Recent Questions - Mathematics Stack Exchange

Prove with holder $\frac{a_1^2}{a_2}+\frac{a_2^2}+\dots+\frac{a_n^2}{a_1}\ge a_1+a_2+\dots a_n$
Torsional free modules , free modules
Surface area under a curve (made by incomplete sprial pattern) on a sphere
Completion of a complete Measure Space
Proof of that $(1+\frac{1}{n})^n$ converges for real $n \to \infty$ [duplicate]
Analog to eigenvalue decomposition that ranks by something other than L2 norm
Prove that $diam_{d_1}([a, b]) = |b - a|$ and $diam_{d_1}((a, b)) = |b -a|$
Need Help with this Question. Exam coming up on this [closed]
Biconditionals and Conjunctions in Truth Tables
If D is a ring with identity such that every unitary D-module is free, then D is a division ring
what is the meaning of the symbol $p(a;b)$?
Help with product rule word problem
How can I find this limit without L'Hopital and is my current solution correct?
How to calculate the minimum of $x^2-xy+y^8$？
Limit of a Riemann Sum.
GCd of a prime number
Spivak uses a property in its own proof?
Discrete derivative condition for martingale?
Are the Rationals under addition isomorphic to any set of numbers?
Intuition behind the method of characteristics
Multivariable Chain Rule: Conditional Independence and "Swapping" Partial Derivatives
Does $P|_{\mathcal{F}}=Q$ imply $E_P[X]=E_Q[X]$ for $\mathcal{F}$-measurable $X$?
Is $L^i(K_4)$ regular for all $i \in \mathbb{Z}^+$? If so, find the general formula for the degree of $L^i(K_4)$?
Connected Cubic Graph Counting
Is there a way to solve this coupled differential equation?
Expected number of rolls until lcm is greater than $2000$?
Density of lines passing through sides of a rectangle?
$E[f(X_1,X_2)|\mathcal{F}_n]=\frac{2}{n(n-1)}\sum_{ 1 \leq p<q \leq n}f(X_p,X_q)$
How many rounds in a tournament with n players
Linear Algebra and planes in Cartesian space

Prove with holder $\frac{a_1^2}{a_2}+\frac{a_2^2}+\dots+\frac{a_n^2}{a_1}\ge a_1+a_2+\dots a_n$

Posted: 13 Oct 2021 09:39 PM PDT

Prove the following inequalities with Holder.

Prove that for all positive real numbers $a,b,c,x,y,z$ $$\frac{a^3}{x}+\frac{b^3}{y}+\frac{c^3}{z}\ge \frac{(a+b+c)^3}{3(x+y+z)} $$
Prove that $$\frac{a_1^2}{a_2}+\frac{a_2^2}+\dots+\frac{a_n^2}{a_1}\ge a_1+a_2+\dots a_n.$$

For the second problem. I think Titu engel form suffices.

We get $$\frac{a_1^2}{a_2}+\frac{a_2^2}+\dots+\frac{a_n^2}{a_1}\ge \frac{(a_1+a_2\dots+a_n)^2}{a_1+a_2+\dots a_n}=a_1+a_2+\dots a_n.$$

Note sure about the Holder proof though. Any solutions?

Torsional free modules , free modules

Posted: 13 Oct 2021 09:39 PM PDT

This particular question was asked in my Abstract Algebra Exam and I couldn't solve it in examination hall. Now, I thought I should try it again but I was unable to.

Prove that every free module F over an arbitrary integral domain with identity is torsion-free but the converse is false.

Module is given free means that a basis set exists (say X).To prove that it is Torsional free i need to show that every element except identity has infinite order OR Assuming it to be not torsional free and then proving that it can't be free module.

Let $f \in F$. Then I am at loss of ideas how it doesn't have any non-identity element of finite order.

On the other hand, let converse holds. So, Every torsion-free module over an arbitrary integral domain is free. But what is the contradiction here?

I am at loss of ideas over this and would very much appreciate help.

Surface area under a curve (made by incomplete sprial pattern) on a sphere

Posted: 13 Oct 2021 09:28 PM PDT

I have an incomplete spiral pattern (as shown) on a sphere of radius R. I want to find the area under the curve made by the highest points of the incomplete sprial. Image showing the incomplete spiral pattern on a sphere

I have tried solving this in these ways:

Thinking about how one finds an area under a curve on a circle. One would make slices of some width w and find the height value of Y in each bin and just multiply Y*w to get that bin contribution. Add all the contributions and get the area under the curve. I am not able to translate this idea for a sphere, how to bin this?

Another way is to consider a disk(in Y-Z plane) of width w, some r distance from the center of the sphere. Now depending on this r, the number of points lying on that disk will vary. One can find the center of this disk and find at which angles are the "top" two points are, find the arc length, and multiply with width w. Do all the disks and add them and that's your area under the curve. This didn't work because there are multiple(which are not the part of the topmost curve) contributing to the area, which is wrong.

Any help is greatly appreciated.

These are the coordinates in X, Y, Z format: https://pastebin.com/Att1Ep1r

P.S.: This is my first time asking questions on stack exchange. Please let me know if I am doing something wrong.

Completion of a complete Measure Space

Posted: 13 Oct 2021 09:36 PM PDT

I am having trouble solving the following:

Let $(X,\mathscr A, \mu)$ be a measure space. Show that $(\mathscr A_\mu)_\bar\mu=\mathscr A_\mu$ and $\bar{\bar \mu}=\bar \mu$.

The fact that $\mathscr A_\mu \subset (\mathscr A_\mu)_\bar\mu$ is obvious. I am stuck on the other direction.

What I have tried. So far, I have noted that if $A\in (\mathscr A_\mu)\bar\mu$, then there exist $E,F\in \mathscr A_\mu$ such that $E\subset A\subset F$ and $\bar\mu(F-E)=0$. This implies that $\bar{\bar\mu}(A)=\bar\mu(F)=\bar\mu(E)$. Now we can notice that continuing this definition, $\exists E_1,E_2,F_1,F_2\in \mathscr A$ such that $E_1\subset E\subset F_1$ and $E_2\subset F\subset F_2$ where $\mu(F_1-E_1)=\mu(F_2-E_2)=0$ so that again $\mu(E_i)=\mu(F_i)$ for $i=1,2$. Furthermore, we say $\bar\mu(E)=\mu(E_1)$ and $\bar\mu(F)=\mu(E_2)$.

So I know is that $\mu(F_2)=\bar\mu(F)=\bar\mu(E)=\mu(E_1)$. However, I believe this isn't enough to conclude that $\mu(E_1-F_2)=0$ and thus that $A\in \mathscr A_\mu$ as desired because we have no limitation on the value of $\mu(E_1)$.

Is there something obvious I am missing here?

Edit

Maybe it wasn't clear, but I have that $E_1\subset E\subset A\subset F\subset F_2$, so if I an show that $\mu(F_2-E_1)=0$ then I am showing that $A\in \mathscr A_\mu$. This uses the definition that if $(X,\mathscr A)$ is a measure space and $\mu$ is a measure on $\mathscr A$, then the completion of $\mathscr A$ under $\mu$ is the collection $\mathscr A_\mu = \{A\subset X\ |\ \exists E,F\in \mathscr A \text{ such that } E\subset A\subset F \text{ and } \mu(F-E)=0\}$.

Proof of that $(1+\frac{1}{n})^n$ converges for real $n \to \infty$ [duplicate]

Posted: 13 Oct 2021 09:14 PM PDT

I've been searching for a rigorious proof of the existence of $\lim\limits_{n\to\infty} (1+\frac{1}{n})^n$ for $n \in \mathbb{R}$, but the proofs I came across only allow $n$ to take integer values (Some examples are What is the most elementary proof that $\lim_{n \to \infty} (1+1/n)^n$ exists? and https://en.wikipedia.org/wiki/Characterizations_of_the_exponential_function#Equivalence_of_the_characterizations)

However, the fact that $\lim\limits_{n\to\infty} f(x)$ exists, where $f:\mathbb{Z} \to \mathbb{R}$, doesn't necessarily imply that $\lim\limits_{n\to\infty} g(x)$ also exists, where $g:\mathbb{R} \to \mathbb{R}$ and $\forall x \in \mathbb{Z}\,(g(x) = f(x))$

An example would be $\sin(\pi x)$, which $\lim\limits_{x\to\infty} \sin{\pi x} = 0$ if we add the restriction $x \in \mathbb{Z}$, but not for $x \in \mathbb{R}$

Therefore is there any way to truly prove $\lim\limits_{n\to\infty} (1+\frac{1}{n})^n$, letting $n$ take real values?

Analog to eigenvalue decomposition that ranks by something other than L2 norm

Posted: 13 Oct 2021 09:03 PM PDT

I am studying a scalar physical quantity $J$ given by $J=\mathbf{h}^\top\mathbf{x}$, where $\mathbf{x}$ is a zero-mean state vector of perturbations (assumed jointly Gaussian) to a physical system and $\mathbf{h}$ is a vector specifying a weighted sum over $\mathbf{x}$. I am interested in the variance of $J$ over realizations of $\mathbf{x}$, which we can write as \begin{align} \text{var}(J)=\mathbf{h}^\top\left<\mathbf{x}\mathbf{x}^\top\right>\mathbf{h} \end{align} where angle brackets denote statistical expectation. In particular, I am interested in determining the leading covarying patterns in $\mathbf{x}$ that are responsible for var($J$).

To find these, I am expanding var($J$) as \begin{align} \text{var}(J)=\mathbf{d}^\top\text{diag}(\mathbf{h})\left<\mathbf{x}\mathbf{x}^\top\right>\text{diag}(\mathbf{h})\mathbf{d} \end{align}

where $\mathbf{d}=[1,1,1,\dots,1]$ is a vector of ones with the dimension of $\mathbf{x}$ and diag designates a matrix that has $\mathbf{h}$ along its diagonal and zeros everywhere else. In this expression, left and right operation by $\mathbf{d}$ serves to sum over the covariance matrix $\mathbf{C}=\text{diag}(\mathbf{h})\left<\mathbf{x}\mathbf{x}^\top\right>\text{diag}(\mathbf{h})$.

To find covarying patterns of $\mathbf{C}$ that maximize the contribution to var($J$), I am then left with the following problem: Find a set of orthogonal vectors $\mathbf{v}_i$ such that

\begin{align} \mathbf{C}=\sum_i \mathbf{v}_i\mathbf{v}^\top_i \end{align}

in a way that maximizes the quantity $\mathbf{d}^\top\mathbf{v}_i\mathbf{v}^\top_i\mathbf{d}$ for each $i$. Thus, analogous to eigenvector decomposition, by ranked by a different property of the vector. Is there a name for this problem and/or algorithms to solve it?

Prove that $diam_{d_1}([a, b]) = |b - a|$ and $diam_{d_1}((a, b)) = |b -a|$

Posted: 13 Oct 2021 08:55 PM PDT

Let $a, b \in R$ such that $a < b$. Show that diam$_{d_1}([a, b]) = |b - a|$ and diam$_{d_1}((a, b)) = |b -a|$. What I know is that diam$_{d}(A)=$ sup $\{d(a,b):a,b\in A\}$. I am not sure if I am allowed to say that the supremum of $d((a,b))$ is the sup$(a,b)-$inf$(a,b)$. I don't know how to start this problem.

Need Help with this Question. Exam coming up on this [closed]

Posted: 13 Oct 2021 08:59 PM PDT

Assistance with the following question

enter link description here

Biconditionals and Conjunctions in Truth Tables

Posted: 13 Oct 2021 09:26 PM PDT

Given that a biconditional $p\iff q$ is True what can be concluded from the statement $\lnot p\land \lnot q$?

In a worded example:

I wear my running shoes if and only if I exercise. (True)

I am not exercising AND I am not wearing my running shoes. (?)

If we set up a truth table, the biconditional is True in two of the four occurrences, but we see that $\lnot p\land \lnot q$ is both True and False, which would mean there is no conclusion, correct?

If D is a ring with identity such that every unitary D-module is free, then D is a division ring

Posted: 13 Oct 2021 08:50 PM PDT

This question is from my module theory assignment and I am struck on this particular problem.

Prove that if D is a ring with identity such that every D-module is free, then D is a divison ring.

Attempt: To prove that D is Divison Ring it is sufficient to show that D has no proper ideals. Let on the contrary D has a proper ideal I. Now , I have to somehow prove that some D-module is not free. But I am not able to get any ideas.

Can you please help?

what is the meaning of the symbol $p(a;b)$?

Posted: 13 Oct 2021 08:44 PM PDT

Sometimes I see the symbol $p(a;b)$. Does it mean the same as $p(a|b)$? If it is not the same, what is its meaning?

Help with product rule word problem

Posted: 13 Oct 2021 09:16 PM PDT

In $1999$, the population of Richmond-Petersburg, Virginia, metropolitan area, was $961{,}400$ and was increasing at roughly $9200$ people per year. The average annual income in the area was $30{,}593$ per capita, and this average was increasing at about $\$1400$ per year. Use the product rule to estimate the rate at which total personal income was rising in the area at this time. Explain the meaning of each term in the product rule.

So what I was thinking was that I could get

$$f(x)=961400+9200x, g(x)=30593+1400x$$

But I don't think it would make that much sense to take the product of these so that

$$f(x)g(x)=f'(x)g(x) + f(x)g'(x) = (9200 \cdot 30593+1400x) + (961400+9200x \cdot 1400)$$

Is this the correct approach? If so, for what each term means, I'd say $f'(x)$ is the instantaneous rate of change of the population, $g(x)$ is the equation for the change in annual income, $f(x)$ is the equation for the change in population, and $g'(x)$ is the instantaneous rate of change of the average annual income. Any help? Thanks.

How can I find this limit without L'Hopital and is my current solution correct?

Posted: 13 Oct 2021 08:43 PM PDT

Find $\lim_{x\rightarrow0} \frac{x^4}{1-2\cos x+\cos^2 x}$

My solution:

Apply L'Hopital

$=\lim_{x\rightarrow0} \frac{4x^3}{2\sin x-\sin 2x}$

Apply L'Hopital again

$=\lim_{x\rightarrow0} \frac{12x^2}{2\cos x-2\cos 2x}$

Apply L'Hopital again

$=\lim_{x\rightarrow0} \frac{24x}{-2\sin x + 4\sin 2x}$

$=\lim_{x\rightarrow0} \frac{24}{-2\cos x + 8\cos 2x}=\frac{24}{-2+8}=4$

Is this correct? How would I approach this problem if I didn't want to use L'Hopital?

How to calculate the minimum of $x^2-xy+y^8$？

Posted: 13 Oct 2021 09:09 PM PDT

If $x^3+y^3=8$, what is the minimum value of $x^2-xy+y^8$？

$x$ and $y$ are positive numbers such that $x^3 + y^3 = 8$. Find the minimum value of $x^2 - xy + y ^8$.
A. $4$
B. $2\sqrt[3]2$
C. $\sqrt[3]2$
D. $\dfrac1{\sqrt[3]2}$
E. $0$

Limit of a Riemann Sum.

Posted: 13 Oct 2021 09:29 PM PDT

I am trying to calculate the limit

$$\lim_{n\to\infty}\sum_{k=n+1}^{2n}\frac{1}{k}$$

Can someone please explain how I can go about doing this?

GCd of a prime number

Posted: 13 Oct 2021 09:41 PM PDT

Calculate $gcd\left(\frac{2^{40}+1}{2^8+1}, 2^8+1 \right)=k$.

Generalization I saw in a book: let $ q=\Pi_{i=1}^{n}p_{i} $ obviously, $ 2^{r}+1|2^{q}+1 $,$ r|q $ here we have $ 2^n $ divisors,and we also have $ \frac{2^{q}+1}{2^{r}+1} $ so,we have $ 4^{n} $ divisors now we only say why do not exist same divisors it equals to say that exists x,y such that $ (2^{x}+1)(2^{y}+1)=2^{q}+1 $ it is same as $ 2^{x+y}+2^{x}+2^{y}=2^{q} $ compare the power of 2 then we have $ x=y=1,q=3 $ ( $ \rightarrow\leftarrow $ )

I don't know if this generalization serves this question. If it fits, how do I apply it?

Spivak uses a property in its own proof?

Posted: 13 Oct 2021 08:53 PM PDT

I'm reading Spivak's Calculus 4th Edition and in Chapter 1: Basic Properties of Numbers I'm having trouble understanding a proof of one of those basic properties.

He first establishes 3 basic properties:

$a + (b + c) = (a + b) + c$
$a + 0 = 0 + a = a$
$a + (-a) = (-a) + a = 0$

Here I quote Spivak:

Property P2 ought to represent a distinguishing characteristic of the number $0$, and it is comforting to note that we are already in a position to prove this. Indeed, if a number $x$ satisfies $$a+x=a$$ for any one number $a$, then $x = 0$ (and consequently this equation also holds for all numbers $a$). The proof of this assertion involves nothing more than subtracting $a$ from both sides of the equation, in other words, adding $-a$ to both sides; as the following detailed proof shows, all three properties P1—P3 must be used to justify this operation. $$ \begin{equation} \begin{split} &\text{If } \qquad\qquad\qquad\: a+x=a, \\ &\text{then }\qquad(-a)+(a+x)=(-a)+a=0; \\ &\text{hence } \:\:\quad ((-a)+a)+x=0; \\ &\text{hence } \:\:\,\,\qquad\qquad 0+x=0; \\ &\text{hence } \qquad\qquad\qquad\,\, x=0. \ \end{split} \end{equation} $$

From what I understood this is a proof of the second property, yet he explicitly states that the second property must be used in the proof. I can't see how this can be justified and I'm finding it frustrating to decipher the logic behind this proof. Can somebody point to where I'm getting this wrong because I'm confident it's a mistake on my behalf rather than an error in the textbook, which is hard to believe considering the many editions the book has been through and is rare for an author of Spivak's caliber.

Also wouldn't a property such as this simply be considered as an axiom of some set of numbers, such as the earlier ones, how does one know whether such a basic property can even be proved?

I found this relevant question Does the given proof show that 0 is the unique additive identity?, but the answer seems to point out that Spivak doesn't prove it but rather stipulates it through P2 which seems odd since Spivak himself states that it is proved.

Apologies in advance if I've made an embarrassing mistake/omission, this book isn't an easy read for me.

Discrete derivative condition for martingale?

Posted: 13 Oct 2021 08:45 PM PDT

For a simple symmetric random walk $S_n$, the process $f(S_n)$ is a martingale if and only if $$D^2 f = f(x+1) - 2f(x) + f(x-1) = 0.$$ Where this condition comes from I have no idea. I have never seen this anywhere and cannot find it anywhere. That being said, I need to find a condition for some function $f(n, S_n)$ using this fact and then prove that $Y_n = \rho ^ {-n} e^{S_n}$ is a martingale where $$\rho = \frac{e + e^{-1}}{2}.$$

Now, I have been looking at this problem for days and have no idea how to do it. I think we can write $f(n,S_n)$ as $$f(n, S_n) = g(n)f(S_n).$$ Then we have that $g(n) D^2 f = 0$, so $g(n)$ can be anything at all. Obviously this is not true.

I truly have no idea. Any help is appreciated.

Are the Rationals under addition isomorphic to any set of numbers?

Posted: 13 Oct 2021 08:51 PM PDT

I've proved that ($\mathbb {Q}$ , +) is not isomorphic to ($\mathbb {Z}$ , +) as part of an assignment:

Let $f : \mathbb {Q} \rightarrow \mathbb {R}$ and $a = f(1)$, then $a=f({1\over2}+{1\over2})=2f({1\over2})=3f({1\over3})=...$ and so on. Hence $\forall_{n\in\mathbb {Z}}$ $n$ divides $a$. Therefore, $a=0$ and so $f(n)=nf(1)=na=0$ $\forall_{n\in\mathbb {Z}}$ and so $f$ is clearly not a bijection.

And so I've been wondering if that same logic could apply to ($\mathbb {Q}$ , +) and any other group? Is there something that makes ($\mathbb {Z}$ , +) special here?

Intuition behind the method of characteristics

Posted: 13 Oct 2021 09:24 PM PDT

I have learnt the method of characteristics in a past PDE course, but it has always been taught as a sort of sequence of steps which lead to a solution. This has always bothered me and I would like to understand the geometric picture behind the method and some of the consequences that follow (i.e. how and what it means by a solution propagating along characteristics).

I have tried going through a few resources including books by Strauss and Evans, lecture notes such as this one by Stanford, and other posts on this website (Explaining the method of characteristics). Despite all this something is still not clicking intuitively.

Here is what I have understood so far, let's take the following PDE as an example: $$a(x,y) u_x + b(x,y)u_y = c(x,y)$$ We first observe that the PDE can be written as $$\bigl \langle a(x,y), b(x,y), c(x,y)\bigr\rangle \cdot \bigl \langle u_x, u_y, -1 \bigr\rangle = 0$$ which tells us that the vector $V = \bigl \langle a(x,y), b(x,y), c(x,y)\bigr\rangle$ must lie in the tangent plane to the graph/surface $S = \{(x,y,u(x,y))\}$. Hence we would like to determine $S$ such that for each point $(x,y,z) \in S$, its tangent plane contains the vector $V$.

This amounts to to finding a curve $\mathcal{C}$ which lies in $S$. We parametrize $\mathcal{C}$ by a variable $s$, so that the tangent vector $V$ is now $$V_s = \bigl \langle a\bigl(x(s),y(s)\bigr), b\bigl(x(s),y(s)\bigr), c\bigl(x(s),y(s)\bigr)\bigr\rangle$$

Thus the curve $\mathcal{C} = \{(x(s), y(s), z(s))\}$ results in the following system of ODEs: $$\frac{dx}{ds} = a\bigl(x(s), y(s)\bigr)\\ \frac{dy}{ds} = b\bigl(x(s), y(s)\bigr)\\ \frac{dz}{ds} = c\bigl(x(s), y(s)\bigr)$$

We finally obtain our characteristic curves by solving the above system of equations. Taking the union of all the resulting characteristic curves results in a solution surface for our original PDE.

Somehow, after solving a first-order PDE using this method, for example taking the transport equation, it becomes apparent that $z(x,t)$ is constant along the lines $x-at = x_0$ but how? Furthermore, how does this imply that $u(x,t) = z(x,t) = f(x-at)$?

My questions:

Why do we parametrize the curve $\mathcal{C}$ to begin with?
What does it mean that characteristic curves propagate discontinuities or solutions? How does one infer this from this picture?
I am still unclear what these curves are ultimately telling us. Does it tell us the "path" that some Cauchy data travels in the $xyz$ plane?
Are there any mistakes in my understanding?
The definition of $S$ seems to come out of nowhere, what is the motivation behind using such a surface?

Multivariable Chain Rule: Conditional Independence and "Swapping" Partial Derivatives

Posted: 13 Oct 2021 09:32 PM PDT

Suppose that we have three functions $f$, $g$, and $h$, such that for all $x$ and $y$, $h(x, y) = g(f(x, y, \cdot))$, where $\cdot$ represents an input where a random variable goes (ignored in the informal version below), and $g$ is the expectation of an expression involving that random variables (this makes $g$ and $h$ deterministic functions). We are given $\frac{\partial h(x, y)}{\partial x}$, and want to find $\frac{\partial h(x, y)}{\partial y}$.

Informal Version (more formal version given below):

We are given the partial derivative of $h$ w.r.t. $x$,

$$\frac{\partial h(x, y)}{\partial x} = \mathbb E \left[\sum_{t=0}^\infty \text{[some stuff involving random variables and } f(x, y)] \frac{\partial f(x, y)}{\partial x}\right],$$

Where the "some stuff" is all independent of $x$ and $y$ given the output of $f(x, y)$. We want to prove that we can substitute $\partial y$ for $\partial x$:

$$\frac{\partial h(x, y)}{\partial y} = \mathbb E \left[\sum_{t=0}^\infty \text{[some stuff involving random variables and } f(x, y)] \frac{\partial f(x, y)}{\partial y}\right].$$

Intuition/Main Question:

Intuitively, we can argue that since $x$ and $y$ only affect $h$ via $f$ (that is, $h$ is conditionally independent of $x$ and $y$ given $f(x, y)$), and we should be able to simply replace $\frac{\partial f(x, y)}{\partial x}$ with $\frac{\partial f(x, y)}{\partial y}$ in the definition of $\frac{\partial h(x, y)}{\partial x}$ to get $\frac{\partial h(x, y)}{\partial y}$ (see the informal chain rule argument in the appendix below for more intuition).

How can we formalize the argument above to prove this property about $\frac{\partial h(x, y)}{\partial y}$? I suspect there some theorem or well-known result that immediately gives us what we need, but I do not know what this theorem is. Thank you.

Appendix (not necessary for answering the question, but may be helpful):

More Formal Version:

We are given that partial derivative of $h$ w.r.t. $x$,

$$\frac{\partial h(x, y)}{\partial x} = \mathbb E \left[\sum_{t=0}^\infty d(f(x, y, Z_t), Z_t) \frac{\partial f(x, y, Z_t)}{\partial x}\right],$$

where, $d$ is some function, the $Z_t$'s are random variables influenced by $f(x, y, \cdot)$ and, for all $t$, $Z_t$ is conditionally independent of $x$ and $y$ given $Z_{t-1}$ and $f(x, y, Z_{t-1})$. We need to find $\frac{dh(x, y)}{dy}.$ We'd like to show that

$$\frac{\partial h(x, y)}{\partial y} = \mathbb E \left[\sum_{t=0}^\infty d(f(x, y, Z_t), Z_t) \frac{\partial f(x, y, Z_t)}{\partial y}\right].$$

More informal intuition involving the chain rule:

Consider the following informal argument, ignoring the sum, expectation, and random variables for a moment:

$$\frac{\partial h}{\partial x} = \frac{\partial h}{\partial f} \frac{\partial f}{\partial x} = d(f(x, y)) \frac{\partial f(x, y)}{\partial x},$$

so $\frac{\partial h}{\partial f} = d(f(x, y)).$ Substituting this definition of $\frac{\partial h}{\partial f}$ to find $\frac{\partial h}{\partial y}$:

$$\frac{\partial h}{\partial y} = \frac{\partial h}{\partial f} \frac{\partial f}{\partial y} = d(f(x, y)) \frac{\partial f(x, y)}{\partial y}.$$

Q.E.D. (except for the troublesome sum, expectation, and random variables).

Does $P|_{\mathcal{F}}=Q$ imply $E_P[X]=E_Q[X]$ for $\mathcal{F}$-measurable $X$?

Posted: 13 Oct 2021 08:45 PM PDT

Consider probability spaces $(\Omega, \mathcal{G}, P)$ and $(\Omega, \mathcal{F}, Q)$ with the properties $\mathcal{F}\subseteq \mathcal{G}$ and $P|_{\mathcal{F}}=Q$.

Let $X$ be a $Q$-integrable (and $\mathcal{F}$-measurable) random variable. Is it true, that

$$E_P[X]=E_Q[X]\quad ?$$

I feel like this should be true. I think there is an approximation of $X$ with simple functions on $\mathcal{F}$ and because of the fact, that $P(F)=Q(F)$ for any $F\in\mathcal{F}$.

Is $L^i(K_4)$ regular for all $i \in \mathbb{Z}^+$? If so, find the general formula for the degree of $L^i(K_4)$?

Posted: 13 Oct 2021 08:44 PM PDT

Let $K_4$ represent the simple, fully connected graph of 4 vertices. Let $L$ represent the operator so that $L(K_4)$ is the line graph of $K_4$. Let $L^1(K_4)=L(K_4), L^2(K_4)=L(L(K_4)),$ etc. Is $L^i(K_4)$ regular for $i \in \mathbb{Z}^+$? If so, find a general formula for the degree of $L^i(K_4)$.

The work I have done thus far is equated the general formula for the number of edges to be reliant on the degree of a given graph. To formalize, let $n$ be the number of vertices and let $k$ be the degree of the graph; the number of edges in $L^i(K_4)$ equates to $n {k \choose 2}$. But this doesn't bring me any closer to answering the first part of the question meaning it cannot yet be used in the proof. And using it to find the degree $k$ of a given graph $L^i(K_4)$ would be algebraic work that I cannot justify combinatorically.

I would appreciate any assistance.

Connected Cubic Graph Counting

Posted: 13 Oct 2021 08:59 PM PDT

I want to count the number of connected and disconnected cubic non-isomorphic graphs with vertices $2n$, but I haven't been able to come up with an algorithm or technique to do so. I've been researching for the past 2 days and haven't found anything convincing.

Techniques I've tried is converting the graphs to adjacency matrices and looking for pattern, using combinatorics, two-colour theory and so on.

I know the sequence of connected cubic graphs for $2,4,6,8,10,12,14,...,2n$ vertices as $0,1,2,5,19,85,509,\ldots$, but how do you derive proceeding graph counts from preceding ones? How do you go from $5$ to $19$ to $85$ and so on?

I have no idea. I need some clarity on this problem and possible solutions.

Is there a way to solve this coupled differential equation?

Posted: 13 Oct 2021 08:50 PM PDT

Say we have a coupled differential equation of the form

$$\sum_{i=1}^p \omega y_1 \frac{\partial q_\mu}{\partial y_i}+g\frac{\partial q_\mu}{\partial y_\mu}+t q_{\mu-1}e^{-g/\omega x_{\mu-1}}-t q_{\mu+1}e^{-g/\omega x_{\mu+1}}=(E+g^2/\omega)q_\mu$$

Where $q_\mu=q_\mu (x_1,x_2,\cdots, x_p)$ and $x_1=y_1-y_2; x_2=y_2-y_3;\cdots;x_{p-1}=y_{p-1}-y_p$ and $x_p=y_p-y-1$ and $\mu=1,\cdots,p$

I was thinking to solve it for the easier cases where say p=2 or 3. The hope was that an ansatz of the form $q_\mu=e^{\lambda x_\mu}$ would solve it, but it did not help because of the different terms having different subscript in x.

Any other ways to solve these coupled differential equations?

Expected number of rolls until lcm is greater than $2000$?

Posted: 13 Oct 2021 09:16 PM PDT

You continually roll a fair $10$ sided dice. What is the expected number of rolls until the lowest common multiple of all numbers that have appeared is greater than $2000$?

The primes in the numbers $1$ to $10$ are $2,3,5,7$. The lowest common multiple of these numbers is $210$. However, different numbers could come in different frequencies (one $9$ is worth two $3$s). How can you deal with this?

I am happy for solutions with approximate answers. The true value by simluation is around $18.8$.

Density of lines passing through sides of a rectangle?

Posted: 13 Oct 2021 09:07 PM PDT

Edit: Reframed the problem to find an actual answer! Though I may well be talking to myself at this point. Answer below the original problem.

The original context of this question is a ridiculous, brainless mobile game. But the geometric question that arises seems quite interesting--I suspect it has already been studied. But I can't quite figure out how to calculate, or even if it can be calculated.

Consider a rectangle $ABCD$ centered on the origin (just for simplicity/symmetry). Now consider two sets of lines: $H$, the set of all lines passing through a point on $\overline{AB}$ and a point on $\overline{CD}$; and $V$, the set of lines passing through a point on $\overline{AD}$ and a point on $\overline{BC}$.

(Or perhaps more simply, all of the lines in $\mathbb{R}^2$ that pass through opposite sides of the rectangle.)

Is there a density function $f(x_0, y_0)$ that computes the proportion of these lines that pass through an infinitesimal area defined as any of the points $(x_0-\delta < x < x_0 + \delta, y_0 - \delta < y < y_0 + \delta)$? If not, can we find such a function that works for a non-infinitesimal area?

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Wednesday, October 13, 2021