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H: Functions cannot be integrated as simple functions
Possible Duplicate:
How can you prove that a function has no closed form integral?
Since I was a college student, I was told there were many functions that cannot be integrated as simple functions.(I'll give the definition of simple functions at the end of the a... |
H: Polynomials with roots having the same module and linear dependent arguments
Is it possible for a polynomial with integer coefficients to have some of its roots:
$$m_1e^{i\theta_1 \pi}, m_2e^{i\theta_2 \pi}, \ldots, m_ke^{i\theta_k \pi}$$
such that there exist nonzero integers $a_1, a_2, \ldots, a_k$ and an integer... |
H: How many elements in a ring can be invertible?
If $R$ is a finite ring (with identity) but not a field, let $U(R)$ be its group of units. Is $\frac{|U(R)|}{|R|}$ bounded away from $1$ over all such rings?
It's been a while since I cracked an algebra book (well, other than trying to solve this recently), so if som... |
H: In an engineering/optimisation context, does set $E$ have any special significance?
I am reading a paper about optimsation and the description, while mostly being a very good description, makes reference to some variables being in some set $E$. For example, it states that parameter $x\in E^n$. However it does not... |
H: Subgroups of $\Bbb Z_5 \times \Bbb Z_5$
Find all subgroups of $\Bbb Z_5 \times \Bbb Z_5$.
I can see that the non-trivial ones are of order $5$. But how do I find them exactly?
Thanks for any help.
AI: We list the subgroups of order $5$. There is the group generated by $(0,1)$. Then there are the groups generated by... |
H: System of equations to represent a matrix
Suppose that there is a $n \times n$ matrix. Then there will be entries $A_{ij}$ where i and j represent row and column.
Equations contain entries of a matrix, $A_{ij}$ and when the equations are solved, we will get each entry.
(Add: By equations I mean like ${A_{11}}^2 + ... |
H: How popular and used were logarithm tables?
I've heard that, for a time, logarithm tables "sold more than the Bible". Can someone produce some reliable documentation about how prevalent they were ? Would a common shopkeep have one ? Would a common merchant ship have one ?
(they would be used to make multiplications... |
H: Normal Distribution Transformation
Suppose we have a normal distribution like
$
f(x) = \mathcal{N}(\mu = 30, \sigma^2=10)
$
and we transform it to another function by multiplying it to
$
g(x) = 2x^2
$
the result would be:
$
f(x).g(x) = h(x) = \frac{2x^2}{\sqrt{2\pi}.\sigma}.e^{-\frac{(x-\mu)^2}{2.\sigma}}
$
h(x... |
H: A trigonometric inequality
How to show that
if $0\le\theta\le2\pi$
$|\sum\limits_{n=1}^{p}{\sin{n\theta}}|\le\csc{\frac{\theta}{2}}$
for all integer p?
AI: $$S_p = \sum_{n=1}^{p} \sin(n \theta) = \csc (\theta/2) \sin(p \theta/2) \sin((p+1) \theta/2)$$
To see why the above is true, multiply $\displaystyle \sum_{n=1... |
H: Issue with combining InEqualities
I have the following two inequalities:
$$\begin{align*}
8x &\gt 12y\\
12y &\gt 15z
\end{align*}$$
Now the book states that we need to line up the inequalities as such
$$\begin{array}{rcccccl}
0 &<& 15z\\
&& 15z &<& 12y\\
& & & & 12y &<&... |
H: $C_0(X)$ is a closed subspace of $C_b(X)$
Can you tell me if my proof is correct? Thank you!
Claim: $C_0(X)$ is a closed subspace of $C_b(X)$
Proof: We have to show that $C_0(X)$ contains all of its limit points. Let $f(x)$ be a limit point of it, then we have a sequence $f_n$ converging to it (in $\|\cdot\|_\infty... |
H: What is so wrong with polynomial hierarchy collapsing
Many computational complexity researchers believe that finite-level collapse of polynomial hierarchy is unlikely. Why do they believe like this?
AI: It's possible to generalize to this conclusion essentially directly from the opinion that P $\neq$ NP. Let's thin... |
H: "Proof" of an Algebraic property of OLS Estimators
I'm having a bit of trouble proving $\sum (x_i - \bar{x})\hat{e_i} = 0$. What I know so far is that the total sum of $\hat{e_i}$'s is zero by property of OLS so when you distribute the $\hat{e_i}$ in, one term "cancels out" and you are left with $\sum x_i\hat{e_i}... |
H: Order of a product of subgroups. Prove that $o(HK) = \frac{o(H)o(K)}{o(H \cap K)}$.
Let $H$, $K$ be subgroups of $G$. Prove that $o(HK) = \frac{o(H)o(K)}{o(H \cap K)}$.
I need this theorem to prove something.
AI: Here is LaTex-ed version of the proof posted in BBred's comment.
I've tried to add details of one plac... |
H: Quibble with terminology
Proposition 5.15 on page 63 in Atiyah-Macdonald goes as follows:
Let $A \subset B$ be integral domains, $A$ integrally closed, and let $x \in B$ be integral over an ideal $ \mathfrak a$ of $A$. Then $x$ is algebraic over the field of fractions $K$ of $A$ and if its minimal polynomial over $... |
H: Is there a residually finite group not finitely presented?
I am looking for a residually finite group which is not finitely presented. Does such a group exist?
AI: Finally, I found a nice example: the lamplighter group $L_2= \mathbb{Z}_2 \wr \mathbb{Z}$. There is a natural morphism from $L_2$ to $\mathbb{Z}_2 \wr \... |
H: Graph theory question involving probabilistic method.
I was trying to prove the following statement using probabilistic methods: Given that $G$ is a graph on $n\geq 10$ vertices, is a graph that has the property: If we draw a new edge, then the number of copies of $K_{10}$ increases. Prove that $|E|\geq 8n-36$.
I a... |
H: proper subgroups of finite p-groups are properly contained in the normalizer
I am trying to prove the following,
Let $G$ be a finite $p$-group and let $H$ be a proper subgroup. Then there exists a subgroup $H'$ such that
$$
H\lneq H'\leq G
$$
and $H\triangleleft H'$.
Obviously, the natural choice for $H'$ ... |
H: How to prove: $S=\frac{4}{3}\sqrt{ m(m-m_a)(m-m_b)(m-m_c)}$
If $$m_a, m_b, m_c$$ are the medians of a triangle and let $$m=\frac{m_a+ m_b+ m_c}{2}$$
then Area $S$ of triangle is given by
$$S=\frac{4}{3}\sqrt{ m(m-m_a)(m-m_b)(m-m_c)}$$ This looks very similar to Heron's formula. How to prove this formula?
AI: $$S=\f... |
H: Compact operator on a hilbert space.
Possible Duplicate:
A compact operator is completely continuous.
I came across this question in a book, it is an exercise, I can't prove it. Can someone please help me.
If $X$ and $Y$ are Banach spaces, we have to prove that a compact linear operator is completely continuous.... |
H: Compact operators between Hilbert spaces
I have the suspect that the following statement is true, but I don't how to prove it. Any suggestion? Thanks to all!
Let $X$, $Y$ be Hilbert spaces and let $T \colon X \to Y$ be a linear continuous injective map. Suppose that for every $\epsilon > 0$ there exists a closed ve... |
H: Probability of a few possible outcomes with certain probability beat the original
Lets say we have a contest of exactly 5 contestants who are all competing against the Original.
The probability that some contestant will win over the Original are as follows:
cont1 = 26.9%
cont2 = 21.3%
cont3 = 20.7%
cont4 = 8.96%
c... |
H: Express $C$ interms of the sets $A_n$
[NBHM_2006_PhD Screening Test_Analysis]
Let $f$ be a real valued function on $\mathbb{R}$ define
$$w_j(x)=\sup\{|f(u)-f(v)|: u,v\in [x-1/j,x+1/j]\}$$ $j\in \mathbb{N}$
and $x\in\mathbb{R}$, Define next
$$A_{j,n}=\{x\in\mathbb{R}:w_j(x)<1/n\}$$ $n=1,2,\dots$ and
$$A_n=\... |
H: Motivation for Topology study in Real Analysis
I'm an engineering student trying to work out some Real Analysis to learn how to write proofs (Needed for my PhD thesis) and just to rekindle my Calculus fires. From what I see, Real Analysis is the study of the basics of Calculus and "constructing" Calculus ground up.... |
H: Find the area enclosed by the curve $r=2+3\cos \theta$.
the question is
Find the area enclosed by the curve:
$r=2+3\cos \theta$
Here's my steps:
since when $r=0$, $\cos \theta=0$ or $\cos\theta =\arccos(-2/3)$.
so the area of enclosed by the curve is 2*(the area bounded by $\theta=\arccos(-2/3)$ and $\theta=0$)
t... |
H: Lebesgue measure on $\mathbb{R}/\mathbb{Z}$
I was reading a (brief) introduction about measure theory today and came across the following statement:
(Lebesgue measure on $\mathbb{R}/\mathbb{Z}$): There is a unique probability measure $\mu$ on $\mathbb{R}/\mathbb{Z}$ such that $\mu((a,b))=b-a$ for all $0\le a<b\le ... |
H: $\chi^2$ test and sampling variance
Let $f(x)$ denote the pdf of a $\chi^2$-distribution with $n\in\mathbb{N}$ degrees of freedom given by
$$f(x) = \frac{2^{-n/2}}{\Gamma(n/2)}\cdot x^{n/2-1}\cdot\mathrm e^{-1/2x}\cdot\textbf{1}_{[0,\infty)}(x),$$
where $\textbf{1}_A(x)=\begin{cases}1,&x\in A,\\0,&\text{else.}... |
H: complexity for $f(x)=n!$ and O($2^n$)
Suppose that algorithm has O($n!$). We all know that $n!$ should be smaller than $2^{2^n}$, but bigger than $2^n$.
So, will O($n!$) be in EXPTIME (EXP)?
Will we able to write O($n!$) as O($2^n$)?
AI: $n!$ is not $O(2^n)$. The function $n\mapsto n!$ grows much faster that $n\... |
H: Fermat's theorem on sums of two squares
There's Fermat's theorem on sums of two squares.
As the prime numbers that are $1\bmod4$ can be divided into the sum of two squares, will the squared numbers be unique?
For example, $41=4^2+5^2$ and the squared numbers will be $4$ and $5$.
AI: Yes, if you don't take into acc... |
H: Epsilon delta proof of a sequence
How can I proof the convergence of
$${\{7^{-n^{-1/5}}\}_{n\geq 1}}$$
?
AI: Just do some computations and use the motonicity of powers and the logarithm to find the $N$ you want:
First note, that $7^{-n^{-1/5}}\le 1$ for every $n$, now let $\epsilon > 0$. We have
\begin{align*}
1... |
H: which of the spaces are Locally Compact
[NBHM_2006_PhD Screening Test_Topology]
which of the spaces are Locally Compact
$A=\{(x,y): x,y \text{ odd integers}\}$
$B=\{(x,y): x,y\text{ irrationals}\}$
$C=\{(x,y): 0\le x<1, 0<y\le 1\}$
$D=\{(x,y): x^2+103xy+7y^2>5\}$
A topological space $X$ is locally compact if e... |
H: Vector Autoregression Algebra, $M_t$, $L$
In the paper here
http://www.ems.bbk.ac.uk/for_students/bsc_FinEcon/fin_economEMEC007U/VAR.pdf
It shows VAR(p) model as
$$
W_t = A_1W_{t-1} + A_2W_{t-2} + ... + A_pW_{t-p} + \epsilon_t
$$
But then it makes a simplification and says the formula above equals to
$$
(I - A_1... |
H: What does the notation "$\Omega \subset \mathbb{R}^n$ is $C^1$" mean?
In my calculus 2 lecture notes, we have the following definition:
A region $\Omega \subset \mathbb{R}^n$ is $C^1$ (or $C_{pw}^1$ or $C^k$ respectively), if for each point $x_0 \in \partial \Omega$ there exist coordinates $(x',x^n) \in \mathbb{R}... |
H: what are the possible values for integral
let $\gamma$ be a closed continuosly differentiable path in the upper half plane not passing through $i$. What are the possible values of the integral
$$\frac{1}{2\pi i}\int_{\gamma}\frac{2i}{z^2+1}dz$$
well the integral can be broken like
$$\frac{1}{2\pi i}\int_{\gamma}\... |
H: If $V \times W$ with the product norm is complete, must $V$ and $W$ be complete?
Let $V,W$ be two normed vector spaces (over a field $K$). Then their product $V \times W$ with the norm $\|(x,y)\| = \|x\|_V + \|y\|_W$ is a normed space.
Using this norm it's easy to show that if $V,W$ are complete then so is $V \time... |
H: maximize the function of three variable
I am completely struck with the problem:
Let $f$ be a function of three variables having continuously partial derivatives. For each direction vector $h=(h_1,h_2,h_3)$ such that $h_1^1+h_2^2+h_3^3=1$, Let $D_hf(x,y,z)$ be the directional derivative of $f$ along $h$ at $(x,y,z... |
H: Finding Product of Scattered Variables
Hi I came across the following question where I need to find $$mk$$ from $$ (x-2) (x+k) = x^2 + mx - 10 $$ The answer is 15. Any suggestions on how I could do that ?
AI: The sum of the roots $(2, -k)$ equals $-m$. The product of the roots $-2k=-10$. Therefore $k=5$ and $m=3$... |
H: Frechet derivative question
I'm trying to show that a map $f$ between Banach spaces $X$ and $Y$ is Frechet differentiable at a point $u$. To do this, it is enough to calculate its Gateaux derivative at $u$ (call it $df(u)$) and show that $df(u)$ is continuous: so for every $\epsilon$, there exists a $\delta$ such t... |
H: Recursive digit-sum
Let the recursive digit-sum(R.D.) be defined as: continue taking the sum of digits until it becomes <10.
For example, the digit-sum of 5987 = 29, the digit-sum of 29 =11
So, R.D. of 5987 is 2.
Prove that the value of R.D. recurs after each 9 numbers i.e., R.D. of any natural numbers of the form ... |
H: The completion of a noetherian local ring is a complete local ring
We have defined the completion of a noetherian local ring $A$ to be $$\hat{A}=\left\{(a_1,a_2,\ldots)\in\prod_{i=1}^\infty A/\mathfrak{m}^i:a_j\equiv a_i\bmod{\mathfrak{m}^i} \,\,\forall j>i\right\}.$$
I have a slight problem trying to understand th... |
H: If $F$ is a formally real field then is $F(\alpha)$ formally real?
Let us call a field $F$ $\textit{ formally real }$ if $-1$ is not expressible as a sum of squares in $F$. Now suppose $F$ is a formally real field and $f(x)\in F[x]$ be an irreducible polynomial of odd degree and $\alpha $ is a root of $f(x)$. Is it... |
H: Calculate the normal unit vector for scalar function
In theory, if I have a certain function I can get his normal unit vector by using the gradient of it.
$$\hat{f} = \dfrac{\nabla f}{|| \nabla f ||}$$
Example (correction from answer):
$$ z = 2 -x -y$$
$$ f(x,y,z)= z + x + y -2 $$
$$ \nabla f(x,y,z)= \hat{i} + \ha... |
H: Is using the - symbol with the Associative Law of multiplication invalid?
I was trying to prove that $-(x + y) = -x - y$ and as you can see in the image below, I took the liberty of using the $-$ symbol as a number and applying the associative law with it. Is it kosher in all rigorousness given the axioms professio... |
H: Possible cup product structures on a manifold
I am studying for a qualifying exam, and I came across this problem:
Let $M$ be a closed orientable connected 4-manifold with $H^1(M) = H^3(M) = 0$ and $H^2(M) \cong H^4(M) \cong \mathbb Z$. What are the possible cup product structures on $H^*(M)$?
My thoughts:
Just u... |
H: Relation between metrics
Let
$$\eqalign{
& d\left( {x,y} \right) = \mathop {\max }\limits_{1 \leqslant i \leqslant n} \left\{ {\left| {{x_i} - {y_i}} \right|} \right\} \cr
& d'\left( {x,y} \right) = \sqrt {\sum\limits_{i = 1}^n {{{\left( {{x_i} - {y_i}} \right)}^2}} } \cr
& d''\left( {x,y} \right) = \sum... |
H: Committees that share exactly one member
A club of $n$ members is organized into four committees following two rules:
Each member belongs to exactly two committees, and
each pair of committees has exactly 1 member in common.
Find all possible values of $n$.
AI: If each member must belong to two committees, he or ... |
H: Is there a sequence in $(0,1)$ such that the product of all its terms is $\frac{1}{2}$?
Is there a sequence in $(0,1)$ such that the product of all its terms is $\frac{1}{2}$?
AI: If you take any sequence $a_1,a_2,a_3,\ldots$ whose sum is $\log_b (1/2)$, then $b^{a_1}, b^{a_2}, b^{a_3},\ldots$ is a sequence whose p... |
H: How to calculate this conditional probability
There's an equation in my script, which I do not understand. Let $(B_t)$ be a Brownian Motion and $\Gamma\in\mathcal{B}(\mathbb{R}^n)$, $t\ge s$ the equation is
$$P(B_t\in\Gamma | B_s)=\frac{1}{\sqrt{(2\pi)^d(t-s)^d}}\int_\Gamma \exp{\left(-\frac{|z-B_s|^2}{2(t-s)}\righ... |
H: Which property should be applied here?
I am having a problem with the following question:
If $$N = 3^P { } $$ and $$M=P-1$$ Then in terms of M what is $$ \frac{3}{N}=?$$
Any suggestions on which properties to apply ?
AI: Simply substitute and use exponents' laws:
$$N=3^P\,,\,M=P-1\Longrightarrow \frac{3}{N}=\f... |
H: Why is the fundamental group of a prime, reducible 3-manifold $\mathbb{Z}$?
I've read in a paper that if $M$ is a prime, reducible $3$-manifold, then $\pi_{1}(M) \cong \mathbb{Z}$. Can anyone explain why this is true?
Thanks in advance.
AI: I'm probably assuming $M$ to be orientable.
Reducible means that there is a... |
H: A maximal ideal is always a prime ideal?
A maximal ideal is always a prime ideal, and the quotient ring is always a field. In general, not all prime ideals are maximal. 1
In $2\mathbb{Z}$, $4 \mathbb{Z} $ is a maximal ideal. Nevertheless it is not prime because $2 \cdot 2 \in 4\mathbb{Z}$ but $2 \notin 4\mathbb{... |
H: Expectation of absolute value of a function
Let x be real valued random variable taking values on $a_1,\ldots, a_n$. Let $\Pr(x=a_i)=p_i$. Let $f$ be real valued function defined on $a_1, \ldots, a_n$
It is known that
$$
E(f(x))=\sum_{i=1}^nf(a_i)p_i.
$$
Would be the same formula true for $E(|f(x)|)$, i. e.
$$
E(|f... |
H: Bound on bounded functions
Let $C[0,1]$ be the space of continuous and nondecreasing functions with the sup norm. Moreover, let $f[0,1]\rightarrow \mathbb{R}$ be continuous and positive, i.e., $f(s)>0,\,s\in[0,1]$. Take any two element $z,h$ in $C$. I would like to put an upper bound on the following expression:
$$... |
H: Element of a set?
I have to say if {2} is an element of the given sets. I'm reading {2} as if it was a subset which would make the problem true for C, D and E only correct?
F it isn't because that is a nested subset and A/B don't contain subsets.
For each of the sets, determine whether {2} is an element of that se... |
H: Estimating the Gamma function to high precision efficiently?
I know there are several approximations of the Gamma function that provide decent approximations of this function.
I was wondering, how can I efficiently estimate specific values of the Gamma function, like $\Gamma (\frac{1}{3})$ or $\Gamma (\frac{1}{4}... |
H: Maximal color difference
I have a picture consisting of a two-dimensional array of ordered triples (red, green, blue) of real numbers from 0 to 1. I'm looking for something like a norm on pictures which expresses the range of colors used.
The idea is that a grayscale image should have norm 0 and an image with pure ... |
H: Can a convergent sum using only integers produce a complex result?
We use this function to define the boundaries for the product in the denominator:
$$f(\text{n$\_$})\text{:=}\frac{1}{8} \left(2 n (n+2)-(-1)^n+1\right)$$
We calculate the infinite sum:
$$\sum _{n=1}^{\infty } \frac{1}{(f(n)+1)_{f(n+1)-f(n)}}$$
We... |
H: Deteriming an angle without Trig. ratios
I am trying to solve the current problem
If O is the center of a circle with diameter 10 and the perimeter of AOB=16 then which is more x or 60
Now I know the triangle above is an isosceles triangle with 2 sides being 5 (since diameter is 10) and third side being 6 thu... |
H: Deriving an equation that satisfies many points
Say I have a collection of points, for example the following:
(1, 167), (2, 11), (3, 255), etc
Is it possible to construct an equation that satisfies all of them? I have a maximum of 32 points.
AI: Given any $n$ points in the plane, none of which lie on the same vert... |
H: How to count the possible ways?
In a bag there are 10 indistinguishable balls. Four of them are white and $6$ are black. The balls are taken out one by one and put on the table as they are taken.
In how many ways we can get at least $2$ consecutive white balls?
When I got this problem, I immediately thought abou... |
H: $f$ strictly increasing does not imply $f'>0$
We know that a function $f: [a,b] \to \mathbb{R}$ continuous on $[a,b]$ and differentiable on $(a,b)$, and if $f'>0 \mbox{ on} (a,b)$ , f is strictly increasing on $[a,b]$. Is there any counterexample that shows the converse fails?
I have been trying to come up with si... |
H: For what value of x will it be less than 2
Is there a value of x for the following equation which will make it less than x ?
The question is which is more $$ \frac{3x+1}{x+1}$$ if $$x\not=-1$$
or simply 2 ?
According to the book there is not enough information to solve this problem, but i think the expression ... |
H: Two questions about continuity of function between topological spaces
Let $X$ and $Y$ be topological spaces and suppose $f: X \to Y$ is continuous. If $f$ is continuous on $U \subset X$, will the restriction $f_U :U \to Y$ be continuous, if we consider $U$ to be a topological space of its own?
My second question i... |
H: Lipschitz continuity of a functional?
Here is my problem. Let $C$ be the space of continuous and nondexcreasing functions defined on $[0,1]$ and endowed with the sup norm. Take any $z\in C$ and consider the following:
$$
U(x;z)=\int_{0}^{x}\left[1-\int_{s}^{1}F(z(\xi))f(\xi)d\xi\right]^{n-1}ds
$$
where $F:[0,1]\rig... |
H: Does (Infer $\phi$ from $\psi$) imply (Infer $\phi^L$ from $\psi^L$)?
I am studying set theory on my own on Drake's famous book and I'm stuck on the (finitary) prove of the relative consistency of the Axiom of Choice.
Is it true that a if we were able to infer $\xi$ from $\zeta$ then we are able to infer $\xi^L$ ... |
H: How do I form this equation?
If $A$ and $B$ are the root of the equation $3x^2-4x-9=0$, what is the equation whose roots are
$(A+3)/(A-3)$ and $(B+3)/(B-3)$
AI: Recall that if $\alpha, \beta$ are roots of $ax^2 + bx + c$, then we have $\alpha + \beta = -\dfrac{b}a$ and $\alpha \beta = \dfrac{c}a$.
Let $\alpha = \df... |
H: what is a general algorithm to find a nonempty integer subset that have integers add up to 0?
what is a general algorithm to compute if a set have nonempty integer subset that have integers add up to 0?
i would like to know one with the least tries and the proof of it.
Example:{−2, −3, 15, 14, 7, −10} have integers... |
H: How to show a subset is part of another set?
I'm not sure how to "show" these two answers.
The small group created from the intersection $A\cap B\cap C$ is a subset of $A\cap B$ since abc is a smaller "portion" of the overall sets.
The difference of $(A-B)-C$ is the same as $A-C$ since part of $A$ was removed with ... |
H: Convergence of $\sum_{n=0}^\infty(-1)^n\frac{4^{n-2}(x-2)}{(n-2)!}$
What theorem should I use to show that $$\sum_{n=0}^\infty(-1)^n\frac{4^{n-2}(x-2)}{(n-2)!}$$ is convergent no matter what value $x$ takes?
AI: Note that while $(-2)!$ and $(-1)!$ are divergent, $1/(-2)! = 1/(-1)! = 0$. Effectively the sum starts a... |
H: Venn diagram for $(\sim A) \cap (\sim B) \cap (\sim C)$
The way I read that it says everything that is not part of $A$,$B$ and $C$. So the answer is $U$ from my diagram?
AI: Recall the De Morgan's law for sets. $$(\sim A) \cap (\sim B) \cap (\sim C) = \sim (A \cup B \cup C)$$
Now you should be able to conclude what... |
H: Why are restrictions important?
When simplifying expressions, why do we add on restrictions for the simplified form if the original form was undefined at a certain point? The simplified form is defined at those points, so why should it be restricted?
An example of what I mean:
$$\frac{x^2−1}{x−1}=x+1, x≠1$$
AI: In ... |
H: Show convergence using CLT
Let $x_1,\ldots,x_n$ be i.i.d. Bernoulli random variables with parameter $1/2$. Let $S=\sum_{i=1}^nx_i$. Using the Central Limit Theorem, show that
$$
\frac{|2S-n|}{\sqrt n}
$$
is convergent to a standard normal random variable.
Thank you.
AI: The expectation of $S$ is $\frac{n}{2}$ and... |
H: Limits and restrictions?
If we assume that the restrictions put on simplified forms of expressions to prevent evaluation at points undefined in the original unsimplified form are important why do we drop them when dealing with limits? For example, consider the following when trying to find the derivative of $f= x^2... |
H: Diagonalizable linear algebraic group is isomorphic to $(\mathbb{C}^*)^r\times A$, for some finite abelian group $A$
I have three questions about algebraic groups.
Let $D$ be a linear algebraic group. Then the following are equivalent:
$D$ is diagonalizable.
$\mathop{Hom}(D,\mathbb{C}^*)$ is finitely generated w... |
H: Combinations - need help clarifying answers
I have the answers to the following two questions, but I'm stumped as to why the answers are calculated this way:
Q.1) There are six comics: A, B, C, D, E, F; How many ways are there to select six comics?
Answer: C(6+6-1, 6-1)
Q.2) There are 20 balls. 6 red, 6 green, 8 p... |
H: Is there a good way to solve for the inverse of $(u^2-u+4)$?
I'm having trouble calculating the inverse of a polynomial. Consider the polynomial $f(x)=x^3+3x-2$, which is irreducible over $\mathbb{Q}$, as it has no rational roots. So $\mathbb{Q}[x]/(f(x))\simeq \mathbb{Q}[u]$ is a field.
How would I calculate $(u^... |
H: Combinations - 2 sets of identical books
In how many ways can 15 identical computer science books and 10 identical psychology books be distributed among five students?
So I'm trying to figure this out: I know how to calculate 15 identical cs books: C(15+6-1, 6-1) and also 10 identical psych books: C(10+6-1, 6-1), ... |
H: Combinations - at least and at most
There are 20 balls - 6 red, 6 green, 8 purple
We draw five balls and at least one is red, then replace them. We then draw five balls and at most one is green. In how many ways can this be done if the balls are considered distinct?
My guess:
$${4+3-1 \choose 3-1} \cdot {? \choos... |
H: Am I finding the Inequality Wrong?
If I am given the inequality $x+6>7>2x$ then can I do the following to find the range of x
since $x+6>7$ so $x>1$ and since $7>2x$ so $\frac{7}{2}$ $>x$
This means
7/2 > x
x > 1
so $\frac{7}{2}$ $>x>1$
Is this correct ?
According to my book "The expression $x+6... |
H: 2012-gon- subsets of vertices.
Can we prove or disprove this?
For a sufficiently large $n$, every set of at least $ n$ points in the plane with no three collinear has a subset that form the vertices of a convex $2012$-gon.
Gerry mentions the Happy Ending theorem but I don't see how it relates. If someone could show... |
H: 1-dim subspace & sphere
I'm reading a book about algebraic topology recently and I have read through this sentence.
"The space of of all one dimensional subspace is equal to the one dimensional circle (that's the circumference)" I don't understand this but there isn't a lot further explanation about this. Can anyon... |
H: When is $\mathbb{F}_p[x]/(x^2-2)\simeq\mathbb{F}_p[x]/(x^2-3)$ for small primes?
I've been considering the rings $R_1=\mathbb{F}_p[x]/(x^2-2)$ and $R_2=\mathbb{F}_p[x]/(x^2-3)$, where $\mathbb{F}_p=\mathbb{Z}/(p)$.
I'm trying to figure out if they're isomorphic (as rings I suppose) or not for primes $p=2,5,11$.
I... |
H: Why is the probability that a prime p is a factor of a number n equal to 1/p
I'm learning some number theory and I can't seem to understand why this is the case.
AI: You ought to specify a distribution before you ask a question like this, because there is no uniform distribution on all of the natural numbers, so th... |
H: Proof of Riemann Integral of an indicator function of an interval on Real
The theorem is as follow:
Let $a<b$ and let $c,d \in [a,b]$ with $c<d$. Then $1_{[c,d)}$ is Riemann Integrable over $[a,b]$ and
$$\int_{a}^{b} 1_{[c,d)} dx = d-c$$
I am using Shroeder's Mathematical Analysis and I came across this proof, but... |
H: generating $\sigma$-field of a set
Let $X=(X_t)$ be a stochastic process and we define the raw filtration by $F=(\mathcal{F}_t)$, where $\mathcal{F}_t:=\sigma (X_s;s\le t)$ Now I want to prove that $\sigma (\mathcal{C})=\mathcal{F}_t$, where $\mathcal{C}:=\{\prod_{k=0}^nf_k(X_{t_k});t_n\le t,f_k:\mathbb{R}\to \math... |
H: proof that there is a random variable for which a function has a zero value
Given a function $h$:
$$
h(x)=af(x)−b[1−F(x)],
$$
where $a$ and $b$ are constants with $b>0$, $f$ is a probability (a generalized) density function and $F$ is its CDF, I want to prove that there exists an $x$ such that $h(x)... |
H: Is $\mathop{Hom}(O(1),\mathbb{C}^*)$ isomorphic to $\mathop{Hom}(\mathbb{C}^*,O(1))$?
I have an elementary question on homomorphisms.
Let $O(1)=\{A: A^t A=1 \}$ and let $\mathbb{C}^*= \{ z\in\mathbb{C}:z\not=0\}$.
Then what is the character group $\mathop{Hom}(O(1),\mathbb{C}^*)$ isomorphic to, and what is $\mat... |
H: "Probability" of a large integer being prime
Someone once told me (rather testily) that we cannot speak of the "probability that a number is prime" because the sequence is deterministic. I think I understood his point but would like to make sure. There is a theorem in Stopple's Primer of Analytic Number Theory (p. ... |
H: Factoring and Exponent rules using "n" notation tripping me up
Hello StackExchange world,
While doing a proof, I encountered:
$$
= S(n-1)+3^n-3^{n-1}
$$
I am focused on the latter part, with the powers of n, apparently it reduces after factoring to:
$$
2*3^{n-1}
$$
and I have no idea why. I tried thinking of n as ... |
H: Fermat's theorem on sums of two squares composite number
Suppose that there is some natural number $a$ and $b$. Now we perform $c = a^2 + b^2$. This time, c is even.
Will this $c$ only have one possible pair of $a$ and $b$?
edit: what happens if c is odd number?
AI: Not necessarily. For example, note that $50=1^2+... |
H: Sword, pizza and watermelon
Suppose that we have a sword and cut a pizza and watermelon. What is the maximum number of pieces of pizza or watermelon obtained after 10 cuts. Is there a general formula
AI: If you are given a plane then, maximum number of pieces after $n$ cuts is given by recursion $$C(n)=C(n-1)+n$$ w... |
H: How to calculate the area of bizarre shapes
I'm looking for an algorithm to calculate the area of various shapes (created out of basic shapes such as circles, rectangles, etc...). There are various possibilities such as the area of 2 circles, 1 triangular and 1 square (intersections possible). As you can see this g... |
H: Why $A_{5}$ has no subgroup of order 20?
Why $A_{5}$ has no subgroup of order 20?
Thanks!
AI: First, yes: it is that simple (ah, how we loathe simple things, uh?)
Second, of course you've used a very special property of $\,A_5\,$: it is a simpe group! Otherwise how could you deduce $\,A_5\,$ is isomorphic to a sub... |
H: $\alpha x^2+\beta y^2=\gamma$ solvable over $\mathbb Q$ iff $ax^2+by^2=z^2$ solvable over $\mathbb Z$ with coprime $x,y,z$?
I want to understand an algorithm from [1] to solve
$$\alpha x^2+\beta y^2=\gamma \text{ over } \mathbb{Q}$$
with $\alpha, \beta, \gamma\in\mathbb{Q}$. As far is I understood the process the f... |
H: Approximating next prime number
Suppose that there is a prime number. Now I want to approximate the next prime number. (It does not have to be exact.) What would be the time-efficient way to do this?
Edit: what happens if we limit the case to the prime number of the form $4k+1$ where k is a natural number?
Edit: i... |
H: Does every Noetherian ring contain at least one maximal ideal?
I want to prove that a noetherian ring $R \neq \{0\}$ contains at least one maximal ideal.
My idea is to consider $\langle 0 \rangle$ and $\langle 1 \rangle$:
If there is no ideal $I$ with $\langle 0 \rangle \subsetneq I \subsetneq\langle 1 \rangle$ t... |
H: Can two topological spaces surject onto each other but not be homeomorphic?
Let $X$ and $Y$ be topological spaces and $f:X\rightarrow Y$ and $g:Y\rightarrow X$ be surjective continuous maps. Is it necessarily true that $X$ and $Y$ are homeomorphic? I feel like the answer to this question is no, but I haven't been... |
H: Circulant vs normal
What is the relationship between the definition for a matrix to be circulant and to be normal? Does one imply the other?
Assume matrix $A$ is symmetric, then $A^T=A$ and clearly it is normal, but not circulant in general. However, if I assume that $A$ is circulant, looks like $A^TA=AA^T$, so is ... |
H: How to find the largest possible rectangle (by perimeter) on the following function?
It's been a while since I last tackled high-school math, and a friend asked me this question which I can't remember how to approach.
I have the following: $ y = -x^2 + 5x $ Which produces an inverse parabola, intersecting with the ... |
H: Predicate Calculus with Sets - Question about use of an axiom
Greets again StackExchange,
I am watching an online lecture, and I believe that my instructor has misused an axiom. Is my concern warranted?
$$\begin{align*}
\text{Given:}& {P \subseteq (Q \cap R)}\\
&{(Q \cup S) \subseteq T}\\
&{x \in (P \cup S) }\\
\... |
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