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
18 views

set functions and relations

$\ M$ is the set of all relations on $\ A = \{1,2,3\}$ $\ K$ is the the following relation on A $\ K=\{(1,1),(2,1),(3,1)\}$ let there be $\ f :M\rightarrow M$ $\ f(R) = RK$ prove that ...
0
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0answers
11 views

Decide coordinates for a vector in a triangle (Image attached)

I have the following triangle. I have to express the line $\overline{AT}$ as a linear combination of $\overline{AC}$ & $\overline{AB}$. A hint was to use the knowledge of $\overline{AT} = ...
2
votes
1answer
18 views

Radius of Convergence of Sum of two Series.

Hi all, I know there are similar questions on here, but none deal with the fact of trying to prove that $T \geq min\{R,S\}$. Intuitively this doesn't make sense to me, If you have, ...
0
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0answers
6 views

Ranks of matrix Lie groups and Lie algebra of SU(1,1), SO(2,1)

I was trying to find out by Googling, but had no luck. Am I right in thinking that for the Lie GROUPS: rank SL(n,R) = n, rank SO(n,R) = n (not sure about this one), rank SU(n,C) = n-1 and ...
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2answers
32 views

Let n be an integer greater than 3. Find a formula for gcd(n, n + 3)

Let $n$ be an integer greater than $3$. Find a formula for $\gcd(n, n + 3)$ for each of the cases : $1)$ $n \equiv 0\mod 3$ $2)$ $n \equiv 1\mod 3$ $3)$ $n \equiv 2\mod 3$ Any help would be greatly ...
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0answers
12 views

Group of numbers with common euler's totient function result

I was asked to find the group of integers, which share the result of euler's function of 84. To be clear: which numbers, when applying eulers function on them, result 84. By calculating I found that ...
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0answers
10 views

Pigeonhole principle - list of fruits and certain restrictions

A bag contains 20 apples, 20 bananas, 20 oranges and 20 pears. In the worst case, how many fruits must one pick in order to be sure that they have a dozen fruits of the same kind? How many in order to ...
2
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3answers
79 views

Analytic Geometry

How does one solve: Find the equation of the circle which has it's center on the line $y= 3-x$ , and which has as tangents the lines $ 2y-x = 22, $ $ 2x+y=11 $ ?
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0answers
24 views

Pigeonhole principle - families with same letter on last name

Argue that if there are 30 families in a colony, at least two of these families have last names that begin with the same letter. What is the minimum number of families for which at least two families ...
1
vote
2answers
191 views

Pigeonhole principle - one pair of numbers add up to 10

How many numbers must be selected from the set {1,2,3,4,5,6,7,8,9} to guarantee that at least one pair of these numbers add up to 10? Justify your answer. Here's my answer. Consider the two sets ...
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0answers
33 views

Direct proof to show that the cube of an odd number is also odd

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 ...
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votes
4answers
105 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 ...
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0answers
7 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: $$gn=g_{n−1}+g_{n−2}+n, g_0=1, g_1=2$$ On that question, I was able to get help to the point of generating this partial fraction ...
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0answers
12 views

Graph Theory - Lower bounds [on hold]

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 ...
1
vote
1answer
27 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: ...
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0answers
24 views

Number of edges of a plane graph isomorphic to its dual [on hold]

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.
3
votes
4answers
61 views

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

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 = ...
0
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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 ...
0
votes
2answers
33 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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1answer
27 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 ...
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0answers
16 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
20 views

How to solve length problems? [on hold]

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
votes
1answer
14 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 ...
2
votes
1answer
21 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 ...
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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
3answers
29 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 ...
1
vote
4answers
41 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
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
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 $$ t = \frac{(b^TAy)-x^TA^TAy}{y^TA^TAy}$$ ...
0
votes
1answer
9 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 ...
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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 $p(y_i | p) = py_i (1 − p)1−y_i , y_i = 0, 1.$ ...
0
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1answer
42 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.
1
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1answer
45 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! $\cos(3θ)$ $= \cos(2θ + θ)$ $= \cos(2θ)\cos(θ) - \sin(2θ)\sinθ$ $= (2\cos^2θ - 1)\cosθ - (2\sinθ\cosθ)\sinθ$ $= ...
1
vote
1answer
35 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?
1
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0answers
20 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 ...
1
vote
0answers
11 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 ...
2
votes
1answer
35 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 ...
2
votes
0answers
21 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 ...
1
vote
1answer
30 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 ...
0
votes
3answers
41 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
3answers
62 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? $$f(x) = \frac{\sqrt{3}\,\sin x}{2 + \cos x}$$
1
vote
2answers
34 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 ...
-3
votes
0answers
72 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.
1
vote
1answer
39 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 ...
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 ...
0
votes
1answer
15 views

Moment Generating Function of the Chi-Squared Distribution

The questions wants us to show that the MGF for the chi-squared distribution is equal to I know that to show that I need to evaluate this integral. I'm not sure where to begin to evaluate it. ...
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 ...
2
votes
0answers
32 views

($\cos^4x$)($\sin^2x$) in terms of first power of cosine

I believe that I have his correct but if someone could check it and see that'd be great. Here's a pic! [IMG]http://i58.tinypic.com/2dgm5ic.jpg[/IMG]
0
votes
0answers
16 views

Find the intersection of three bisection lines

Let $p_1 = (a_1, b_1), p_2 = (a_2, b_2), p_3 = (a_3, b_3)$ be three, non-colinear points in the plane. For each pair of these points, let $L_{ij}$ denote the line segment from $p_i$ to $p_j$. (a) For ...
2
votes
0answers
18 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, ...