0
votes
0answers
2 views

Do disjoint cycles commute?

When a given set is finite it is clear. I'm asking the general case. Let $X$ is an arbitrary set. Let $\sigma,\tau$ be disjoint cycles on $X$. Then do they commute?
0
votes
0answers
2 views

Probabilistic model of parallel web servers

Note: The following probabilistic model of parallel web servers is abstracted from an engineering project. I am not good at probability theory and I am seeking some evaluations and suggestions. ...
1
vote
0answers
14 views

Computing the inverse explicitly (real analysis)

I have a function $\ f:\mathbb R\to\mathbb R$ such that $\ f(x,y)=(xe^y,xe^{-y}) $ Let $\ a=(1,0), b=(1,1) $ and let $\ g$ be the continuous inverse of $\ f$ such that $\ g(b)=a$. Compute $\ g$ ...
0
votes
1answer
11 views

How to Simplify Sin/tan problem.

I am trying to simplify $\displaystyle\frac{\sin^2}{\tan^2}$ but I don't know how to go about it. Any help is appreciated.
0
votes
0answers
5 views

Inverse of the function $\frac{(1+x)^2-i(1-x)^2}{(1+x)^2+i(1-x)^2}$

It can be proved that the function $f:[-1,1]\to \mathbb{C}$ defined by $$f(x)=\frac{(1+x)^2-i(1-x)^2}{(1+x)^2+i(1-x)^2}$$ maps the interval $[-1,1]$ one to one onto the lower part of the unit circle. ...
0
votes
1answer
18 views

Express $a^5$ in terms of $c_0+c_1a+c_2a^2.$

Let $F=\mathbb Z_2,f(x)=x^3+x+1\in F[x].$ Suppose $a$ is a zero of $f(x)$ in some extension of $F.$ Then $F(a)\simeq F[x]/\langle f(x)\rangle$ and there is an isomorphism $$\phi:F[x]/\langle ...
0
votes
1answer
10 views

Direct Products Help Abstract

Let Z be the additive group of integers and S = {-1,1} be a group under multiplication. Is the product Z x S cyclic? Why or why not? I am really confused on this question and have no idea where to ...
0
votes
2answers
10 views

Probability of weather on consecutive days.

Probability of a cloudy day is .55 Probability of a sunny day is .45 A)What is the probability of three consecutive cloudy days, followed by a sunny day? B)What is the probability that exactly 1 out ...
0
votes
1answer
26 views

Does the series $\sum\limits_{n=1}^\infty\frac{\sin(n)n!}{n^n}$ converge?

$\sum\limits_{n=1}^\infty\frac{\sin(n)n!}{n^n}$ Please let me know how you did it. Thank you.
0
votes
0answers
3 views

How do I draw a Hasse Diagram for the given PO-set?

Copied from my homework: Draw a Hasse Diagram for the PO-set: ({$p, r, p \lor r, p \land r, p \to r$}, $\Rightarrow$) where {$p, r, p \lor r, p \land r, p \to r$} is a set of propositions and ...
0
votes
2answers
13 views

Probability, chose two skittles, out of 2 skittles left from a bag of skittles with 5 colors.

so me and my friend are studying statistics but we are just stuck on this stupid skittle question we made up ourselves when we tried to guess the colors of the two last skittles so we can see who will ...
0
votes
1answer
7 views

If $r_n\to r$ and $s_n\to s$, then $(r \star s)_M/M \to rs$.

I was going to ask this question, but I think I figured it out, so I thought I'd post my answer: In this question of mine, a user's answer makes the following claim: Suppose $r_n$ and $s_n$ are ...
0
votes
1answer
27 views

Maximum among $1, 2^{1/2}, 3^{1/3}, 4^{1/4},…$

What is maximum value among $1, 2^{1/2}, 3^{1/3}, 4^{1/4},....$ ? My approach: let $f(x)=x^{1/x}$ then I found out the derivative of $f$. Since $f(x)$ is maximum where $f'(x)=0$ and $f''(x)<0$ ...
0
votes
0answers
12 views

Double integral help

I'm having difficulty with a question. It says By putting $x=r\cos(\theta), y=r\sin(\theta)$, prove that $$\int_0^{\infty}\int_0^{\infty}e^{-(x^2 + 2xy\cos(\alpha)+y^2)}dx\ ...
0
votes
2answers
6 views

Proving that a set is denumerable without using a particular theorem

this question may seem like a duplicate of another one that I asked, but it is not. In this question, I am not allowed to use the Theorem which states: Every infinite subset of a denumerable set is ...
0
votes
0answers
4 views

Finite subcover of pairwise disjoint open intervals

I have the following exercise: Prove that if $X$ is a countable compact subset of $ \mathbb{R}$, then for any $\varepsilon>0$ there is a finite collection of pairwise disjoint open intervals ...
0
votes
0answers
5 views

this function is in some Holder space?

Consider $\Omega$ a open, bounded, and smooth domain of $R^N$ with $N \geq 3.$ And let $f: \Omega \times R \rightarrow$ a Caratheodory function . Supoose that $f$ is locally Lipschitz. Supose that ...
0
votes
0answers
5 views

How the extension of complex plane with complex infinity $\tilde{\infty}$ coexists with extension of real line with positive infinity $\infty$?

How the extension of complex plane with complex infinity $\tilde{\infty}$ coexists with extension of real line with positive infinity? Are there any paradoxes arizing? What are the rules when the ...
1
vote
0answers
8 views

Show that the $n$th cyclotomic polynomial is in $\mathbb Z [t] $

For any $ n$ we define the $n$th primitive root of unity to be an element $ z \in \mathbb C$ such that $z^n =1 $ but $z^r \neq 1$ for all $ 1 \le r < n$. We have proofed there are $\varphi(n)$ ...
0
votes
1answer
11 views

Proving some probability identities

I was hoping someone could help me with the following. Suppose $A$ and $B$ are two events such that $\Pr(B)\neq 1$ I want to prove that $\Pr(A\cap B)\geq \Pr(A)+\Pr(B)-1$ $\Pr(A)>\Pr(A|B)\quad ...
0
votes
0answers
22 views

Trigonometry Limit Question

I am wondering if 1 get 1/0 for a limit i am calculating, is the limit necessarily infinity? How to prove that the limit of $\frac{1}{sinx}$ as x tends to 0 is is infinity or does not exist? and the ...
-1
votes
0answers
8 views

Integral Evaluation with MATLAB-Mupad (triple and lesser degree integrals)

https://www.wolframalpha.com/input/?i=integral+of+2c%28x%5E2%2By%5E2%29%28√%28a%5E2+-+x%5E2+-+y%5E2%29%29+with+respect+to+y+from+-√%28a%5E2+-+x%5E2%29++to+√%28a%5E2+-+x%5E2%29 Here is a link to the ...
0
votes
1answer
9 views

Abstract Direct Product Proof Help

Let G = G1 x G2. Let H = {(x1, e2) : x1 ∈ G1} and K = {(e1, x2) : x2 ∈ G2}. (a) Prove H ≤ G and K ≤ G. (b) Prove that HK = KH = G (c) Prove that H ∩ K = {(e1, e2)} (d) Show that G/H is isomorphic to ...
0
votes
1answer
6 views

Different Polynomial Expansions of Natural Logarithm

I was recently Taylor-expanding ln around $(1,0)$. I noticed that this polynomial will have a range of input that converges between $0$ and $2$ regardless of Taylor ...
0
votes
2answers
13 views

Show that a linear operator P is orthogonal

inner product (A|B) = tr(A B^t) linear operator A(X) = X^t Is P skew or self adjoint? self = P = P* neg is skew
0
votes
1answer
20 views

Preparation for a graduate commutative algebra course based on Eisenbud

I am an undergraduate with two semesters of algebra(groups,rings, Galois theory, etc) under my belt and I am planning on going through Atiyah and MacDonald's book over the summer. Is this sufficient ...
0
votes
0answers
16 views

Very slow convergence of a particular series?

I've read that $$ \sum_{k=2}^{\infty} \frac{1}{k (\log k)^2} = 2.1097\ldots $$ However when I compute the partial sums it looks like a lot of terms are needed to even get the first decimals right. My ...
0
votes
1answer
7 views

Probability and Standard Deviation

Hey I'm confused about how to do this kind of problem. I can't figure out how to find the standard deviation. There are on average 4 tetanus cases reported in the US each month. What is the ...
0
votes
0answers
5 views

Show that there exists a satisfactory assignment for the unstandard language of arithmetic $\{\textbf{0}, ', <_1\}$

Consider: $A1: \textbf{0} \not = x'$ $A2: x'=y' \rightarrow x = y$ $A3: \neg x < \textbf{0}$ $A4: x < y' \leftrightarrow (x < y \vee x = y)$ $A5: \textbf{0} < y ...
0
votes
0answers
6 views

Covariance of two values

A fair die is rolled twice (independently). Let X1 and X2 be the numbers resulting from the first and second rolls, respectively. Define Y=X1+X2 and Z=4⋅X1−X2. Find the covariance between Y and Z. ...
0
votes
1answer
8 views

Help in Stats, Joint p.d.f

Let $X$ and $Y$ be random variables that have a joint p.d.f., which is given by the formula $\displaystyle p_{X,Y}(x,y)=\frac{5e^{−5x}}{x}$ when $0< y < x < \infty$, and $p_{X,Y}(x,y)=0$ for ...
3
votes
0answers
12 views

Is every smooth function from $\mathbb{R}^n \to \mathbb R$ with compact support the laplacian of some function?

Given a smooth function $g:\mathbb R^n \to \mathbb R $ with compact support, is it true that there exists a function $u$ such that $g=\Delta u$?
0
votes
1answer
9 views

Equation with a sum for the prime-counting function involving the Mobius function

I have come across the statement that $$ \sum_{n\leq x}\sum_{d\mid(n,P_z)}\mu(d) = \sum_{d\mid P_z}\mu(d) \left[\frac{x}{d}\right], $$ where $P_z=\prod_{p\leq z}p$ where $p$ is prime, $\mu(d)$ is the ...
0
votes
2answers
17 views

Unusual 3D Packing Problem

I made up this interesting problem playing with wire sculptures: If I have a $10 \times 10 \times 10$ clear box and inside I can put wireframe unit cubes, what's the maximum number of unit edges (or ...
0
votes
2answers
13 views

proving a natural projection is linear and finding its kernel

Let $V_i = 1,...,N$ be a collection of vector spaces over a field $F$. Consider the Cartesian product $V=V_1 \times V_2 \times ... \times V_N$ with the natural projections $\pi=V \rightarrow V_i$. ...
0
votes
0answers
6 views

Gauss-Laguerre quadrature and error estimation

While using Gauss-Laguerre quadrature of varying orders, n, to estimate the value of integral what happens to the error as the value of n increases. would it increase or decrease and why?
0
votes
1answer
12 views

Expected Value for coin flips for five heads.

Consider a coin that comes up heads with probability $p$ and tails with probability $1-p$ We flip this coin (independently) and stop as soon as it comes up heads for the 5th time. Let X be the random ...
1
vote
2answers
13 views

uniformly continuous and bounded derivative

When I learn uniformly continuous function, I used to view uniformly continuous function as a function with bounded derivative. Most of time up to now it seems right. But can we prove or disprove ...
0
votes
0answers
5 views

Why is $LG(n) \cong U(n)/O(n)$?

Let $LG(n)$ be a Lagrangian Grassmanian manifold. That is, $LG(n)$ is the set of Lagrangian subspaces of a symplectic vector space of dimension $2n$. Why is $LG(n)$ can be identified with $U(n)/O(n)$? ...
0
votes
0answers
13 views

Show that $ (\forall x)(A \lor B) \rightarrow A \lor (\forall x)B $ is, in general, NOT a theorem.

Show that $$ (\forall x)(A \lor B) \rightarrow A \lor (\forall x)B $$ is, in general, NOT a theorem. My answer: First, I got the abstraction of the formula which is $ p \rightarrow A \lor q$ then ...
-1
votes
0answers
8 views

Using BCR experiment

consider a random experiment of observing a mechanical or electrical unit consisting of five components and determining which components are working and which have failed. Use the BCR to find the ...
0
votes
0answers
10 views

Which probability to calculate?

I was wondering if someone could help me with the probability question stated below. The probability that a particular moth trap $A$, collects $r$ moths overnight is given by $(1-\alpha)\alpha^r$ for ...
0
votes
1answer
13 views

Tangent plane that passes through a point

How would I find the (a,b) that satisfies that the tangent plane to $f(x,y) = (x^2) + 2xy + (y^2)$ passes through the point $(2,1,0)$ ? I know that $f(x)= 2x + 2y$, and $F(y): 2x + 2y$. Therefore ...
0
votes
1answer
18 views

prove following properties of self-adjoint operator

$A: V \rightarrow V$ self-adjoint; $b$ is a real number. Show 1) the minimal polynomial has distinct roots; 2) $\ker(L) = \ker(L^k)$ for $k\geq1$; 3) $\text{im}(L) = \text{im}(L^k)$ for k bigger ...
0
votes
0answers
5 views

show that exist a left quasi-inverse of any element in J(R)

I studying for math. but this problem, I don't understand. J(R): the radical of R, written as J(R),is the set of all elements of R which annihilate all the irreducible R-module. and show that exist a ...
0
votes
0answers
4 views

Counting processes question

Let's say that arrivals at a counter come at times of a Poisson process with rate $\lambda$. A ball that arrives to an unlocked counter is registered and then locks the counter for an amount of time ...
2
votes
0answers
8 views

Interesting association between tangent lines of slope one and ellipses

Why is it that a tangent line with slope $1$ to an ellipse centered at the origin will have a transformation of $\pm \sqrt{a^2 +b^2}$ where $a$ and $b$ are the major and minor axis of the ellipse? ...
0
votes
0answers
4 views

measurability restriction operator

Let $M\subset \mathbb{R}^k$ compact. For every $x\in M$ we define $L(x): \mathbb{R}^m \rightarrow \mathbb{R}^m $ a linear isomorphism Let $G_n (\mathbb{R}^m)=\{ W: W\ \mbox{is subspace of} \ ...
0
votes
0answers
8 views

Lambda calculus logical operators

Define the and operator in lambda calculus and prove your definition Define the exclusive or operator in lambda calculus, and prove your definition My answer for #1 is: AND $\equiv$ ...
0
votes
0answers
4 views

Bayes factor and Posterior odds

Consider the following posterior odds \begin{equation*} \frac{P(H|D_1,D_2)}{P(\overline{H}|D_1,D_2)}=\frac{P(D_2|H,D_1)\times P(D_1|H)P(H)}{P(D_2|\overline{H},D_1)\times ...

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