0
votes
1answer
113 views

How do I find these bounds using Chebychev's inequality and the Central Limit Theorem?

Let $X$ have gamma distribution with parameters $\alpha=7$ and $\lambda=1$. Investigate the value of $F_X(10)$ using these methods: Find a lower bound using Chebychev's inequality. Approximate the ...
0
votes
2answers
40 views

Ratio for home work

In the year 2000 a rich aunty won £12000 and decided to share her money between her 3 nieces in the ratio of their age 14 10 18. she died in 2006 how much will each niece get?
0
votes
1answer
486 views

posterior density for bayesian estimations

Suppose that the number of accidents occurring daily in a certain plant has a Poisson distribution with an unknown mean $\lambda$. Based on previous experience in similar industrial plants, suppose ...
8
votes
1answer
171 views

Hom between 2 schemes

Why is the set $Hom(X,Y)$ between 2 schemes $X$ and $Y$ a scheme as well? Where can I read the construction? For example, $Hom(\mathbb{A}^1,\mathbb{A}^1)$ is the set of all polynomial, and what is the ...
0
votes
1answer
235 views

subgame perfect nash equilibrium for war of attrition

the question is as follow: suppose that two players are playing war of attrition, that means both of them could choose either to fight or quit, if either one of them quit, the game ends, and if ...
1
vote
3answers
285 views

Power series representation of $x$?

This may not be a very good question, but I'm totally stumped. I need to know the power series representation of $x$, or if there even is one. I'll show you why: I am trying to solve $y''+2xy'-y=x$ ...
0
votes
1answer
90 views

Prove min(L) = all words in L that they don't have any prefix of themselves in L

We define the minimal words language of $L, \min(L)$, to be the language of all words in $L$ that don't have any prefix in $L$. Assume $L$ is regular language. I need to prove by building an ...
4
votes
3answers
1k views

arithmetic mean of a sequence converges

We had a theorem that the means of a sequence also converges: Let $(a_n)_{n\in\mathbb N}$ be a convergent sequence. Then $\displaystyle \overline a_n=\sum_{k=1}^n \frac{a_k}n$ also converges. ...
2
votes
5answers
278 views

Why is $3 \cdot 3^k = 3^{k+1}$ and not $9^k$?

Why is $3 \cdot 3^k = 3^{k+1}$ and not $9^k\;$? I'm aware that $3 = 3^1$ but I would expect $3\cdot 3^k\;$ to be $\;9^k$ or $\;9^{k+1}$.
4
votes
2answers
244 views

How do Flatlanders represent Möbius strips?

There are 3D representations of Klein bottles that give people in our 3D universe a pretty good idea of how one is constructed: We can sort of see how this thing needs to be 'twisted' in the fourth ...
4
votes
1answer
83 views

Removing one edge from Graph

How many new graphs that are not isomorphic will I have by removing any of its edges (but only one!) ? I did following: Where the numbers mean which graph will I get by removing corresponding ...
3
votes
2answers
65 views

$(a_{2n})$ and $(a_{2n+1})$ converges then $(a_n)$ converges

Whe had the following theorem in class: If $(a_{2n})_{n\in\mathbb N}$ and $(a_{2n+1})_{n\in\mathbb N}$ are convergent sequences with the same limit $a$, then the sequence $(a_{n})_{n\in\mathbb N}$ ...
2
votes
1answer
138 views

Find a subring of $\mathbb{ R }$ isomorphic to $\mathbb{ Q }[ x ] / \langle x^3 - 2 \rangle$.

I need to find a subring of $\mathbb{ R }$ isomorphic to $\mathbb{ Q }[ x ] / \langle x^3 - 2 \rangle$. I considered using the Isomorphism Theorem to try to find a homomorphism $\theta: \mathbb{ Q ...
0
votes
1answer
52 views

Problem with one step in a proof of fundamental theorem of curves

Let $k$, $l$ be smooth functions from an interval $I$ into $\mathbb R$ and $k>0$. Let's consider system of differential equations $$ t'=k n, $$ $$ n'=-k t-l b, $$ $$ b'=ln $$ with unknow ...
4
votes
0answers
201 views

There are $n$ horses. At a time only $k$ horse can run in the single race. How many minimum races are required to find the top $m$ fastest horses?

There are $n$ horses. At a time only $k$ horses can run in the single race. How many minimum races are required to find the top $m$ fastest horses? Please explain your answer. PS: There is no timer.
3
votes
2answers
74 views

Symmetry in ordinary differential equations

Suppose I am given an ode $${dy\over dx}={1\over x^2}f(xy)$$ where $f$ is some arbitrary function. How then does doing the following help solve the equation? : First I have a vector field ...
1
vote
1answer
290 views

Uniform convergence of continuous functions with Lipschitz limit

Let $K \subset \mathbb R^d$ be a compact. Let $\phi_{\varepsilon} \colon K \rightarrow \mathbb R$ be continuous and converge uniformly to $\phi$. Suppose further that $\phi$ is Lipschitz continuous. ...
1
vote
2answers
90 views

Is $\omega_{\alpha}$ sequentially compact?

For an ordinal $\alpha \geq 2$, let $\omega_{\alpha}$ be as defined here. It is easy to show that $\omega_{\alpha}$ is limit point compact, but is it sequentially compact?
2
votes
2answers
124 views

Proof : $2^{n-1}\mid n!$ if and only if $n$ is a power of $2$.

I want to prove that: $2^{n-1}\mid n!$ if and only if $n$ is a power of $2$.
7
votes
1answer
285 views

Order topology and partial order

Let $(X,\leq)$ be an ordered space and define on $X$ the order topology form the subbasis $\{(x,y), (x,\to), (\leftarrow,x) : x,y \in X \}$. Can you read the fact that $\leq$ is a total order on the ...
3
votes
2answers
235 views

Show that if $f( 0 ) = 0$ and $f'( x ) > f( x )$ for all $x \in \mathbb{ R }$ then $f( x ) > 0$ for all $x > 0$.

I need to show that if $f$ is a $C^1$ function with $f( 0 ) = 0$ and $f'( x ) > f( x )$ for all $x \in \mathbb{ R }$ then $f( x ) > 0$ for all $x > 0$. I think I need to show that $f( x ) ...
1
vote
1answer
81 views

Finding generators for an ideal consisting of the set of functions vanishing on a subset of Spec

Let $U$ be the union of the $x,y,z$ axes in complex affine 3-space. The set of functions that vanish on $U$ is an ideal. Can we neatly express their generators? There are a lot of similar such ...
2
votes
3answers
501 views

Use three 11's and various math symbols to make an equation equal to 6

The puzzle is to use the following symbols $$+,\;-,\;*,\;/,\;(\;,\;),\;!, \;\sqrt(\cdot)$$ in order to make a valid equation out of $$11~~~~~~11~~~~~~~11 = 6.$$ (There are three elevens with space in ...
2
votes
3answers
127 views

How to show that: $ \gcd \Big(2^{2^a}+1 , 2^{2^b}+1\Big)=1$

How to show that: $$ \gcd \Big(2^{2^a}+1 , 2^{2^b}+1 \Big)=1$$
6
votes
1answer
576 views

Functions of bounded variation as the dual of $C([a,b])$

I am trying to understand this proposition about the dual of $C([a,b])$. I would like some help with the following: (1) What does the integral with respect to a function of bounded variation mean? ...
2
votes
1answer
163 views

Intersection of monomial ideals

Let $ [n] = \lbrace 1,2,\dots,n \rbrace $, and $ F \subset [n] $. We denote by $ P_F \subset K[X_1,\dots,X_n] $ the monomial ideal generated by the variables $ X_i $ with $ i \in F $. Given an ...
0
votes
1answer
355 views

Probability of ace of spades on 21st selection

A deck of cards is shuffled and the cards are then turned over one at a time until the first ace appears. Given that the first ace is the 20th card to appear, what is the conditional prob that the ...
2
votes
6answers
225 views

How to show that $f\left( \frac{ x + y }{ 2 }\right ) \leq \frac{ f( x ) + f( y ) }{ 2 }$ when $f''(x) \geq 0$.

I need to show that if $f: (a,b) \to \mathbb{ R }\text{ with}\;\; f''( x ) \geq 0$ for all $x \in (a,b)$, then $f\left( \frac{ x + y }{ 2 } \right) \leq \frac{ f( x ) + f( y ) }{ 2 }$. I know that ...
3
votes
2answers
157 views

Men on a boat problem

There is the usual question of some men on a boat- various men have various speeds, the boat has a capacity of 2 men, and the boat takes on the speed of the slowest man in the boat at any given time. ...
2
votes
1answer
100 views

dimension of a quotient of a local ring

Let $\mathcal{O}_{k^2,(0,0)}$ be the local ring at origin, i.e. $k[x,y]_{(x,y)}$. I want to show that $\dim_k \mathcal{O}_{k^2,(0,0)}/(y-x^2,x^3)=3$. My rough argument is the following, but I feel ...
1
vote
1answer
86 views

Does $X\overset{f}{\rightarrow} Y\rightarrow0$ being exact imply $\operatorname{coker}f=0$?

In a category with zero object, it is easy to see that if $0\rightarrow X\overset{f}{\rightarrow} Y$ is exact then $\ker f=0$, since $0\rightarrow X$ is monic and hence is its own image. However, when ...
1
vote
0answers
96 views

Limit question: $\lim_{x\to \infty}f(x),\;\; \lim_{x\to -\infty}f(x)$?

Assuming $$f(x)=ax^3+bx^2+cx+d$$ show that $\lim_{x\rightarrow\infty}f(x)$ and $\lim_{x\rightarrow -\infty}f(x)$ exists, find the limits.$a,b,c,d\in\mathbb{R}$ Well, I think the limits exists in ...
9
votes
1answer
901 views

Losing at Spider Solitaire

Spider Solitaire has the property that sometimes none of the cards in the final deal can "go" and so you lose, regardless of how much progress you have made beforehand. You would have known that you ...
2
votes
1answer
105 views

Am I doing something wrong with these moment generating functions?

I'm told to find the moment generating function of the pdf $6x(1-x)$, $0<x<1$, and I found $$\frac{6te^t-12e^t+6t+12}{t^3}$$ Then it asks to find the expected value, and as anyone else would ...
2
votes
1answer
98 views

The sufficient and necessary condition for a function approaching a continuous function at $+\infty$

Problem Suppose $f:\Bbb R^+\to\Bbb R$ satisfies $$\forall\epsilon>0,\exists E>0,\forall x_0>E,\exists\delta>0,\forall x(\left|x-x_0\right|<\delta): ...
1
vote
3answers
267 views

Probability we get a king on the nth card draw when drawing from a pack of 52

I'm looking for the probability that we first get a king on the nth card draw when drawing from a pack of 52 cards. Here's what I have done - Let $A_i$ be the event that we don't get a king on card ...
1
vote
2answers
101 views

set of all points where $|f(x)|>\epsilon$ is finite

$f:\mathbb{R}\rightarrow \mathbb{R}$ is function such that $\forall\epsilon>0$, the set $\{x:|f(x)|>\epsilon\}$ is finite. We need to show $\{x:f(x)=0\}$ is uncountable. Could any one give me ...
1
vote
2answers
830 views

Is “product” of Borel sigma algebras the Borel sigma algebra of the “product” of underlying topologies?

A Borel sigma algebra is the smallest sigma algebra generated by a topology. The "product" of a family of Borel sigma algebras is to first take the Cartesian of the Borel sigma algebras, and then ...
3
votes
2answers
251 views

Proving that $\sum_{|j| < n} (n-|j|) \exp(ij\lambda)= \frac{\sin^2(\frac 1 2 n\lambda)}{\sin^2(\frac 1 2 \lambda)}$

I want to show that $\sum_{|j| < n} (n-|j|) \exp(ij\lambda)= \dfrac{\sin^2(\frac 1 2 n\lambda)}{\sin^2(\frac 1 2 \lambda)}$. I know from Proving ...
0
votes
2answers
489 views

How to find the inverse of a unit in a polynomial quotient ring.

I am supposed to find the inverse of $1 + x^2$ in the division ring $\mathbb{ Z }_{ 11 }[ x ] / \langle x^3 + 1 \rangle$. I know that that $x^3 = -1 = 10$. I have tried solving for $a$, $b$, and ...
1
vote
2answers
3k views

Drawing two cards from 52, what is the probability that the second card has a higher face value than the first?

So if I draw two cards from 52, what is the probability that the second card has a higher face value than the first? The values of the cards are Ace = 1, Two = 2,..., King = 13. I got as far as ...
1
vote
1answer
120 views

Non-Isomorphic induced representations (from the same representation of a subgroup)

I believe that it is true that if we have a group $G$, and two copies $H_1$, $H_2$ of some group $H$ as subgroups of $G$, we can fix a representation $V$ of $H$ and have the situation: ...
0
votes
2answers
72 views

showing existence of some continuous function

Let $X$ be a compact metric space, $\mathcal{B}(X)$ be the borel-sigma algebra. Assume $\mu(G)>0$ for all nonempty open sets $G\subseteq X$. Show that for each open set $G$, exists a closed ...
2
votes
1answer
50 views

for $\epsilon\gt0$ find $\delta_\varepsilon\gt0$ such that $\left|x-x_0\right|\lt\delta_\varepsilon\Rightarrow \left|f(x)-f(x_0)\right|<\varepsilon$

For the following function, for each $\varepsilon\gt0$ find $\delta_\varepsilon\gt0$ such that $\left|x-x_0\right|\lt\delta_\varepsilon\Rightarrow \left|f(x)-f(x_0)\right|<\varepsilon$, where ...
0
votes
1answer
115 views

How to move a one 3D line from three 3d parallel lines

I have 3 parallel line segments (say AB, CD, and EF are line segments and they are nearly horizontal) lay on 2 slanted planes which have been intersected through the CD. If I projected all the line ...
2
votes
2answers
60 views

Non-isomorphic combinatorial classes with growth rate equal to the golden ratio?

In a couple of weeks, I'm giving a talk about the growth rates of some combinatorial classes. In the introduction, I thought I'd present the class of tilings of a $n\!\times\!2$ strip with dominos, ...
5
votes
1answer
94 views

Lower Bound of countable order-types of linearly ordered sets

On Page 44, Set theory, Jech(2006) (Show that) there are at least $\mathfrak{c}$ countable order-types of linearly ordered sets. [For every sequence $a = \{ a_n : n \in N\}$ of natural numbers ...
0
votes
1answer
92 views

Is the relation $\{ (a,b) \in\Bbb Z^2 : |a-b| \le 10 \}$ an equivalence relation or not ??

Is the relation ↖ an equivalence relation or not ?? I know we are supposed to prove that it is reflexive , transitive and symmetric . I found that it is an equivalence relation but i am not sure , so ...
1
vote
1answer
148 views

Question on inclusion-exclusion principle when $n=2$

Using the inclusion exclusion principle - http://www.proofwiki.org/wiki/Inclusion-Exclusion_Principle - if I set $n=2$ I get the following - $$P(A_1 \cup A_2) = P(A_1) + P(A_2) - P(A_1 \cap A_2) + ...
0
votes
1answer
50 views

Parallel computing : scheduling processors for large sums

does anyone know if there exists in the literature an algorithm that solves the following problem? I have M different and indipendent sums, and P processors. The size of sums are in ascending order ...

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