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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0answers
222 views

(Improved; Kindly assist )CSI Forensic Maths

In a square room with 7 m-long sides, five people have shot at random people. They all had two bullets and killed two people in the crowd. Other people in the room were not hurt. CCTV recordings show ...
2
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1answer
76 views

What is $\lim\limits_{k \to 0}{f(k) = 2 + k^{\frac{3}{2}}\cos {\frac{1}{k^2}}}$

Just want to check this one: I got: $$\displaystyle \lim_{k \to 0}{f(k) = 2} \;+\; \lim_{k \to 0}{k^{\frac{3}{2}}\cos {\frac{1}{k^2}}}$$ Since $\lim\limits_{k \to 0}\cos{\frac{1}{k^2}} = 0$, using ...
0
votes
2answers
133 views

Successes or failures in a row

Let random variable $X$ be defined as the number of independent Bernoulli(p) trials required until we observe either two successes or two failures in a row. Find $P(X=n)$ for $n=2,3,4, \ldots$ Then ...
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1answer
1k views

prove that the set of rational numbers is not connected on the real line

Could someone help me through this problem? Prove that the set of rational numbers is not connected on the real line
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1answer
461 views

help me prove that a matrix with a trivial nullspace must always be invertible

I'm proving that $A^tA$ will be positive definite iff $A$ is invertible. I guess that there are ways to show this with determinants, eigenvectors. But I've just gone with positive definites must ...
3
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0answers
652 views

Solving geodesic problems with Euler-Lagrange equation

This is the question: Problem B.1 Two cities - Tel-Aviv, Israel and SanDiego, CA - have the same latitude 32 ◦ N, but, different longitudes: Tel-Aviv is 34 ◦ E and San-Diego is 117 ◦ W. What is the ...
0
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1answer
48 views

How to approach this combinatorics programming question?

This question is asking for the pseudocode mainly: Design an algorithm that would play "Word Game." In Word Game an English word is supplied, then one is supposed to form as many words that ...
2
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1answer
58 views

If $\langle x,u\rangle = \langle x, v\rangle = \langle x,w\rangle = 0$ then $x=0$

I'm given that $$\begin{align*} u &= (1, 2, 3)\\ v &= (2, -1, 1)\\ w &= (3, 1, 0) \end{align*}$$ And I'm asked to verify if $\langle x,u\rangle = \langle x, v\rangle = \langle x,w\rangle ...
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5answers
2k views

give an example of an infinite class of closed sets whose union is not closed.

give an example of an infinite class of closed sets whose union is not closed. Thanks for your help
3
votes
3answers
692 views

Using the definition of a concave function prove that $f(x)=4-x^2$ is concave (do not use derivative).

Let $D=[-2,2]$ and $f:D\rightarrow \mathbb{R}$ be $f(x)=4-x^2$. Sketch this function.Using the definition of a concave function prove that it is concave (do not use derivative). Attempt: ...
4
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2answers
1k views

Find all connected 2-sheeted covering spaces of $S^1 \lor S^1$

This is exercise 1.3.10 in Hatcher's book "Algebraic Topology". Find all the connected 2-sheeted and 3-sheeted covering spaces of $X=S^1 \lor S^1$, up to isomorphism of covering spaces without ...
0
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2answers
650 views

Absolute value and distance traveled

I don't know what to do again. I can get the displacement (but don't know what it means) but I can't get the distance traveled. The velocity function is given for a particle moving along a lone. Find ...
4
votes
2answers
759 views

Showing Unit sphere is convex

Good evening guys! I have to show that the unit sphere represented by is convex. A set is said to be convex when $sx + (1 - s)y \in M$, where $x, y \in M$ and $s \in (0,1)$ I've read on wikipedia ...
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1answer
326 views

Sets and limit points

Give an example of a set which $\ \ \ $a) contains a point which is not a limit point of the set $\ \ \ $b) contains no point which is not a limit point of the set In part b), I think it might be ...
5
votes
1answer
95 views

Integral of an $L^2$ function

My problem is: Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a measurable function (w.r.t. the Lebesgue measure) that is in $L^2$. Show that the function $$F(x)=\int_0^x f(t)\,dt$$ ...
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2answers
2k views

Finding time traveled from acceleration function

I don't know how to do this, it doesn't make sense to me and there are no examples in the book. The acceleration function and the initial velocity are given for a particle moving along a line: $\ \ ...
3
votes
1answer
179 views

Linear algebra - Dual, functionals

Given three functionals $f_1(p) = \int_0^1 p(t)\,dt$ $f_2(p) = \int_0^2 p(t)\, dt$ $f_3(p) = \int_0^{-1} p(t)\, dt$ defined on $V = P_2$, the space of all polynomials over $\mathbb R$ of degree ...
4
votes
4answers
4k views

Probability that a coin lands on tails an odd number of times when it is tossed $100$ times

A coin is tossed 100 times , Find the probability that tail occurs odd number of times! I do not know the answer, but I tried this, that there are these $4$ possible outcomes in which tossing of ...
1
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1answer
175 views

Find the different equivalence classes of this relation on A and show their connection to a partition of A.

Define a relation $R$ on the set $A = \{n \mid n \in \mathbb{N} \textrm{ and } 0 < n < 14\}$ such that $R$ is an equivalence relation on $A$. (You can either define a property on $A$ or simply ...
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3answers
694 views

How to find $h'(x)$, if $h(x) = f(g(x))$.

Let $f'(x) = \sqrt{3x + 4}$ and $g(x)=x^2-1$. Find $h'(x)$, if $h(x) = f(g(x))$. I know that $g'(x) = 2x$, but I don't know how to do further. The answer is $h'(x) = 2x \sqrt{3x^2 + 1}$.
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1answer
194 views

Problem with continuity of a piecewise function

I am stuck with a homework problem which is about the modelling of tumor growth by ODEs. The function $A(t)$ is the amount of drugs in the patients blood. During some intervals (namely, $(n,n+\tau)$, ...
3
votes
2answers
184 views

Is the $ L^{p}$$[0,1]$ norm continuous in p?

I ran into the following problem when I was doing my homework, and I have no thoughts on where I should start with: (1) If $f\in L^{2}$, show that $\displaystyle \lim_{p \rightarrow ...
2
votes
4answers
241 views

Find $g'(3)$ if $g(x)$

if $f(3)=-2$ and $f'(3)=5$, find $g'(3)$ if, $g(x)=3x^2-5f(x)$ the answer is -7, I find that very hard to understand the question. thanks
1
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1answer
1k views

Average and instantaneous rates of change

For $y=x^2+1$ Find the average rate of change of $y$ with respect to $x$ over the interval [3,5]. I did it $5=3^2+1$ =5 but the answer is 8 I think there must be formula for average rate of ...
3
votes
3answers
365 views

Set Theory - Subset of set

I have labelled this $$ \{\{1\}\}\subseteq\{1,2,\{1,2\}\} $$ as true, it is a subset of the third element, is this true?
1
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0answers
138 views

Understanding Linear Regressions with Least Squares

I am currently trying to understand the linear regression fit by least squares for my machine learning homework, where I implement it and have to plot the result: I have given two data sets, ...
3
votes
1answer
140 views

Find $\frac{dy}{dx}$ when $y=\frac{x^2-1}{x^4-1}$

Find $\frac{dy}{dx}$ $$\begin{align*} y&=\frac{x^2-1}{x^4-1}\\ &=\frac{x^4-1(2x)-x^2-1(4x^3)}{(x^4-1)^2}\\ &=\frac{2x^5-2x-4x^5-4x^3}{(x^4-1)^2} \end{align*}$$ but the right answer is ...
0
votes
1answer
50 views

What qualifies as a non constant part of a power series?

In a power series $$\sum c_n (x-a)^n$$ What qualifies to be in the $(x-a)$ part? eg. $2-x$ $5x-2$ $\frac{x-3}{2}$ A combination of the above? $\frac{3-2x}{6}$ In a question I am working on, ...
4
votes
1answer
920 views

How to calculate transition matrix to get Jordan form?

Suppose $A$ is a linear transformation of a 3-dim vector space $V$, defined as $$A(\epsilon_1,\epsilon_2,\epsilon_3)=(\epsilon_1,\epsilon_2,\epsilon_3) \begin{pmatrix} -10 & 12 & 7\\ -3 & ...
0
votes
1answer
85 views

Find restrictions on $a>0$ and $b>0$ that ensure that $f(x_1,x_2)$ is concave.

Let $f:\mathbb{R}_{+}^2 \rightarrow \mathbb{R}$ be $f(x_1,x_2)=x_1^a x_2^b$ for $a>0$ and $b>0$. Find restrictions on $a>0$ and $b>0$ that ensure that $f(x_1,x_2)$ is concave. I ...
1
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1answer
652 views

Closure, boundary and interior

Describe the interior, closure and boundary of the following sets in the real line: the set of all integers the set of all rationals the set of all irrationals $(0,1)$ $[0,1]$ ...
3
votes
1answer
268 views

Isomorphism between homology group and reduced homology group of mapping cone

Given a map $f : X \to Y$, the mapping cone $C(f)$ is the space obtained from the mapping cylinder $M(f)$ by identifying the subspace $X \times \{0\}$ to a single point. How can I construct an ...
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2answers
2k views

Integral of an absolute value

I am trying to evaluate the integral $x-2|x|$ from -1 to 2 I know that this should give me $x^2 /2 - x^2$ for the antiderivative I then evaluate at 2 which gives me $2 - 4 = -2$ Then evaluate ...
0
votes
2answers
710 views

Cantor intersection theorem

I have Cantor intersection theorem: Let $X$ be a complete metric space, and let $\{F_n\}$ be a decreasing sequence of non-empty closed subsets of X such that $d(F_n)$ converges to $0$. Then ...
5
votes
2answers
863 views

Radius of convergence of power series or geometric series

To find radius of convergence of geometric series $$\sum_{n=1}^\infty a_n$$ I need to use ratio/root test to find $|L|<1$ To find radius of convergence of power series $$\sum_{n=1}^\infty c_n ...
4
votes
5answers
208 views

Help me prove the identity $\overline{f(0)} = \frac{1}{2\pi}\int_0^{2\pi}\frac{e^{i\phi}}{e^{i \phi}-z}\overline{f(e^{i\phi})}d\phi$

Let f be an analytic function defined in an open set containing the closed unit disk and let z in ℂ be fixed. I've simplified a more complicated expression down to this identity, and as implausible ...
1
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3answers
171 views

Difficulties in a proof by mathematical induction (2) [duplicate]

Possible Duplicate: proof by induction: n/(n+1) Continuing from here, I got a splendid answer that helped a lot. I'm tackling one now, but I've run into problems. Prove by mathematical ...
4
votes
1answer
394 views

How can I find the infinite sum of this non-conventional geometric series?

There's something about a geometric series that makes it easily verifiable. Series like $\sum\frac{10^n}{9^n}$ or $\sum\frac{1}{2^n}$ aren't too bad; the variables $n$ are simple and easily reachable, ...
1
vote
2answers
181 views

how much gross profit does the retailer earns

The question is A retailer sells widgets for $\$120$ each, which is $20\%$ more than they cost from wholesales. How much gross profit does the retailer earn after selling $8$ widgets $\$ ...
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votes
5answers
544 views

Deduce the next term in this sequence: m,n,a,z,l,o,b,y,k,p,c,x,j

This is the question : m,n,a,z,l,o,b,y,k,p,c,x,j In the letter series above, which one of the following choices logically follows d k m q r I have no ideas about ...
0
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1answer
157 views

how many of relatives are female

I am working on this problem but I am not so sure my answer is right. The question is below with multiples choices but there is only is right. ...
2
votes
2answers
104 views

Can someone check my work on this integral?

$$ \begin{align} \int_0^{2\pi}\log|e^{i\theta} - 1|d\theta &= \int_0^{2\pi}\log(1-\cos(\theta))d\theta \\ &= \int_0^{2\pi}\log(\cos(0) - \cos(\theta))\,d\theta\\ &= ...
0
votes
1answer
37 views

Find a circumference with center on a line

I have a set of circumferences $$x^2 + y^2 + \alpha_1 x + \beta_1 y + \gamma_1 + k(x^2 + y^2 + \alpha_2 x + \beta_2 y + \gamma_2) = 0$$ $\alpha_1, \alpha_1, \beta_1, \beta_2, \gamma_1, \gamma_2$ ...
1
vote
6answers
185 views

Taking the derivative of $y = \dfrac{x}{2} + \dfrac {1}{4} \sin(2x)$

Again a simple problem that I can't seem to get the derivative of I have $\frac{x}{2} + \frac{1}{4}\sin(2x)$ I am getting $\frac{x^2}{4} + \frac{4\sin(2x)}{16}$ This is all very wrong, and I do not ...
0
votes
2answers
766 views

Finding the derivative of a square root

I am trying to find the derivative of a pretty simple problem but I just can not force the answer to match the one provided by the book. $ - \frac {(1+x^2)^\frac{1}{2}}{x}$ I mean it is a very ...
3
votes
2answers
494 views

Riemannian metric in the projective space

Let $A: \mathbb{S}^n \rightarrow \mathbb{S}^n$ be the antipode map ($A(p)=-p$) it is easy to see that $A$ is a isometry, how to use this fact to induce a riemannian metric in the projective space such ...
3
votes
2answers
358 views

Sum of cubes of binomial coefficients

I reduced a homework problem in combinatorics to giving an asymptotic estimate for $\sum_{k=0}^n{n \choose k}^3$. I assume Stirling's approximation can help, but I'm not experienced with making ...
3
votes
4answers
210 views

Does $|x|^p$ with $0<p<1$ satisfy the triangular inequality on $\mathbb{R}$?

I am curious about whether $|x|^p$ with $0<p<1$ satisfy $|x+y|^p\leq|x|^p+|y|^p$ for $x,y\in\mathbb{R}$. So far my trials show that this seems to be right... So can anybody confirm whether this ...
9
votes
2answers
203 views

How many automorphisms of $S_n$ take transpositions into transpositions?

I need to show that an automorphism of $S_n$ which takes transpositions to transpositions is an inner automorphism. I thought it could be done by showing that such automorphisms form a subgroups ...
1
vote
2answers
224 views

cardinality of a set X in a topological space

This is my extra credit homework problem and I have no idea how to prove this please help me. Thank you. We denote by #S the cardinality of a set S. $\aleph_0$ = #$\mathbb{N}$, $c=2^{\aleph_0}$ = ...