Thursday, December 2, 2021

Recent Questions - Mathematics Stack Exchange

Recent Questions - Mathematics Stack Exchange

Checking subspaces for independence

Posted: 02 Dec 2021 01:16 PM PST

Assume: $V:= \mathbb{Q}[\sqrt{2}] = \left \{ a+b \sqrt{2} \right \}$ My task is to check if the following subset is linear independent: $\\$

$\left \{ 10, 3+ \sqrt{2} \right \}$

My approach: $\alpha 10 + \beta ({3+ \sqrt{2}})=0$.
I am looking for a non trivial solution. Therefore I am searching for at least one facor $\alpha$ , $\beta$ such that one term cancels out and I can set the other factor to zero therefore the hole equation is zero. But I have no idea how to solve it..

Check if $\sum_{n=1}^{\infty}\left|\frac{\ln(n+1)}{\ln(n^2+1)}-\frac{1}{2}\right|$ converges

Posted: 02 Dec 2021 01:10 PM PST

This is the series: $$\sum_{n=1}^{\infty}\left|\frac{\ln(n+1)}{\ln(n^2+1)}-\frac{1}{2}\right|$$

We can drop the absolute value, because; $$\frac{\ln(n+1)}{\ln(n^2+1)} > \frac{1}{2}$$

I've tried using inequalities below to form comparison tests, but to no avail.

$$\frac{n}{n+1} < \ln(n+1) < n$$ $$\frac{1}{n^2} < \frac{1}{\ln(n^2+1)} < \frac{n^2+1}{n^2}$$

I also tried d'Alembert's ratio test, but it resulted in limit of 1, which is inconclusive. I have no clues on how to approach this.

Spivak Partitions of Unity

Posted: 02 Dec 2021 01:04 PM PST

On page 65 of Spivak's Calculus on Manifolds, he states enter image description here

Why does $\int_A\phi \cdot |f|$ exist even if $A$ is not bounded? Thanks.

How to find the analytical expression for the measured step response of a dynamic system?

Posted: 02 Dec 2021 01:03 PM PST

let's say I have following situation. I have recorded a step response of a dynamic system in following form

enter image description here

The data have been gathered with the sampling period $100\,\mu s$. My goal is to find the analytical expression of the step response. Based on the dynamic model of the system I expected that the system should have following transfer function

$$\frac{A}{\tau\cdot s + 1}\cdot e^{-s\cdot T_d}.$$

Based on that I supposed the Laplace transform of the step response in following form

$$A\cdot\frac{\frac{1}{\tau}}{s\cdot(s + \frac{1}{\tau})}\cdot e^{-s\cdot T_d}.$$

i.e. I supposed that I have been looking for a function in following form

$$A\cdot\left[1 - e^{\left(\frac{-(t-T_d)}{\tau}\right)}\right].$$

Then I have used the least square method for finding the unknown parameters $A, T_d, \tau$. Unfortunately this approach resulted in following poor conformity between the measured data and fitted curve.

enter image description here

It seems that the dynamic of the system is more complex than I expected. Can anybody give me an advice how to find the analytical expression for the measured step response?

Complement of the $3$-dimensional Cube $\overline {Q_3}$ is nonplanar

Posted: 02 Dec 2021 01:02 PM PST

I was asked the question

Prove that the compliment of the $3$-dimensional Cube, $\overline {Q_3}$ is not planar.

I tried using $e\leq 3v-6$ but that doesn't work in this case since $e=16$ and $v=8$. Also, I tried finding a $K_{3,3}$ or a $K_5$ in it which I couldn't.

Any ideas how to do it?

Limit property of second derivative of bounded monotone function

Posted: 02 Dec 2021 01:02 PM PST

Suppose $f:\mathbb R \rightarrow \mathbb R$ is twice continuously differentiable, bounded and monotone. Is it possible to show that $\lim\inf_{x\rightarrow \infty} x^{2}f''(x)\leq 0$?

This is established in the following question when $\lim_{x\rightarrow \infty} xf'(x)= 0$:

Limit property of second derivative of bounded function.

The proof offered here relies on this property of the first derivative. I am wondering if it is possible to show without this property?

proving 1/(x^a + 1) is convex using the convex definition with 0<a<=1 (without differentiating)

Posted: 02 Dec 2021 01:01 PM PST

I want to show that for $\lambda, a \in [0,1]$ and $x>0$

$f \left( \lambda x_1+(1-\lambda)x_2 \right) \leq \lambda f(x_1) + (1-\lambda) f(x_2)$

In other words,

$1/(\left( \lambda x_1+(1-\lambda)x_2 \right)^a + 1) \leq \lambda/(x_1^a + 1) + (1-\lambda)/(x_2^a + 1)$.

I know that this function is convex, but I found tough to solve it with the convex definition.

Any hints will help!

Exercise 1.6 from Stein's Real Analysis

Posted: 02 Dec 2021 12:52 PM PST

This is an exercise from Chapter 1 of Stein and Shakarchi's Real Analysis. This has been posted in different forms on StackExchange. However, I have a different question than what is provided elsewhere. The problem states:

Let $\delta = (\delta_1, \dots, \delta_d)$ be a d-tuple of positive numbers; for a subset $E \subset \mathbb{R}^d$ define $$\delta E = \big\{(\delta_1x_1,\dots, \delta_dx_d ):(x_1,\dots, x_d) \in E\big\}$$ If $E$ is measurable, then show that $\delta E$ is measurable as well and $$m(\delta E) = \delta_1\cdots \delta_dm(E)$$

I have proven that $\delta E$ is measurable. Thus, $m(\delta E) = m_*(\delta E)$, where $m_*$ is the exterior measure. For the second assertion, do the following deductions then hold?

$$m(\delta E) = m_*(\delta E) = \inf \Big\{\sum_{j=1}^\infty m(Q_j) : \delta E \subset \bigcup_{k=1}^\infty Q_k\Big\} = \inf \Big\{\sum_{j=1}^\infty m(\delta Q_j) : E \subset \bigcup_{k=1}^\infty Q_k\Big\}$$ $$= \inf \Big\{ \delta_1 \cdots \delta_d \sum_{j=1}^\infty m(Q_k) : E \subset \bigcup_{k=1}^\infty Q_k \Big\} = \delta_1\cdots \delta_d m(E)$$

Here the infimum is taken over all coverings $\big\{Q_i\big\}$ of $\delta E$ by cubes in $\mathbb{R}^d$.

Understanding Why we Integrate joint density function with opposite bounds to get marginal density

Posted: 02 Dec 2021 01:13 PM PST

I have a function $f_{x,y}(x,y)$ which represents the joint density function. In order to get marginal density function in terms of $x$, I need to integrate using $y$ bounds. Why is that? I assumed that we would integrate with $x$ bounds.

Is a Poisson process (in law) a.s. *strictly* increasing?

Posted: 02 Dec 2021 12:48 PM PST

Let $\operatorname{Poi}(\lambda)$ denote the Poisson measure with parameter $\lambda>0$, i.e. it is the measure with density $$\mathbb N_0\to(0,\infty)\;,\;\;\;n\mapsto\frac{\lambda^n}{n!}e^{-\lambda}\tag1$$ with respect to the counting measure $\zeta$ on $(\mathbb N_0,2^{\mathbb N_0})$.

We usually say that $(N_t)_{t\ge0}$ is a Poisson process (in law) on a probability space $(\Omega,\mathcal A,\operatorname P)$ if $N_0=0$ and $N_t$ is $\operatorname{Poi}(t\lambda)$-distributed for all $t\ge0$.

It is easy to show that such a process is nondecreasing outside a $\operatorname P$-null set (see, for example, https://math.stackexchange.com/a/1672217/47771).

However, in order to show (for example) that the $n$th-arrival time $$\tau_n:=\inf\underbrace{\{t\ge0:N_t=n\}}_{=:\:I_n}$$ is a stopping time (with respect to the filtration genated by $(N_t)_{t\ge0})$ satisfying $N_{\tau_n}=n$, we even need that $(N_t)_{t\ge0}$ is (strictly!) increasing. Can we show that this is actually the case; outside a $\operatorname P$-null set?

Any way to determine the applied sequence of a matrix chain multiplication? (or simpler cases tetrahedron attachment, finite field, rotation)

Posted: 02 Dec 2021 12:47 PM PST

General problem
Given a starting matrix $S$ and an ending matrix $E$ and $m$ ($>1$) transforming square matrices $M_i$.
It is know that $E$ can be constructed out of $S$ by applying a sequence of matrix multiplications like: $$E = ((SM_{i_1})M_{i_2})M_{i_3}... = S \prod_{j=1}^{N_M} M_{i_j}$$ $$i_j \in \{1,..,m\}$$

Given $E$, $S$ and all $M_i$ is there any (efficient) way to determine any sequence of matrix multiplications which is able to construct $E$ out of $S$?
($N_M$ is unknown. If there are multiple solutions just one is sufficient. In best case this with smallest $N_M'$)

Example
Let $m = 4$ and $E = SM_3M_2M_1M_4M_1M_4M_3M_2M_4$
Given $E,S,M_1,M_2,M_3,M_4$. I'm looking for a way to efficiently determine the sequence $[3,2,1,4,1,4,3,2,4]$ (or any other sequence with same result) without testing any possible multiplication.

Possible simplification for tetrahedrons
More specific I'm looking for a solution related to attached regular tetrahedrons (triangular pyramid)
Given a the vertices of regular tetrahedron $A,B,C,D$ an attached tetrahedron (attached to one side triangular face) can be constructed by reflection one vertex at the plane defined by the three other vertices. E.g. to construct A' out of A this would be: $$A' = (B+C+D)\cdot \frac{2}{3}-A$$ The starting matrix $S$ would be defined as: $$S = [A\text{ }B\text{ } C\text{ } D] = \begin{bmatrix} A_x & B_x & C_x & D_x\\ A_y & B_y & C_y & D_x\\ A_Z & B_z & C_z & D_y\\ \end{bmatrix}$$ $m=4$ transformation matrices $M_1,M_2,M_3,M_4$ would exist which are very similar to each other. E.g. $$M_1 =M_A = \begin{bmatrix} -1 & 0 & 0 & 0\\ \frac{2}{3} & 1 & 0 & 0\\ \frac{2}{3} & 0 & 1 & 0\\ \frac{2}{3} & 0 & 0 & 1\\ \end{bmatrix}$$

As those $M_i$ are similar to each other and much more limited than a random $M$ would this make finding a multiplication chain in between $S$ and $E$ easier?

Simplification 2 or even harder in finite field?
Even more specific I'm looking for a solution applied to attached regular tetrahedrons in a finite field $\mathbb{F}_P$ with $P$ a prime.
Not much would change at the formula above. Just the the division need to be replaced by the inverse (for $M_i$ as well). $$A' \equiv (B+C+D)\cdot 2 \cdot (3)^{-1}-A \mod P$$

Would this make finding a sequence for given $S$, $E$ easier or even harder?

Simplification 3 rotation
Instead of storing the tetrahedron position just storing the rotation would be enough for target application. The tetrahedron center would be: $$ T = (A+B+C+D)\cdot (4)^{-1} \mod N$$ And the starting rotation would be: $$S = [A\text{ }B \text{ } C \text{ }D]-T \mod N $$ This would define the direction vector from center to each vertex.
($M_1,M_2,M_3,M_4$ need to be changed as well.)

Finding a sequence from $S$ to $E$ would reducing the problem to finding a tetrahedron with same rotation. (This won't be to be the same solution as for above including the position)

Or is this problem equivalent to another (better studied) problem?

Question regarding baby do Carmo's definition of a regular surface.

Posted: 02 Dec 2021 12:54 PM PST

The following is part of baby do Carmo´s definition of a regular surface (at the beginning of the second chapter):

First part of do Carmo's definition Second part of do Carmo's definition

This seems to imply that any differentiable function $f:U\rightarrow \mathbb{R}^3$ where $U$ is an open subset of $\mathbb{R}^2$ must have continuous partial differential derivatives of all orders. Is this true? If so, how can it be shown?

What I do know is that if a function $f:\mathbb{R}^n\rightarrow \mathbb{R}^m$ is differentiable at a point $x\in U$, then its partial derivatives exist at $x$. Also, if the partial derivatives are continuous at $x$, then the function $f$ is (total) differentiable at $x$.

How to show that $\sqrt{\sum_{i=1}^n a_i} \leq \sum_{i=1}^n \sqrt {a_i}, \quad a_i \in [0,\infty)$? [closed]

Posted: 02 Dec 2021 01:05 PM PST

$\sqrt{\sum_{i=1}^n a_i} \leq \sum_{i=1}^n \sqrt {a_i}, \quad a_i \in [0,\infty)$

Optimisation Question- can someone please check this for me?

Posted: 02 Dec 2021 01:06 PM PST

Can someone please check the general logic for part (i) and check the part (ii)

Question:

enter image description here

My Answer:

enter image description here

Gaussian random variables and change of variables

Posted: 02 Dec 2021 01:02 PM PST

I have the following situation

Let $N_1, N_2 \sim \mathcal{N}(0,1)$ two independent r.v. Let $X = \frac{N_1}{\sqrt{N_{1}^{2} + N_2^2}}$ and $Y = \frac{N_2}{\sqrt{N_{1}^{2} + N_2^2}}$.

Now I know how show to that $X$ and $Y$ are not independent, but I don't know how to show $X$ and $Y$ are uncorrelated. Can anybody helps me?

Thanks in advance

Evaluate: $\lim_{n\to\infty} \int_a^{\infty}\frac{n^2xe^{-n^2x^2}}{1+x^2}\,dx.$

Posted: 02 Dec 2021 12:54 PM PST

Show that $\displaystyle\lim_{n\to\infty}\displaystyle\int_a^{\infty}\frac{n^2xe^{-n^2x^2}}{1+x^2}\,dx=0$ for $a>0$ but not for $a=0.$

My work: Let the value of the given limit be $L.$ Put $n^2x^2=t$ in the integrand. Then we have: $$L=\frac12\cdot\lim_{n\to\infty}\int_{n^2a^2}^{\infty}\frac{e^{-t}}{1+t/n^2}\,dt.\tag1$$ Now plugging in $t=n^2a^2+z$ yields: $$L=\frac12\cdot\lim_{n\to\infty}\int_0^{\infty}\frac{e^{-n^2a^2}\cdot e^{-z}}{1+a^2+z/n^2}\,dz.$$ The last integrand is less than $e^{-z},$ an integrable function in $[0,\infty).$ Also as $n\to\infty,$ the same integrand approaches $0.$ Hence by dominated convergence theorem, we obtain that $L=0.$

The way I have solved the problem, I am not noticing any effect of the sign of $a$ on the value of $L.$ However, if I consider $a=0$ then from $(1)$ using a similar approach I found that $L=\frac12.$ So sign of $a$ does matter. So my main query is what is wrong with my approach? Please give some insights.

Thanks in advance.

Find subgroup $\langle a,b\rangle$ of $\Bbb Z_{20}^*$ which is not cyclic.

Posted: 02 Dec 2021 01:00 PM PST

Find subgroup $\langle a,b\rangle$ of $\Bbb Z_{20}^*$ which is not cyclic.

I know that $\mathbb{Z}_{20}^{*}$ is not cyclic

But how can I find subgroup which is not cyclic?

e.g why is $\langle 3,11\rangle$ not cyclic?

Thanks and sorry if I have English mistakes :)

Bayes Theorem - Discrete Math

Posted: 02 Dec 2021 01:16 PM PST

Suppose that one person in 10,000 people has a rare genetic disease. There is an excellent test for the disease; 99.5% of people with the disease test positive and only 0.06% who do not have the disease test positive.
What is the probability that someone who tests positive has the genetic disease?

$P(A) = 0.0001 ( 1/10000)$ (People having the disease among population)
$P(B|A) = 99.5/100 = 0.995$ (People who are positive)
$P(\bar{A}) = 1 - 0.0001 = 0.9999$ (People who dont have the disease)
$P(B|\bar{A}) = 0.06/100 = 0.0006$ (People who show up positive but dont have the disease)

Using the formula :
$$P(A|B)=\frac{P(A)P(B|A)}{P(A)P(B|A)+P(\bar{A})P(B|\bar{A})}$$

We determine that the probability of some who tests positive and has the genetic disease is :

$$\frac{0.0001 * 0.995}{0.0001*0.995 + 0.9999*0.0006}$$
= 0.1422

Is this the right approach to this problem, please advise.

Find formulas for the entries of $��^{\,��}$, where �� is a positive integer.

Posted: 02 Dec 2021 12:49 PM PST

Let

$$M = \begin{pmatrix}\ \ 3&6\\-3& 12\end{pmatrix}$$

Find formulas for the entries of $M^n$ where 𝑛 is a positive integer. I found the eigenvalues and eigenvectors but I keep getting the answer wrong.

In what sense does a number "exist" if it is proven to be uncomputable?

Posted: 02 Dec 2021 12:46 PM PST

Uncomputable functions: Intro

The last month I have been going down the rabbit hole of googology (mathematical study of large numbers) in my free time. I am still trying to wrap my head around the seeming paradox of the existence of natural numbers that are well-defined but uncomputable (in the sense that it has been proven that they can never be calculated by a human / a Turing machine). Let me give two of the most famous examples:

Busy beaver function $\Sigma(n,m)$

$\Sigma(n,m)$ "is defined as the maximum number of non-blank symbols that can be written (in the finished tape) with an $n$-state, $m$-color halting Turing machine starting from a blank tape before halting." It has been shown that $\Sigma$ grows faster than all computable functions and, thus, is uncomputable. Calculating $\Sigma$ for sufficiently large inputs would require an oracle Turing machine as it would literally be a solution to the halting problem. Thus, it is uncomputable, although the forumlation of $\Sigma$ in set theory is precise and clear. More details here.

Rayo's number $\text{Rayo}\left(10^{100}\right)$

Rayo's number was the record holder in the googology community for a long time and it is defined as "the smallest positive integer bigger than any finite positive integer named by an expression in the language of first-order set theory with googol symbols or less." It is defined in the language of an (unspecified) second-order set theory here. (Its well-definedness is thusly a bit controversial but it would outgrow $\Sigma$ by a huge marigin if resolved.)

My mathematical / existential questions

  • Does a number like $x=\Sigma\left(10^{100},10^{100}\right)$ "exist" in set theory in the same sense like the number $4$? Does it even make sense to include it in a mathematical operation like $(x$ mod $4)$ or $x^x$ if we cannot even write it down in a decimal expansion?

  • I am well aware of Gödel's incompleteness theorems and the existence of unprovable statements like the continuum hypthesis, which can neither be proven nor disproven by ZFC axioms in any finite number of steps. Is there some parallel between that and the existence of numbers that cannot be computed in any finite amount of time?

  • Is there some version of mathematics or system of axioms which resolves this problem? (i.e. where well-definedness of an object is equivalent to computability?)

I would be very happy if anyone could answer or point me in the right direction.

What´s the value of the area of the triangle below?

Posted: 02 Dec 2021 12:57 PM PST

For reference: The sides of an acute-angled triangle measure $3\sqrt2$, $\sqrt{26}$ and $\sqrt{20}$. Calculate the area of ​​the triangle (Answer:$9$)

My progress... Is there any way other than Heron's formula since the accounts would be laborious or algebraic manipulation for the resolution?

$p=\frac{\sqrt{18}+\sqrt{20}+\sqrt{26}}{2}\implies S_{ABC} =\sqrt{p(p-\sqrt{20})(p-\sqrt{26})(p-\sqrt{18})}$

Is there a name for this differential equation?

Posted: 02 Dec 2021 12:55 PM PST

I am studying the temperature $T$ of a wire that generates heat due to current flow. The electric resistivity is temperature dependent, so I end up with the equation:

$$\Delta T + \lambda T = f $$

If $\lambda = 0$ then it's the Poisson equation, but is there a name for this PDE in the case of $\lambda\neq 0$ ?

To compute $\lim_n (1+n)^{\frac1{\ln n}}$ without L'Hospital

Posted: 02 Dec 2021 12:56 PM PST

We suppose that have this limit of a succession $n\in \Bbb N$ with real terms $$\lim_n (1+n)^{\frac1{\ln n}}$$ Now this limit have the value $e$. In fact I have done these steps remebering that $f^g=e^{g\ln f}$

$$\lim_n (1+n)^{\frac1{\ln n}}=\lim_n e^{ \frac{\ln (1+n)}{\ln n}}$$

Without L'Hospital's rule, I do not remember the result of $$\lim_n \frac{\ln (1+n)}{\ln n}$$ (related here - I have seen this now Limit of $\underset{n\to \infty }{\text{lim}}\frac{\ln (n+1)}{\ln (n)}$ without L'Hôpital) hence I had due to compute the $$\frac{\ln (1+n)}{\ln n}$$ when $n$ is more larger than 1. In fact if I choose $n=1000$, for example, I will have $\frac{\ln (1+n)}{\ln n} \to 1$ and $$\lim_n (1+n)^{\frac1{\ln n}}=\lim_n e^{ \frac{\ln (1+n)}{\ln n}}=e^1=e.\,\square$$

This way I do not like very much; hence I have tried another way knowing that

$$\lim_n (1+n)^{\frac 1n}=e$$

Thus, if I remember well, $\lim\limits_n \frac n{\ln n}=+\infty$ and $\lim\limits_n \frac{\ln n}n=0$, and

$$\lim_n (1+n)^{\frac1{\ln n}}=\lim_n \left[(1+n)^{\frac1{n}}\right]^{\frac n{\ln n}}=e^\infty=+\infty\neq e.$$ Actually I do not see my mistake. Thank you very much everyone.

Application of Rolle's

Posted: 02 Dec 2021 01:02 PM PST

Suppose $q$ is a nonzero function of a real-variable such that $$u^2q''(u)+uq'(u)=u^2q(u)+q(u)$$ for all $u$.

Assume there exist $x,y$ such that $q(x)=q(y)=0$. By Rolle's there exists $x<z<y$ such that $q'(z)=0$. Plugging $z$ into the above equation is $$z^2q''(z)+zq'(z)=z^2q(z)+q(z)\iff z^2q''(z)=z^2q(z)+q(z).$$ And plugging $x,y$ in the equation is $$x^2q''(x)+xq'(x)=0\land y^2q''(y)+yq'(y)=0\implies xq''(x)+q(x)=0\land yq''(y)+q(y)=0.$$ I try to deduce contradiction from above but I don't know the next step. One strategy can be to show one of three equalities is actually is strict inequality. Another one can be to pursue second derivatives in the intervals $(x,z)$ and $(z,y)$ by mean value theorem. But none of these seem to show anything important. I would appreciate a hint.

Probability with card deck flips

Posted: 02 Dec 2021 01:00 PM PST

Here is my problem : we flip cards from a 52-card standard well-shuffled deck until the first club appears. I am looking to calculate the probability that the next card at the $k+1$th flip is also a club given that the $k$th flip is a club. Let $T$ be the flip on which we encounter the first club. Thanks to this answer I get $$\mathbb{E}[T]=\frac{53}{14} \approx 3.7857$$ Now let $Y_n=1$ if we flip a club on the $n$th flip and $Y_n=0$ if we flip another suit. The number of clubs flipped amongst the first $n$ flips would be $$C_n=\sum_{k=1}^n Y_k$$ with $C_T=1$. After the $n$th flip, we have $\tilde{X}_n$ clubs remaining in the deck with proportion $X_n$: $$X_n =\frac{\tilde{X}_n}{52 - n}, \ \tilde{X}_n = 13 - C_n$$ with $\tilde{X}_T=12$. So $$X_T = \frac{13-C_T}{52 - T} = \frac{12}{52 - T}$$ We get $$\mathbb{E}[X_T] = \frac{12}{52 - 3.7857} \approx 0.2489$$ the probability that the next card is a club. Can I use $\mathbb{E}[T]$ in the denominator like this? Thanks!

Deriving Finsler geodesic equations from the energy functional

Posted: 02 Dec 2021 12:59 PM PST

I'm struggling to derive the Finsler geodesic equations. The books I know either skip the computation or use the length functional directly. I want to use the energy. Let $(M,F)$ be a Finsler manifold and consider the energy functional $$E[\gamma] = \frac{1}{2}\int_I F^2_{\gamma(t)}(\dot{\gamma}(t))\,{\rm d}t\tag{1}$$evaluated along a (regular) curve $\gamma\colon I \to M$. We use tangent coordinates $(x^1,\ldots,x^n,v^1,\ldots, v^n)$ on $TM$ and write $g_{ij}(x,v)$ for the components of the fundamental tensor of $(M,F)$. We may take for granted (using Einstein's convention) that $$F^2_x(v) = g_{ij}(x,v)v^iv^j, \quad \frac{1}{2}\frac{\partial F^2}{\partial v^i}(x,v) = g_{ij}(x,v)v^j, \quad\frac{\partial g_{ij}}{\partial v^k}(x,v)v^k = 0.\tag{2} $$

Setting $L(x,v) = (1/2) F_x^2(v)$, and writing $(\gamma(t),\dot{\gamma}(t)) \sim (x(t),v(t))$, the Euler-Lagrange equations are $$0 = \frac{{\rm d}}{{\rm d}t}\left(\frac{\partial L}{\partial v^k}(x(t),v(t))\right) -\frac{\partial L}{\partial x^k}(x(t),v(t)),\quad k=1,\ldots, n=\dim(M).\tag{3}$$It's easy to see (omitting application points) that $$\frac{\partial L}{\partial x^k} = \frac{1}{2}\frac{\partial g_{ij}}{\partial x^k}\dot{x}^i\dot{x}^j\quad\mbox{and}\quad \frac{\partial L}{\partial v^k} = g_{ik}\dot{x}^i,\tag{4}$$so $$\frac{\rm d}{{\rm d}t}\left(\frac{\partial L}{\partial v^k}\right) = \frac{\partial g_{ik}}{\partial x^j}\dot{x}^j\dot{x}^i +{\color{red}{ \frac{\partial g_{ik}}{\partial v^j} \ddot{x}^j\dot{x}^i }}+ g_{ik}\ddot{x}^i\tag{5}$$ Problem: I cannot see for the life of me how to get rid of these $v^j$-derivatives indicated in red, even using the last relation in (2), as the indices simply don't match. I am surely missing something obvious. Once we know that this term does vanish, then (4) and (5) combine to give $$ g_{ik}\ddot{x}^i + \left(\frac{\partial g_{ik}}{\partial x^j} - \frac{1}{2}\frac{\partial g_{ij}}{\partial x^k}\right)\dot{x}^i\dot{x}^j =0\tag{6}$$as in the Wikipedia page.

Recurrence relation for Kravchuk polynomials

Posted: 02 Dec 2021 01:03 PM PST

I'm reading this article, where they use the following equivalent expressions for the Kravchuk polynomials: \begin{equation} \begin{split} K_j(i) &= \sum_{h=0}^j (-1)^h(q-1)^{j-h} \binom{i}{h}\binom{d-i}{j-h} \\ &= \sum_{h=0}^j (-q)^h(q-1)^{j-h} \binom{i}{h}\binom{d-h}{j-h} \\ &= \sum_{h=0}^j (-1)^h q^{j-h} \binom{d-i}{j-h}\binom{d-j+h}{h} \end{split} \end{equation} In the paper, they claim that the following recurrence relation holds for $i,j\geq 1$ without any proof: \begin{equation} (q-1)(d-i)K_j(i+1) - (i+(q-1)(d-i)-qj)K_j(i) + i K_j(i-1) = 0 \end{equation} Could anyone help me figuring out how to prove this?

Summation by Part and Integration by part.

Posted: 02 Dec 2021 01:11 PM PST

How is summation by part similar to integration by parts? How are they different?

I was doing some questions on integration by parts and summation by parts and now i am curious to know the similarities and differences between both.

How to find probability distribution function given the Moment Generating Function

Posted: 02 Dec 2021 01:03 PM PST

After searching, I found two questions like mine, but didn't see my answer to my question.

  1. Finding a probability distribution given the moment generating function
  2. Finding probability using moment-generating functions

My question is how to find any probability distribution function, given its moment generating function. In particular, how to find this from First Principles (and not memorizing a table).

Let's try an example:

Let $ X \perp Y$. Define the moment generating functions for $X, Y$ respectively as $$M_X(t)=\exp(2e^t-2), M_Y(t)=\left(\frac{3}{4}e^t+ \frac{1}{4}\right)^{10}$$ Find $P(X+Y = 2)$.

First, the problem doesn't tell us whether the distributions are continuous or discrete, so I assume continuous. Now, how do we solve the following for $f_X(x)$?

$$M_X(t)= \int_{-\infty}^{\infty}e^{xt} f_X(x) \ dx = \exp ( 2\ e^t - 2)\tag{1}$$

Next, can we take the derivative with respect to $x$ to both sides, to bring us closer to the solution $f_X(x)$?

I read that a m.g.f. $m_X(t)$ is characteristic to and unique to the distribution of $X$. I saw something about Laplace Transforms in another question, but we have learned nothing of that sort in this course.

If the graph of a function $f: A \rightarrow \mathbb R$ is compact, is $f$ continuous where $A$ is a compact metric space?

Posted: 02 Dec 2021 01:09 PM PST

I have seen answers to this question, which go beyond my understanding of compactness and continuity. I was wondering whether we can cook up a proof using sequential compactness and certain equivalent definitions of continuity such as the inverse image of any closed set is closed.

Here is what I have been able to conjure up so far.

Assume that the graph of $f$ is compact. This means that it is also closed and bounded. The graph is a closed and bounded subset of $A \times f(A)$. All we need to show is that $f(A)$ is compact, and we are are home free, right? (since continuous functions take compact sets to compact sets).

Question is: how do we show that $f(A)$ using the fact that the graph is compact. Can we claim that $f(A)$ is closed and bounded (since by Heine-Borel, any closed and bounded subset of $\mathbb R$ is compact)?

I feel like I am really close. Can anyone help me out?
Posted by at

No comments:

Post a Comment

Subscribe to: Post Comments (Atom)