $[1,\cdots,1]$. Take $A_n,\cdots A_{2n-2}$ upper triangular with $2$ bands.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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What is the measure-theoretic definition of the conditional Wiener measure? The Wiener measure $W$ on the space of (continuous, a posteriori) curves defined on $[0,t]$ is uniquely characterized by being Borel and having prescribed pushforwards (that I shall not write here). It is immediate that $W$ is concentrated on t... | [In your integration formula, $dp$ should be replaced by $(2\pi t)^{-n/2}\exp(-|p|^2/2t)\,dp$.]
One way to present $W_p$ is as the image of $W$ under the transformation sending the path $\{x(s), 0\le s\le t\}$ to the path $[0,t] \ni s\mapsto x(s)-(s/t)[x(t)-p]$. This choice makes the disintegration formula true, and $... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Does independence of each $Y, Z$ from $ X$ imply independence of $f(Y,Z)$ from $X$? I'm trying to figure out a proof for the following statement.
If $X$ and $Y$ are independent, and $X$ and $Z$ are independent,
then $X$ and $f(Y,Z)$ are also independent, for any $f(\cdot, \cdot)$
Is there any counter-example against th... | I like this question. Your statement sounds very credible AND is wrong. One cannot see enough examples of statements like that!
I like Joriki's counterexample. Here is another one. I throw a die, the outcome is A. We take $X$ to be the event $A \in \{1, 2, 3\}$, $Y$ is the event that $A \in \{1, 5\}$ and $Z$ is the eve... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Finding the sum of a series with an n term in the numerator Sum the series:
$$\sum_{n=1}^\infty\frac{2n}{7^{2n-1}}$$
I know it converges, but it's not a geometric series nor is it power/telescoping/alternating. I think having the n term in the numerator makes it difficult to solve.
I took calculus BC a number of years ... | For your specific problem, rewrite $$\sum_{n=1}^\infty\frac{2n}{7^{2n-1}}=2 \times 7\sum_{n=1}^\infty\frac{n}{7^{2n}}=14\sum_{n=1}^\infty\frac{n}{49^{n}}$$ Now, consider
$$\sum_{n=1}^\infty n x^n=x\sum_{n=1}^\infty n x^{n-1}=x\left(\sum_{n=1}^\infty x^{n}\right)'$$
Finish and, when done, make $x=\frac 1 {49}$
| {
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"timestamp": "2023-03-29T00:00:00",
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Find the inverse Laplace transform of $e^{-3s} \frac {3s+1}{s^2-s-6}$ Problem: Find the inverse Laplace transform of $e^{-3s} \frac {3s+1}{s^2-s-6}$
My attempt:
Using the method of partial fractions $\frac {3s+1}{s^2-s-6}=\frac {2}{s-3} + \frac {1}{s+2}$, so that $L^{-1}(e^{-3s}\frac {3s+1}{s^2-s-6})=2L^{-1}(e^{-3s}\fr... | $$f(t)=\mathcal{L^{-1}}( e^{-3s} \frac {3s+1}{s^2-s-6})=\mathcal{L^{-1}}( e^{-3s} \frac {3s+1}{(s+2)(s-3)})$$
$$f(t)=\mathcal{L^{-1}}( e^{-3s} \frac {2s+4+s-3}{(s+2)(s-3)})$$
$$f(t)=\mathcal{L^{-1}}( e^{-3s} \left (\frac {2}{(s-3)}+\frac {1}{(s+2)} \right ))$$
$$f(t)=2\mathcal{L^{-1}} \left (e^{-3s}\frac {1}{s-3}\right... | {
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"timestamp": "2023-03-29T00:00:00",
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Why is the set ${1,x^2,x^3...}$ linearly independent as functions? I know that the set is a basis for the vector space of polynomial over some fields.
but why are they linearly independent?
namely, as function from the polynomial to the base field, what is a proof that they cannot be linearly dependent as functions?
| You are having trouble proving this because it's not true. Over the $p$ element field the polynomial $x^p -x$ is identically $0$ as a function, so $\{x, x^p\}$ is a dependent set.
Those polynomials are in fact dependent over any finite field, since
there are only finitely many functions from a set to itself and the l... | {
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"timestamp": "2023-03-29T00:00:00",
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Does this pattern continue $\lfloor\sqrt{44}\rfloor=6, \lfloor\sqrt{4444}\rfloor=66,\dots$? By observing the following I have a feeling that the pattern continues.
$$\lfloor \sqrt{44} \rfloor=6$$
$$\lfloor \sqrt{4444} \rfloor=66$$
$$\lfloor \sqrt{444444} \rfloor=666$$
$$\lfloor \sqrt{44444444} \rfloor=6666$$
But I'm un... | Hint :
We have $$\left(\frac{6\cdot (10^n-1)}{9}\right)^2=\frac{4\cdot (10^{2n}-1)}{9}-\frac{8\cdot (10^n-1)}{9}$$
Try to find out why this proves that the pattern continues forever.
| {
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"url": "https://math.stackexchange.com/questions/2849609",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "45",
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Convergence of $\int_0^{\pi/2}(\tan x)^p\,dx$
For what values of $p$ the integral is converge/diverge?
$$\int_{0}^{\pi/2} (\tan x)^p ~{\rm d}x$$
I tried use the fact that $\displaystyle{\tan x=\frac{\sin x}{\cos x}}$ but it didn't work.
| You can use the substitution $\tan (x) = t$ to get
$$\int \limits_0^{\pi/2} (\tan(x))^p \, \mathrm{d} x = \int \limits_0^\infty \frac{t^p}{1+t^2} \, \mathrm{d} t \, .$$
Now think about how the integrand behaves for small and large values of $t$ and you should find that $-1 < p < 1$ must hold for the integral to be fini... | {
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Path to Riemann Surfaces and Complex Geometry Right now I'm looking at finding a textbook on complex analysis that will be sufficient enough to prepare me for Riemann Surfaces and Complex Geometry. I'm currently looking at Zill's "A First Course in Complex Analysis", Joseph Bak's "Complex Analysis", Gamelin's "Complex ... | You can download the following text for free. I found it very well written for the beginners.
A first course in Complex Analysis Version 1.53 by Matthias Beck, Gerald Marchesi, Dennis Pixton, Lucas Sabalka.
The book is available at the website
http://math.sfsu.edu/beck/complex.html
| {
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"url": "https://math.stackexchange.com/questions/2849782",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Prove $ \min \left(a+b+\frac1a+\frac1b \right) = 3\sqrt{2}\:$ given $a^2+b^2=1$ Prove that
$$ \min\left(a+b+\frac1a+\frac1b\right) = 3\sqrt{2}$$
Given $$a^2+b^2=1 \quad(a,b \in \mathbb R^+)$$
Without using calculus.
$\mathbf {My Attempt}$
I tried the AM-GM, but this gives $\min = 4 $.
I used Cauchy-Schwarz to get $\q... | Calling $a = \cos u, b = \sin u\;\;$ we have
$$
\left(\cos u + \frac{1}{\cos u}\right)+\left(\sin u + \frac{1}{\sin u}\right)\ge 2\sqrt{\left(\cos u + \frac{1}{\cos u}\right)\left(\sin u + \frac{1}{\sin u}\right)} = 2\sqrt{\frac{(\cos^2 u+1)(\sin^2 u + 1)}{\sin u\cos u}}
$$
Now examining
$$
f(u) = \frac{(\cos^2 u+1)}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2850000",
"timestamp": "2023-03-29T00:00:00",
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Weak convergence implies stochastic boundedness I'm trying to prove the following implication:
Every sequence of random variables which converges weakly is also stochastically bounded.
I know that this is an implication of Prohorov's Theorem but I would prefer a direct approach. The overall aim is to deduce tightness ... | Let $\epsilon >0$. There exists $M$ such that $P\{|X| >M\} <\epsilon$. [ Because the events $\{|X| >M\}$ decrease to the empty set]. There exists $M_1 >M$ such that $M$ and $-M$ are continuity points for the distribution of $X$. [ Because there are at most countably many points where the distribution function of $X$ i... | {
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Prove that the elements of X all have the same weight I have had this problem on my mind for weeks and haven't been able to find a solution. Let $X$ be a set with $2n+1$ elements, each of which has a positive "weight" (formally, we could say there exists a weight function $w:X\to \mathbb{R}^+$). Suppose that for every ... | The weights $w_i$ span a vector space $V:=\langle w_1,w_2,\ldots,w_{2n+1}\rangle$ over ${\mathbb Q}$ of dimension $d\leq2n+1$. Let $(\xi_1,\ldots,\xi_d)$ with $\xi_k\in{\mathbb R}$ be a basis of $V$. Then there are rational numbers $a_{ik}$ such that
$$w_i=\sum_{k=1}^d a_{ik}\,\xi_k\qquad(1\leq i\leq 2n+1)\ .$$
Any int... | {
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"url": "https://math.stackexchange.com/questions/2850197",
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what should the bounds of integration be?
I'm confused if the region on xy-plane is a rectangle $R = [-4,4]*[-2,2]$ or something like $-2\le y\le2$ and $-4+y^2 \le x\le4-y^2$. If someone could provide me with an explanation that would be great.
| Your region on the $xy$-plane is bounded by the horizontal parabolae
$$4-y^2-x=0, \quad 4-y^2+x=0.$$
This is because you know that the three-dimensional region $R$ whose volume you are after is defined as
$$R=\{(x,y,z)\in\mathbb R^3\ |\ 0 \leq z \leq f(x,y)\}$$
and the intersection of $R$ with the $xy$-plane happens wh... | {
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Fibres of a Holomorphic Function Let $f$ : $U\rightarrow V$ be a proper holomorphic map where $U$ and $V$ are open subsets of $\mathbb{C}$ with $V$ connected. Show that the cardinality of the fibres of $f$, i.e. $f^{-1}(\{z\})$ counted with the multiplicities are the same for each $z \in$ $V$. This looks like the prope... | This is very simple using some complex analysis.
Since $f$ is proper, given $p\in V$ and small enough $\epsilon>0$ there exists a cycle $\Gamma\subset U$ such that if $|p-q|<\epsilon$ then all the zeroes of $f-q$ lie "inside" $\Gamma$, and in fact such that if $z$ is a zero of $f-q$ then the index of $\Gamma$ about $z$... | {
"language": "en",
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Parameters of Weierstrass Elliptic Function I am currently studying the Chen-Gackstatter surface, and in the link it uses Enneper-Weierstrass parameterization of the surface.
A function called Weierstrass elliptic function is used to define the parametrization, and I have seen the wiki page of the Weierstrass elliptic... | To make Somos's comments more explicit: it is not very hard to use the classical results of the "lemniscatic case" of the Weierstrass $\wp$ function to derive the required invariants $g_2,g_3$ (the proper term for what OP termed the "parameters", as they enter in the defining cubic $4u^3-g_2 u -g_3$). I gave a derivati... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2850495",
"timestamp": "2023-03-29T00:00:00",
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Lipschitz Continuity of $f(t,y)=e^{-t(t^2+y^2)}-(t^2+y^2)^{\frac{1}{4}}-sin(t)$ Prove that the $$f(t,y)=e^{-t(t^2+y^2)}-(t^2+y^2)^{\frac{1}{4}}-sin(t)$$
where
$t^2+y^2 \leq 1$ is not Lipschitz continuous with respect to $y$ in any region around $(0,0)$ This $f(t,y)$ is part of an o.d.e problem im trying to solve
$y... | Generally, one proves Lipschitz continuity by taking the derivative with respect to $y$ and showing it is bounded.
This strategy will fail here because of the term $(t^2+y^2)^{\frac{1}{4}}$, which has unbounded $y$-derivative (and therefore is not Lipschitz continuous) in a neighborhood of $(0,0)$. In fact, this can b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2851592",
"timestamp": "2023-03-29T00:00:00",
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Find $E[X\mid Y]$ where $Y$ is uniform on $[0,1]$ and $X$ is uniform on $[1,e^Y]$ Let $Y$ a uniform random variable in $[0,1]$, and let $X$ a uniform variable in $[1,e^Y]$
My work
By definition
$E(X\mid Y)=\int_{x\in A}xp(x\mid y)\,dx$
Now, I need the function $f(x\mid y)$.
By definition, $f(x\mid y)=\frac{f(x,y)}{f_y... | My 2 cents... and I'm not an expert so see if this makes sense to you.
$E[X|Y]$ is a random variable and not a constant. So for my attack I'll first get $E[X|Y=k]$ and then once I get that switch $k$ for $Y$.
$$E[X|Y=k] = \int_{1}^{e^k} x f_{X|Y}(x|y=k) dx$$
We know that $X|Y=k \sim Uniform(1,e^k)$ which means $f_{X|Y}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2851733",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Radius of a circumference having two chords In the image, the lenghts of the chords are $6$ and $8$, and the gap between the chords is $1$. Then the radius of the circumference is?
I drew the perpendicular diameters to the chords and tried to apply power of a point, but i didn't find anything. I need some hints.
| By Pythagoras,
$$\sqrt{r^2-3^2}-\sqrt{r^2-4^2}=1$$
then
$$(r^2-3^2)-2\sqrt{r^2-3^2}\sqrt{r^2-4^2}+(r^2-4^2)=1,$$
$$(2r^2-26)^2=4(r^2-9)(r^2-16),$$
$$4r^2-100=0.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2851872",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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what is the probability that in a random group of seven people two were born on Monday and two on Sunday? If people can be born with the same probability any day of the week, what is the probability that in a random group of seven people two were born on Monday and two on Sunday?
My analysis:
7 people can be born in 7... | There are only $^7C_4\cdot \frac{4!}{2!\cdot 2!}\cdot 5^3$ ways to have four out of seven people with two birthdays on Monday and two on Tuesday.
$P(\text{mmtt}) = \frac{^7C_4\cdot \frac{4!}{2!\cdot 2!}\cdot 5^3}{7^7} = .03187$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2851973",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
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Help showing aMarkov chain with a doubly-stochastic matrix has uniform limiting distribution I have a lot of difficulty with proofs; could someone help me with this question that really can not solve? I would also like some indication of material to get through with this kind of question and some hint of material about... | $X_1=X_0P$, where $P$ is the transition matrix. As $X_0=[1/k,\ldots,1/k]$, one would have $X_1^i=\frac{1}{k}(P_{2i}+P_{1i}+\ldots+P_{ki})=\frac{1}{k}$ by the double stochastic property (sum of the entries on colums are $1$). The general result follows by induction.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2852052",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
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} |
Group Action of $SO(3)$ on unit sphere We know that $SO(3)$ acts transitively on $S^2$. That is, given any two points $p_1,p_2\in S^2$, one can find an element in $SO(3)$ mapping $p_1$ to $p_2$. My question is, given two sets of points $(a_1,a_2)$ and $(b_1,b_2)$ on $S^2$ such that the induced distance on sphere are th... | Yes. Take $a_3 = a_1 \times a_2$ and $b_3 = b_1 \times b_2$. Define the map $T: \mathbb R^3 \to \mathbb R^3$ by the formula $T(a_i) = b_i$. This map is a oriented isometry of $\mathbb R^3$ and therefore belongs to $SO(3)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2852148",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Prove that $\left|\left\{ 1\leq x\leq p^{2}\ :\ p^{2}\mid\left(x^{p-1}-1\right)\right\} \right|=p-1$ Let $p$ be a prime number and to simplify things lets denote
$$
A=\left\{ 1\leq x\leq p^{2}\ :\ p^{2}\mid\left(x^{p-1}-1\right)\right\}
$$
and we have to show that $\left|A\right|=p-1$.
For every $x\in A$ we know that $... | Let's fix an integer $a$ in ther range $1\le a<p$. By Little Fermat we know that $a^{p-1}\equiv1\pmod p$. We use this to study the number of solutions $x\in A$ such that $x\equiv a\pmod p$.
So let $x=a+kp$ for some $k$, $0\le k<p$. The binomial theorem tells us that
$$
\begin{aligned}
x^{p-1}&=(a+kp)^{p-1}\\
&=a^{p-1}+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2852257",
"timestamp": "2023-03-29T00:00:00",
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