Homework questions are welcome as long as they are asked honestly, explain the problem, and show sufficient effort. Please do not use this as the only tag for a question. For the answers on homework questions, helpful hints or instructions are preferred to a complete solution. Please do not add ...

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3 views

Using Combinators in Lambda Calculus

K $\equiv$ $\lambda$xy.x S $\equiv$ $\lambda$xyz.((xz)(yz)) Prove that the identify function I $\equiv$ $\lambda$x.x is equivalent to ((S K) K) I have no clue where to even start for ...
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1answer
11 views

Covariance of Sum and Differences of Dice Values

Question: A fair die is rolled twice (independently). Let X 1 and X 2 be the numbers resulting from the first and second rolls, respectively. Define Y=X 1 +X 2 and Z=4⋅X 1 −X 2 . Find the ...
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1answer
44 views

Proving a relation is transitive

I am trying to understand transitive relations. I understand given that a set may have $\{(a,b)(b,c)\}$ it must contain $(a,c)$ for it to be transitive. But for longer sets I am getting confused in ...
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1answer
25 views

Show that the entries of a matrix are:

For a regression model $y=\beta x$ (note there is no intercept term), show that entries of the matrix $\bf{H} = \bf{X}[\bf{X'}\bf{X}]^{-1}\bf{X'}$ are $h_{ij} = ...
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3answers
30 views

what are the equilibrium points of the following: [on hold]

where $x$ represents susceptible individuals, $y$ represents infected individuals. Find the two biologically meaningful equilibria. $$ \frac{\mathrm{d}x}{\mathrm{d}t} =12−3xy−3x $$ $$ ...
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1answer
23 views

Question about a closed subspace of a complete space

Let $J$ be a closed interval. Let $C(J)$ be space of continuous functions on $J$. We know $C(J)$ is a complete metric space with metric $d(x(t),y(t)) = \max_{t \in J} |x - y | $. Consider now $$ K(J) ...
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1answer
8 views

Convex Subset Projection

Suppose that C is a closed convex subset of $\mathbb R^n$ and $x \in \mathbb R^n$. The projection of $\mathbf x$ onto C is the closest point $\mathbf y \in C : \mathbf z = \mathbf y$ minimizes ...
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1answer
28 views

Calculate depth using triginometry

I was asked a question like this on an exam today and I'm wondering if I got it right or not. ...
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0answers
25 views

Matching Birthday Probability Question (Answered)

The question: A large number of people are waiting in line at Espresso Royale (as it often happens at around noon). The barista announces that he will start asking for each person's birthday, one ...
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0answers
12 views

Finding the distance from a parabola (ballistic trajectory) to a point (for use in collision detection)

I need to have some form of collision detection / prevention for an object moving along a ballistic trajectory and a second stationary object on the same plane plane. The ballistic trajectory is ...
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1answer
45 views

Finding Truth Values Of Nested Quantifiers

I'm looking at for example, $∃x∀y,P(x≥y+1)$ I'm told in order to prove that this is true I can us the technique that follows: Find one value of $x∈X$(only needs to be one) that has the property that ...
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0answers
29 views

An application of Sylow theorems in p-groups!

If $G$ is a finite group of order $p^{n}$ (which $p$ is a prime number) and have only one subgroup of order $p^{n-1}$ ,namely $H$ ,then $G$ is cyclic ! My "proof" is as follows: suppose $$x\in G-H$$ ...
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0answers
10 views

Showing Integrability

Suppose that $I,J$ are intervals in $\mathbb{R}$ and that $F:I\to\mathbb{R}$ and $G:J\to\mathbb{R}$ are integrable. Prove that $H:I\times J\to \mathbb{R}$ defined by $H(x,y) = F(x)+G(y)$ must also be ...
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0answers
15 views

Groups - Compositions

If the f is written to the right of its argument does that mean the composition of $f g$ is actually $g(f(x))$ instead of being $f(g(x))$ which is the notation I'm used to. I ask this because I read ...
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1answer
17 views

Please give feedback to my answers(sets-t)

Prove or find a counter-example to the claim that for all sets $A,B,C$ if $A∩B=B∩C=A∩C=∅$ then $A∩B∩C≠∅$. Solution False. Let $A=\{1,4,6\}, B=\{2,3,5\}, C=\{7,8,9\}$ Then : $A∩B$ is equal to an ...
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0answers
23 views

finding maximum likelihood estimators

Let $Y_1, Y_2,\cdots,Y_n$ be independent, identically distributed (iid), each with the following probability density function: $\displaystyle ...
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3answers
33 views

Identify the basic function f(x)

Given $f(x) = \sqrt x$, $g(x)=4\sqrt{x+2}-7$ Describe the sequence of transformations from f to g.
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0answers
8 views

Using Implicit Function Theorem to show that F has a differentiable local inverse

Suppose that $F$ is continuously differentiable, with domain and range nonempty open set in $\mathbb{R}^n$, and that the derivative matrix of $F$ is invertible at $a$. Use the Implicit function ...
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0answers
45 views

Please give feedback to my answers (sets)

Prove or find a counter-example to the claim that for all sets $A, B,C$ if $A\cap B = B\cap C = A\cap C = \varnothing$ then $A\cap B\cap C \neq\varnothing$. Solution False. Let $A = ...
2
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1answer
32 views

Show that the follow function is Riemann integrable on $[0 , 2]$, and use te definition to find $\int_0^2f.$

Show that the follow function is Riemann integrable on $[0 , 2]$, and use te definition to find $\int_0^2f.$ $$ f(x) = \left\{ \begin{array}{c} -1, &0 \le x < 1 \\ 2, &1 \le x \le 2 ...
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0answers
31 views

Understanding underlying algebra for calculus convergence problem

I'm working on series convergence/divergence problems in my Calc 2 class, and (as has happened often), I'm hung up on some underlying algebra. The first step in the solution manual for a problem I'm ...
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10answers
1k views

How to solve 4sin θ +3cos θ equals 5

Another problem that i already wasted hours on. Given $$4\sinθ +3\cosθ = 5$$ Find $$4\cosθ -3\sinθ$$ Help me guys (PS:I'm not that good in maths)
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0answers
10 views

showing minimum and relative efficiency of 2 independent samples

Let Ȳ1 and Ȳ2 be the means of independent random samples of size n1 and n2 from a normal population with mean µ and variance σ2. a) Show that the variance of the unbiased estimator xȲ1 + (1-x)Ȳ2 ...
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1answer
12 views

unbiased and consistent estimators of 2 independent populations

Let Ȳ1 be the mean of a random sample of size n from a normal population with mean µ and variance σ12 and let Ȳ2 be the mean of a random sample of size n from a normal population with mean µ and ...
3
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1answer
17 views

Show that $\lambda \in \sigma(A),$ $\lambda$ not an eigenvalue, implies that $\lambda \in \sigma(A + K)$ where $K$ is compact.

Let $A : H \rightarrow H$ be a bounded linear map where $H$ is a Hilbert space with $\dim H = \infty$. Suppose that $\lambda \in \sigma(A)$ but $\lambda$ is not an eigenvalue. Let $K : H \rightarrow ...
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0answers
14 views

finding method of moments estimator for beta distribution

Let Y1, Y2,...,Yn be a random sample from a population that follows a beta distribution with β=3. Find the method of moments estimator of α. Please show all steps and work So far I have equations ...
2
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1answer
119 views

Does series converge or not?

$$\sum_{n=1}^\infty~\left|\frac{\cos2^n}{n}\right|$$ I just confused what to do.
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4answers
74 views

If $d=\gcd(a+b,a^2+b^2)$, with $\gcd(a,b)=1$, then $d=1$ or $2$

Suppose $\gcd(a,b)=1$. Let $d=\gcd(a+b,a^2+b^2)$. I want to prove that $d$ equals $1$ or $2$. I get that $d\mid2ab$ but I can't find a linear combination that will give me some help to use the fact ...
0
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0answers
11 views

finding maximum likelihood estimators for gamma distribution

Let Y1, Y2,...,Yn be a random sample of size n from a gamma population with α=2. a) Find the maximum likelihood estimator for β. b) Show that the maximum likelihood estimator for β is an unbiased ...
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1answer
32 views

Exercise 3.6: Elementary Functional Analysis By Barbara [on hold]

Let $X=\ell_\Bbb R^\infty$ denote the space of bounded sequences with real entries, in the supremumnorm. Consider the operator $T$ defined on $X$ by $T(x_1, x_2, . . .) = (x_2, x_3, . . .)$; this is ...
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1answer
8 views

Finding all continuous solutions to an integral

I need help with both parts of this problem. Part (i) seems obvious, because the integrand $f(t)$ would become $F(t)$, which is obviously differentiable because its derivative is $f(t)$ by ...
1
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1answer
36 views

Fredholm operators in Hilbert spaces

Suppose $T_r$ and $T_l$ are the left and the right translations in $l_2$. $T_l$ maps $(x_1,x_2,x_3,...)$ to $(x_2,x_3,x_4,...)$, $T_r$ maps $(x_1,x_2,x_3,...)$ to $(0,x_1,x_2,...)$. It can be easily ...
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2answers
63 views

Prove that integral of continuous function is continuously differentiable

Lots of things going on here. I immediately know that $F(x)$ does exist since $f$ is riemann integrable due to the fact that it is continuous. First I need to show that $F$ is continuous, then find ...
4
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1answer
68 views

Solving integral $ \int \frac{x+\sqrt{1+x+x^2}}{1+x+\sqrt{1+x+x^2}}\:\mathrm{d}x $

there is integral $$ \int \frac{x+\sqrt{1+x+x^2}}{1+x+\sqrt{1+x+x^2}}\:\mathrm{d}x$$ i am trying to separate this : $$=\int \mathrm{d}x -\int \frac{\mathrm{d}x}{1+x+\sqrt{1+x+x^2}} $$ but have no idea ...
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0answers
8 views

Find maximum and minimum values of the function on the rectangular region by looking at level curves and gradients

$$f(x,y)=x+y+3$$ $$Region:-4 \le x \le 4, \space -5 \le y \le 5$$ I know how to find max and min the regular way (partial derivatives, determinant), and by checking the regions I obtained max$=12$ ...
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2answers
25 views

The zoo car park

In the zoo car park there were 780 vehicles. Most of the vehicles were cars,but one section contained coaches only.the ratio of cars to coaches was 11:2. How many vehilces were cars and how many were ...
0
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3answers
51 views

Induction, show that something is smaller then …

I have to show the following by induction. $1 \cdot 2 \cdot 3 ... (n - 1) \leq (\frac{n}{2})^{n -1}$ As it is homework I "only" need a push in the right direction. my thought is that is something ...
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2answers
25 views

Calculate QR and PR

The measurements for isosceles triangle $PQR$ are: angle $P$ = angle $Q$ = $42^\circ$ $PQ = 6$cm Find the length of the other two sides. Give your answers correct to one decimal ...
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4answers
27 views

Groups - Inversions

Above is just an example I'm trying to work from as I have the solutions. I've seen lots of definitions of what inversions are but they use signs like sigma, and it doesn't really explain what the ...
0
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1answer
37 views

how does this converges? Sequence and series convergence

Consider the following problem- Converges or Diverges? $$(1-2)-(1-2^{1/2})+(1-2^{1/3})-(1-2^{1/4})+....$$ I said it converges but then my work i showed in paper got wrong How would you prove that ...
2
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1answer
43 views

Simple planar graph

What's the best way of proving this? ...
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2answers
22 views

Finding the probability that a joint distribution is less than a certain value, given the correlation coefficient.

For this problem, we are told that X and Y are jointly normally distributed variables, both being standard normal. We're given their correlation coefficient. So, how do I get from there to finding the ...
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2answers
63 views

linear differential equation problem [closed]

Consider the following system of linear differential equations: $$\begin{split} \frac{dx}{dt}&=−3x+y\\ \frac{dy}{dt}&=x−3y \end{split}$$ Find the eigenvalues and eigenvectors associated ...
2
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4answers
31 views

Calculate the height of a building

This question I really need help with, I simply do not know where to start! Anyone can help, all I can offer is supreme thanks. Please include method. I don't want simple answers which don't help me ...
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0answers
26 views

$|G'/G''| \geq p^3$ where $G$ is $p$ group [on hold]

Let $G$ be a $p$ group. Commutator subgroup of $G$ is denoted by $G'$ prove that 1. If $G'$ has order $p^3$ ,then $G'$ is abelian 2. If $G''$ is not identity then $|G'/G''| \geq p^3$
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0answers
17 views

Working out how many units per hour

I'm having trouble working this out in my own head, I'm trying to work out how to efficiently collect resources in a browser game I play, assuming I collect them every hour. Here's the situation: ...
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2answers
25 views

Pyramid height square base.

A pyramid is made from $4$ equilateral triangles and a square base. The sides of each shape is $20$cm long. Calculate the height.
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2answers
83 views

Contraction of compact sets

I am trying to solve the following problem. Let $X$ be a compact Hausdorff space and let $f:X\to X$ be continuous. Show that there exists a non-empty set $A\subset X$ such that $f(A)=A$. ...
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1answer
23 views

Solve an Angle-Side-Angle special case triangle if it has an obtuse angle?

I've seen this type of problem multiple times on homework, and it's confusing me like mad. The scenario: We have a triangle. It is a special case triangle, with one angle, one side, and another ...
1
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1answer
28 views

Expected number of rolls when repeatedly rolling an $n$-sided die

Suppose I roll an $n$-sided die once. Now you repeatedly roll the die until you roll a number at least as large as I rolled. What is the expected number of rolls you have to make? I know the answer ...