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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1answer
11 views

Quantified Logic with miltuple variables

Problem: ∀y¬∃x¬(Fxy ∨ Fyx) ⊢ ∀y∀z(Fyz→Fzy) I don't really understand how to deal with multiple variables in instances like this. So far I have: ...
0
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0answers
3 views

Using a reverse polynomial for a partial fraction decomposition in a recurrence relation problem

I recently asked this question about finding the formula for: $$gn=gn−1+gn−2+n, g_0=1, g_1=2$$ On that question, I was able to get help to the point of generating this partial fraction ...
0
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0answers
2 views

A proof by contraposition problem

Use a direct proof to show that the cube of an odd number is also odd. Proof: Assume that a is odd. Therefore, a = 2p + 1, where p is a non-negative integer. Therefore, a^3 = (2p + 1)^3 ...
0
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0answers
3 views

Is $\sum_{n=1}^\infty {1\over 3^{\sqrt{n}}}$ convergent?

Is $\sum_{n=1}^\infty {1\over 3^{\sqrt{n}}}$ convergent ? I use it to compare with $1/n^2$, and then I used LHôpitals rule multiple times. Finally , I can solve it. However,I think we have other ...
0
votes
1answer
25 views

Proving If $\int^{\frac{\pi}{2}}_{0} f(x) \cos x dx \lt f \left(\frac\pi2\right),$ then $ \int^{\frac{\pi}{2}}_{0} f'(x) \sin x dx \gt 0. $

how do i tackle such problems? Let f(x) be a function differentiable on the interval$ $$\Bigl[$$ 0, \frac\pi2 $$\Bigr]$$ $ such that f'(x) is integrable on this interval. Prove the following ...
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0answers
13 views

Expected number of feed-forward/backward triangles in a random graph with internal nodes.

Suppose we have a graph with N* nodes (these are internal nodes. they all have at least one child). Every directed link in the network exists with probability p. What would be the expected number of: ...
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1answer
83 views

Good, satisfied and bad numbers

Set of natural numbers is divided in to the following three set 1) Good numbers 2) Satisfied numbers 3) Bad numbers The conditions given are : (I) Y is a function on X and X is either $0$ or ...
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0answers
7 views

Graph Theory - Lower bounds

I am trying to solve for the following problem: Find (and justify) a lower bound for 0(G) in terms of X'(G) and E|(G)| and alpha'(G). (where alpha'(G) represents the maximum size of a matching in ...
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4answers
37 views

Prove this identity: $\sin^4x = \dfrac{1}{8}(3 - 4\cos2x + \cos4x)$.

The problem reads as follows. Prove this identity: $\sin^4x = \dfrac{1}{8}(3 - 4\cos2x + \cos4x)$. I started with the right side and used double angles identities for $\cos2x$ and a sum and then ...
0
votes
1answer
552 views

How to describe minimization of L1 norm error using linear programming?

Given a set of $n$ pair points $(x_1, y_1), ..., (x_n, y_n)$ in the plane, I need to find a line $ax + by = c$ that fits the points of the L1 norm error points as closely as possible. I need a linear ...
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0answers
16 views

Number of edges of a plane graph isomorphic to its dual

I am having trouble proving the following statement: Suppose that $G$ is a plane graph which is isomorphic to its dual. Prove that $G$ has $2n-2$ edges.
2
votes
1answer
26 views

$\mathbb{C} [G] \longrightarrow \prod_{\rho} \text{End}(V_{\rho})$ an intertwining isomorphism

Consider the vector space of functions $f: G \longrightarrow \mathbb{C}$ where $G$ is a finite group, or equivalently a vector space of all formal linear combinations of elements of $G$ over the ...
0
votes
2answers
26 views

If $\sin B = −1/2$ with $B$ in QIII, find $\cos B/2$

For the following, assume that all the given angles are in simplest form, so that if A is in QIV you may assume that 270° < A < 360°. If $\sin B = −1/2$ with B in QIII, find $\cos B/2$ Here's ...
0
votes
1answer
12 views

If $\sin A = 4/5$ with $A$ in QII, find $\cos A/2$

For the following, assume that all the given angles are in simplest form, so that if A is in QIV you may assume that 270° < A < 360°. If $\sin A = 4/5$ with A in QII, find $\cos A/2$ I keep ...
3
votes
4answers
46 views

$\lim_{n\to \infty}\left(1 - \frac {1}{n^2}\right)^n =?$

Can you give any idea regarding the evaluation of the following limit? $\lim_{n\to \infty}\left(1 - \frac {1}{n^2}\right)^n$ We know that $\lim_{n\to \infty}\left(1 - \frac {1}{n}\right)^n = ...
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2answers
30 views

If $\sin (B) = − \frac 1 2 $ with $B$ in third quadrant, then find $\cot (B/2)$

If $\sin (B) = − \frac 1 2 $ with $B$ in third quadrant, then find $\cot (B/2)$ I'm getting $-\sqrt{3}-2$
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3answers
45 views

Sequence problem.

Help for this problem would be much appreciated, as I am expected to solve it without being properly taught how. Suppose a single cell of bacteria divides into three every 12 hours. Suppose that the ...
0
votes
1answer
13 views

System of inequalities. Points of intersection?

$x^2+y^2<=81$ $y<x$ Is this correct? My answer: (-9sqrt(2)/2,-9sqrt(2)/2), (9sqrt(2)/2,9sqrt(2)/2)
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votes
2answers
423 views
+50

Ratios in big-O notation?

Hi can anyone give me a counter example of the following claim: f(n) = O(s(n)) and g(n)=O(r(n)) imply f(n)/g(n) = O(s(n)/r(n)) Thank you
1
vote
1answer
44 views

Prove $\cos 3\theta = 4 \cos^3\theta − 3 \cos \theta$

$\cos 3θ = 4 \cos^3 θ − 3 \cos θ$ Here's my attempt. Is it correct? Thanks! $\cos(3θ)$ $= \cos(2θ + θ)$ $= \cos(2θ)\cos(θ) - \sin(2θ)\sinθ$ $= (2\cos^2θ - 1)\cosθ - (2\sinθ\cosθ)\sinθ$ $= ...
0
votes
2answers
31 views

Finding Distinct Elements and Permutation in Partitioned Set

I am having a hard time figuring out where to start on a homework problem. The question is: A set of $nk$ elements is partitioned into $k$ subsets in two ways, each subset having size $n$: one ...
2
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0answers
18 views

Need help finding probability distribution [on hold]

In Cairo $30\%$ of residents listen to the local fm radio. $10$ residents are chosen at random: a) state the distribution of the random variable b) find the smallest value of $s$ so that $\Pr(X \ge ...
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0answers
6 views

how to calculate cagr of investment in dividend reinvestment option?

Assume Mr.X has bought equity mutual fund (dividend reinvestment option) on 5th july 2005 $10000 and fund declared maiden dividend of 20% on 7th sep 2006 (post div NAV 13.79) next dividend declared ...
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votes
3answers
28 views

Show that there are exactly two lines through a point p outside the circle that are tangent to the circle C

Let $C$ be a circle of radius $r$ in the plane. Let $p$ be a point in the plane that lies outside of $C$. Show that there are exactly two lines through $p$ that are tangent to $C$. It is one of ...
0
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0answers
15 views

Evaluate $\int_\gamma z^n e^{1/z} dz$ where $\gamma$ is the circle of radious 1 centered at 0 and traveled once in the counterclockwise direction

I am struggling to find the residue of the laurent expansion for $z^n e^{1/z}$. All I have so far is $z^n e^{1/z} = z^n \sum_{k=0}^{\infty} \frac{(1/z)^k}{ (k!)} = \sum_{k=0}^{\infty} ...
0
votes
1answer
15 views

Find t which minimizes ||A(x+ty)-b||$^2_2$

Let, f(t) = ||A(x+ty)-b||$^2_2$ = $(A(x+ty)-b)^T(A(x+ty)-b))$ ... $$= x^TA^TAx + 2tx^TA^TAy+t^2y^TA^TAy-2b^TA(x+ty)$$ Then, letting $f'(t) = 0$, we have $$ t = \frac{(b^TAy)-x^TA^TAy}{y^TA^TAy}$$ ...
1
vote
1answer
25 views

Euler Equation and Marginal Rate of Substitution

I was wondering if someone could help me clarify a result from my lecture notes. I have put them as a picture. It concerns the result on the last slide (the other three slides are included as well ...
2
votes
1answer
20 views

Prove this identity: $ \tan(2x)-\sec(2x) =\tan(x-\pi/4)$

I've been having a time with this problem. I tried to start with the left side but I hit a dead end quick... I then tried the right side and had a little more luck but I've hit a block. I first used ...
0
votes
1answer
16 views

How to solve length problems?

Newbie question please bear. A 12ft rope is cut into three pieces so that the second piece is 1ft longer than the first and the third piece is 1 ft longer than the second. How long are the pieces? ...
0
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0answers
23 views

vector field problems

I'm trying to review some problems on vector fields for the final, and would appreciate if someone can tell me whether my answers are right, so I know if I'm doing it correctly: $f$ is a vector ...
0
votes
1answer
8 views

Derive the Cramer-Rao lower bound (CRLB) for any unbiased estimator of $\mu^2$.

Let $Y_1, Y_2, . . . , Y_n$be a random sample from a normal distribution with mean μ and variance 1. Derive the Cramer-Rao lower bound (CRLB) for any unbiased estimator of $\mu^2$. Could anyone ...
3
votes
2answers
5k views

Sum of independent Gamma distributions is a Gamma distribution

If $X\sim \mathrm{Gamma}(a_1,b)$ and $Y \sim \mathrm{Gamma}(a_2,b)$, I need to prove $X+Y\sim(a_1+a_2,b)$ if $X$ and $Y$ are independent. I am trying to apply formula for independence integral and ...
1
vote
2answers
36 views

Recurrence relation with generating function problem

I've got a recurrence problem that I'm close to solving, but having trouble with finishing up. Solve the following recurrence relation using generating functions: $$g_n = g_{n-1} + g_{n-2} + ...
0
votes
1answer
41 views

Real Analysis question about polygons and derivatives [on hold]

So.. I honestly have no idea what to do here. Any help at all is appreciated.
0
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0answers
8 views

Questions related to Rao–Blackwell theorem

In this exercise, we illustrate the direct use of the Rao–Blackwell theorem. Let $Y_1, Y_2, . . . , Y_n$ be independent Bernoulli random variables with $p(y_i | p) = py_i (1 − p)1−y_i , y_i = 0, 1.$ ...
1
vote
1answer
33 views

Find Determinant of A

I've tried creating a triangular matrix, tried row reducing but can't figure it out as I keep on having c-unknown in my answer. How would I do this?
2
votes
1answer
33 views

How to find the inverse of this particular symmetric matrix

Basically, I have a $n \times n$ symmetric matrix, which looks like this: $$ \begin{bmatrix} 1 & \alpha & \cdots & \alpha \\ \alpha & 1 & \cdots &\alpha \\ \vdots &\vdots ...
1
vote
0answers
11 views

Is the complement of the inversion relation (in the context of permutations) transitive?

I'm studying from An Invitation to Discrete Mathematics where I came upon an exercise which confuses me. Let $\pi$ be a permutation of the set $\{1,2,\dots,n\}$ and let $I(\pi)$ denote the set of ...
0
votes
0answers
8 views

find E($\bar{Y^4})$ by using moment generating function for a normal distribution with mean μ and variance 1.

Let $Y_1, Y_2, . . . , Y_n$be a random sample from a normal distribution with mean μ and variance 1. I would like to find E($\bar{Y^4})$ by using moment generating function. The setup I have right ...
1
vote
3answers
61 views

Range of f(x) = $\frac{\sqrt3\,\sin x}{2 + \cos x}$ [duplicate]

Can you give any idea about the range of the following function? $$f(x) = \frac{\sqrt{3}\,\sin x}{2 + \cos x}$$
0
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3answers
40 views

Prove this identity? $\cos t ⋅ \cos u ⋅ \cos v = \frac14(\cos(t + u + v)+ \cos(t + u - v)+cos(t-u-v))$

The problem reads as follows. Prove the identity $$\cos t⋅\cos u⋅\cos v =\frac14\big(\!\cos(t + u + v)+\cos(t + u - v)+\cos(t-u-v)\big)$$ Hint: begin with the right side and use cosine sum identity ...
1
vote
1answer
38 views

a question about convergence of sequecce!I have tried cauchy method, but it doesn't work

suppose $a_n>0$,and$\sum_{i=0}^\infty a_i$ is convergent,so we need to prove $\sum_{n=1}^\infty{ {1\over n}(a_n+a_{n+1}+\cdots+a_{2n})}$ is also convergent! I have tried cauchy method, but maybe ...
1
vote
2answers
33 views

How to show that the $\phi $ and $\varphi$ satisfy the Cauchy Riemann equation

when u=(u,v)=($\frac{\partial\varphi}{\partial y},-\frac{\partial\varphi}{\partial x}$) and u(x)=grad$\phi$(x)=$\nabla\phi$(x) how can you use the equations above to prove that the $\phi$ and the ...
2
votes
0answers
17 views

Fundamental Solution of a Nonlinear ODE (using Riccati Transformation & Wronskian)

I am given the differential equation: \begin{equation*} y^{\prime}(t) = y(t)^{2} + 2\sin(t)\cos(t) - \sin^{4}(t) \end{equation*} and one solution $y_{1}(t) = \sin^{2}(t)$. I wish to find a second ...
0
votes
1answer
18 views

Constructing nontrivial $\mathbb{Z}$-bilinear map from $\mathbb{R} \times (\mathbb{R} / \mathbb{Z})$

I'm trying to show $\mathbb{R} \otimes_\mathbb{Z} (\mathbb{R} / \mathbb{Z})$ is nontrivial, where my tensor product is defined using the universal property. This problem can be easily reduced ...
2
votes
0answers
13 views

Characteristic curves of 2nd-order PDEs under invertible coordinate transformations

First off, I'm not very experienced with the subject and English is also not my first language, so if there are any inaccuracies in the following text, let me know. Given a linear, scalar, ...
0
votes
0answers
20 views

How to show uniqueness of $\nabla\phi$ using Green's theorem when the value of Neumann problem exists.

It says getting function $\phi(x,y,z)$ that satisfies the following conditions $\frac{\partial\phi(x)}{\partial n}$=h(x), x$\in$S is called the Neumann problem. The problem is to show the uniqueness ...
1
vote
1answer
22 views

2 dimensional Laplace's equation in polar coordinates

The problem asks you to get Laplace's equation in 2 dimensions in polar coordinates using the fact that $\operatorname{div}(\cdot)$ in two dimensional vector field could be written as $$\nabla \cdot u ...
-3
votes
0answers
68 views

Reflexive. What does it mean? [on hold]

I would like to know the definition for reflexive. I have not found anything on the internet or in my book.
13
votes
5answers
765 views

A simple way to evaluate $\int_{-a}^a \frac{x^2}{x^4+1} \, \mathrm dx$?

I am currently trying to show that $\int_{-\infty}^\infty \cos(x^2) \, \mathrm dx = \sqrt{\frac{\pi}{2}}$ and the last integral I have to evaluate is $$\int_{-a}^a \frac{x^2}{x^4+1} \, \mathrm dx.$$ ...