Wednesday, May 4, 2022

Recent Questions - Mathematics Stack Exchange

Recent Questions - Mathematics Stack Exchange

Game Theory - What is the equilibria?

Posted: 04 May 2022 07:08 PM PDT

I think i'm braindead from the amount of time i've stared at this. What is the equilibria in each of the scenarios in this? and preferably the subgame perfect equilibrium? Is there even any, as player 1's choice to bribe can pose different payoffs at the end?

GAME THEORY - 3 player sequential game

A levels Statistics 2 Poisson Why am I getting two different answers?

Posted: 04 May 2022 07:06 PM PDT

Why am I getting different answers in the second part using two different approaches which should give the same answer. What am I doing wrong? The second method is given in the marking scheme.

enter image description here

Integrating Using Stokes's Theorem

Posted: 04 May 2022 07:00 PM PDT

Let $S$ be the first-octant portion of $x^2+z^2=196$ over the circle $x^2+y^2=196$. Use Stokes's Theorem to evaluate $$\int_C \mathbf{F}\cdot d\mathbf{r}$$ if $C$ is the boundary of $S$, and $\mathbf{F}$$(x,y,z)=(11-6x)\mathbf{i}+8xz\mathbf{j}+(x^2+y^2)\mathbf{k}$.

I'm still not sure how to apply Stokes's theorem and I was looking for someone to help me to learn the process on how to solve problems like these.

Distribution of $f(X)X$ when $X$ is Uniform?

Posted: 04 May 2022 06:41 PM PDT

Suppose that $f$ is the probability density function of a probability measure $P$ which is absolutely continuous with respect to Lebesgue measure.

Suppose X is a uniform random variable. Is it possible to characterize the distribution of the random variable $Y := f(X) X$?

One example: suppose that $P$ is the uniform law on $[0, 1/t]$, $t \ge 1$. Then $Y$ is $0$ with probability $1-1/t$ and otherwise it is $tZ$ where $Z$ is distributed according to $P$.

Hoeffding's lemma proof using this entropy definition

Posted: 04 May 2022 06:39 PM PDT

For a positive random variable $Y$ let $H(Y)=E[Y \log Y]-E[Y] \log E[Y]$. And for random variable $X$ let $m_{X}(\lambda)=E\left[e^{\lambda X}\right]$. Assume that $H\left(e^{\lambda X}\right) \leq \frac{\lambda^{2} \sigma^{2}}{2} m_{X}(\lambda), \quad \forall \lambda \in \mathbb{R}_{+}$, for some constant $\sigma^{2}<\infty$. I want to show the Hoeffding's lemma using this entropy definition.

$$ E\left[e^{\lambda(X-E[X])}\right] \leq e^{\frac{\lambda^{2} \sigma^{2}}{2}}, \quad \forall \lambda \in \mathbb{R}_{+} . $$

I tried quite a lot but none worked. Have no clue how to show this, can I have some hints.

Complex Analysis - Entire and Bounded Condition in Liouvilles's Theorem

Posted: 04 May 2022 06:37 PM PDT

The Liouvilles' Theorem states that if f is entire and bounded in the complex plane, then f(z) is constant in that plane. For example, if there exists a polar function f(z)=(r-2pi,theta), I want to apply the theorem to prove f(z) is constant (or not). Can I first verify Cauchy Riemann Equations' conditions to make sure they are differentiate everywhere in C, thus it must be entire? Can I directly conclude that entire function is bounded and thus by L's Thm it must be a constant?

Lagrange dual function for multivaribale optimiziation problem

Posted: 04 May 2022 06:38 PM PDT

I wish to apply the duality theorem to the optimization problem: $$\text{minimize}~~s$$ $$\text{subject to}~~g_j(x)\leq{s},~~\text{for all}~~j=1,...,r,~~x\in{X},~~s\in{\mathbb{R}},$$ where $X\subseteq{\mathbb{R}^n}$ is a nonempty convex set, and $g_j:\mathbb{R}^n\rightarrow\mathbb{R},~j=1,...,r$ are convex functions.
I want to use the result of this optimization problem for showing that a set of strict inequalities $g_j(x)<0$ (for all $j=1,...,r$) has no solution within the set $X$ if and only if there exists a vector $\nu\in\mathbb{R}^r$ such that: $$\nu\geq0,~~\sum_{j=1}^r\nu_j=1,~~\langle\nu,g(x)\rangle\geq0~~\text{for all}~x\in{X},$$ and $g(x)=[g_1(x),...,g_r(x)]'$.
From my understanding, both $s$ and $x$ are the optimization variables and the Lagrange dual function should be formed using both of them. I wrote: $$g(\mu,\lambda)=\inf\limits_{s\in\mathbb{R},~x\in{X}}\{s+\sum_{j=1}^r\mu_j(g_j(x)-s)\}.$$ When I take derivative w.r.t $x$, I end up with $\sum_{j=1}^{r}\mu_j\nabla{g_j}(x)=0$. When I take derivative w.r.t $s$, I get $\sum_{j=1}^r\mu_j=1$. After this point, I don't know what to do with derivatives that I have.
1- I do not know if I am doing the right thing for the dual function or not.
2- Also, I would appreciate some explanations on how to mathematically relate this dual problem to the problem related to the existence of $\nu$.

How to find the password space given several restrictions?

Posted: 04 May 2022 07:10 PM PDT

The I am trying to determine all valid passwords (the password space) that fulfill this list of requirements.

  1. Password is exactly 15 characters long.
  2. Must contain at least 2 lowercase letters (26 in total).
  3. Must contain at least 2 uppercase letters (26 in total).
  4. Must contain at least 2 numbers 0-9 (10 in total).
  5. Must contain at least 2 special characters (like !, @, # ...) (35 in total).

I know that the unrestricted password space would be (26+26+10+35)^15 = 6.33*10^29 so the final answer must be less than that.

My first approach was to find all of the invalid passwords and subtract that from the unrestricted password space. However, even this step proved to be too complicated to for me. I broke down all invalid passwords into a matrix of how they fulfill the 4 "contain" requirements.

requirements fulfillment table

I thought that by finding the number of passwords that fit in each row excluding the first and last row, I'll have all invalid passwords and thus I can subtract that from the unrestricted password space. The first row is all valid passwords which what we are trying to find. The last row is impossible because it is impossible to fill 15 characters without using any lowercase, uppercase, numbers, and/or special characters at least twice. Order does matter, so permutations are used somewhere but I'm not sure how.

Any ideas on how to approach this is a simpler way? Thanks.

Is it possible to write a vector form ODE to an explicit form?

Posted: 04 May 2022 07:01 PM PDT

Any explicit differential equation of order n, $F\left(x,y,y',y'',\ \ldots ,\ y^{(n-1)}\right)=y^{(n)}$ can be written as a system of n first-order differential equations.

Conversely, can any vector form be rewritten into an explicit form?

How do I rigorously compute $\lim_{x\rightarrow0} a^x$ for $a \in \mathbb{R}$?

Posted: 04 May 2022 07:03 PM PDT

How do I rigorously compute $$\lim_{x\rightarrow0} a^x$$ for $a \in \mathbb{R}$?

I can intuitively and graphically get the answer of $\delta_{a\neq0}$ (Kroenecker delta), and I think also by using the $(\delta, \epsilon)$ definition, but I'm not sure if I'm doing it hand-wavely or rigorously.

If $f \in I(\mathbb{R}^2)$ and $T_c$ is a translation, then ${T_c}^{-1} \cdot f \cdot T_c$ is the same kind of isometry as $f$.

Posted: 04 May 2022 07:09 PM PDT

I want to know if this statement is true.

If $f \in I(\mathbb{R}^2)$ and $T_c$ is a translation, then ${T_c}^{-1} \cdot f \cdot T_c$ is the same kind of isometry as $f$.

This question arises from a proof of the statement "If $f \in I(\mathbb{R}^2)$, then $f$ is either translation, rotation, reflection, or glide."

The proof that I'm reading (I have no link since it is from my lecture note) uses the strategy as follows.

$f(x)=Ax+b$ for some $A \in O(2)$ and $b \in \mathbb{R}^2$.

i) $1$ is an eigenvalue of $A$, then ${T_c}^{-1} \cdot f \cdot T_c=A$ is either reflection or glide. Thus, $f$ is either reflection or glide. ($A=Id$ gives translation).

ii) $1$ is not an eigenvalue of $A$, then ${T_c}^{-1} \cdot f \cdot T_c=A$ is either rotation or reflection. Thus, $f$ is either rotation or reflection.

Here, $c$ (for translation $T_c, {T_c}^{-1}$) depends on each case, but I believe this wouldn't matter for this question.

If we can conclude as above, I think it must be true that "if $f \in I(\mathbb{R}^2)$ and $T_c$ is a translation, then ${T_c}^{-1} \cdot f \cdot T_c$ is the same kind of isometry."

For example, if ${T_c}^{-1} \cdot g \cdot T_c$ is some glide, then $g$ is some glide.

Is my assumption correct?

Find a simple path in given tree with minimum number of edges

Posted: 04 May 2022 07:09 PM PDT

Suppose given a Tree $T=(V,E)$. Each nodes in $T$ has a degree at most two. Also, edges in $T$ has weight distinct and positive natural. Suppose $|V|=n$, our goal is find a simple path with length given input number $k$ and has minimum number of edges. Also we try find a divide and conquer approach that has running time $O(n\log^2 n)$.

I think we must find a $v\in V$ such that $v$ subtrees have at most $\frac{n}{2}$ nodes. Then we divide $v$ into subtrees and then we solve our problem recursively .

Can we test whether a density is bounded away from $0$ at a point given sample data?

Posted: 04 May 2022 07:00 PM PDT

Suppose $X$ is a random variable with unknown support $S$ and unknown density $f$. Given a finite iid sample $D$, is there a test to check whether the density $f$ is bounded away from $0$ at a particular point in the support?

I feel like this is something not testable, since if you observe a point in the sample, you know the density at that point is positive; but if you don't observe a particular point, then there's no way to tell. Existing kernel methods to approximate the density use smoothing so even if the true density is $0$ but continuous (or discontinuous but the jump is very small) at a particular point, it will get smoothed out by nearby observations.

I'm not sure if this is the right intuition. Searched online and couldn't find any relevant work/statistical tests.

Combinatorics related math problem...need suggestion

Posted: 04 May 2022 07:05 PM PDT

I am trying to solve the following question: "In a programming class of 7 students, the instructor wants each student to modify the program from a previous assignment; however, no student should work on his or her own assignment. In how many ways can the instructor assign programs to the students."

I was trying to solve this with the following thoughts.

Each student among 7 students have to choose from other 6 previous assignment excluding his/her own assignment. So, 6 ways an instructor can assign to one student. Therefore, 6.5*.4*.3*.2*.1 = 6! ways can the instructor assign programs to the students. Answer: 6!*

Is the solution I am thinking is correct or there is some problem..please let me know

Probability mass function and expected value in lottery

Posted: 04 May 2022 06:25 PM PDT

I can buy one lottery ticket out of two available. In the first lottery I can win \$100 with probability 0.1, and the price of ticket is 10. In the second lottery, I can win \$50 with probability 0.1 and $500 with probability 0.01. The price of ticket is \$20. To decide which ticket to buy I toss a fair coin once. I chose first ticket in case of head and second otherwise. Let X be random variable that denotes my net payout (taking into account price of a ticket). Find probability mass function of X (hint: use law of total probability). Show that expected value of X is an average of expected values of net payouts for each of two lotteries. Explain, why. Will it still hold if lotteries has different payouts or probabilities? Prove it.

Could you please check if my calculations and assumptions are correct?

Let L be a lottery. After tossing a coin once we have P(L1)=0.5 and P(L2)=0.5

L1 L2 L2
X 100-10 50-20 500-20
P 0.1 0.1 0.0.1

According to the law of total probability:

$pmf(X)=P(L1) \cdot P(x)+P(L2) \cdot P(x)+P(L2) \cdot P(x)$

$pmf(X)=0.5 \cdot 0.1+0.5 \cdot 0.1+0.5 \cdot 0.01=0.105$

Expected value:

$E(X) = \sum_{i=1}^n x_ip_i$

$E(X) = (0.1 \cdot 90) \cdot 0.5 + (0.1 \cdot 30) \cdot 0.5 + (0.01 \cdot 480) \cdot 0.5 = 8.4$

I don't quite understand what should I do next concerning the last questions.

Show that expected value of X is an average of expected values of net payouts for each of two lotteries. Explain, why. Will it still hold if lotteries has different payouts or probabilities? Prove it.

Prove: If A intersection B = A union B, then the Powerset, A union B, equals the Powerset of A.

Posted: 04 May 2022 07:00 PM PDT

Prove or disprove the following:

If $A\cap B=A\cup B$, then $\mathscr P(A\cup B)=\mathscr P(A)$.

I have no clue where to start...

Category Theory: Functors as Diagrams

Posted: 04 May 2022 07:06 PM PDT

I have began classes on Category Theory, I do not understand the concept of 'functors as diagrams' the book I am using is Category Theory and Applications by Marco Grandis. I have been asked to prove the following statements:

  1. Every functor defined on a preordered set $S$ is automatically a commutative diagram.

  2. A functor $1 \longrightarrow C$ amounts to an object of $C$ while a functor $2 \longrightarrow C$ amounts to a morphism of C.

  3. A functor $X : (\mathbb{N},\leq) \to C$ defined on the ordered set of natural numbers (i.e. the ordinal $\omega$), amounts to a sequence of consecutive morphisms of $C$ \begin{equation*} X_0 \longrightarrow X_1 \longrightarrow X_2 \longrightarrow \dots \longrightarrow X_n \longrightarrow \dots \end{equation*}

    while the functor $X : (\mathbb{N},+)\to C$, defined on the additive monoid of natural numbers, amounts to an endomorphism $u_1 : X_* \to X_*$ $(\text{and its powers} \quad u_n=(u_1)^n$, including $u_0=id(X_*))$

Question from isi previous years

Posted: 04 May 2022 07:06 PM PDT

(a) Show that $\left(\begin{array}{l}n \\ k\end{array}\right)=\sum_{m=k}^{n}\left(\begin{array}{c}m-1 \\ k-1\end{array}\right)$.

(b) Prove that $$ \left(\begin{array}{l} n \\ 1 \end{array}\right)-\frac{1}{2}\left(\begin{array}{l} n \\ 2 \end{array}\right)+\frac{1}{3}\left(\begin{array}{l} n \\ 3 \end{array}\right)-\cdots+(-1)^{n-1} \frac{1}{n}\left(\begin{array}{l} n \\ n \end{array}\right)=1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n} $$

I was thinking mathematical induction for the second part that is,,,

Let $$ \begin{aligned} P(n):\left(\begin{array}{l} n \\ 1 \end{array}\right)-\left(\frac{1}{2}\right)\left(\begin{array}{l} n \\ 2 \end{array}\right) &+\frac{1}{3}\left(\begin{array}{c} n \\ 3 \end{array}\right) \cdots+(-1)^{n-1} \frac{1}{n}\left(\begin{array}{l} n \\ n \end{array}\right) \\ &=1+\frac{1}{2}+\cdots \frac{1}{n} \end{aligned} $$ $P(1):\left(\begin{array}{l}1 \\ 1\end{array}\right)=1$. $P(1)$ true. $$ \begin{aligned} P(2):\left(\begin{array}{l} 2 \\ 1 \end{array}\right)-\frac{1}{2}\left(\begin{array}{l} 2 \\ 2 \end{array}\right) \\ =& 2-\frac{1}{2}=1+\frac{1}{2} \end{aligned} $$ $P(2)$ is also true. Let $p(k)$ tque $\Rightarrow\left(\begin{array}{l}k \\ 1\end{array}\right)-\frac{1}{2}\left(\begin{array}{l}k \\ 2\end{array}\right)+\frac{1}{3}\left(\begin{array}{l}k \\ 3\end{array}\right) \cdots+(-1)^{k+\frac{1}{k}}$ Now we try to show $p(k+1)$ will be true $1+\frac{1}{2}+\cdots \frac{1}{k}$ $P(k+1)=\left(\begin{array}{c}k+1 \\ 1\end{array}\right)-\frac{1}{2}\left(\begin{array}{c}k+1 \\ 2\end{array}\right)+\cdots(-1)^{k} \frac{1}{k+1}\left(\begin{array}{l}k+1 \\ k+1\end{array}\right)$

I cannot argue from here,,, please help me for both part. Thank you.

multi-robot race winning strategy leading to nash equilibrium

Posted: 04 May 2022 06:41 PM PDT

Jane Street released a puzzle this month that kind of builds upon one that they released a few months ago. The one a few months ago was pretty straightforward, but for this one I'm not even really sure how to go about it. If anyone knows of some similar problems that I should look at or could point me in the right direction here, that'd be great.

https://www.janestreet.com/puzzles/current-puzzle/

I tried to write a little bit of Python to get the chances of winning based on how many other people are using the discrete strategy assuming you are one of the robots and are also using the discrete strategy, but it does not account for the cases where a randomly selected robot wins the races where no one enters and thus is removed from other races so my code does not output the correct percentages (if that makes sense). I'll know I have the right output when the code outputs exactly 1/3 for i == 23 because that means that all robots are using the discrete strategy so the chances of winning should be exactly 1/3.

def func():    for i in range(24):      num = (7 / 8) ** i      for j in range(1, i + 1):        num += comb(i, j) * (7 / 8) ** (i - j) * (1 / 8) ** j * (1 / (j + 1))      print(i, "\t {:.6f}".format(num))

My general thought process was that if I know the chances of winning if a robot is using the discrete strategy for all different scenarios of other robots using or not using discrete, I'll be able to calculate P because the frequency of each different scenario of what strategy each robot is using is known. For example, the frequency of every robot using the discrete strategy is P^24, the frequency of one robot not using discrete and the rest using discrete is (1 - P) * P^23, and so on and so on....

Evaluating $\lim_{n \to \infty} \sum_{i=0}^n \left( i \right) \cdot \left( \frac {e ^ \frac{i}{n}}{n ^ 2}\right)$

Posted: 04 May 2022 06:57 PM PDT

$$\lim_{n \to \infty} \sum_{i=0}^n \left( i \right) \cdot \left( \frac {e ^ \frac{i}{n}}{n ^ 2}\right)$$

The sum is an AGP, and can readily be solved if one has the AGP formula at hand. Refer :

https://brilliant.org/wiki/arithmetic-geometric-progression/#:~:text=Now%20let%27s%20derive%20a%20general%20formula%20for%20the,in%20the%20above%20example%20to%20prove%20this%20theorem%3A

What other way is there to evaluate this sum?

How do you calculate the magnitude of a directly represented flat in conformal geometric algebra?

Posted: 04 May 2022 06:38 PM PDT

A flat connected to the origin point $o$ is represented by $X=o \wedge \textbf{A}_k \wedge \infty$, where $\textbf{A}_k$ is a k-blade of directions in the Euclidean subspace, and $\infty$ is the point at infinity. I want to calculate $$||X||^2 = X \widetilde{X} = (o\wedge \textbf{A}_k \wedge \infty)(\infty \wedge \widetilde{\textbf{A}_k} \wedge o)$$

I am not sure how to simplify the expression. The answer is supposed to be $X\widetilde X = -||\textbf{A}_k||^2$.

It is scalar, so $X\widetilde X=\langle (p\wedge \textbf{A}_k \wedge \infty)(\infty \wedge \widetilde{\textbf{A}_k} \wedge p) \rangle$.

In this CGA, we have $\infty \cdot \infty=0$, $o\cdot o=0$, $o\cdot \infty = -1$.

If there was a left contraction between $X$ and $\widetilde X$, then the solution would follow, but how would that be justified?

apply theorem of differentiation through an integral sign

Posted: 04 May 2022 06:41 PM PDT

I am working with the Morse Lemma, which is on page 5 of 'Morse Theory' by Milnor. There is one little calculus thing in its proof, in lemma 2.1 regarding the function $g_i$, which I fail to see. It comes down to this :

If I have f which is C² ($R^n \rightarrow R$), $h(t) = f(tx)$ and $t \in [0,1]$, then
$f(x) = h(1)-h(0) = \int_0^1 h'(t)dt = x \int_0^1 f'(tx)dt = \sum_{j}x_j \int_0^1 {\partial f \over \partial x_j} (tx) dt $.
Now $\int_0^1 {\partial f \over \partial x_j}(tx)dt$ is C¹ , but I fail to see why exactly:

You would show this using the theorem on differentiation under an integral sign:
$t \rightarrow {\partial f \over \partial x_j}(tx)$ can be integrated by construction
$x \rightarrow{\partial f \over \partial x_j}(tx)$ is C¹ because f is C² is and tx is linear
$t \rightarrow {\partial^2 f\over \partial x_j\partial x_k}(tx)$ is continuous because f is C²
what I am left with, is whether ${\partial^2 f\over \partial x_j\partial x_k}(tx) $ is dominated by an integrable function for all x and t. For every x, ${\partial^2f \over \partial x_j\partial x_k} (tx)$ is bounded because f'' is continuous and t is from a compact set. But how do I know that $|{\partial^2f\over \partial x_j\partial x_k}(tx)|$ is dominated for all t AND x ??

I realize that there is an embarassingly little thing that I am not seeing here. That derivation under the integral sign is used, I have seen confirmed in a handout I found on stanford.edu, without any further comment. So it must be something extremely basic.

Thanks for any tips

Exercise 12, Section 30 of Munkres’ Topology

Posted: 04 May 2022 06:57 PM PDT

Let $f \colon X \rightarrow Y$ be a continuous open map. Show that if $X$ satisfies the first or the second countability axiom, then $f(X)$ satisfies the same axiom.

My attempt: Let $y\in f(X)$. Then $\exists x\in X$ such that $f(x)=y$. Since $X$ is first countable, $\exists \{U_n \in \mathcal{N}_x|n\in \Bbb{N}\}$ with the following property: $\forall U\in \mathcal{N}_x$, $\exists p\in \Bbb{N}$ such that $U_p \subseteq U$. Claim: $\{f(U_n)\in \mathcal{N}_y|n\in \Bbb{N}\}$ have the following property: $\forall V\in \mathcal{N}_y$, $\exists q\in \Bbb{N}$ such that $f(U_q)\subseteq V$. Proof: Since $U_n \in \mathcal{T}_X$, $\forall n\in \Bbb{N}$ and $f$ is open, we have $f(U_n)\in \mathcal{T}_Y$, $\forall n\in \Bbb{N}$. And $x\in U_n$, $\forall n\in\Bbb{N}$. So $f(x)=y \in f(U_n)$, $\forall n\in \Bbb{N}$. Thus $f(U_n)\in \mathcal{N}_y$, $\forall n\in \Bbb{N}$. let $V\in \mathcal{N}_y$. Then $f(x)\in V\in \mathcal{T}_Y$. Since $f$ is continuous, $f^{-1}(V) \in \mathcal{N}_x$. So $\exists m\in \Bbb{N}$ such that $U_m\subseteq f^{-1}(V)$. By elementary set theory, $f(U_m)\subseteq f(f^{-1}(V))\subseteq V$. Thus $\exists m\in \Bbb{N}$ such that $f(U_m)\subseteq V$. $f(X)$ have countable basis at $y$. Hence $f(X)$ is first countable.

Let $\mathcal{B}_X=\{B_n|n\in \Bbb{N}\}$ be a countable basis of $\mathcal{T}_X$. Claim: $\mathcal{B}'=\{ f(B_n)|n\in \Bbb{N}\}$ is countable basis of $\mathcal{T}_{f(X)}$. Proof: (1) Since map $f$ is open, $f(B_n)\in \mathcal{T}_Y$, $\forall n\in \Bbb{N}$. So $f(B_n)\cap f(X)=f(B_n)\in \mathcal{T}_{f(X)}$, $\forall n\in \Bbb{N}$. Thus $\mathcal{B}'\subseteq \mathcal{T}_{f(X)}$. (2) Let $V\in \mathcal{T}_{f(X)}$ and $y\in V$. By definition of subspace topology, $V=f(X) \cap U$; $U\in \mathcal{T}_Y$. $y\in f(X)$ and $y\in U$. So $\exists x\in X$ such that $f(x)=y\in U$. $x\in f^{-1}(U)$. Since $x\in f^{-1}(U)\in \mathcal{T}_X$, $\exists m\in \Bbb{N}$ such that $x\in B_m \subseteq f^{-1}(U)$. So $f(x)=y \in f(B_m)$ and $f(B_m)\subseteq f(f^{-1}(U)) \subseteq U$. Which implies $f(B_m)=f(B_m) \cap f(X) \subseteq U\cap f(X)=V$. Hence $\exists f(B_m)\in \mathcal{B}'$ such that $y\in f(B_m)\subseteq V$. By lemma 13.2, $\mathcal{B}'$ is basis of $\mathcal{T}_{f(X)}$. Is this proof correct?

Find volume between $x^2+y^2+z^2 \le 2(y+z)$ and $ x^2+y^2 \le z^2$

Posted: 04 May 2022 06:55 PM PDT

Recently I've been proposed the next problem at my calculus course.

Find the volume of the following corps: $$\left\{(x,y,z) \in \mathbb{R}^3 : \; x^2+y^2+z^2 \le 2(y+z), \; x^2+y^2 \le z^2\right\}$$

On first place, I attempted to plot the functions via Mathematica. This is a screenshot of the plot

enter image description here

I've thought to divide the figure in 4 shards like this and this, so I can find the volume of each one separately. (sorry for only having the images, I don't know how to explain it)

For the first part, I tried to find the intersection of the sphere ($x^2+y^2+z^2≤2(y+z)$) and the cone ($x^2+y^2 \le z^2$) and try to find a parabolic cylinder that goes trough that intersection and is normal to the XY plane, so I can have $y$ in function of $x$ so I'm able to do the integral, but I'm struggling here, so I hope you can help me.

How to prove that the image is a submanifold?

Posted: 04 May 2022 07:07 PM PDT

Here is the question I am trying to tackle:

Show that the map $$f : \mathbb R P^n \to \mathbb R P^{n + 1},$$ defined by $$[p] = [p_0, \dots, p_n] \mapsto [p,0] = [p_0, \dots, p_n, 0]$$

is an embedding.

I am having a hard time in proving that it is a submanifold, what exactly should we do to prove that it is a submanifold? I think I do not understand the definition very well.

Here is the definition of a submanifold I am using:

enter image description here enter image description here

Here is the definition of an embedding:

enter image description here

The hint I have for the solution of this problem:

enter image description here

Could someone help me please?

Can a quintic equation be solved without needing a "piece of paper the size of a large asteroid"(!)?

Posted: 04 May 2022 07:05 PM PDT

I remember from a long time ago reading a paper regarding the solution of quintic polynomial equations using hypergeometric functions. In particular, the methods are based around the solution of the so-called "Bring quintic form"

$$t^5 - t - \rho = 0$$

which has a solution

$$t = -\rho\ _4F_3\left(\frac{1}{5}, \frac{2}{5}, \frac{3}{5}, \frac{4}{5}; \frac{1}{2}, \frac{3}{4}, \frac{5}{4}; \frac{3125}{256} \rho^4\right).$$

However, the paper also mentioned that to solve the general quintic form

$$x^5 + ax^4 + bx^3 + cx^2 + dx + e = 0$$

you need "a piece of paper as big as a large asteroid" to write down all the formula, which is extraordinary, given that the first four degrees, while large, are not that large. Now I don't know how big a "large asteroid" is supposed to be, but if presumably that's, say, 10 km across, and treating that asteroid as a sphere, we're looking at a surface area of about 523 km^2 or somewhere around 8 billion sheets of A4 paper, using one side only, so 4 billion sheets if using both sides and, thus, if the latter were bound into 800-page (400 sheet), thus rather thick, books, 10 million such books, which is also over three times as many books as in the whole Library of Congress, most of which are not so big and thick! And not even my computer's hard disk could store that much - it sounds like quite a few TB of data, and I have only about 1-2 TB total of hard disk storage on my machine with all drives pooled. And certainly no easily affordable computer could have stored it at the time that paper was written - a small disk farm would have been needed.

And what I am wondering about is: can one do better than this? I note that the method commonly employed to reduce the quintic to the Bring form, which involves taking a "resultant" of the quintic with a quartic, can also be used to solve a cubic, and when that is done (using a quadratic, to reduce to a perfect cube), the resulting formula is considerably more "wordy" than the traditional Cardano cubic formula. And thus that suggests to me a sort of inefficiency in the method. So is there a way to improve upon it and make it maybe, if not necessarily "small", then at least small enough that it could fit in, say, one book, with everything plugged together?

It would also be acceptable, by the way, to not entirely plug it together, but to just write $\rho$ as a function of $a$, $b$, $c$, $d$, and $e$, together with $t$ (already given) and how to derive $x$ from $t$ - so long as both of those (for $\rho$ and $x$) are all, of course, plugged together themselves. Though if the Bring form must be abandoned altogether, that is fine too, so long as we don't need to introduce functions beyond a suitable complexity level, e.g. equivalent to inverting fixed polynomials with no free parameters.

First Order Differential Equation -Variation of Parameters Method

Posted: 04 May 2022 07:02 PM PDT

I'm trying to solve arbitrary current equation for neuron membrane from Gerstner's book.

Here it begins:

Question: Assuming that before a given time t0 the current is null and the membrane potential is at rest, derive the general solution to Eq. (1) for arbitrary I(t).

My Solution: Variation of Parameters method is used to solve differential equation:

$$\tau\frac{du}{dt} = \ - (u(t) - u_{\text{rest}}) + RI(t)\quad \tag1 $$ $$\tau\frac{du}{dt} +u(t) =0 \implies \tau\frac{du}{dt}=-u(t)\implies \int \frac{du}{u}=-\int\frac{dt}{\tau}$$ $$u(t)= e^{-t/{\tau}}\cdot C \implies u(t)=C(t)\cdot e^{-t/{\tau}}$$ $$ u^{'}(t)=C^{'}(t)\cdot {e^{-t/{\tau}}-\frac{1}{\tau}e^{-t/\tau}}\cdot C(t) $$ Inserting into Eq. $(1)$: $$ \tau\left(C^{'}(t)\cdot {e^{-t/{\tau}} - \frac{1}{\tau}e^{-t/\tau}}\cdot C(t)\right)= -C(t)e^{-t/\tau} +u_{\text{rest}}+RI(t) $$ $$ \tau\cdot C^{'}(t)\cdot e^{-t/{\tau}} =u_{\text{rest}}+RI(t) $$ $$ \int C^{'}(t)=\int \frac{1}{\tau} \left(u_{\text{rest}}+RI(t)\right)\cdot e^{t/{\tau}}dt \implies C(t)= C_{1}+ \frac{1}{\tau}\int_{t_{0}}^t (u_{\text{rest}}+RI(t))e^{t/{\tau}}dt $$

$$ U(t)= e^{-t/{\tau}}\left(C_{1} +\frac{1}{\tau}\int_{t_{0}}^t (u_{\text{rest}}+RI(t))e^{t/{\tau}}dt \right) $$

In solution, it says before a given time $t_{0}$ the current is zero and the $ut_{0} = u_{\text{rest}}$. With this information I can't apply it to find $C_{1}$. In some resources, they say applying initial condition results that integration part of equation equals to zero. So I tried that: $$ U(t)= e^{-t/{\tau}}\left(C_{1} +\frac{1}{\tau}\int_{t_{0}}^t (u_{\text{rest}}+RI(t))e^{t/{\tau}}dt \right) $$ Initial condition $Ut_{0} = U_{0}$ $$U_{0} =e^{-t0/{\tau}}(C_{1} + \text{ (they specified integration equals to zero)) } \implies C_{1}=U_{0}\cdot e^{t0/{\tau}} $$ $$U_{t}=e^{(t0-t)/{\tau}}\cdot U_{0} + \frac{e^{-t/{\tau}}}{\tau}\int_{t_{0}}^t(u_{\text{rest}}+RI(t))e^{s/{\tau}}ds $$

I couldn't proceed from that point.

Book's Solution: $$\tau\frac{du}{dt} +u(t) =0 \iff \frac{du/dt}{u}=-\frac{1}{\tau} \implies u(t)=ke^{-t/{\tau}} $$ A particular solution can be obtained by the "variation of parameters" method: we write $u(t)=k(t)e^{-t/{\tau}}$ and replace it in Eq. (1). $$\tau\left(\frac{dk(t)}{dt}-\frac{1}{\tau}k(t)\right)e^{-t/{\tau}} + k(t)e^{-t/{\tau}} =u_{\text{rest}}+RI(t) $$ $$\frac{dk(t)}{dt}=\frac{1}{\tau}(u_{\text{rest}}+RI(t))e^{t/{\tau}} $$ Integrating, we find $k(t) = k_{2}+\frac{1}{\tau}\int_{t_{0}}^t (u_{\text{rest}}+RI(s))e^{s/{\tau}}ds $ where k2 is a new integration constant. Denoting the initial condition by $u_{0} = u_{to}$, we obtain: $$u(t) = u_{\text{rest}} + (u_{0}-u_{\text{rest}})\cdot e^{-(t-t0)/{\tau}} +\frac{1}{\tau}\int_{t_{0}}^t RI(s)e^{-(t-s)/{\tau}}ds $$ Using the particular initial condition $u(t_{0}) = u_{\text{rest}}$, the equation simplifies to: $$ \bbox[yellow]{ u(t) = u_{\text{rest}} + \frac{1}{\tau}\int_{t_{0}}^t RI(s)e^{-(t-s)/{\tau}}ds \implies \text{The real solution}} $$

As you can realize, my solution is wrong. How can I achieve the correct solution. Also some sources say, these equations can be solved by laplas and convolution methods. I searched this, but this method is used solve second degree diff equations. If someone wish to solve by laplas, I will be appreciated. Pls try to help me, I am very desperate right now. Best regards.

Some questions on "A Necessary and Sufficient Condition for Edge Universality of Wigner Matrices"

Posted: 04 May 2022 06:58 PM PDT

I'm currently reading the paper "A Necessary and Sufficient Condition for Edge Universality of Wigner Matrices" by Ji Oon Lee and Jun Yin. I'm trying to understand the proof of the main theorem, i.e., Theorem $1.2$ on page $3$. In sections $1$ to $3$, they basically introduce some notations and theorems/lemmas that will be used to prove the main theorem. Section $4$ is the proof of the main theorem, and the last two sections (Section $5$ and $6$) are the proofs of those theorems/lemmas that are used to show the main theorem.

I'm not trying to go over all the proofs, but only the proof of the main theorem (Section $4$). I have a question on $(4.38)$ and $(4.39)$. enter image description here To understand these two statements, we need the following: enter image description here enter image description here enter image description here where $$ G^{S}(\mu):=\frac {1}{H^{S}-\mu}. $$ $H^{S}$ is defined as follows: enter image description here where $X$ is a symmetric matrix with zero mean on both diagonal and off-diagonal entries. For off-diagonal entries, the variance is $1$, and for diagonal entrie, the variance is finite. What I have so far for $(4.38)$ is the following: Given that $\tilde{z}=2+iN^{-2/3}$, we have $$ m_{sc}(\tilde{z})=\frac {-\tilde{z}+\sqrt{\tilde{z}^2-4}}{2}=-\frac {i}{2N^{2/3}}+\frac {1}{2}\sqrt{-\frac {1}{N^{4/3}}+\frac {4i}{N^{2/3}}}-1\sim -1\text{ as }N\to \infty. $$ By $(3.3)$ and $(2.12)$, we have \begin{align*} \max_{1\leq i\leq 2s+t}\left|G^{S}_{ii}(\tilde{z})+1\right|&\leq \varphi^{C}\left(\frac {1}{q}+\sqrt{\frac {\text{Im }m_{sc}(\tilde{z})}{N\eta}}+\frac {1}{N\eta}\right)\\ &\leq q\left(\frac {1}{q}+\sqrt{\frac {\text{Im }m_{sc}(\tilde{z})}{N\eta}}+\frac {1}{N\eta}\right)\\ &= 1+2N^{-1/3 +\epsilon} \end{align*} Here $q=N^{\epsilon}$ since $|h_{ij}|=|x_{ij}|/\sqrt{N}\leq N^{1/2 -\epsilon}/\sqrt{N}$. However, $1+2N^{-1/3 +\epsilon}\geq N^{-\epsilon/2}$. And I have no idea how to show $(4.39)$.

Ray through Polytope boundary(vertex/surface)?

Posted: 04 May 2022 07:02 PM PDT

This is a follow up problem to the question I asked sometime earlier also a similar problem to this, please what's the best way to quickly determine if a ray originating from a point intersects a n-polytope externally given it passes through a common boundary(vertex/surface) of the polytope. Here are some visual examples of the problem:

here the ray $\vec{r}$ intersects $abcd$ externally at $d$ enter image description here here the ray $\vec{r}$ intersects $abcd$ internally at $d$ enter image description here here the ray $\vec{r}$ intersects $abcdefgh$ externally at $d$ enter image description here here the ray $\vec{r}$ intersects $abcdefgh$ internally at $d$ enter image description here here the ray $\vec{r}$ intersects $abcdefgh$ externally at $dh$ enter image description here here the ray $\vec{r}$ intersects $abcdefgh$ internally at $dh$ enter image description here

With some vector algebra how can one discern this for any given ray/n-polytope scenario?

UPDATE: The n-polytope is convex, all its vertices are known, also the intersection of the ray on the n-polytope is known and it could be a vertex/edge/surface/hyperface

How to rearrange multiple regression equation to solve for an independent in excel

Posted: 04 May 2022 07:01 PM PDT

Is there a way to solve this equation for $F_2$? I am using excel to do this if that matters. Thanks for any help.

$Y=B_2+B_3F_2+B_4G_2+B_5H_2+B_6I_2+B_7J_2+B_8K_2$

Y=INTERCEPT+B1*X1+B2*X2+B3*X3+B4+X4+B5*X5+B6*X6
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