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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0
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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
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 ...
-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
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 $\equiv$ ...
-1
votes
2answers
34 views

In tossing 5 6-sided fair dice, what is the probability of at least one 2 if the dice are indistinguishable?

I know that the answer is .4 because it is given. I just do not know how to get there. The answer would be .598 if the dice were distinguishable (ordered).
0
votes
1answer
20 views

Question about Symmetric matrix

Ok my book says this matrix $A = \left ( \array{ -2 & 1 \\ 1 & -3 } \right )$is symmetric. But, I don't understand b/c if it were a symmetric matrix, wouldn't it be ...
1
vote
2answers
19 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 ...
0
votes
1answer
21 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 ...
3
votes
1answer
31 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. ...
0
votes
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) ...
0
votes
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 ...
0
votes
0answers
29 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 ...
0
votes
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 ...
1
vote
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 ...
1
vote
0answers
31 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 $$x\in G-H$$ ...
0
votes
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.
0
votes
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 ...
0
votes
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 ...
0
votes
1answer
47 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 ...
0
votes
0answers
48 views

Please give feedback to my answers (sets) [duplicate]

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
votes
1answer
33 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 ...
2
votes
0answers
32 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 ...
2
votes
1answer
143 views

Does series converge or not?

$$\sum_{n=1}^\infty~\left|\frac{\cos2^n}{n}\right|$$ I just confused what to do.
0
votes
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 ...
0
votes
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} = ...
3
votes
1answer
18 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 ...
0
votes
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$=12$ ...
0
votes
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 ...
0
votes
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
votes
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 ...
0
votes
2answers
24 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 ...
4
votes
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 ...
0
votes
1answer
9 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 ...
-3
votes
1answer
34 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 ...
0
votes
3answers
31 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. $$ \frac{\mathrm{d}x}{\mathrm{d}t} =12−3xy−3x $$ $$ ...
2
votes
4answers
32 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 ...
0
votes
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 $|G'/G''| \geq p^3$
1
vote
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.
0
votes
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: ...
0
votes
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 ...
3
votes
1answer
27 views

Logarithmic Functions algebra question

This is my first post and I honestly just want a second opinion on my answer to a question I got incorrect on an exam before I go arguing over it with my professor. Basically, is this mathematically ...
3
votes
1answer
10 views

Meromorphic and even

I would like to do the following exercise : Let $f$ be a meromorphic function and $\mathcal{P}$ the set of its poles. We also assume that $f$ is even ($\forall z \in \mathbb{C}, \; ...
0
votes
0answers
12 views

Substitution in multiple integrals, rewriting variables

I have this problem that seems almost laughably simple, but has stumped me for some time. For the variables $ u = x^2 - y^2 $ $ v = x$ $y$ Under the condition $ u > 0 $ We're simply asked to ...
1
vote
1answer
26 views

Ho To Perform U-Substitution On Given Intergal

$\int{x^2\sqrt{2+x}}{dx}$ I haven't been able to find what u should be in this intergal, where should I start? I've gotten as far as: let $u = 2 + x$; $du=\frac{1}{x}dx$
1
vote
8answers
105 views

Calculate the last digit of $3^{347}$

I think i know how to solve it but is that the best way? Is there a better way (using number theory). What i do is: knowing that 1st power last digit: 3 2nd power last digit: 9 3rd power last digit: ...
2
votes
2answers
40 views

Trigonometric substitution

Been out of touch with trigonometry for some time now. Need help proving this expression. $$\sin^{2}\left(\frac{x}{2}\right) = \frac{1}{2}(1-\cos\left(x\right))$$ Any help will be appreciated. ...
1
vote
1answer
28 views

A question about a continuous function that satisfy certain limits at $\pm\infty$

I got this question: Let $f:\mathbb{R}\to\mathbb{R}$ be a continuous function such that $\lim_{x\to\infty}\frac{f(x)}{x^2}$ and $\lim_{x\to -\infty}\frac{f(x)}{x^2}$ exist and are real numbers. ...
0
votes
1answer
8 views

Uniform convergence of functions and Hausdorff convergence of their graphs

Consider a sequence of continuous functions $f_n:[a,b] \to \mathbb{R}$. If their graphs $G_n$ converge to the graph $G$ of a continuous function $f$ (in the Hausdorff metric $d_H$), prove that $f_n$ ...
-2
votes
0answers
28 views

Equilibrium question [on hold]

Consider the differential equation $$x' = x^3 − x^2 − 6x.$$ (a) Find all equilibria. (b) Determine the stability of each equilibrium analytically (not from the phase line diagram). (c) Sketch ...
1
vote
1answer
24 views

Continuous function $f:\mathbb{R}\to\mathbb{R}$ that got no extrema must be one to one

I got this question: Prove that if $f:\mathbb{R}\to\mathbb{R}$ is a continuous function that got no extrema then $f$ is one to one. I tried to prove it but I don't know how to proceed. I started by ...