Thursday, April 21, 2022

Recent Questions - Mathematics Stack Exchange

Recent Questions - Mathematics Stack Exchange

Show that ∑a(n)logn converges if and only if ∑a(p) converges.

Posted: 21 Apr 2022 08:05 AM PDT

I am studying analytic number theory at the time and i came across with this. I was wondering if is there is a way to prove the following without using prime number theorem or any summation formula : Let a(n) be a nonincreasing sequence of positive numbers. Show that ∑a(p) -sum over all primes- converges if and only if ∑a(n)logn -sum from 2 to infinity- converges. I have seen 2 questions here about this where they use prime number theorem and summation formulas, but I have a strong feeling that it can be proven without these, i have tried it but nothing seems to work out well.My idea is to work somehow like proving the cauchy condensation theorem for series and maybe use somehow the fact that there are constants a,b positive such that anlnn<p(n)<bnlnn for all n from some N and after. Any help or ideas whould be much appreciated.

Continuous distribution

Posted: 21 Apr 2022 08:05 AM PDT

An up and coming fashion designer plans to expand her business and is looking for a seamstress to add in her team. She was able to narrow down the applicants to two, Marcela and Lorenza. Help the fashion designer decide on who to hire as new seamstress by answering the following.

  1. Marcela and Lorenza agreed to go under one month observation period. During the time, the fashion designer noticed that to cut a basic clothing pattern, Marcela takes 20 to 40 minutes while Lorenza takes 15 to 45 minutes. Suppose the pattern for a simple garment was requested on a rush. If both seamstresses started cutting at 8:00 AM, who has a better chance of completing the task by 8:25 AM?
  2. During the observation period, it was observed that on the average, the number of simple garments completed per day by Marcela and Lorenza is 6 and 4, respectively. This is considering that they are only allowed to work on one garment at a time. a. Suppose Marcela and Lorenza started working on the same garment at 10:00 AM and 30 minutes have passed since. What is the probability that each seamstress is still working on the garment one hour later? b. Who has the higher probability of completing her 12th garment within two days?
  3. From their individual interview, Marcela claims that no more than 5% of the garments she sewed did not pass quality check while Lorenza claims her to be no more than 3%. During the observation period, Marcela completed 120 garments of which 5 needs to be refined and Lorenza completed 90 garments of which 3 needs to be refined. a. Who has the higher probability of telling the truth? b. The fashion designer implements a salary deduction whenever a seamstress produces 10 subpar garments. Based on the proportion of garments completed by Marcela and Lorenza that needed refining and using approximation, what is the probability that each seamstress gets salary deduction before completing her 100th garment?
  4. Based on the information provided and probabilities computed in (1) to (3), which seamstress will you recommend for hiring? Justify your answer.

Atwood system of two pulleys

Posted: 21 Apr 2022 08:05 AM PDT

A light string passes over a light fixed pulley carrying a mass P at one extremity and a light pulley at the other. Another light string passes over this second pulley and has masses Q and R at its extremities. If the system starts from rest and R remains at rest throughout then show that

$$4/P + 1/Q = 3/R$$

I have drawn the free body diagram of the problem and assigned tension $T$ corresponding to the rope of $R$; ie $T=Rg$. Now the other end of same string has equation $T-Qg = Qa$ where $a$ is the acceleration due to gravity of the mass $Q$. However I am facing trouble wrt the string carrying $P$. Here the tension is $2T$ but I have trouble assigning acceleration and related parameters. How do I solve on from here? Thank you in advance.

Intuitive explanation for $P(A \cap B) - P(A)P(B) = P(\bar{A} \cap \bar{B}) - P(\bar{A}) P(\bar{B})$

Posted: 21 Apr 2022 08:03 AM PDT

For two events $A$, $B$ and a probability measure $P$, one has \begin{equation*} P(A \cap B) - P(A)P(B) = P(\bar{A} \cap \bar{B}) - P(\bar{A}) P(\bar{B}), \end{equation*} where the bars denote complements. The probability measure is not special as the identity remains true for measurable sets of an arbitrary measure (including infinite ones).

The proof by direct calculation is of course easy and elementary, what I'm looking for is an intuitive, hopefully somewhat visual argument, in particular clearly describing the inherent symmetry.

Can these two matrices (composed of fifth roots of unity) satisfy the following determinant relations?

Posted: 21 Apr 2022 08:01 AM PDT

I have two 4x3 complex matrices whose elements are composed by fifth roots of 1. There is no zero entry. Let $\gamma=e^{2i \pi /5}$, then

$A=\begin{pmatrix} \gamma^{a_1} & \gamma^{a_2} & \gamma^{a_3} \\ \gamma^{b_1} & ... & ... \\ \gamma^{c_1} & ... & ... \\ \gamma^{d_1} & ... & ... \end{pmatrix}$ and $B=\begin{pmatrix} \gamma^{e_1} & \gamma^{e_2} & \gamma^{e_3} \\ \gamma^{f_1} & ... & ... \\ \gamma^{g_1} & ... & ... \\ \gamma^{h_1} & ... & ... \end{pmatrix}$.

If any row is removed from either $A$ or $B$, the determinant of the resulting 3x3 is always non-zero. Furthermore, I have a somewhat odd condition: let $|A_x|$ ($|B_x|$) be the determinant of the 3x3 submatrix of $A$ ($B$) obtained by removing the $x$-th row, then

$\frac{|A_1|}{|B_1|}=\frac{|A_2|}{|B_2|}=\frac{|A_3|}{|B_3|}$.

My question is: when can this actually happen? What are the conditions on the integers (mod $5$) $a_1,...,h_3$ such that these properties are satisfied?

Calculate $k\int_{-\infty}^{\infty}\frac{x^r}{(1+x^2)^m}dx$

Posted: 21 Apr 2022 08:00 AM PDT

Calculate $$\mu_r = k\int_{-\infty}^{\infty}\frac{x^r}{(1+x^2)^m}dx$$

What I have tried:

For $m>1$, making the substitution $z = \frac{1}{1+x^2} \implies x = z^{-\frac{1}{2}}(1-z)^{\frac{1}{2}} \implies dx = -\frac{1}{2z^{\frac{3}{2}}(1-z)^{\frac{1}{2}}}$

Plugging these in $$ \begin{align} \mu_{2r} = k\int_{-\infty}^{\infty}\left(\frac{x^{r/m}}{1+x^2}\right)^mdx &= k\int_0^{1}\left(z^{-\frac{r}{2m}+m}(1-z)^{\frac{2}{2m}}\right)^m\left(-\frac{1}{2z^{\frac{3}{2}}(1-z^{\frac{1}{2}})}\right)dz \\ &= -k\int_0^1\frac{1}{2}(1-z)^{\frac{r}{2}-\frac{1}{2}}z^{m-\frac{r}{2}-\frac{3}{2}}dz \end{align}$$

However, when working through the book it shows instead

$$k\int_0^1z^{m-r-3/2}(1-z)^{r-1/2}dz$$.

How do I get it in that form but also where do the integral bounds $\int_0^1$ come from? I understand that the $r/2$ cancels when we have $2r$ which gets the bit inside the integral. However, I do not know how to rid $-1/2$, I'm guessing it had something to do with the integral bounds.

I know that I can do $$\int_{\infty}^{\infty} = \int_{-\infty}^1+\int^\infty_1$$

and that $\frac{1}{1+\infty^2} \to 0$

So something like $$-2\int^0_1=2\int_0^1 \implies k\int_0^1z^{m-r-3/2}(1-z)^{r-1/2}dz$$

Which would remove $-1/2$. However, what's stopping me from using $\int_{-\infty}^1$? Does it become an odd function with this integral, and hence equal to 0?

a generalized Jensen's equality for Nevanlinna class

Posted: 21 Apr 2022 08:00 AM PDT

Define the Nevanlinna class of holomorphic functions on unit disk by $\displaystyle N(\mathbb{D})=\left\{f\in\text{Hol}(\mathbb{D}):\sup_{0<r<1}\int_\mathbb{T}\log^+|f_r|dm<\infty\right\}$ where $\log^+=\max(0,\log t)$ for $t>0$ and $f_r(z)=f(rz)$ and $m$ is the normalized Lebesgue measure on unit circle $\mathbb{T}$ .

Let $f\in N(\mathbb{D})$ , $f(0)\neq0$ and $(\lambda_n)_{n\geq1}$ are the zeroes of $f$ in $\mathbb{D}$ counting multiplicity and a real singular measure $\mu$ w.r.t. $m$ satisfies $\displaystyle V_\mu(z)=\exp\left(-\int_\mathbb{T}\frac{\xi+z}{\xi-z}d\mu(\xi)\right)$ . Show that $$\log|f(0)|+\sum_{n\geq1}\frac{1}{\lambda_n}+\mu(\mathbb{T})=\int_\mathbb{T}\log|f|dm$$

A hint says to use the inner-outer factorization of Nevanlinna class like $f=\lambda BV_\mu[h]$ where $\lambda\in\mathbb{C}$ with $|\lambda|=1$ , $B$ is the Blaschke product of zeroes of $f$ and $\displaystyle[h](z)=\exp\left(\int_\mathbb{T}\frac{\xi+z}{\xi-z}\log|h(\xi)|dm(\xi)\right)$ and $h=|f|$ . Taking absolute and log both sides give $$\log|f/B|=\log|V_\mu|+\log|[h]|$$ $$\implies\log|f/B|=-\int_\mathbb{T}\frac{1-|z|^2}{|\xi-z|^2}d\mu(\xi)+\log|h|$$ $$\implies\log|(f/B)(0)|+\int_\mathbb{T}d\mu(\xi)=\log|h(0)|=\log|f(0)|$$ $$\implies\log|f(0)|+\sum_{n\geq1}\log\frac{1}{\lambda_n}+\mu(\mathbb{T})=\int_\mathbb{T}\log|f|dm$$ a.e. on $\mathbb{T}$ . Everything is alright except the log is appearing inside summation part , and it is evident by mean value theorem , can someone throw some light what is possibly wrong here ? Thanks in advance .

Multivariate Time Series Clustering with Affinity Propagation

Posted: 21 Apr 2022 07:54 AM PDT

I'm trying to implement the clustering procedure described in the paper that follows at the link https://www.hindawi.com/journals/wcmc/2021/9915315/, but I get to the algorithm 2 described and I get lost. As I understand it, steps S1-S3 are nothing more than obtaining the similarity (cost) matrices from the DTW algorithm for each dimension of the time series. In S4, it does nothing more than apply the affinity propagation algorithm to find the clusters of these similarity matrices and the MTS similarity matrix (considering all its dimensions). In S5 I got lost, what would this R = C2R(C) be? So far I'm interpreting it as if it were simply a correlation matrix, like Pearson's. If you can help me, I will be very grateful.

Give an O(n+m) algorithm that returns a node for which if we delete we get a maximum number of components

Posted: 21 Apr 2022 07:51 AM PDT

I have this homework :

Give an O(n+m) algorithm that returns a node of an undirected graph for which if we remove it, the remaining graph will consistent of the maximum number of components.

My idea: First, i thought if we delete the node with max degree we get the result. But since it says O(n+m) i thought maybe i should use DFS. I read that DFS can help to look for bridges, but it doesn't give back a node but an edge instead. So my question, can i use DFS and look for bridges and for all vertices with bridge i use the one with max degree and that's the answer.

Is it correct? Or looking for vertices with bridges is a bad idea? I would appreciate any help.

Let $A$ be any ring, $a \in A$ and $f \in A[T]$. Prove that there exists an expression $f=(T-a)q+r$ with $q \in A[T]$ and $r \in A$.

Posted: 21 Apr 2022 08:00 AM PDT

Let $A$ be any ring, $a \in A$ and $f \in A[T]$. Prove that there exists an expression $f=(T-a)q+r$ with $q \in A[T]$ and $r \in A$. (Hint: subtract off a suitable multiple of $(T-a)$ to cancel the leading term and use induction on $\deg f$.)

Let $f \in A[T]$, then $f=c_nT^n + \dots c_0$. Now by suitable multiple I think they mean $f(a)$ so what I'll have is that $$f(T)-f(a)=c_n(T^n-a^n)+ \dots+c_1(T-a)$$ but this does not cancel any leading terms nor do I get anything about the degree of $f$? The degree is just $n$?

Why does this limit not exist when visually the function approaches 1?

Posted: 21 Apr 2022 07:53 AM PDT

The limit from this question Why does L'Hopital's rule fail in calculating $\lim_{x \to \infty} \frac{x}{x+\sin(x)}$?

doesn't exist. But in Desmos, tye function surely approaches 1 as $x$ gets larger and larger. Why does the limit not exist and how to check such one?

Please explain in layman's terms for I am only an O level student

Justification for $\operatorname{dim} \wedge^k(V)$ is $n\choose k$ the alternating $k$-linear form. [duplicate]

Posted: 21 Apr 2022 07:57 AM PDT

I was told that: $\operatorname{dim} \wedge^k(V)$ is $n\choose k.$ and I am trying to justify this by computing it for different dimensions of $V.$

If $\operatorname{dim}(V) = 1,$ then $v_1 \wedge v_1 = 0$. Therefore, \operatorname{dim} $\wedge^1(V)$ is $1\choose 1$

If $\operatorname{dim}(V) = 2,$ then we will have the following cases:

\begin{align*} v_1 \wedge v_1 &= 0\\ v_1 \wedge v_2 &= - v_2 \wedge v_1\\ v_2 \wedge v_1 &= - v_1 \wedge v_2\\ v_2 \wedge v_2 &= 0 \end{align*}

But then I can not conclude what is my $k$ in this case and how many variables actually I am choosing as I have the second case is just the same as the third case. Could anyone help me in this please?

Also, I know that

\begin{align*} v_1 \wedge v_1 \wedge v_1 &= 0\\ v_1 \wedge v_2 \wedge v_3 &= - v_2 \wedge v_1 \wedge v_3\\ &= - v_3 \wedge v_2 \wedge v_1\\ &= - v_1 \wedge v_3 \wedge v_2\\ &= v_2 \wedge v_3 \wedge v_1\\ &= v_3 \wedge v_1 \wedge v_2\\ v_2 \wedge v_2 \wedge v_2 &= 0\\ v_3 \wedge v_3 \wedge v_3 &= 0 \end{align*}

But I do not know how to translate this into $n \choose k.$ Could someone help me in this also?

Thanks in advance!

EDIT:

Note that I was told that my question maybe a duplicate of this If $V$ is $k$-dimensional then show that $\dim \wedge^n V = \binom {k} {n}.$ but I do not agree with this as I need a small and concrete example to be calculated not an abstract procedure as in the previous link and the other one is asking about a specific case when $k<n$ in $k \choose n$ which is not my case.

Finding all integer solutions.

Posted: 21 Apr 2022 07:49 AM PDT

I need to try to find and of the integer solutions for the equation given

(xˆ2 - 7x + 11)ˆ(xˆ2 - 11x + 30) = 1

I'm not to sure where to start so any help with the solutions would be nice.

How to prove : $A ∩ A^c ⊂ \emptyset$

Posted: 21 Apr 2022 07:59 AM PDT

I would like you to clear this doubt.

I know how to prove the inverse, but to prove how these sets are contained in the empty set, I don't know how to do it.

$$A ∩ A^c ⊂ \emptyset$$

$A^c$ is the complementary of the set $A$.

I thought about

Assuming $A ∩ A^c ⊄ \emptyset$, then it means $x ∈ A$ and $x ∈ A^c$ by the definition of intersection. However, it is impossible for the element $x$ to belong to the set $A$ and to the set $A^c$ at the same time by the definition of complement. Soon $A ∩ A^c ⊂ \emptyset$ is true.

I thought about it but I don't know if it's correct.

A question about Comparison Principle in Nonlinear Systems?

Posted: 21 Apr 2022 08:01 AM PDT

A question about Comparison Principle

For a general system, we have $$ V=x^{2}+y^{2} $$ where $x \in \mathbb{R}$ and $y \in \mathbb{R}$ are two independent states, and $V$ is a Lyapunov function. There are two cases: Case 1 is very common, based which I formulate the question in Case 2 .

Case 1. If $$ \dot{V} \leqslant-K\left(x^{2}+y^{2}\right)+\beta $$ holds for $K>0$ and $\beta>0$, we have $$ \dot{V} \leqslant-K V+\beta $$ then, according to Comparison Principle, $$ \limsup _{t \rightarrow \infty} V=\limsup _{t \rightarrow \infty}\left(x^{2}+y^{2}\right) \leqslant \frac{\beta}{K} $$ Case 2. If $$ \dot{V} \leqslant-K\left(x^{2}+100 y^{2}\right)+\beta $$ holds for $K>0$ and $\beta>0$, will we have $$ \limsup _{t \rightarrow \infty}\left(x^{2}+100 y^{2}\right) \leqslant \frac{\beta}{K} ? $$

Integrating with cos

Posted: 21 Apr 2022 07:54 AM PDT

I am trying to show that $\int_{- \pi}^{\pi} cos(t)^n \mathrm{dt} = 4\int_{0}^\frac{\pi}{2} cos(t)^n \mathrm{dt}$ if $n=2m$ but I don't understand the transition? Should I apply u substitution?

Markov chain that is "increasing" on average

Posted: 21 Apr 2022 07:53 AM PDT

Suppose that $S = \mathbb{Z}_{\geq 0}$ and that $X_n$ is an irreducible, aperiodic Markov chain satisfying that for all $x \in S$ that $E_x[X_n] \geq x$ so that in expectation $X_n$ is not decreasing. Is it true that $X_n$ is not positive recurrent? There are simple examples of such $X_n$ that are transient or null recurrent, but I can't think of any that are positive recurrent.

I tried using the expectation condition to examine the expected return time to say $x = 0$ to try to show the expectation is infinite but did not have any success. Does anyone have any hints or insights?

Bijective morphism between varieties is birational

Posted: 21 Apr 2022 07:46 AM PDT

Let $X$ and $Y$ be reduced separated schemes of finite type over a field $k$ with characteristic zero. How can we show that any bijective morphism $f:X\rightarrow Y$ induces an isomorphism $U\cong f(U)$ for some dense open subset $U$ of $X$?

Question on tangent and normal for the curve ${x^3 \over a}+{y^3 \over b} =xy$

Posted: 21 Apr 2022 07:47 AM PDT

Find at what point on the curve $${x^3 \over a}+{y^3 \over b} =xy,$$ the tangent is parallel to one of the coordinate axes.

If $E,F\subseteq X$ are homeomorphic set then is $E$ open/closed if $F$ is open/closed?

Posted: 21 Apr 2022 08:00 AM PDT

Let be $X$ a topological space and we suppose that $E$ and $F$ are homeomorphic though a map $f$ from $F$ to $E$. So I ask to me if $E$ is open/closed when $F$ is open/closed but unfortunatley I was not able to prove or to disprove this so that I thought to put a specific question where I ask some clarification: in particular if the result is generally false I'd like to know if it can be true we additional hypotesis, e.g. Hausdorff separability, First Countability, Metric Topology, etc....

So could someone help me, please?

Proving $\operatorname{span}\{h_{nm}(x,y)=f_n(x)g_m(y)\mid n,m\in\mathbb{N}\}$ is dense in $C([0,1]^2)$ w.r.t $L^2([0,1]^2)$ norm

Posted: 21 Apr 2022 07:46 AM PDT

I am given two orthonormal bases for $L^2[0,1]$, $\{f_n\}_{n=1}^\infty$ and $\{g_m\}_{m=1}^\infty$, that consist of continuous functions (but not necessarily all continuous functions).

I'm trying to prove that $$\operatorname{span}\bigg\{h_{nm}(x,y)=f_n(x)g_m(y)\mid n,m\in\mathbb{N}\bigg\}$$ is dense in $C([0,1]^2)$ with respect to the $L^2([0,1]^2)$ norm defined by the following inner product:

$$\langle f, g\rangle = \int_{[0,1]}\int_{[0,1]}f(x,y)\overline{g(x,y)}dxdy $$

I'm supposed to use an earlier result where we have proven that if $X,Y$ are compact metric spaces, then for all $h\in C(X,Y)$ and for all $\epsilon>0$ there are $f_1,\dots f_n \in C(X)$ and $g_1,\dots g_n \in C(Y)$ such that $$ \Big |\Big | h(x,y)-\sum_{i=1}^nf_i(x)g_i(y) \Big |\Big |_{\infty} < \epsilon $$

Ideally, $\{f_n\}_{n=1}^\infty$ and $\{g_m\}_{m=1}^\infty$ would be precisely $C(X)$ and $C(Y)$, because then the result would immediately follow (it is easy to show that the sup norm is stronger than the $L^2([0,1]^2)$ norm), but I was not able to prove that they are (it also makes sense that this would have been too strong, but was worth a shot).

Next, I tried using the fact that $span\{f_i \}$ and $span\{g_i \}$ are dense in $C(X), C(Y)$, respectively (w.r.t the $L^2$ norm) but this got very messy with multiple sums and eventually didn't work.

As far as I can tell - there isn't much else I could try. Am I missing something trivial here?

Edit: Since this is part of a larger problem where I am trying to prove that the set is dense in $L^2([0,1]^2)$ (in fact, that what's inside the span is a basis), I cannot use this as a means to prove the above.

Problem about error term

Posted: 21 Apr 2022 08:00 AM PDT

So the question is there are three random variables $X$, $Y$ and $\epsilon$, which the relation can be shown by:

$Y = \beta_0 + \beta_1 X + \epsilon$

Hence see if the following is right and prove why if it is wrong or right

[1] If $E(X\epsilon) = 0$, then $E(X^2\epsilon) = 0$

[2] If $E(\epsilon|X) = 0$, then $E(X^2\epsilon) = 0$

[3] If $E(\epsilon|X) = 0$, then $X$ and $\epsilon$ are independent

[4] If $E(X\epsilon) = 0$, then $E(\epsilon|X) = 0$

[5] If $E(\epsilon|X) = 0$ and $E(\epsilon^2|X) = \sigma^2$, then $X$ and $\epsilon$ are independent

My professor did not really went through error terms in lecture but he sent this as task... The only thing that was written on the lecture note was that

Y and X can be described $Y = E(Y|X) + \epsilon$ ,where

1.$\epsilon$ is mean-independent of X that is $E(\epsilon|X) = 0$

  1. $\epsilon$ is uncorrelated with any function of X

I have no idea what this means at all, but I tried by:

for [1]

$Cov(X,X\epsilon) = E(X^2\epsilon) - E(X)E(X\epsilon)$

Since $E(X\epsilon) = 0$....... Really got stuck here

[2]

This one I think I kind of proved it

$E(X^2\epsilon) = E(E(X^2\epsilon|X))$

$E(X^2E(\epsilon|X)) = 0$ ??

Hence, this is true but I am kind of stuck.. here. I don't even understand the concept. I need some help.

Solving a PDE with a positively valued characteristic coefficient

Posted: 21 Apr 2022 07:49 AM PDT

I want to confirm that I get the solution of the PDE with positive coefficient right.

I have the initial value problem:

\begin{equation} \begin{cases} u_t=\alpha u_xx \ \ \ \ \ 0<x<L, t>0\\ u_x(0,t)=u_x(L,t)=0 \\ \end{cases}\\ u(x,0)= \begin{cases} 0\ \ \ \ 0<x<L/2\\ 1\ \ \ \ L/2<x<L \end{cases} \end{equation}

Set $\alpha=1$, and using separation of variables, we get:

\begin{equation} X_{xx}\pm k^2X=0\\ T_t=\pm k^2 \end{equation}

I would like to control the case of $k^2>0$, this gives with IC, that $k=\frac{1}{L}$

\begin{equation} X(x)=A\cosh\frac{1}{L}x \end{equation}

But when I use the second IC, in order to find A I get $A=0$ on $0<x<L$ and $\frac{1}{\cosh\frac{1}{L}x}$ on $L/2<x<L$. This gives

\begin{equation} u(x,t)= \begin{cases} 0, \ \ \ \ 0<x<L \\ e^{\frac{1}{L^2}t} , \ \ \ \ L/2<x<L \end{cases} \end{equation}

But this is not a complete function of two variables. What is wrong here?

Solution of multi variables dependent first order differential equation?

Posted: 21 Apr 2022 08:00 AM PDT

I am solving first order differential equation: $$(x_0x_1)' =-8 \dfrac{x_0'}{R},$$ where $x_0=f_0(z)$, $x_1=f_1(z)$, $R=r_1-z(r_1-1)=f_2(z)$ (they are all dependent on $z$, but that is not the same dependency).

Also, here first derivative of all variables is derivative along $z$ axis, for example: $x_0' = \dfrac{\mathrm{d}x_0}{\mathrm{d}z}$. These are my steps in solving equation: $$\dfrac{\mathrm{d}}{\mathrm{d}z}(x_0x_1) = -\dfrac{8}{R}\dfrac{\mathrm{d} x_0}{\mathrm{d}z} \bigg/*\mathrm{d}z$$

$$\mathrm{d}(x_0x_1) = -\dfrac{8}{R}\mathrm{d} x_0 \bigg/ integration$$ $$x_0 x_1 = - \dfrac{8}{R} x_0 + C$$ $$z=1: x_0 = 1, x_1 = 0\qquad\Rightarrow \qquad0 = - \dfrac{8}{R} + C$$ $$C=\dfrac{8}{R}$$ $$x_1 = \dfrac{8}{R} \left( \dfrac{1}{x_0}-1\right)$$ Did I miss something in solving this equation? Because, when I compare numerical and this solution there is some difference.

Difference between this-analytical and numerical solution.

Action of a map on homology

Posted: 21 Apr 2022 08:00 AM PDT

I'm currently studying algebraic topology from Hatcher's text, and I came across the following problem from an old qualifying exam:

The coefficient sequence $0 \rightarrow \mathbb{Z} \xrightarrow{p} \mathbb{Z} \xrightarrow{r} \mathbb{Z}/p \rightarrow 0$ induces a long exact sequence with boundary map $\overline{\beta}_p:H_k(X;\mathbb{Z}/p) \rightarrow H_{k-1}(X;\mathbb{Z})$. We define $\beta_p = r_{\ast}\overline{\beta}_p:H_k(X;\mathbb{Z}/p) \rightarrow H_{k-1}(X;\mathbb{Z}/p)$. Compute the action of $\beta_2$ on $H_{\ast}(\mathbb{R}P^{\infty};\mathbb{Z}/2)$.

I'm not quite sure how to approach this problem; namely, from my reading of Hatcher's text, I'm not familiar with what's meant by computing the action of a map on the homology groups. Hatcher discusses related ideas such as the action of $\pi_1$ on higher homotopy groups, but how can I compute the action specified in this problem?

Any thoughts would be appreciated.

Thanks!

Calculate the $g_{ij}$ of the following metric.

Posted: 21 Apr 2022 07:55 AM PDT

I have the following problem:

Let $\varphi:\mathbb{R^n}\rightarrow \mathbb{R}^{n+k}$ be such that $(M,\varphi)$ is a differentiable manifold, where $M=\varphi(\mathbb{R}^n)\subseteq\mathbb{R}^{n+k}$. In other words, $\varphi$ is a global parametrization for $M$. Let $\langle,\rangle_{\varphi(0)}$ be the inner product in $T_{\varphi(0)}M$ such that $\{\frac{\partial}{\partial x_1}\vert_{\varphi(0)},\ldots,\frac{\partial}{\partial x_n}\vert_{\varphi(0)}\}$ is an orthonormal basis. Now, for $a\in \mathbb{R}^n$, we consider $L_a:M\rightarrow M$, defined by $L_a(\varphi(x))=\varphi(x+a)$. We choose $\langle,\rangle_{\varphi(a)}$ an inner product in $T_{\varphi(a)}M$ such that $$DL_a(\varphi(0)):T_{\varphi(0)}M\rightarrow T_{\varphi(a)}M$$ is an euclidean isometry. Find the components $g_{ij}$ of $(M,\langle,\rangle)$.

My attempt:

The first thing I did was calculating the localization $f$ of $L_a$: $$x\overset{\varphi}{\longmapsto}\varphi(x)\overset{L_a}{\longmapsto}\varphi(x+a)\overset{\varphi^{-1}}{\longmapsto}x+a$$ So $f:\mathbb{R}^n\rightarrow \mathbb{R}^n$ is defined by $f(x)=x+a$. Thus, the jacobian matrix of $f$ is the identity $I$ and it doesn't depend on the point $\varphi(x)$ nor $a$. Now I want to calculate $$g_{ij}=\langle\frac{\partial}{\partial x_i}\vert_{\varphi(x)},\frac{\partial}{\partial x_j}\vert_{\varphi(x)}\rangle_{\varphi(x)}$$ As we know that $$DL_x(\varphi(0)):T_{\varphi(0)}M\rightarrow T_{\varphi(x)}M$$ is an euclidean isometry, then: $$\langle u,v\rangle_{\varphi(0)}=\langle DL_x(\varphi(0))(u),DL_x(\varphi(0))(v)\rangle_{\varphi(x)}$$ for all $u,v\in T_{\varphi(0)}M$.

But the jacobian of the localization of the previous differential is the identity, so $DL_x(\varphi(0))(\frac{\partial}{\partial x_i}\vert_{\varphi(0)})=\frac{\partial}{\partial x_i}\vert_{\varphi(x)}$, and we can conclude that: $$g_{ij}=\langle\frac{\partial}{\partial x_i}\vert_{\varphi(x)},\frac{\partial}{\partial x_j}\vert_{\varphi(x)}\rangle_{\varphi(x)}=\langle\frac{\partial}{\partial x_i}\vert_{\varphi(0)},\frac{\partial}{\partial x_j}\vert_{\varphi(0)}\rangle_{\varphi(0)}=\delta_{ij}$$

But this sounds strange for me. Any help or suggestions are welcome. Thanks!

Finding "growth" rate of a sum

Posted: 21 Apr 2022 07:56 AM PDT

Question

Let $n$ be a positive integer. The sum in question is: $$S_n=\sum_{i=1}^{ n}\frac{1}{2^i}\bigg(1-\frac{1}{2^{i}}\bigg)^{n}.$$ Clearly this sum is convergent, and it seems like the sum goes to $0$ as $n$ goes to infinity. Testing for $n<2000$ it seems like $S_n$ "grows" (decays) like $1/n$ asymptotically. So I am currently trying to upper bound $S_n$ by $c/n$ for some constant $c>0$.

Attempt

Pick a constant $d \in(0,1)$. Then for $i \leq d \log_2(n)$ one can easily show that: $$\big(1-2^{-i}\big)^n$$ tends to $0$ exponentially quickly with respect to $n$. So we can disregard the first bit of the sum and focus on: $$\sum_{i=\lfloor d\log_2(n)\rfloor+1}^{ n}\frac{1}{2^i}\bigg(1-\frac{1}{2^{i}}\bigg)^{n}.$$ We also clearly have that: $$\sum_{i=\lfloor \log_2(n) \rfloor}^n\frac{1}{2^i} \leq d'/n$$ for some $d'>0$.

But now we have to upper bound that middle portion of the sum, and using the methods I have used so far can give an upper bound that decays slower than $1/n$.

How do I proceed from here?

Calculate $\int_0^{\infty} \frac{\cos 3x}{x^4+x^2+1} dx$

Posted: 21 Apr 2022 07:49 AM PDT

Calculate $$\int_0^{\infty} \frac{\cos 3x}{x^4+x^2+1} dx$$

I think that firstly I should use Taylor's theorem, so I have:$$\int_0^\infty \frac{1-\frac{x^2}{2!}+\frac{x^4}{4!}-\dots}{(x^2+1)^2}dx$$ However I don't know what I can do the next.

Number of integer solutions to $x^2 + xy + y^2 = c$

Posted: 21 Apr 2022 07:58 AM PDT

Inspired by this question on Mathematica StackExchange:

Consider the set of integer solutions $x, y \in \mathbb{Z}$ to the equation $x^2 + xy + y^2 = m$, for $m \in \mathbb{N}$. Conjecture: the number of distinct solutions to this equation is divisible by 6 for all integer $m$.

I have done a brute-force calculation in Mathematica for $m \leq 10^4$, and have not found any counterexamples. For example, there are:

  • 0 integer solutions for $m = 2$;
  • 6 distinct solutions for $m = 3$ [$(x,y) = \pm (2, -1)$, $(x,y) = \pm (-1, 2)$, and $(x,y) = \pm (1, 1)$];
  • 12 distinct solutions for $m = 7$;
  • 18 distinct solutions for $m = 49$;

and so forth. The largest number of solutions Mathematica found was 54 solutions, for $m = 8281$. All of these are divisible by 6.

Is there a counterexample to this conjecture for some larger value of $m$? Or can the conjecture be proven?

I suspect that a proof will involve some kind of hidden symmetry of the polynomial $x^2 + xy + y^2$ that maps integer solutions to integer solutions; but I haven't been able to nail it down. It's not hard to see that the number of solutions must be even (if $(x,y)$ is a solution, then so is $(-x, -y)$, and these are distinct unless $x = y = 0$); but the divisibility by 6 is much more mysterious to me.

Maxima & Minima Word Problem

Posted: 21 Apr 2022 08:03 AM PDT

Problem: Given the following profit-versus-production function for a certain commodity: $P=200000-x-(\frac{1.1}{1+x})^8$. Where P is the profit and x is the unit of production. Determine the maximum profit.

Solution: Taking its first derivative, $\frac{dP}{dx} = -1-8(\frac{1.1}{1+x}^7) * (\frac{-1.1}{(1+x)^2})$, then equate to $0$, the value of x would be equal to $0.371$. Then substituting it to the original equation would result to $199,999.46$ which is the maximum profit.

Question:

  1. How to solve if instead, the problem asked for the minimum profit?
  2. In some problems, the minimum is the value of x (example: the 0.371 in the problem above) after differentiating the given equation and equating it to 0. But in some problem the minimum is the value after substituting that x, so in some problem, that 199,999.46 is the minimum instead. So how can I know which is which?

Any help or tip would be appreciated.
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